A Synopsis of Elementary Results in Pure and Applied Mathematics: Volume 1+2 1108050670, 1108050689

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La

PAY

A

SYNOPSIS

OF

ELEMENTARY

RESULTS

IN

PURE

MATHEMATICS :

CONTAINING

PROPOSITIONS, FORMULE, AND METHODS OF ANALYSIS ,

WITH ABRIDGED DEMONSTRATIONS.

SUPPLEMENTED BY AN INDEX TO THE PAPERS ON PURE MATHEMATICS WHICH ARE TO BE FOUND IN THE PRINCIPAL JOURNALS AND TRANSACTIONS OF LEARNED SOCIETIES , BOTH ENGLISH AND FOREIGN, OF THE PRESENT CENTURY.

BY G.

S.

CARR ,

M.A.

LIBRARY OF TEF UNIVERSITY

OF CALIFORNIA LONDON : FRANCIS HODGSON, 89 FARRINGDON STREET, E.C. CAMBRIDGE : MACMILLAN & BOWES .

1886 .

(All rights reserved )

Q A41

C 3 Engineering Library

LONDON : PRINTED BY C. F. HODGSON AND SON, GOUGH SQUARE, FLEET Street . 76606

9A41 C= Evni ľ

OF TRY NIVERSITY

PREFACE

TO

PART I.

THE work, of which the part now issued is a first instal ment, has been compiled from notes made at various periods of the last fourteen years , and chiefly during the engagements of teaching.

Many of the abbreviated methods and mnemonic

rules are in the form in which I originally wrote them for my pupils. The general object of the compilation is , as the title indicates, to present within a moderate compass the funda mental theorems , formulæ , and processes in the chief branches of pure and applied mathematics . The work is intended , in the first place, to follow and supplement the use of the

ordinary text-books , and it is

arranged with the view of assisting the student in the task of revision of book-work.

To this end I have , in many cases ,

merely indicated the salient points of a demonstration , or merely referred to the theorems by which the proposition is proved .

I am convinced that it is more beneficial to the

student to recall demonstrations with such aids , than to read and re-read them.

Let them be read once , but recalled often .

The difference in the effect upon the mind between reading a mathematical demonstration, and originating one wholly or

PREFACE .

iv

partly, is very great.

It may be compared to the difference

between the pleasure experienced , and interest aroused , when in the one case a traveller is passively conducted through the roads of a novel and unexplored country, and in the other case he discovers the roads for himself with the assistance of

a map. In the second place , I venture to hope that the work, when completed, may prove useful to advanced students as an aide-mémoire and book of reference.

The boundary of

mathematical science forms , year by year, an ever widening circle , and the advantage of having at hand some condensed statement of results becomes more and more evident. To the original investigator occupied with abstruse re searches in some one of the many branches of mathematics, a work which gathers together synoptically the leading propo sitions in all, may not therefore prove unacceptable .

Abler

hands than mine undoubtedly , might have undertaken the task of making such a digest ; but abler hands might also , perhaps , be more usefully employed , -and with this reflection I have the less hesitation in commencing the work myself.

The design

which I have indicated is somewhat comprehensive , and in relation to it the present essay may be regarded as tentative . The degree of success

which it may meet with , and the

suggestions or criticisms which it may call forth , will doubt I

; have their effect on the subsequent portions of the work. With respect to the abridgment of the demonstrations , I

may remark, that while some diffuseness of explanation is not only allowable but very desirable in an initiatory treatise , conciseness is one of the chief requirements in a work intended

PREFACE .

for the purposes of revision and reference only.

In order ,

however, not to sacrifice clearness to conciseness , much more labour has been expended upon this part of the subject- matter of the book than will at first sight be at all evident .

The only

palpable result being a compression of the text , the result is so far a negative one.

The amount of compression attained

is illustrated in the last section of the present part , in which more than the

number

of propositions

usually

given in

treatises on Geometrical Conics are contained , together with the figures and demonstrations , in the space of twenty- four pages . The foregoing remarks have a general application to the work as a whole .

With the view, however , of making the

earlier sections more acceptable to beginners , it will be found that, in those sections, important principles have sometimes been more fully elucidated and more illustrated by examples , than the plan of the work would admit

of in subsequent

divisions . A feature to which attention may be directed is the uni form system of reference adopted throughout all the sections . With the object of facilitating such reference, the articles have been numbered progressively from the

commencement in

large Clarendon figures ; the breaks which will occasionally befound in these numbers having been purposely made , in order to leave room for the insertion of additional matter , if it should 389 be required in a future edition , without disturbing the original numbers and references . With the same object, demonstrations and examples have been made subordinate to enunciations and formulæ, the former being printed in small, the latter in bold

vi

PREFACE .

type .

By these aids , the interdependence of propositions is

more readily shown , and it becomes easy to trace the connexion between theorems in different branches of mathematics , with out the loss of time which would be incurred in turning to separate treatises on the subjects . The advantage thus gained will, however, become more apparent as the work proceeds. The Algebra section was printed some years ago , and does not quite correspond with the succeeding ones in some of the particulars named above .

Under the pressure of other

occupations , this section moreover was not properly revised before going to press .

On that account the table of errata

will be found to apply almost exclusively to errors in that section ; but I trust that the list is exhaustive.

Great pains

have been taken to secure the accuracy of the rest of the volume.

Any intimation of errors will be gladly received .

I have now to acknowledge some of the sources from which the present part has been compiled .

In the Algebra , Theory

of Equations , and Trigonometry sections , I am largely in debted to Todhunter's well- known treatises , the accuracy and completeness of which it would be superfluous in me to dwell

upon . In the section entitled Elementary Geometry, I have added to simpler propositions a selection of theorems from Town send's Modern Geometry and Salmon's Conic Sections . In Geometrical Conics , the line of demonstration followed agrees , in the main, with that adopted in Drew's treatise on the subject.

I am inclined to think that the method of that

author cannot be much improved . 1 .

portant properties

It is true that some im

of the ellipse, which are arrived at in

vii

PREFACE .

Drew's Conic Sections through certain intermediate proposi tions, can be deduced at once from the circle by the method of orthogonal projection .

But the intermediate propositions can

not on that account be dispensed with, for they are of value in themselves .

Moreover, the method of projection applied to

the hyperbola is not so successful ; because a property which has first to be proved

true in the case of the equilateral

hyperbola, might as will be proved at once for the general case . I have introduced the method of projection but sparingly, always giving preference to a demonstration which admits of being applied in the same identical form to the ellipse and to the hyperbola.

The remarkable analogy subsisting between

the two curves is thus kept prominently before the reader. The account of the C. G. S. system of units given in the preliminary section , has been compiled from a valuable con tribution on the subject by Professor

Everett ,

published by the Physical Society of London. *

of Belfast ,

This abstract,

and the tables of physical constants , might perhaps have found a more appropriate place in an after part of the work.

I have ,

however, introduced them at the commencement , from a sense of the great importance of the reform in the selection of units of measurement which is embodied in the C. G. S. system , and from

a

belief that the

student cannot be too

early

familiarized with the same. The Factor Table which follows is , to its limited extent, a reprint of Burckhardt's " Tables des diviseurs ," published in

* " Illustrations of the Centimetre- Gramme-Second System of Units.” London: Taylor and Francis. 1875.

viii

PREFACE .

1814-17 , which give the least divisors of all numbers from 1 to 3,036,000 .

In a certain sense, it may be said that this is

the only sort of purely mathematical table which is absolutely indispensable, because the information which it gives cannot be supplied by any process of direct calculation . rithm of

a number ,

formula .

Not so its

The loga

for instance, may be computed by a prime factors .

These

can only be

arrived at through the tentative process of successive divisions by the prime numbers , an operation of a most deterrent kind when the subject of it is a high integer. A table similar to and in continuation of Burckhardt's has recently been constructed for the fourth million by J. W. L. Glaisher, F.R.S. , who I believe is also now engaged in com pleting the fifth and sixth millions . The factors for the seventh , eighth , and ninth millions were calculated previously by Dase and Rosenberg, and published

in

1862-65 , and the

million is said to exist in manuscript.

tenth

The history of the

formation of these tables is both instructive and interesting. *

As , however, such tables are necessarily expensive to pur chase, and not very accessible in any other way to the majority of persons ,

it

seemed to me that a small portion of them

would form a useful accompaniment to the present volume. I have , accordingly, introduced the first eleven pages of Burckh ardt's tables , which give the least factors of the first 100,000 integers nearly.

Each double page of the table here printed is

* See " Factor Table for the Fourth Million." London : Taylor and Francis. 1880. Pt. IV. , and Nature, No. 542 , p. 462.

By James Glaisher, F.R.S.

Also Camb. Phil. Soc. Proc., Vol . III. ,

ix

PREFACE .

an exact reproduction , in all but the type, of a single quarto page of Burckhardt's great work. It may be noticed here that Prof. Lebesque constructed a table to about this extent, on the plan of omitting the multiples of seven, and thus reducing the size of the table by about one- sixth. *

But a small calculation is required in

using the table which counterbalances the advantage so gained . The values of the Gamma- Function , pages 30 and 31 , have been taken from Legendre's table in his " Exercices de Calcul Intégral," Tome I.

The table belongs to Part II. of this

Volume, but it is placed here for the convenience of having all the numerical tables of Volume I. in the same section . In addition to the authors already named, the following treatises have been consulted- Algebras, by Wood , Bourdon , and Lefebure de Fourcy ; Snowball's Trigonometry ; Salmon's Higher Algebra ; the Geometrical Exercises in Potts's Euclid ; and Geometrical Conics by Taylor , Jackson , and Renshaw. Articles 260 , 431 , 569 , and very nearly all the examples , are original.

The latter have been framed with great care, in

order that they might illustrate the propositions as completely as possible. G. S. C.

HADLEY, MIDDLESEX ; May 23, 1880.

"Tables diverses pour la décomposition des nombres en leurs facteurs premiers." Par V. A. Lebesque. Paris . 1864. b

ERRATA.

Art. 13, for - a2b2 "" 56, Line 1 , 39 3 99 66 , 99 5, " X "" 90, 99 4, "9 numerators 1 , 1, 1 "" 99, "" 1, 99 denominator r - 1 "" 107, "9 1, "" taken "" 108, "9 2, 99 (196) " 131, 99 1, 2, "9 5 "" 5, 99 99 99 (-1) "" 133, "9 3,6,7, 99 6x "" "" 99 8, 99 4 "" 9, 29 "" 99 204, 459 "" 29 "" 10, "" 459 10.9.8 "" 138, ,, 4, 99 1.2.3 "" "" "" 99 ་ནོ

5, 4, 6, 4, 11 , 5, 11, 4, 4, 8, 13, 14, 3, 7, 4, 2, 17,

"" 99 :::

,, 99 99 "" 99 99 "" 29 99 " "" "" "" "" " "" 2

6464E

99 140, ,, 182, "" 191, ,, 220, "" 221, "" *237, "" 238, ,, 239, "" 248, "" 267, "" 274, "" 276, "" 99 ,, 283, "" 288, 99 289, "" 290, ,, 325, "" 99 "" 333,

"" "" 99 99 "" 29 "" "" 99

(g + 1)* un -1 in numerator (163) (x + y + z)² (1) x2 = 1 (x²-4x + 8) on left side (234) (29) (267) 11

read 39 "" "" "" 99 "" "" 22 "" "" "" ""

+ a²b². 3.9 x2. 1, a, a². n - 1. taken m at a time. (360). 6. (-1)23. 3x. 34. 102, 306. 9n.

99

7.8.9.10 .

"" "" "" 99 "" "" ""

(q + 1) (*) Notation of ( 96) . 2 26"-1°

19 29 "" P+2 "" (p - 1) "" x=1 "" N-1 "" H (r, n - 1) "" H (r + 1 , n − 1) "" P2 "" PPP3, last line but one ""

(164). 2 (x +y + z)². square of (1). x² = -1 . (x²- 4x + 8)³. Dele. (28). (266). 2 11.

p + 1. p- 1. a = b. n + 1. H (n, r − 1). H (n, r). P. Q1Q₂Q3.

"" 3, 99 ( a +b) "" (a+b)™. 3 6 1, "" 7, "" "" 3528 "" 10284. ,, 481 , "" 6, "" n- 3 "" n - 1. "" 514, 99 4, Dele. applying Descartes' rule 99 517, 99 3, "3 a3 "" x3. "" 544, 99 Transpose F and f. "" 551 , "" 1, "" B₁ "" B. "9 "" 99 "" a - n 93 α- k. ,, 704, "" ( 11 , 12) "" (9, 10, 1) . 22 729, "" (940) "" (960). Article 112 should be as follows : 1 1 +2 /3 + √2 = 1 + 2 /3 + / 2 = (1 + 2 /3 + /2) (11-4 / 3) 1 +2/3- √√2 (1 +2 /3) -2 73 11 +4 /3

^ Tr UNIV.

TABLE

OF

CONTENTS .

»

PART

I.

SECTION I.-MATHEMATICAL TABLES.

Physical Constants and Formulæ ...

...

... ...

... ...

TABLE I.-English Measures and Equivalents in C. G. S. Units II.-Pressure of Aqueous Vapour at different temperatures III.-Wave lengths and Wave frequency for the principal ... lines of the Spectrum ... ... ... IV. THE PRINCIPAL METALS ― Their Densities ; Coeffi cients of Elasticity, Rigidity, and Tenacity ; Expan sion by Heat ; Specific Heat ; Conductivity ; Rate of conduction of Sound ; Electro- magnetic Specific Resistance ... ... V. THE PLANETS -Their Dimensions, Masses, Densities , and Elements of Orbits ... ... ... ... ... ... VI.-Powers and Logarithms of π and e ... ... VII.-Square and Cube Roots of the Integers 1 to 30 VIII. Common and Hyperbolic Logarithms of the Prime numbers from 1 to 109 ... ... IX.-FACTOR TABLE ... Explanation of the Table The Least Factors of all numbers from 1 to 99000 ... X.-VALUES OF THE GAMMA- FUNCTION

...

SECTION II. -ALGEBRA. 100

...

...

...

Newton's Rule for expanding a Binomial .... MULTIPLICATION AND DIVISION ... ... INDICES ... ... ... ... ... HIGHEST COMMON FACTOR ... ... ... ...

⠀⠀⠀ :

FACTORS

... ... ...

... ... ... ...

1 2 4. 4

74

INTRODUCTION . THE C. G. S. SYSTEM OF UNITS Notation and Definitions of Units ... ...

Page

4

5 5 6 6 6

7 8 30

No. of Article 1 12 28 29 30

xii

LOWEST COMMON MULTIPLE ...

CONTENTS .

...

...

:

... ...

:

:

:

... ... ... ... THEORY OF QUADRATIC EXPRESSIONS ... ... EQUATIONS IN ONE UNKNOWN QUANTITY. - EXAMPLES Maxima and Minima by a Quadratic Equation ... SIMULTANEOUS EQUATIONS AND EXAMPLES ... ... ... RATIO AND PROPORTION ... The Theorem ... ... ... ... ... ... Duplicate and Triplicate Ratios ... ... ... ... Compound Ratios ... ... VARIATION ... ... ... ARITHMETICAL PROGRESSION ... ... ... ... ... ... GEOMETRICAL PROGRESSION ... ... ... ... ... HARMONICAL PROGRESSION ... ... ... PERMUTATIONS AND COMBINATIONS ... ... SURDS ... ... ... .. . Va+ √b and Simplification of √ a + √b b ... ... ... ... ...

No. of Article 33

:

:

:

EVOLUTION Square Root and Cube Root Useful Transformations ... QUADRATIC EQUATIONS

...

...

: : :

:

... ... ... ... ... ...

35 38 45 50 54 58 59 68 70 72 74 76 79

:

83 87 94 108

: ...

121

... ... ... ... ... ... ... ... ... ...

... ... ***

124 125 137 142 149 160 167 182 186 188 195 197 199

Special cases in the Solution of Simultaneous Equations ... ... ... ... Method by Indeterminate Multipliers ... ... Miscellaneous Equations and Solutions ... ... ... On Symmetrical Expressions IMAGINARY EXPRESSIONS ... ... ... METHOD OF INDETERMINATE COEFFICIENTS ... ... ... METHOD OF PROOF BY INDUCTION ... ... PARTIAL FRACTIONS . - FOUR CASES ... CONVERGENCY AND DIVERGENCY OF SERIES ... ... ... ... ... General Theorem of (x) ... ... ... ...

211 213 214 219 223 232 233 235 239 243

... Simplification of Va + b. BINOMIAL THEOREM ... ... ... ... ... ... MULTINOMIAL THEOREM LOGARITHMS ... ... …… . ... ... ... ... EXPONENTIAL THEOREM ... CONTINUED FRACTIONS AND CONVERGENTS ... ... General Theory of same To convert a Series into a Continued Fraction

... ... ... ... ... ... ... ...

A Continued Fraction with Recurring Quotients ... INDETERMINATE EQUATIONS TO REDUCE A QUADRATIC SURD TO A CONTINUED FRACTION ... ... ... To form high Convergents rapidly General Theory ... *** …… . ――― EQUATIONS

:

: : : :

:

CONTENTS .

... ... ...

... ... ... ... ... ... ... ... ... Case of Quadratic Factor with Imaginary Roots ... ... ... ... ... ... Lagrange's Rule SUMMATION OF SERIES BY THE METHOD OF DIFFERENCES ... ... ... ... ... Interpolation of a term DIRECT FACTORIAL SERIES ... ... ... ... ... INVERSE FACTORIAL SERIES ... ... ... ... ... SUMMATION BY PARTIAL FRACTIONS ... ... ... ... COMPOSITE FACTORIAL SERIES ... ... ... MISCELLANEOUS SERIES Sums of the Powers of the Natural Numbers ... ... Sum of a + (a + d) r + (a + 2d) r³ + &c. ... Sum of n - n (n - 1) + &c. ... ... POLYGONAL NUMBERS ... ... ... ... ... FIGURATE NUMBERS ... ... HYPERGEOMETRICAL SERIES ... ... ... m incommensur Proof that e" is able ... ... ... INTEREST ... ... ... ANNUITIES ... ... ... ... ... ... ... PROBABILITIES ... ... ... ... ... ... ... ... ... ... INEQUALITIES ... ... ... Arithmetic Mean > Geometric Mean

... ... ... ... ... ... ... ... ... ... ...

No. of Artic'e 248 251 257 258 263 264 267 268 270 272 274

:

EXPANSION OF A FRACTION ... RECURRING SERIES ... ... The General Term ...

xiii

276 279 285 287 289 291

... ... ... ... ... ... Arithmetic Mean of mth powers > mth power of A. M. ... SCALES OF NOTATION ... ... ... ... ... ... Theorem concerning Sum or Difference of Digits THEORY OF NUMRERS ... ... ... ... ... ... Highest Power of a Prime p contained in | m

295 296 302 309 330 332 334 342 347 349 365

:

... ... ... ... ... ...

:

:

: : :

Fermat's Theorem ... Wilson's Theorem Divisors of a Number S, divisible by 2n + 1

... ... ...

... ... ... ...

... ... ... ...

... ... ... ...

:

:

: :

SECTION III .- THEORY OF EQUATIONS . FACTORS OF AN EQUATION ... ... ... ... ... To compute f(a) numerically Discrimination of Roots ... ... DESCARTES' RULE OF SIGNS ... ... ... ... THE DERIVED FUNCTIONS OF f (x) ... To remove an assigned term To transform an equation EQUAL ROOTS OF AN EQUATION Practical Rule

... ... ... ...

369 371 374 380

... ... ... ...

400 403 409 416 424 428 430 432 445

... ... ...

CONTENTS .

...

: :

:

... ...

:

:

RECIPROCAL EQUATIONS BINOMIAL EQUATIONS ..

:

:

:

LIMITS OF THE ROOTS Newton's Method Rolle's Theorem NEWTON'S METHOD OF DIVISORS

:

xiy

... ... ...

:

: :

:

COMMENSURABle Roots

:

BIQUADRATIC EQUATIONS Descartes' Solution ... Ferrari's Solution Euler's Solution

:

: : :

:

Solution of " +1 = 0 by De Moivre's Theorem... • ... ... · CUBIC EQUATIONS Cardan's Method ... ... Trigonometrical Method

No. of Article. ... 448 ... 452 ... 454 459 466 472 480 483 484 489

...

...

...

492 496 499 502

:

... Lagrange's Method of Approximation ... Newton's Method of Approximation Fourier's Limitation to the same Newton's Rule for the Limits of the Roots ... Sylvester's Theorem ... Horner's Method ... ... ...

:

:

INCOMMENSURABLE ROOTS Sturm's Theorem Fourier's Theorem ...

... ... ... ... ... ... ...

SYMMETRICAL FUNCTIONS OF THE ROOTS OF AN EQUATION Sums of the Powers of the Roots Symmetrical Functions not Powers of the Roots ... The

... ... ... ... ... ...

Equation whose Roots are the Squares of the ... Differences of the Roots of a given Equation.

Sum of the mth Powers of the Roots of a Quadratic ... ... ... ... Equation Approximation to the Root of an Equation through the Sums of the Powers of the Roots ... ... ... EXPANSION OF AN IMPLICIT FUNCTION OF 2 ... ... DETERMINANTS -

:

Definitions ... ... ... General Theory To raise the Order of a Determinant

...

... ...

: :

Analysis of a Determinant ... ... ... Synthesis of a Determinant Product of two Determinants of the nth Order ... ... ... ... Symmetrical Determinants .. ... ... ... Reciprocal Determinants ... ... Partial and Complementary Determinants

... ... ... ... ... ... ... ...

506 518 525 527 528 530 532 533 534 538 541 545 548 551

554 556 564 568 569 570 574 575 576

XV

CONTENTS .

Theorem of a Partial Reciprocal Determinant Product of Differences of n Quantities ...

... ...

... ... ...

:: ::

Product of Squares of Differences of same Rational Algebraic Fraction expressed as a Determinant ELIMINATION ... ... ... Solution of Linear Equations ... ... Orthogonal Transformation Theorem of the n- 2th Power of a Determinant ... ... Bezout's Method of Elimination ... ... ... ... ... ... Sylvester's Dialytic Method ... ... ... Method by Symmetrical Functions ... ELIMINATION BY HIGHEST COMMON FACTOR ...

No. of Article. 577 578 579 581 582 584 585 586 587 588 593

SECTION IV.- PLANE TRIGONOMETRY .

ANGULAR MEASUREMENT TRIGONOMETRICAL RATIOS

... ... ... ... ... ... ... ...

600 606 613 627 674 690 700 704 709

:

... ... ... ... ... ... ... ... Formulæ involving one Angle ... Formulæ involving two Angles and Multiple Angles Formulæ involving three Angles ... ... RATIOS OF 45°, 60°, 15°, 18°, &c. ... ... PROPERTIES OF THE TRIANGLE ... ... ... ... ... The s Formulæ for sin 4, &c. 440 ... ... The Triangle and Circle ... ―― SOLUTION OF TRIANGLES ... ... Right-angled Triangles Scalene Triangles.-Three cases ... ... Examples on the same Quadrilateral in a Circle ... Bisector of the Side of a Triangle ... Bisector of the Angle of a Triangle

:

... ... ... ... ... ... Perpendicular on the Base of a Triangle ... REGULAR POLYGON AND CIRCLE ... ... ... SUBSIDIARY ANGLES ... ... ... ... LIMITS OF RATIOS ... ... ... ... DE MOIVRE'S THEOREM ... ...

... ... ... ... ... ... ... ... ... ...

Expansion of cos no, &c. in powers of sin and cos 0 ... Expansion of sin 0 and cos in powers of Expansion of cos" and sin" in cosines or sines ... ... ... ... ... multiples of Expansion of cos no and sinne in powers of sin 0 Expansion of cos no and sin no in powers of cos 0 Expansion of cos no in descending powers of cos 0 ... Sin a + c sin (a +b) + &c., and similar series

... " ... ... ... ... ... ... ... ...

... ... ... of ... ……. ……… ... ...

718 720 859 733 738 742 744 746 749 753

756 758 764

772 775 779 780 783

CONTENTS .

xvi

Gregory's Series for 0 in powers of tan 0... ... Formula for the calculation of ... Proof that is incommensurable ... ... Sinan sin (x + a ) .- Series for æ

No. of Article. ... 791 ... 792 ... 795 ... 796 ... 800 ... 807 ... 808 ... 815 ... 817 ... 819 ... 821 ... 823 ... 832 ... 835 ... 838

... ...

Sum of sines or cosines of Angles in A. P. Expansion of the sine and cosine in Factors Sin no and cos no expanded in Factors Sin and cos in Factors involving e* −2 cos 0 + e¯ * expanded in Factors De Moivre's Property of the Circle ... ... Cotes's Properties ... ... ... ADDITIONAL FORMULE ...

... ... ... ... ... ... ...

Properties of a Right-angled Triangle ... Properties of any Triangle ... ... ... ... Area of a Triangle Relations between a Triangle and the Escribed, and Circumscribed Circles.

... ... ... ... ... ... ... ...

Inscribed, ... ...

841

Other Relations between the Sides and Angles of a Triangle ... ... Examples of the Solution of Triangles ...

850 859

SECTION V.-SPHERICAL TRIGONOMETRY . INTRODUCTORY THEOREMS ... Definitions ... ... Polar Triangle RIGHT-ANGLED TRIANGLES Napier's Rules

... ...

...

... ...

... ...

870 871

...

...

...

...

881

... ***

... ...

...

882 884

...

...

894

... ... ...

895 896 897

...

898 900

... ... ...

... ... ... ...

... ... ... ...

⠀⠀⠀⠀

Area of Spherical Polygon ... ... Cagnoli's Theorem ... Lhuillier's Theorem

...

... ...

...

::⠀⠀⠀

...

... ...

⠀⠀⠀

SPHERICAL AREAS Spherical Excess

::

SPHERICAL TRIANGLE AND CIRCLE Inscribed and Escribed Circles ... Circumscribed Circles

⠀⠀⠀ :

OBLIQUE-ANGLED TRIANGLES. ... Formula for cos a and cos A The S Formula for sin 4, sin la, &c . sin C sin B sin A = = ... ... sin c sin a sin b cos b cos C cot a sin b - cot A sin C ... Napier's Formulæ ... Gauss's Formulæ ... ...

::

...

:

... ...

... ...

... ...

902 903 904 905

:

xvii

CONTENTS .

POLYHEDRONS

...

...

... ... ...

... ... ...

The five Regular Solids The Angle between Adjacent Faces Radii of Inscribed and Circumscribed Spheres ...

... ... ... ...

No. of Article. 906 907 909 910

SECTION VI.-ELEMENTARY GEOMETRY . MISCELLANEOUS PROPOSITIONS— Reflection of a point at a single surface ... do. at successive surfaces do. Relations between the sides of a triangle, the segments of the base, and the line drawn from the vertex ... ... Equilateral triangle ABC ; PA²+ PB² + PC Sum of squares of sides of a quadrilateral Locus of a point whose distances from given lines or points are in a given ratio ... To divide a triangle in a given ratio Sides of triangle in given ratio . Harmonic division of base

Locus of vertex

Triangle with Inscribed and Circumscribed circles THE PROBLEMS OF THE TANGENCIES ... ... Tangents and chord of contact . By = u² To find any sub-multiple of a line Triangle and three concurrent lines ; Three cases Inscribed and Escribed circles ; 8, s - a, &c. NINE-POINT CIRCLE ... ... ... CONSTRUCTION OF TRIANGLES ...

...

Locus of a point from which the tangents to two circles have a constant ratio ... ... COLLINEAR AND CONCURRENT SYSTEMS ... ... Triangle of constant species circumscribed or inscribed .".. ... to a triangle RADICAL AXIS

⠀⠀⠀⠀

Inversion of a point ... circle do. ... do. right line POLE AND POLAR ... COAXAL CIRCLES ... CENTRES AND AXES OF SIMILITUDE

⠀⠀

... ...

⠀⠀ ⠀⠀⠀⠀⠀

... ... :

Of two Circles Of three Circles INVERSION

... ...

⠀⠀

Homologous and Anti-homologous points do. do. chords C

920 921

922 923 924

926 930 932 933 935 937 948 950 951 953 954 960 963 967 977

984 997

... ... ...

...

1000 1009 1012 1016 1021 1037 1038

CONTENTS .

xviii

: ... ...

:

Projections of the Sphere ... ADDITIONAL THEOREMS

...

:.

Orthogonal Projection

... ... ... :

Gergonne's Theorem ANHARMONIC RATIO AND PENCIL HOMOGRAPHIC SYSTEMS OF POINTS INVOLUTION PROJECTION ... On Perspective Drawing ...

:

Constant product of anti-similitude Circle of similitude Axes of similitude of three circles

...

... ... ... ...

... ... ... ... ...

Squares of distances of P from equidistant points on a ... circle ... Squares of perpendiculars on radii, &c. ... Polygon with inscribed and circumscribed circles . Sum ... of perpendiculars on sides, & c....

No. of Article. 1043 1045 1046 1049 1052 1058 1066 1075 1083 1087 1090

1094 1095 1099

SECTION VII.- GEOMETRICAL CONICS . SECTIONS OF THE CONE

Defining property of Conic PS = ePM ... Fundamental Equation ... ... Projection from Circle and Rectangular Hyperbola JOINT PROPERTIES OF THE ELLIPSE AND HYPERBOLA— Definitions ... CS : CA CX ... ... ' ... PSPS = AA AC² + BC² ... CS ... SZ bisects 4 QSR .. ... If PZ be a tangent PSZ is a right angle Tangent makes equal angles with focal distances Tangents of focal chord meet in directrix CN.CT = AC ... GS : PS ... e NG : NC BC : AC

Auxiliary Circle PN: QNBC : AC PN ": AN.NA = DC : 10 BC² Cn . Ct SY.SY BC² ... PE = AC ... To draw two tangents ...

...

... ... ...

... ... ... ... ...

...

1151 1156 1158 1160 1162 1163 1164 1166 1167 1168 1169 1170 1171 1172 1173 1174 1176 1177 1178 1179 1180 1181

...

Tangents subtend equal angles at the focus :

CONTENTS .

xix No. of Article. 1183 1184 1186 1187 1188 1189 1191 1192

...

:

:

ASYMPTOTIC PROPERTIES OF THE HYPERBOLA— RN² – PN² = BC² ... PR.Pr BC² CE AC ...

:

... PD is parallel to the Asymptote QR = qr ... ... ... PL Pl and QV = qV... QR. Qr = PL2 = RV² - QV² 4PH.PK = CS ... ... ... JOINT PROPERTIES OF ELLIPSE AND HYPERBOLA RESUMED. JUGATE DIAMETERS QV² : PV . VP ' = CD² : CP² PF.CDAC . BC ... PF.PG BC2 and PF.PG ′ = AC² PG.PG CD2 ... ... ... :

...

:

... ...

...

... ...

...

Properties of Parabola deduced from the Ellipse THE PARABOLA Defining property PS = PM Latus Rectum = 4AS ...

...

... ...

:

If PZ be a tangent, PSZ is a right angle Tangent bisects 4 SPM and SZM ST SP SG ... ... Tangents of a focal chord intersect at right directrix ... ... ... ... ANAT ... NG = 24S ... ... 4AS . AN ... PN2 SA : SY : SP ... To draw two tangents

1193 1194 1195 1197 1198 1201 1202 1203 1205 1207 1211 1213 1214 1216 1217 1219 1220 1222 1223 1224 1225

1226 1227 1228 1229 1231 1232 1233 1234 1235 1238

... ...

:

: ::

: :

... ... ***

angles in ... ...

:

:

... SQ : SO : SQ 4OSQOSQ and QOQ = Q SQ DIAMETERS ... The diameter bisects parallel chords

... ... ... ... ...

:

Diameter bisects parallel chords ... Supplemental chords Diameters are mutually conjugate ... ... CV.CT = CP : ... CN dR, CR = pN CN² ± CR² = AC², DR² ± PN² = BC² CP² ± CD² = AC² ± BC² ... ... PS . PS CD² ... ... ... = CD : CD' ... OQ . Oq OQ . Oq = ... ... SR : QL = e Director Circle

CON

CONTENTS .

XX

QV = 4PS . PV

... ... OQ . Oq OQ Oq' = PS : P'S ... Parabola two-thirds of circumscribing METHODS OF DRAWING A CONIC ... ... To find the axes and centre ...

……. ... parallelogram ... ... ... ... ... ... ... To construct a conic from the conjugate diameters CIRCLE OF CURVATURE ... ... Chord of curvature = QV ÷ PV ult. CD CD CD³ Semi -chords of curvature, ... CP' PF' AC ... In Parabola, Focal chord of curvature = 4SP ... do. Radius of curvature = 2SP² ÷ SY ... Common chords of a circle and conics are equally in clined to the axis ... ... To find the centre of curvature ... ... MISCELLANEOUS THEOREMS ... ... ... ... ...

No. of Article. 1239 1242 1244 1245 1252 1253 1254 1258

1259 1260 1261 1263 1265 1267

:

INDEX

TO

PROPOSITIONS

OF

EUCLID

REFERRED TO IN THIS WORK.

The references to Euclid are made in Roman and Arabic numerals ; e.g. (VI . 19).

BOOK

I.

I. 4.-Triangles are equal and similar if two sides and the included angle of each are equal each to each. I. 5. - The angles at the base of an isosceles triangle are equal. I. 6. The converse of 5. I.

8.-Triangles are equal and similar if the three sides of each are equal each to each .

I. 16.-The exterior angle of a triangle is greater than the interior and opposite. I. 20. I. 26.

Two sides of a triangle are greater than the third. Triangles are equal and similar if two angles and one corres ponding side of each are equal each to each.

I. 27.

Two straight lines are parallel if they make equal alternate angles with a third line. I. 29. -The converse of 27. I. 32.

The exterior angle of a triangle is equal to the two interior and opposite ; and the three angles of a triangle are equal to two right angles.

COR. 1. - The interior angles of a polygon of n sides = (n - 2) π. COR. 2. -The exterior angles = 2π. I. 35 to 38. - Parallelograms or triangles upon the same or equal bases and between the same parallels are equal. I. 43. The complements of the parallelograms about the diameter of a parallelogram are equal . I. 47.

The square on the hypotenuse of a right- angled triangle is equal to the squares on the other sides . I. 48. The converse of 47.

xxii

INDEX TO PROPOSITIONS OF EUCLID

BOOK II. II. 4.- If a, b are the two parts of a right line, (a +b) = a² + 2ab + b². If a right line be bisected, and also divided, internally or externally, into two unequal segments, then II. 5 and 6. The rectangle of the unequal segments is equal to the difference of the squares on half the line, and on the line between the points of section ; or (a + b) (a - b) = a - b². II. 9 and 10. - The squares on the same unequal segments are together double the squares on the other parts ; or (a + b ) ² + (a − b) ² = 2a² + 2b³. II. 11. To divide a right line into two parts so that the rectangle of the whole line and one part may be equal to the square on the other part.

II. 12 and 13.- The square on the base of a triangle is equal to the sum of the squares on the two sides plus or minus (as the vertical angle is obtuse or acute) , twice the rectangle under either of those sides, and the projection of the other upon it ; or a² = b² +c² - 2bc cos A (702) .

BOOK III. III .

3.— If a diameter of a circle bisects a chord, it is perpendicular to it : and conversely.

III. 20. - The angle at the centre of a circle is twice the angle at the circumference on the same arc. III. 21.-Angles in the same segment of a circle are equal . III. 22.- The opposite angles of a quadrilateral inscribed in a circle are together equal to two right angles . III. 31. -The angle in a semicircle is a right angle. III. 32. -- The angle between a tangent and a chord from the point of contact is equal to the angle in the alternate segment . III. 33 and 34. - To describe or to cut off a segment of a circle which shall contain a given angle. III. 35 and 36.— The rectangle of the segments of any chord of a circle drawn through an internal or external point is equal to the square of the semi -chord perpendicular to the diameter through the internal point, or to the square of the tangent from the external point. III. 37. -The converse of 36. If the rectangle be equal to the square, the line which meets the circle touches it.

xxiii

REFERRED TO IN THIS WORK .

BOOK IV . IV. IV. IV. IV.

2. To 3.- To 4. To 5. To

inscribe a triangle of given form in a circle. describe the same about a circle. inscribe a circle in a triangle. describe a circle about a triangle.

IV. 10. — To construct two-fifths of a right angle. IV. 11. To construct a regular pentagon.

CALIFOSS BOOK VI. VI. 1.-Triangles and parallelograms of the proportional to their bases.

same altitude are

VI. 2.—A right line parallel to the side of a triangle cuts the other sides proportionally ; and conversely. VI. 3 and A. - The bisector of the interior or exterior vertical angle of a triangle divides the base into segments proportional to the sides. VI. 4.-Equiangular triangles have their sides proportional homo logously. VI. 5. -The converse of 4. VI. 6.

Two triangles are equiangular if they have two angles equal, and the sides about them proportional.

VI. 7. -Two triangles are equiangular if they have two angles equal and the sides about two other angles proportional , provided that the third angles are both greater than, both less than , or both equal to a right angle. VI. 8.-A right-angled triangle is divided by the perpendicular from the right angle upon the hypotenuse into triangles similar to itself. VI. 14 and 15.— Equal parallelograms, or triangles which have two angles equal, have the sides about those angles reciprocally proportional ; and conversely, if the sides are in this pro portion, the figures are equal . VI. 19.— Similar triangles are in the duplicate ratio of their homo logous sides. VI. 20. Likewise similar polygons . VI. 23. —Equiangular parallelograms are in the ratio compounded of the ratios of their sides .

VI. B. - The rectangle of the sides of a triangle is equal to the square of the bisector of the vertical angle plus the rectangle of the segments of the base.

xxiv

INDEX TO PROPOSITIONS OF EUCLID.

VI. C.- The rectangle of the sides of a triangle is equal to the rect angle under the perpendicular from the vertex on the base and the diameter of the circumscribing circle. VI. D.- Ptolemy's Theorem. The rectangle of the diagonals of a quadrilateral inscribed in a circle is equal to both the rectangles under the opposite sides .

BOOK

XI.

XI.

4.-A right line perpendicular to two others at their point of intersection is perpendicular to their plane.

XI.

5.- The converse of 4. If the first line is also perpendicular to a fourth at the same point, that fourth line and the other two are in the same plane.

XI. XI.

6.—Right lines perpendicular to the same plane are parallel. 8.- If one of two parallel lines is perpendicular to a plane, the other is also.

XI. 20.- Any two of three plane angles containing a solid angle are greater than the third . XI . 21.— The plane angles of any solid angle are together less than four right angles .

TABLE

OF

CONTENTS .

PART II. SECTION VIII . —DIFFERENTIAL CALCULUS .

... ...

... ...

...

:

INTRODUCTION ... ... ... ... Successive differentiation ... Infinitesimals. Differentials ... DIFFERENTIATION.

No. of Article. 1400 1405 1407 1411-21

:

:

Methods ... ... SUCCESSIVE DIFFERENTIATION

... ... ... ... Derivatives of the nth order ( see Index) PARTIAL DIFFERENTIATION ... ... ... THEORY OF OPERATIONS ... Distributive, Commutative, and Index laws ... EXPANSION OF EXPLICIT FUNCTIONS— Taylor's and Maclaurin's theorems ... Symbolic forms of the same ... ... ... ... ... f(x + h, y + k), &c. Leibnitz's theorem

... ...

1460 1461-71

... ...

1480 1483 1488

... 1500, 1507 1520-3 ... 1512-4 ...

Methods of expansion by indeterminate coefficients. ... ... ... Four rules ... ... ... ... ... ... Method by Maclaurin's theorem ... Arbogast's method of expanding ø (z) Bernoulli's numbers ... ... ... ... Stirling and Boole Expansions of (x + h ) −p (x) . EXPANSIONS OF IMPLICIT FUNCTIONS

1527-34 1524 1536 1539 1546-7

:

:

Lagrange's, Laplace's, and Burmann's theorems, 1552 , 1556-63 1 1555 Cayley's series for (z) 1572 Abel's series for ... ... (x + a) INDETERMINATE FORMS ... 1580 ... ... ... JACOBIANS 1600 ... ... ... 1604 Modulus of transformation ... ...

d

xxvi

CONTENTS .

... QUANTICS Euler's theorem

... ...

...

...

Eliminant, Discriminant, Invariant , Covariant , Hessian ... Theorems concerning discriminants ... Notation Abc -f², &c. Invariants ... ... ... ... ...

Cogredients and Emanents IMPLICIT FUNCTIONS

... ... ...

:

: :

:

:

...

...

:

: :

:

Contragredient and Contravariant Notation %, p, &c., q, r, s, t ... MAXIMA AND MINIMA ... One independent variable

...

: :

... One independent variable ... Two independent variables ... n independent variables CHANGE OF THE INDEPENDENT VARIABLE Linear transformation ... ... ... Orthogonal transformation

: :

: :

:. :

:

...

:

Continuous maxima and minima

1626-30 1635-45 1642 1648-52 1653-5 1700 1725 1737 1760 1794 1799 1813 1815 1830 1841

...

... Two independent variables. Three or more independent variables... ... ... Discriminating cubic ... Method of undetermined multipliers ...

No. of Article. 1620 1621

1852 1849 1862 1866

: SECTION IX.—INTEGRAL CALCULUS .

INTRODUCTION

...

Multiple Integrals METHODS OF INTEGRATION―――――

...

1900 1905

... ...

:

:

By Substitution, Parts, Division , Rationalization, ... Partial fractions, Infinite series ... STANDARD INTEGRALS ... ... VARIOUS INDEFINITE INTEGRALS Circular functions ... ... ... Exponential and logarithmic functions ... ... Algebraic functions ... ... Integration by rationalization ... Integrals reducible to Elliptic integrals

: ... : :

... ... ...

Logarithm of an imaginary quantity ...

:

:

Elliptic integrals approximated to SUCCESSIVE INTEGRATION ... ... HYPERBOLIC FUNCTIONS Cosh x, sinh x, tanh x Inverse relations ... ... ... Geometrical meaning of tanh S

1908-19 1921 1954 1998 2007 2110 2121-47 2127 2148 2180 2210 2213 2214

E

xxvii

CONTENTS .

No. of Article. 2230 2233 2245 2253 2258 2261 2262 2280 2294 2317 2341 2391 2451 2571

:

DEFINITE INTEGRALS Summation of series ... ... THEOREMS RESPECTING LIMITS OF INTEGRATION

:

METHODS OF EVALUATING DEFINITE INTEGRALS (Eight rules) DIFFERENTIATION UNDER THE SIGN OF INTEGRATION ... ... ... Integration by this method Change in the order of integration ... APPROXIMATE INTEGRATION BY BERNOULLI'S SERIES ... THE INTEGRALS B (l, m) and (n) ... log F (1 +n) in a converging series ...

Numerical calculation of F (x) INTEGRATION OF ALGEBRAIC FORMS ... ... INTEGRATION OF LOGARITHMIC AND EXPONENTIAL FORMS INTEGRATION OF CIRCULAR FORMS ... ... INTEGRATION OF CIRCULAR LOGARITHMIC AND EXPONENTIAL FORMS MISCELLANEOUS THEOREMS Frullani's, Poisson's, Abel's, Kummer's, and Cauchy's formulæ ... ... ... ... ... FINITE VARIATION OF A PARAMETER ... ... Fourier's formula ... ... ... THE FUNCTION ( ) ... ... (x) ... Summation of series by the function ... (a) as a definite integral independent of (1 ) NUMERICAL CALCULATION OF LOG T (x) ... ... CHANGE OF THE VARIABLES IN A DEFINITE MULTIPLE INTEGRAL MULTIPLE INTEGRALS EXPANSIONS OF FUNCTIONS IN CONVERGING SERIES ... Derivatives of the nth order ... Miscellaneous expansions ... ...

2700-13 2714 2726 2743 2757 2766 2771 2774

2852 2911 2936 2955 2991 2992-7

...

Legendre's function X, EXPANSION OF FUNCTIONS IN TRIGONOMETRICAL SERIES APPROXIMATE INTEGRATION ... ... ... Methods by Simpson, Cotes, and Gauss

SECTION X.—CALCULUS OF VARIATIONS . ... ... ... ... ...

FUNCTIONS OF ONE INDEPENDENT VARIABLE Particular cases... ...

:

Other exceptional cases FUNCTIONS OF TWO DEPENDENT VARIABLES Relative maxima and minima ... ... Geometrical applications

3028 3033 3045 3051 3069 3070

:

:

xxviii

CONTENTS .

FUNCTIONS OF TWO INDEPENDENT VARIABLES ... Geometrical applications APPENDIX

No. of Article. 3075 3078

...

:

On the general object of the Calculus of Variations ... ... ... Successive variation ... Immediate integrability ... ...

3084 3087 3090

SECTION XI. - DIFFERENTIAL EQUATIONS . GENERATION OF DIFFERENTIAL EQUATIONS ... DEFINITIONS And Rules ….. ... ... SINGULAR SOLUTIONS ... FIRST ORDER LINEAR EQUATIONS

... ...

Integrating factor for Mda + Ndy = 0 ... ... Riccati's Equation FIRST ORDER NON- LINEAR EQUATIONS ... ... ... ... Solution by factors

... ... ... ...

: : :

Linear Equations with Constant Coefficients HIGHER ORDER NON- LINEAR EQUATIONS ... ... ... Depression of Order by Unity ... EXACT DIFFERENTIAL EQUATIONS ... ... MISCELLANEOUS METHODS

3214 3221 3222

... ...

: :

: :

Solution by differentiation HIGHER ORDER LINEAR EQUATIONS

3150 3158 3168 3184 3192

...

Approximate solution of Differential Equations by ... ... ... ... Taylor's theorem ... ... SINGULAR SOLUTIONS OF HIGHER ORDER EQUATIONS ... :..

EQUATIONS WITH MORE THAN TWO VARIABLES ... SIMULTANEOUS EQUATIONS WITH ONE INDEPENDENT VARIABLE ... PARTIAL DIFFERENTIAL EQUATIONS ... ... ... Linear first order P. D. Equations ... ... ... Non-linear first order P. D. Equations

Non-linear first order P. D. Equations with more ... than two independent variables ... SECOND ORDER P.D. EQUATIONS LAW OF RECIPROCITY ... ... SYMBOLIC METHODS ... ... ... ... ... SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS BY SERIES SOLUTION BY DEFINITE INTEGRALS ... ... P. D. EQUATIONS WITH MORE THAN two IndepenDENT VARIABLES DIFFERENTIAL RESOLVENTS OF ALGEBRAIC EQUATIONS

3236 3237 3238 3251 3262 3270 3276 3289 3301 3320 3340 3380 3381 3399 3409 3420 3446 3470 3604 3617 3629 3631

xxix

CONTENTS .

No. of Article. SECTION

XII. ―――― CALCULUS

OF FINITE

DIFFERENCES . FORMULE FOR FIRST AND nth DIFFERENCES ...

:

... ... :

The operations E, A, and d Herschel's theorem ...

:

Expansion by factorials Generating functions

...

:

: : :

A theorem conjugate to Maclaurin's INTERPOLATION ... ... ... ... ... Lagrange's interpolation formula MECHANICAL QUADRATURE ... ... ... Cotes's and Gauss's formulæ ... ... Laplace's formula SUMMATION OF SERIES ... APPROXIMATE SUMMATION ...

... ... ... ...

... ... ...

...

:

...

:

: :

:

...

3706 3730 3732 3735 3757 3759 3762 3768 3772 3777 3778 3781 3820

:

:

SECTION XIII.- PLANE COORDINATE GEOMETRY .

SYSTEMS OF COORDINATE-S Cartesian, Polar, Trilinear, Areal, Tangential, and ... Intercept Coordinates

4001-28

ANALYTICAL CONICS IN CARTESIAN COORDINATES .

LENGTHS AND AREAS ... ... TRANSFORMATION OF COORDINATES THE RIGHT LINE . ... ...

... ... ...

... ... ...

:

:

Equations of two or more right lines ... GENERAL METHODS ... ... ... ... Poles and Polars ... ... ... THE CIRCLE ... ... ... ... Co-axal circles ... ... ... THE PARABOLA ……. ... ... ... ... THE ELLIPSE AND HYPERBOLA ... ... ... ... Right line and ellipse ... ... ... Polar equations of the conic ... ... ers ate diamet Conjug ... Determination of various angles THE HYPERBOLA REFERRED TO ITS ASYMPTOTES

... ... ... ... ... ... ... ... ... ...

:

... :

... ...

:

: :

...

... ... ... ... ... ...

:

The rectangular hyperbola

... ...

...

4032 4048 4052 4110 4114 4124 4136 4161 4200 4250 4310

4336 4346 4375 4387 4392

CONTENTS.

XXX

THE GENERAL EQUATION

... :

:

...

:

Rules for the analysis of the general equation Right line and conic with the general equation Intercept equation of a conic ... SIMILAR CONICS ... ... ... CIRCLE OF CURVATURE Contact of Conics CONFOCAL CONICS ...

:

The ellipse and hyperbola ... Invariants of the conic ... ... para ... bola The Method without transformation of the axes ...

No. of Article. 4400 4402 4417 4430 4445 4464 4487 4498 4522

:

:

4527 4550

ANALYTICAL CONICS IN TRILINEAR COORDINATES .

4601

THE RIGHT LINE

: :

Inscribed conic of the trigon ... Inscribed circle of the trigon

:

:

:

:. :

Equations of particular lines and coordinate ratios ... of particular points in the trigon ... ANHARMONIC RATIO ... ... ... ... The complete quadrilateral ... ... THE GENERAL EQUATION OF A CONIC ... ... ... ... Director- Circle ….. PARTICULAR CONICS ... ... Conic circumscribing the trigon

:

General equation of the circle... Nine-point circle ... Triplicate-ratio circle ... ... Seven-point circle ... CONIC AND SELF- CONJUGATE TRIANGLE ... On lines passing through imaginary points ... CARNOT'S, PASCAL'S, AND BRIANCHON'S THEOREMS ... THE CONIC REFERRED TO TWO TANGENTS AND THE CHORD OF CONTACT Related conics .... ANHARMONIC PENCILS OF CONICS ... ... CONSTRUCTION OF CONICS .. ..% ... Newton's method of generating a conic Maclaurin's method of generating a conic ... THE METHOD OF RECIPROCAL POLARS ... *** TANGENTIAL COORDINATES ... ... Abridged notation

4628 4648 4652 4656 4693 4697 4724 4739 4747 4751 4754 47546 4754c 4755 4761 4778-83

4803 4809 4822

: :

:

: :

:

:

:

4829 4830 4844 4870 4907

xxxi

CONTENTS .

No. of Article. ON THE INTERSECTION OF TWO CONICS ... ... ... ....

4916 4921

:

Geometrical meaning of ( -1 ) THE METHOD OF PROJECTION ... INVARIANTS AND COVARIANTS ... ... To find the foci of the general conic

... ...

4936 5008

... ...

THEORY OF PLANE CURVES .

...

... ...

... ... ... ...

The companion to the cycloid ... Prolate and curtate cycloids

... ...

... ... ...

:

... ... ... ... ... ... ... CURVES ... ...

...

:

... ... ...

... ... ...

: : :

:

...

... :

... ... ...

:

:

Epitrochoids and hypotrochoids Epicycloids and hypocycloids ... The Catenary ... ... ... The Tractrix ... ... ... The Syntractrix ... The Logarithmic Curve ... The Equiangular Spiral ... The Spiral of Archimedes

5174 5175-87 5188 5192 5196 5212 5220 5229 5230-5 5239 5243 5244 5246 5247 5248 5249

:

Holditch's theorem ... Trajectories Curves of pursuit Caustics ... ... Quetelet's theorem TRANSCENDENTAL AND OTHER ... The cycloid

... ... ... ... ... ... ... ...

⠀⠀

Area, length, and radius of curvature... ... The envelope of a carried curve Instantaneous centre ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ...

5100 5134 5160 5167 5172

:

... ...

&c. ... ... ... ... ... ...

: :

... ...

:

PEDAL CURVES ... ROULETTES ...

points, ... ... ... ... ... ...

: :

:

INVERSE PROBLEM AND INTRINSIC EQUATION ... ASYMPTOTES ... ... ... ... ... ... ... Asymptotic curves SINGULARITIES OF CURVES--... Concavity and Convexity Points of inflexion, multiple CONTACT OF CURVES ... ... ENVELOPES ... ... ... INTEGRALS OF CURVES AND AREAS INVERSE CURves ... ... ...

: ::

TANGENT AND NORMAL ... ... RADIUS OF CURVATURE AND EVOLUTE

:

:

The Hyperbolic or Reciprocal Spiral...

... ... ... ...

...

5250 5258 5260 5262 5266 5273 5279 5282 5284 5288 5296 5302

:

xxxii

CONTENTS.

:

:

Kempe's five- bar linkage. Eight cases Reversor, Multiplicator, and Translator Peaucellier's linkage ... ... ... The six-bar invertor ... ...

The eight-bar double invertor The Quadruplane or Versor Invertor

... ... ... ... ...

The Pentograph or Proportionator The Isoklinostat or Angle- divider

:

The Versiera (or Witch of Agnesi) ... ... ... The Quadratrix... ... ... ... The Cartesian Oval ... ... The semi- cubical parabola ... The folium of Descartes ... ... LINKAGES AND LINKWORK ...

:

...

:

... ... ... ... ... ... ... ...

:

... Involute of the Circle Cissoid ... Cassinian or Oval of Cassini ... Lemniscate ... Conchoid ... ... ... ... The Limaçon ...

The The The The The

No. of Article. 5306 5309 5313 5317 ... ... 5320 5327 5335 ... ... 5338 5341 ... 5359 ... 5360 5400 ... ... 5401-5417 5407 ... 5410 ... 5419 ... 5420 ... 5422 ... 5423 5425 5426

A linkage for drawing an Ellipse A linkage for drawing a Limaçon, and also a bicir ... ... ... ... cular quartic ... ... A linkage for solving a cubic equation ... ... On three-bar motion in a plane The Mechanical Integrator ... ... The Planimeter ... ...

5427 5429 5430 5450 5452

SECTION XIV.- SOLID COORDINATE GEOMETRY .

SYSTEMS OF COORDINATES THE RIGHT LINE ... THE PLANE ... ... TRANSFORMATION OF COORDINATES

... ... ...

... ... ... ... ... ...

... ...

... ... ... :

THE SPHERE ... ... ... The Radical Plane ... Poles of similitude ... CYLINDRICAL AND CONICAL SURFACES ... Circular Sections ... ... ELLIPSOID, HYPERBOLOID, AND PARABOLOID

... ...

...

5501 5507 5545 ... 5574 5582 ... 5585 5587 5590 ... 5596 ...5599-5621

:

xxxiii

CONTENTS .

No. of Article.

... ... ...

:

:

: :

:

:

:

:

:

: :

... ...

:

:

General equation of a surface ... Tangent line and cone at a singular point ... The Indicatrix Conic ... ... Euler's and Meunier's theorems Curvature of a surface... ...

... ... :

:

Reciprocal and Enveloping Cones THE GENERAL EQUATION OF A QUADRIC RECIPROCAL POLARS ... ... ... THEORY OF TORTUOUS CUrves ... ... The Helix ... ... GENERAL THEORY OF SURFACES

... ... ...

:

CENTRAL QUADRIC SURFACE Tangent and diametral planes... Eccentric values of the coordinates CONFOCAL QUADRICS ... ... ... ...

: :

...

: :

INTEGRALS FOR VOLUMES AND SURFACES Guldin's rules ... CENTRE OF MASS ... MOMENTS AND PRODUCTS OF INERTIA

:

Osculating plane of a line of curvature GEODESICS ... ... ... INVARIANTS ... ...

:

Momental ellipsoids ... Momental ellipse ... ... Integrals for moments of inertia PERIMETERS, AREAS, VOLUMES, CENTRES OF MASS , AND MOMENTS OF INERTIA OF VARIOUS FIGURES : :

Rectangular lamina and Right Solid ... The Circle ... ... ... ... The Right Cone ... Frustum of Cylinder ... The Sphere ... ... The Parabola ... ... ... The Ellipse ... Fagnani's, Griffith's, and Lambert's theorems ... The Hyperbola ... The Paraboloid ... ... ... The Ellipsoid ... Prolate and Oblate Spheroids...

е

5626 5638 5656 5664 5673 5704 5721 5756

5780 5783 5795 5806-9 5826 5835 5837-48 5856 5871 5879 5884 5903 5925-40 5953 5978

6015 6019

6043 6048 6050 6067 6083 ...6088-6114 6115 6126 6142 ... 6152-65

PREFACE

TO

PART

II.

APOLOGIES for the non -completion of this volume at an earlier period are due to friends and enquirers .

The labour

involved in its production , and the pressure of other duties , must form the author's excuse. In the

compilation

of

Sections

VIII. to

XIV . ,

the

following works have been made use of : Treatises on the Differential and Integral Calculus, by Bertrand , Hymer, Todhunter, Williamson, and Gregory's Examples on thes e same subjects ; Salmon's Lessons on Higher Algebra. Treatises on the Calculus of Variations, by Jellett and Tod hunter ; Boole's Differential Equations and Supplement ; Carmichael's Calculus of Operations ; Boole's Calculus of Finite Differences, edited by Moulton. Salmon's Conic Sections ; Ferrers's Trilinear Coordinates ; Kempe on Linkages (Proc. of Roy. Soc. , Vol. 23) ; Frost and Wolstenholme's Solid Geometry ; Salmon's Geometry of Three Dimensions . Wolstenholme's Problems.

The Index which concludes the work, and which , it is hoped, will supply a felt want, deals with 890 volumes of 32 serial publications : of these publications , thirteen belong to Great Britain, one to New South Wales, two to America, four to France,

five to Germany,

three to Italy, two to

Russia , and two to Sweden.

As the volumes

only date

from the

year

1800 ,

the

xxxvi

PREFACE .

important contributions of Euler to the " Transactions of the

Petersburg

St.

excluded .

Academy,"

It was, however,

in the last century,

are

unnecessary to include them ,

because a very complete classified index to Euler's papers , as well as to those of David Bernoulli , Fuss , and others in the same Transactions , already exists . The titles

of this

Index,

and of the

works of Euler

therein referred to , are here appended , for the convenience of those who may wish to refer to the volumes .

Tableau

général des publications de l'Académie Impériale de St. Pétersbourg depuis sa fondation . 1872. [ B.M.C.:* R.R. 2050, e.]

I. Commentarii Academiæ Scientiarum Imperialis Petropolitanæ. 1726-1746 ; 14 vols. [B. M. C.: 431 , f. ] II . Novi Commentarii A. S. I. P. 1747-75, 1750-77 ;

21 vols .

[ B. M. C.: 431 ,f. 15-17, g.1-16, h.1 , 2. ] III. Acta A. S. I. P. 1778-86 ; 12 vols . [ B. M. C : 431 , h.3-8 ; or T.C. 8, a. 11. ] IV. Nova Acta A. S. I. P.

1787-1806 ; 15 vols .

[B. M. C .:

431 ,

h.9-15, i . 1-8 ; or T.C. 8, a . 23. ] V. Leonhardi Euler Opera minora collecta, vel Commentationes Arith meticæ collectæ ; 2 vols. 1849. [ B. M. C.: 8534, ee . ] VI . Opera posthuma mathematica et physica ; 2 vols . 1862. [ B.M.C .:

8534, f. ] VII . Opuscula analytica ; 1783-5 ; 2 vols . [ B. M. C.: 50, i. 15. ] Analysis infinitorum . [B. M. C.: 529,6.11 . ]

G. S. C. ENDSLEIGH GARDENS, LONDON, N.W. , 1886.

* British Museum Catalogue.

C) T Vre

ERRATA CONTINUED FROM PAGE x. (Corrections which are important are marked with an asterisk.)

Page

1,

""

6,

*Art. "" 99 29 "" 99

"" 99 99

123, 259 , 276, 291, 292, 322, 361 , 459, 470, 489,

555, 99 593, "" 604, 99 713, "" 897, 99 29 922ii., 949, "" 1076, 99 1158, 99 1178, "" 1241 , "" "" 1413, 1491 , "" 1849, "" 39 1903, 1925, "" 19 1954-6, ,, 2030-2 , 2035, "" * "" 2140, "" 2136, 19 2354, 2392, 39 99

99 99 29 "" "9 99 "" 99

2465, 3237, 3751, 4678, 4680, 4692, 4903, 5154, 5330,

Line 7 , "" 10, 29 5, "" 6, "" 2, "" 2, "" 6, 29 1, ", 3 & 4, "" 9, 99 7, "" 3 & 9, "9 1, "" 9,

for volume ,, gramme-million "" 1.4971499 "" .6679358 "" 215 "" a²+ B2 99 3n² + n - 1 "" δ "" a "" 45 and 13 99 3528 99 -6 99 Xm

14, "" 11 & 12,,, 2, 99 2, "" 5, 99 "" "" last, "" "" 2, 29 last, "" "" 4, "" "" 1, "" 99 "" 3, "" 3, "" "" "" 1, footnote,,,

99 "" "" "" 99

"" 1 , "" 1, 99 last,

19 "" "" 99 ""

,, 1,

"" 99 99 ""

5, 3, 4, 2,

""

read weight. "" gramme-six. "" ⚫4971499. 39 1.1447299 . "" 2/15. "" (a² + B²)n. "" 3n² + 3n - 1 . "" 7. "" a. "" 35 and 10. "" 8584. "" -16. "" x . ""

a number of rows "2 two columns. R and R "" R₁ and R2. one-sixtieth "" one-ninetieth . II. "" III. cos c 99 sin c. b2+ 2c² "" 262+ c². D "" C. dele " The projections ... are parallel." 1201 99 1217. PS "" PS'. parallel "" conjugate. -du dv "" + du dv. "" = + "" -f(x). ,, (x) supply dx. supply x. erroneous, because 7 in ( 1427 ) is necessarily an integer. ax 22 a. ✓ applies to the whole denominator. 2294 99 2293. 99 + xp 99 x -1 . T p "" 2 2 99 (n − 1) . (n − 1 ) x supply ur. dele 2 in the second term . supply the factor 4 on the left. dele 2 in the second term. supply the factor 4 on the left. 3155 "" 5155 . m "" b.

"" "" and refer to Fig. 129, Art . 5332 on the cardioid is wanting.

OF TH UNIVERSITY OF ALFORHIN

MATHEMATICAL

TABLES .

INTRODUCTION.

The Centimetre- Gramme- Second system of units. NOTATION .― The decimal measures of length are the kilo metre, hectometre, decametre, metre , decimetre, centimetre, millimetre. The same prefixes are used with the litre and gramme for measures of capacity and volume. weight Also , 107 metres is denominated a metre- seven ; 10-7 metres , a seventh-metre ; 1015 grammes , a gramme-fifteen ; and so on . A gramme-million is also called a

megagramme ; and a

millionth-gramme , a microgramme ; and similarly with other measures . DEFINITIONS . -The C. G. S. system of units refers all phy sical measurements to the Centimetre (cm. ) , the Gramme (gm. ) , and the Second (sec. ) as the units of length , mass , and time . The quadrant of a meridian is approximately a metre seven. More exactly, one metre 3.2808694 feet 39-370432 inches . The Gramme is the Unit of mass , and the weight of a gramme is the Unit of weight, being approximately the weight of a cubic centimetre of water ; more exactly , 1 gm . = 15.432349 grs . The Litre is a cubic decimetre : but one cubic centimetre is the C. G. S. Unit of volume. 1 litre = 035317 cubic feet =

2200967 gallons .

The Dyne (dn .) is the Unit offorce , and is the force which , in one second generates in a gramme of matter a velocity of one centimetre per second . The Erg is the Unit of work and energy, and is the work done by a dyne in the distance of one centimetre . The absolute Unit of atmospheric pressure is one megadyne per square centimetre = 74.964 cm . , or 29.514 in. of mercurial column at 0° at London , where g 98117 dynes .

Elasticity of Volumek , is the pressure per unit area upon a body divided by the cubic dilatation . B

evverte

2

MATHEMATICAL TABLES.

Rigidityn, is the shearing stress divided by the angle of the shear. Young's Modulus = M, is the longitudinal stress divided by the elongation produced , = 9nk ÷ (3k +n) . Tenacity is the tensile strength of the substance in dynes per square centimetre. The Gramme-degree is the Unit of heat, and is the amount of heat required to raise by 1° C. the temperature of 1 gramme of water at or near 0°. Thermal capacity of a body is the increment of heat divided by the increment of temperature. When the incre ments are small, this is the thermal capacity at the given temperature. Specific heat is the thermal capacity of unit mass of the body at the given temperature .

The Electrostatic unit is the quantity of electricity which repels an equal quantity at the distance of 1 centimetre with the force of 1 dyne . The Electromagnetic unit of quantity := 3X1010 electro static units approximately . The Unit ofpotential is the potential of unit quantity at unit distance . The Ohm is the common electromagnetic unit of resistance , and is approximately = 10° C. G. S. units. The Volt is the unit of electromotive force, and is = 109 C. G. S. units ofpotential. The Weber is the unit of current, being the current due to an electromotive force of 1 Volt , with a resistance of 1 Ohm. It is = C. G. S. unit. = Resistance ofa Wire Specific resistance × Length ÷ Section .

Physical constants and Formulæ. In the latitude of London, g = 32-19084 feet per second. = 981.17 centimetres per second. In latitude A, at a height h above the sea level, g = = (980· 6056—2: 5028 cos 2A - 000003 h ) centimetres per second. Seconds pendulum = ( 99-3562—2536 cos 2A - 0000003 h) centimetres. THE EARTH.- Semi-polar axis, 20,854895 feet* = 635411 x 108 centims . 20,926202 99 * = 6 37824 × 108 Mean semi- equatorial diameter, 99 39-377786 × 107 inches* = 1 · 000196 × 107 metres. Quadrant of meridian, Volume, 1.08279 cubic centimetre-nines . Mass (with a density 5 ) = Six gramme- twenty- sevens nearly.

These dimensions are taken from Clarke's " Geodesy," 1880 .

MATHEMATICAL TABLES.

Velocity in orbit = 2933000 centims per sec. Obliquity , 23° 27′ 16″.* Angular velocity of rotation = 1 ÷ 13713. Precession, 50"-26. * Progression of Apse, 11 "-25 . Eccentricity, e01679. Centrifugal force of rotation at the equator, 3.3912 dynes per gramme. Force of attraction upon moon, 2701. Force of sun's attraction, 5839. Ratio of g to centrifugal force of rotation, g : rw³ = 289. Sun's horizontal parallax, 8"-7 to 9'.* Aberration, 20" 11 to 20"-79. * Semi-diameter at earth's mean distance, 16′ 1 ″·82 . * Approximate mean distance, 92,000000 miles, or 1.48 centimetre-thirteens .† Tropical year, 365-242216 days , or 31,556927 seconds. Sidereal year, 365-256374 31,558150 "" 99 Anomalistic year, 365-259544 days. Sidereal day, 86164 seconds. THE MOON.- Mass Earth's mass × 011364 = 6· 98 1025 grammes. Horizontal parallax. From 53′ 56″ to 61′24″. * Sidereal revolution , 27d . 7h. 43m . 11 ·46s . Lunar month, 29d . 12h . 44m . 2 · 87s. Greatest distance from the earth, 251700 miles, or 4:05 centimetre- tens, 3.63 Least 225600 99 99 19 Inclination of Orbit, 5° 9' . Annual regression of Nodes, 19° 20′. RULE. (The Year + 1 ) ÷ 19 . The remainder is the Golden Number. (The Golden Number - 1 ) x 11 ÷ 30 . The remainder is the Epact. mm' GRAVITATION.- Attraction between masses = dynes. m, m' at a distance l 1 × 1· 543 × 107 }

The mass which at unit distance ( 1 cm.) attracts an equal mass with unit force ( 1 dn. ) is = √ ( 1 · 543 × 107) gm. = 3928 gm . WATER.-Density at 0° C., unity ; at 4°, 1000013 (Kupffer) . Volume elasticity at 15°, 2-22 × 10¹º. Compression for 1 megadyne per sq. cm., 4.51 × 10-5 (Amaury and Descamps ). The heat required to raise the temperature of a mass of water from 0° to to is proportional to t + 00002 +0000003 (Regnault) . GASES.-Expansion for 1° C., 003665 1 ÷ 273. Specific heat at constant pressure = 1.408. Specific heat at constant volume Density of dry air at 0° with Bar. at 76 cm . = 0012932 gm. per cb. cm. (Regnault) . At unit pres. (a megadyne) Density = 0012759. Density at press . p = px 1.2759 × 10-". Density of saturated steam at tº °, with p take * 7936098p n} = from Table II., is approximately (1 + 003667) 109° (elasticity of medium ÷ density) . SOUND. -Velocity = Velocity in dry air at t° = 33240 √(1 + 00366t) centimetres per second. 99 "" Velocity in water at 0° = 143000

LIGHT.- Velocity in a medium of absolute refrangibility μ = 3·004 × 10¹º ÷ µ (Cornu ) . If P be the pressure in dynes per sq. cm., and t the temperature, μ- 12903 x 10 - P ÷ (1 + 003661) (Biot & Arago). * These data are from the " Nautical Almanack" for 1883. + Transit of Venus, 1871, " Astrom. Soc. Notices," Vols. 37 , 38.

MATHEMATICAL TABLES.

TABLE I. Various Measures and their Equivalents in C. G. S. units. Dimensions. = 2.5400 = 30-4797 = 160933 = 185230

Pressure.

1 1 1 1

inch foot mile nautical do.

1 1 1 1

sq. inch = 6·4516 sq. cm. 99 sq. foot == 929.01 sq. yard = 8361.13 99 2.59 × 10¹º,, sq . mile cb. inch = 16.387 cb. cm. Force of Gravity. cb. foot = 28316 99 981 upon 1 gramme dynes 764535 29 = 63.56777 cb. yard "" "" 1 grain 99 = 2.7811 × 10" 99 gallon = 4541 "" 1 oz . = 277-274 cb. in. or the vo = 4.4497 x 105 "" 99 1 lb. lume of 10 lbs. of water = 4.9837 x 107 99 99 1 cwt. at 62° Fah., Bar. 30 in. = 9.9674 × 108 99 "" 1 ton

1 1 1 1

cm. "" 99 ""

Mass.

1 grain 1 ounce 1 pound 1 ton 1 kilogramme 1 pound Avoir. 1 pound Troy

06479895 gm. = 28.3495 29 = 453.5926 "" = 1,016047 = 2.20462125 lbs. = 7000 grains 99 =€ 5760

981 dynes per sq.cm. 1 gm. per sq. cm. "" 1 lb. per sq . foot = 479 99 1 lb. per sq . in. = 68971 76 centimetres of mercury = 1,014,000 99 at 0° C. lbs. per sq. in. 1 = 70·307 = ⚫014223 gms. per sq. cm.

Work (g = 981 ) . 1 gramme -centimetre = 981 ergs . 1 kilogram -metre = 981 × 105 99 = 1.937 × 10³ 99 1 foot-grain = 1.356 × 107 "" 1 foot-pound 1 foot-ton = 3· 04 × 1010 "" 1 'horse power' p. sec. = 7.46 × 10° ""

Heat. Velocity. 1 gramme-degree C. = 42 x 10° ergs. = 191 × 44.704 cm. per sec. 1 pound-degree 1 mile per hour X 108 99 1 kilometre 99 = 27.777 1 pound-degree Fah. 106 x 108 "9 ""

TABLE II .

TABLE III .

Pressure of Aqueous Vapour in dynes per square centim .

Values for the principal Lines of the Spectrum in air at 160° C. with Bar. 76 cm .

Pressure. 1236 1866 2790 4150 6133 8710 12220 16930 23190 31400 42050

Temp . 40° 50° 60° 80° 100° 120° 140° 160° 180° 200°

Pressure. 73200 122600 198500 472900 1014000 1988000 3626000 6210000 10060000 15600000

Wave-length in centims. 7.604 x 10-5 6.867 99 6.56201 99 D (mean) 5.89212 "" 5.26913 39 4.86072 22 4.30725 39 3.96801 29 3.93300 ""

ABCDEFGAB

Temp . -20° -15° -10° - 5° 0° 5° 10° 15° 20° 25° 30°

No.ofvibrations per second . 3· 950 × 10¹4 4.373 "" 4.577 "" "" 5:097 5.700 29 6.179 "" 6.973 "" 7.569 "" 7.636 ""

*

M.

Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

Rigidity

2:40

0.46669 0.72826 1.01678 1.66578 :45378 5 10.07328 20.07612 30.29888

― S 0.415

0.964 1.456 1.841

30750 0 0.71840 0.98322 1.38160

.m h ."

of Time .Rotation

381 1000

888000 3000 7700 7926 4500 92000 75000 36000 35000

. Density

―

13360 5690

9827

9158 2081 96190 19850 1521 1615

Elec t M . agn of Rate Conduction Specific Resistance Sound of °C. at 0 per cm .in sec

354936 0.25 2:01 0.118 0.883 0.97 1:00 1.000 0.132 0.72 338.034 0.24 101.064 0.13 14.789 0.15 24.648 0.27

Diameter . Mass in . Miles

000875 ⚫0335 001483 ⚫0330

8. 600 0 87.9697 24 2 5 8 0 8 23 21 33 31 224-701 2 23 56 365-256 4 22 37 24 51 5 1 686-980 55 26 1 4332-585 4.95182 8 40 9 17 29 10 29 10759-2202 9-00442 28 1 18.2891630686-821 30 46 46 1 60126-722 29-77506 59

O

Expansion Linear Specific Relative Heat be of ExpansionConduc Volume between degree tween 0 per tivity . C. 100 & 100 C. 0&

1 ×05 2.69 1.74 "" 1.23 180 "" 2.61 96 973 0 000061001 · 557 "" 898 0949 75 3.74 000054001 "" 3.56 00005400193 357 4.32 "" 000033 001111 19 5.06 374 .1098 001258 000040 99 5.22 001260― 000037 "" 304 7 0 · 000630022 ·0927 94363 000088002 4.53 •1770 81000015000 99

Inclination of Orbit Ecliptic .to

TABLE V.

58.6 "" 79.3 "" 3.17 99 5.17 99 ―

Sidereal Revolution Days .in

99

""

""

C. 000027 000045 0 · 00180 861 000086002 1×08 2.28

Tenacit y .

41.4 33.8 "" "9

Elasticity volume of

10.542 ×0¹2

. k

Least distance Sun. from

99

1011 x 4-47 1.684 3.66 99 5.32 99 1-963 7-677 7.69 "" 8.19 99

. n

distance Greatest Sun. from Earth's mean distance 1. =

Platinum 21 19.26 Gold Mercury 13.596 59 0 · 0¹2 11:35 1× Lead Silver 10.47 Copper 8.843 1.234 99 d , rawn Brass 8.471 1.075 "" ,cast Iron 1.349 7.235 "" w , rought Iron 99 Steel 2.139 7.849 "" ,cast Tin 7.29 Zinc ,cast 7.19 fGlass , lint 0.603 2.942 ""

.1=

Young's Density Water s Modulu

39

C. .IV units S. G. TABLE in

MATHEMATICAL TABLES. с5т

6

MATHEMATICAL TABLES.

TABLE VI.-Functions of π and e.

ta

c

a rr

π π = 3.1415926 ⚫3183099 π2 = 9.8696044 •1013212 π = π-3 = 31.0062761 π = ⚫0322515 200º ÷ π = 6396619772 √π = 1 · 7724539 180°. T= 57°2957795 log104971499 = 206264" 8 loge π = 06679358 +-144729876 TABLE VII.

TABLE VIII,

No.

Square root.

Cube root.

N.

234

1.4142136 1.7320508 2.0000000 2.2360680 2.4494897 2.6457513 2.8284271 3.0000000 3.1622777 3.3166248 3.4641016 3.6055513 3.7416574 3-8729833 4.0000000 4.1231056 4-2426407 4.3588989 4-4721360 4.5825757 4.6904158 4.7958315 4.8989795 5·0000000 5.0990195 5.1961524 5.2915026 5.3851648 5:4772256

1.2599210 1-4422496 1.5874011 1.7099759 1.8171206 1.9129312 2.0000000 2-0800837 2.1544347 2.2239801 2.2894286 2: 3513347 2-4101422 2: 4662121 2.5198421 2.5712816 2.6207414 2.6684016 2.7144177 2-7589243 2-8020393 2.8438670 2.8844991 2.9240177 2.9624960 3:0000000 3:0365889 3: 0723168 3: 1072325

23

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

e = 2.71828183 e² == 738905611 e-1 = 0.3678794 e - ² = 0 · 1353353 log10 e = 0· 43429448 log, 10 = 2 · 30258509

5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109

log10 N.

log, N.

69314718 •3010300 •4771213 1.09861229 *6989700 1.60943791 ⚫8450980 1.94591015 1.0413927 2:39789527 1.1139434 2.56494936 1.2304489 2.83321334 1.2787536 2.94443898 1.3617278 3.13549422 1.4623980 3.36729583 1.4913617 3.43398720 1-5682017 3.61091791 1.6127839 3.71357207 1.6334685 3.76120012 1.6720979 3.85014760 1.7242759 3.97029191 1.7708520 4:07753744 1.7853298 4: 11087386 1.8260748 4.20469262 1.8512583 4-26267988 1.8633229 4.29045944 1.8976271 4-36944785 1.9190781 4-41884061 1.9493900 4-48863637 1.9867717 4.57471098 2-00432144-61512052 2 : 0128372 4:63472899 2:0293838 4-67282883 2.0374265 4.69134788

NOTE. The authorities for Table IV. are as follows :-Columns 2, 3, and 4 (Mercury excepted) , Everett's experiments ( Phil. Trans., 1867) ; g is here taken 981-4. The densities in these cases are those of the specimens Col. 6, Watt's Dict. of Chemistry. employed. Cols. 5 and 7, Rankine. Col. 8, Dulong and Petit . Col. 10, Wertheim. Col. 11 , Matthiessen. Table V. is abridged from Loomis's Astronomy. The values in Table III. are Angström's.

ང་ BURCKHARDT'S

FACTOR

TABLES .

FOR ALL NUMBERS FROM 1 TO 99000 .

EXPLANATION. The tables give the least divisor of every number from 1 up to 99000 : but numbers divisible by 2 , 3 , or 5 are not printed . All the digits of the number whose divisor is sought , excepting the units and tens , will be found in one of the three rows of larger figures . The two remaining digits will be found in the left-hand column . The least divisor will then be found in the column of the first named digits , and in the row of the units and tens . If the number be prime, a cipher is printed in the place of its least divisor. The numbers in the first left-hand column are not conse cutive. Those are omitted which have 2 , 3 , or 5 for a divisor. Since 22.3.5 = 300 , it follows that this column of numbers will re-appear in the same order after each multiple of 300 is reached . MODE OF USING THE TABLES .-If the number whose prime factors are required is divisible by 2 or 5 , the fact is evident upon inspection , and the division must be effected . The quotient then becomes the number whose factors are required . If this number, being within the range of the tables , is yet not given , it is divisible by 3. Dividing by 3 , we refer to the tables again for the new quotient and its least factor , and so on. EXAMPLES. -Required the prime factors of 310155 . Dividing by 5, the quotient is 62031. This number is within the range of the tables. But it is not found printed. Therefore 3 is a divisor of it. Dividing by 3, the quotient is 20677. The table gives 23 for the least factor of 20677. Dividing by 23, the quotient is 899. The table gives 29 for the least factor of 899. Dividing by 29, the quo tient is 31 , a prime number. Therefore 3101553.5.23.29.31 . Again, required the divisors of 92881. The table gives 293 for the least divisor. Dividing by it, the quotient is 317. Referring to the tables for 317, a cipher is found in the place of the least divisor, and this signifies that 317 is a prime number. Therefore 92881 293 x 317, the product of two primes. It may be remarked that, to have resolved 92881 into these factors with out the aid of the tables by the method of Art. 360 , would have involved fifty-nine fruitless trial divisions by prime numbers.

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MATHEMATICAL TABLES.

TABLE VI .- Functions of

11

a

at

r ér

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TABLE VII .

and e. e = 2.71828183 e² = 738905611 e-¹0:3678794 e-2 = 0·1353353 log10 e = 0·43429448 log, 10 = 2 ·30258509

TABLE VIII ,

No.

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Cube root.

N.

log10 N.

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69314718 1.09861229 1-60943791 1.94591015 2:39789527 2.56494936 2-83321334 2.94443898 3.13549422 3:36729583 3:43398720 3.61091791 3.71357207 3-76120012 3-85014760 3.97029191 4-07753744 4: 11087386 4-20469262 4-26267988 4-29045944 4-36944785 4-41884061 4-48863637 4: 57471098 4-61512052 463472899 4: 67282883 4: 69134788

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NOTE. The authorities for Table IV. are as follows : -Columns 2 , 3, and 4 (Mercury excepted ) , Everett's experiments ( Phil . Trans., 1867 ) ; g is here taken 981-4. The densities in these cases are those of the specimens employed. Cols . 5 and 7, Rankine. Col. 6 , Watt's Dict. of Chemistry. Col. 8, Dulong and Petit . Col. 10, Wertheim. Col. 11 , Matthiessen . Table V. is abridged from Loomis's Astronomy. The values in Table III. are Angström's .

7

BURCKHARDT'S

FACTOR

TABLES .

FOR ALL NUMBERS FROM 1 To 99000 .

EXPLANATION.

The tables give the least divisor of every

number from 1 up to 99000 : but numbers divisible by 2 , 3, or 5 are not printed . All the digits of the number whose divisor is sought , excepting the units and tens , will be found in one of the three rows of larger figures . The two remaining digits will be found in the left- hand column . The least divisor will then be found in the column of the first named digits , and in the row of the units and tens . If the number be prime , a cipher is printed in the place of its least divisor . The numbers in the first left-hand column are not conse Those are omitted which have 2 , 3 , or 5 for a divisor. cutive. Since 22.3.52 = 300 , it follows that this column of numbers will re-appear in the same order after each multiple of 300 is reached . MODE OF USING THE TABLES .-If the number whose prime factors are required is divisible by 2 or 5 , the fact is evident upon inspection , and the division must be effected . The quotient then becomes the number whose factors are required . If this number, being within the range of the tables , is yet not given, it is divisible by 3. Dividing by 3 , we refer to the tables again for the new quotient and its least factor , and so on.

I

EXAMPLES . - Required the prime factors of 310155 . Dividing by 5 , the quotient is 62031. This number is within the range of the tables. But it is not found printed. Therefore 3 is a divisor of it. Dividing by 3 , the quotient is 20677. The table gives 23 for the least factor of 20677. Dividing by 23, the quotient is 899. The table gives 29 for the least factor of 899. Dividing by 29, the quo tient is 31 , a prime number. Therefore 3101553.5.23.29.31 . Again, required the divisors of 92881. The table gives 293 for the least divisor. Dividing by it, the quotient is 317. Referring to the tables for 317, a cipher is found in the place of the least divisor, and this signifies that 317 is a prime number. Therefore 92881 = 293 × 317 , the product of two primes. It may be remarked that, to have resolved 92881 into these factors with out the aid of the tables by the method of Art. 360, would have involved fifty-nine fruitless trial divisions by prime numbers.

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1

2

3

4

5

6

7

8

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1.00 97497 950019251290030 87555 8508782627 80173 77727 1.01 9.9975287 72855 70430 68011 65600 63196 60799 58408 56025 53648 1.02 51279 48916 46561 44212 41870 39535 37207 34886 32572 30265 1.03 27964 25671 23384 21104 18831 16564 14305 12052 09806 07567 1.04 05334 03108 00889 98677 96471 94273 92080 89895 87716 85544 1.05 9.9883379 81220 79068 76922 74783 72651 70525 68406 66294 64188 1.06 62089 59996 57910 55830 53757 51690 49630 47577 45530 43489 1.07 41469 39428 37407 35392 33384 31382 29387 27398 25415 23449 1.08 21469 19506 17549 15599 13655 1171709785 07860 0594104029 1.09 02123 00223 98329 96442 94561 92686 90818 88956 87100 85250 1.10 9.9783407 81570 79738 77914 76095 74283 72476 70676 68882 1.11 65313 63538 61768 60005 58248 56497 54753 53014 51281 1.12 47834 46120 44411 42709 41013 39323 37638 35960 34288 1.13 30962 29308 27659 26017 24381 22751 21126 19508 17896 1.14 14689 13094 11505 09922 08345 06774 052090365002096 1.15 9.9699007 97471 95941 94417 92898 91386 89879 88378 86883 83910 82432 80960 79493 78033 76578 75129 73686 72248 1.16 1.17 69390 67969 66554 65145 63742 62344 6095259566 58185 1.18 55440 54076 52718 5136650019 48677 47341 46011 44687 1.19 42054 40746 39444 38147 36856 35570 34290 33016 31747

67095 49555 32622 | 16289 00549 85393 70816 56810 43368 30483

1.20 29225 27973 26725 25484 24248 23017 21792 2057319358 18150 1.21 16946 15748 14556 1336912188 1101109841 08675 07515 06361 1.22 05212 04068 02930 01796 00669 99546 98430 97318 96212 95111 1.23 9.9594015 92925 91840 9076089685 886168755386494 85441 | 84393 | 1.24 83350 82313 81280 80253 79232 78215 77204 76198 75197 | 74201 1.25 73211 72226 71246 702716930168337 67377 66423 6547464530 1.26 63592 62658 61730 6080659888 58975 58067 57165 56267 55374 1.27 54487 53604 52727 51855 50988 50126 49268 48416 4757046728 1.28 45891 45059 44232 43410 42593 41782 40975 40173 39376 38585 1.29 37798 37016 36239 35467 34700 33938 33181 32439 3168230940

1.30 30203 29470 28743 28021 27303 1.31 23100 22417 21739 21065 20396 1.32 16485 15850 15220 14595 13975 1.33 10353 09766 09184 0860608034 1.34 04698 04158 03624 03094 | 02568 | 1.35 9.9499515 99023 98535 98052 97573 1.36 94800 94355 93913 93477 93044 1.37 90549 90149 89754 89363 88977 1.38 86756 86402 86052 85707 85366 1.39 83417 83108 82803 82503 82208 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

80528 78084 76081 74515 73382 72677 72397 72539 73097 74068

80263 80003 7786477648 7590575733 7438274254 73292 73207 72630 72587 7239372392 72576 72617 73175 73258 74188 | 74312

26590 25883 25180 24482 23789 1973219073 18419 1777017125 13359 12748 12142 11540 | 10944 07466 0690306344 05791 | 05242 02048 01532 0102100514 | 00012 97100 966309616695706 | 95251 92617 92194 91776 91362 90953 88595 88218 87846 87478 87115 85030 84698 84371 84049 83731 81916 81630 81348 81070 80797

79748 79497 79250 79008 78770 77437 77230 77027 76829 76636 75565 75402 75243 7508974939 74130 | 7401073894 7378373676 73125 73049 72976 7290872844 72549 72514 72484 72459 72437 72396 7240472416 72432 72452 72662 72712 72766 72824 72886 73345 73436 735317363073734 744407457274708 74848 74992

78537 78308 76446 76261 74793 | 74652 73574 73476 72784 72728 72419 72406 72477 72506 72952 | 73022 7384173953 | 75141 | 75293

NOTE. This table is taken from Vol . II. of Legendre's work, and not from Vol. I., as stated in the Preface : the numbers given in Vol. I. being inaccurate in the seventh decimal place. In Vol II. the values are given to twelve places of decimals . The figure here printed in the seventh place is

31

Log г (n). n

0

1

2

3

1.509.9475449 | 7561075774 75943 1.51 7723777438 77642 77851 1.52 79426 796677991280161 1.53 82015 82295 82580 82868 1.54 84998 85318 85642 85970 1.55 88374 88733 | 8909689463 92139 925379293893344 1.56 1.57 96289 96725 97165 97609 1.58 9.9500822 0129601774 02255 1.59 05733 062450676007280

1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69

11020 16680 22710 29107 35867 42989 50468 58303 66491 75028

4

5

6

7

8

9

76116 76292 76473 76658 76847 77040 78064 78281 78502 78727 78956 79189 80414 80671 80932 81196 81465 81738 83161 83457 83758 84062 84370 84682 8630286638 86977 87321 87668 88019 89834 90208 90587 90969 91355 91745 93753 94166 94583 95004 9542995857 980569850898963 9942299885 00351 0274103230 | 03723 04220 04720 | 05225 07803083300886009395 09933 | 10475

11569 | 12122 | 12679 13240 13804 14372 14943 15519 17267 17857 18451 19048 19650 20254 20862 21475 23333 23960 24591 2522525863 | 26504 27149 27798 29767 30430 31097 31767 32442 33120 | 33801 | 34486 36563 37263 37966 3867339383 40097 40815 41536 43721 44456 45195 45938 46684 47434 48187 48944 51236 52007 52782 5356054342 | 55127 55916 56708 59106 599136072361536 62353 63174 63998 64826 67329 68170 69015 69864 70716 715717243073293 75901 76777 7765778540 79427 80317 81211 82108

16098 22091 28451 35175 42260 49704 | 57504 65656 74159 83008

1.70 83912 84820 85731 86645 87563 88484 89409 90337 91268 92203 1.71 93141 94083 95028 95977 96929 97884 98843 99805 00771 01740 1.72 9.96027120368804667 05650 06636 07625 08618 09614 10613 11616 1.73 12622 13632 14645 15661 16681 17704 18730 19760 20793 21830 1.74 22869 23912 24959 26009 27062 28118 29178 30241 31308 32377 1.75 33451 34527 3560736690 37776 38866 39959 41055 42155 | 43258 1.76 44364 45473 46586 47702 48821 49944 51070 52200 53331 | 54467 1.77 55606 56749 57894 59043 6019561350 6250963671 64836 66004 1.78 67176 68351 69529 70710 71895 73082 74274 75468 76665 | 77866 1.79 79070 80277 81488 82701 83918 85138 86361 87588 88818 90051

91287 1.80 1.81 9.9703823 16678 1.82 29848 1.83 43331 1.84 57126 1.85 71230 1.86 85640 1.87 1.88 9.9800356 15374 1.89

92526 05095 17981 31182 44697 58522 72657 87098 01844 16893

93768 06369 19287 32520 46065 59922 74087 88559 03335 18414

95014 07646 20596 33860 47437 61325 75521 90023 04830 19939

96263 97515 08927 10211 21908 23224 35204 36551 4881250190 62730 64140 76957 78397 9149092960 06327 07827 21466 22996

98770 00029 11498 12788 24542 25864 37900 39254 51571 52955 6555166966 79839 81285 94433 95910 09331 10837 24530 26066

30693 32242 33793 35348 36905 38465 40028 1.90 46311 47890 49471 51055 52642 54232 55825 1.91 62226 63834 65445 670586867570294 71917 1.92 78436 80073 81713 83356 85002 8665188302 1.93 1.94 94938 96605 98274 99946 01621 03299 04980 1.95 9.9911732 13427 15125 16826 18530 20237 21947 1.96 28815 30539 32266 33995 35728 37464 39202 1.97 46185 47937 49693 51451 53213 54977 56744 1.98 63840 65621 67405 69192 70982 72774 74570 | 1.99 81779 83588 85401 87216 89034 90854 92678

41595 57421 73542 89957 06663 23659 40943 58513 76368 94504

01291 2555 14082 15378 27189 28517 40610 41969 54342 55733 6838469805 8273484186 97389 98871 12346 13859 27606 | 29148 43164 44736 5902060622 75170 76802 91614 93275 08350 10039 25375 27093 42688 44435 60286 62062 | 78169 79972 96333 98165

the one nearest to the true value whether in excess or defect. This table, and the table of Least Factors, have each been subjected to two complete and in dependent revisions before finally printing off.

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1

2

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78

1.00 97497 95001 92512 90030 87555 85087 82627 80173 1.01 9.9975287 72855 70430 68011 65600 63196 60799 58408 56025 1.02 51279 48916 46561 44212 41870 39535 37207 34886 32572 1.03 27964 25671 23384 21104 18831 16564 14305 12052 09806 1.04 05334 03108 00889 98677 96471 94273 92080 89895 87716 1.05 9.9883379 81220 79068 76922 74783 72651 70525 68406 66294 1.06 62089 59996 57910 55830 53757 51690 49630 47577 45530 1.07 41469 39428 37407 35392 33384 31382 29387 27398 25415 1.08 21469 19506 17549 15599 13655 11717 09785 07860 05941 1.09 02123 00223 98329 96442 94561 92686 90818 88956 87100

9 77727 53648 30265 | 07567 85544 64188 43489 23449 04029 85250

1.10 9.9783407 81570 79738 77914 76095 74283 72476 70676 68882 67095 1.11 65313 63538 61768 60005 582485649754753 53014 51281 49555 1.12 47834 46120 44411 42709 41013 39323 37638 35960 34288 32622 1.13 30962 29308 27659 26017 24381 22751 21126 19508 17896 16289 1.14 14689 13094 11505 09922 08345 067740520903650 0209600549 1.15 9.9699007 97471 95941 94417 92898 91386 89879 88378 86883 85393 1.16 83910 82432 80960 79493 78033 76578 75129 73686 72248 70816 | 1.17 69390 67969 66554 65145 63742 62344 6095259566 58185 56810 1.18 55440 54076 52718 51366 50019 48677 47341 46011 4468743368 1.19 42054 40746 39444 38147 36856 35570 34290 33016 31747 30483

1.20 29225 27973 26725 25484 24248 23017 21792 20573 1.21 16946 15748 14556 1336912188 11011 09841 08675 1.22 05212 04068 02930 01796 00669 99546 98430 97318 1.23 9.9594015 92925 91840 9076089685 88616 87553 86494 1.24 83350 82313 81280 80253 79232 78215 77204 76198 1.25 73211 72226 71246 70271 69301 68337 67377 66423 1.26 63592 62658 61730 6080659888 58975 58067 57165 1.27 54487 53604 52727 51855 50988 50126 49268 48416 1.28 45891 45059 44232 43410 42593 41782 40975 40173 1.29 37798 37016 36239 35467 34700 33938 33181 32439

1935818150 07515 06361 96212 95111 85441 84393 75197 74201 65474 64530 56267 55374 47570 46728 39376 38585 31682 30940

1.30 30203 29470 28743 28021 27303 26590 25883 25180 24482 23789 1.31 23100 22417 21739 2106520396 | 19732 19073 18419 1777017125 1.32 16485 15850 15220 14595 13975 13359 12748 12142 11540 | 10944 1.33 10353 09766 09184 0860608034 07466 06903 06344 0579105242 1.34 04698 04158 03624 03094 02568 02048 015320102100514 | 00012 1.35 9.9499515 99023 98535 98052 97573 97100 96630 96166 95706 95251 1.36 94800 94355 93913 93477 93044 92617 9219491776 9136290953 1.37 90549 90149 89754 89363 88977 88595 88218 87846 87478 87115 1.38 86756 8640286052 857078536685030 84698 84371 84049 83731 1.39 83417 83108 82803 8250382208 81916 81630 81348 81070 80797 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

80528 78084 76081 74515 73382 72677 72397 72539 73097 74068

80263 80003 77864 77648 75905 75733 74382 74254 73292 73207 72630 72587 7239372392 72576 72617 73175 73258 74188 74312

79748 79497 79250 77437 77230 77027 755657540275243 74130 | 7401073894 73125 73049 72976 72549 72514 72484 723967240472416 72662 72712 72766 73345 73436 73531 74440 74572 74708

79008 76829 75089 73783 72908 72459 72432 72824 73630 74848

78770 76636 74939 73676 72844 72437 72452 72886 73734 74992

78537 78308 76446 76261 74793 74652 73574 73476 72784 72728 7241972406 72477 72506 72952 73022 7384173953 75141 75293

NOTE. This table is taken from Vol. II. of Legendre's work, and not from Vol. I., as stated in the Preface : the numbers given in Vol. I. being inaccurate in the seventh decimal place. In Vol II. the values are given to twelve places of decimals. The figure here printed in the seventh place is

31

Log г (n). n

0

1

2

3

4

5

6

7

8

9

1.50 9.9475449 75610 75774 75943 76116 76292 76473 76658 76847 77040 1.51 77237 774387764277851 78064 78281 78502 78727 78956 79189 1.52 79426 796677991280161 80414 80671 80932 81196 8146581738 1.53 82015 822958258082868 83161 83457 83758 84062 84370 84682 84998 85318 85642 85970 863028663886977 87321 87668 88019 1.54 1.55 88374 887338909689463 89834 90208 90587 9096991355 91745 1.56 92139 9253792938 93344 93753 94166 94583 95004 95429 95857 1.57 96289 96725 97165 97609 980569850898963 99422 9988500351 1.58 9.9500822 | 01296 0177402255 02741 03230 | 0372304220 | 04720 | 05225 1.59 05733 062450676007280078030833008860093950993310475 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69

11020 16680 22710 29107 35867 42989 50468 58303 66491 75028

11569 17267 23333 29767 36563 43721 51236 59106 67329 75901

1212212679 17857 18451 | 23960 | 24591 30430 31097 37263 37966 44456 45195 5200752782 59913 60723 68170 69015 76777 77657

13240 13804 14372 14943 1551916098 19048 1965020254 20862 21475 22091 25225 25863 26504 27149 27798 28451 31767 32442 33120 | 33801 34486 35175 38673 39383 40097 40815 4153642260 45938 46684 47434 48187 48944 49704 5356054342 55127 55916 56708 57504 61536 62353 63174 63998 64826 65656 69864 70716 71571724307329374159 78540 79427 80317 81211 82108 83008

1.70 83912 84820 85731 86645 87563 88484 89409 90337 1.71 93141 94083 95028 95977 96929 97884 98843 99805 1.72 9.9602712 03688 04667 056500663607625 08618 09614 1.73 12622 13632 14645 15661 16681 17704 1873019760 1.74 22869 23912 24959 26009 27062 28118 29178 30241 1.75 33451 34527 35607 36690 37776 38866 39959 41055 1.76 44364 45473 46586 47702 48821 49944 51070 52200 1.77 55606 56749 57894 59043 6019561350 6250963671 1.78 67176 68351 69529 70710 71895 73082 7427475468 1.79 79070 80277 81488 82701 83918 85138 86361 87588

91268 92203 00771 01740 10613 11616 20793 21830 31308 32377 42155 | 43258 53331 54467 6483666004 7666577866 88818 90051

91287 92526 93768 95014 96263 97515 98770 00029 01291 02555 1.80 1.81 9.9703823 05095 06369 07646 08927 10211 11498 12788 14082 15378 16678 17981 19287 20596 2190823224 24542 25864 27189 28517 1.82 29848 31182 32520 33860 35204 36551 37900 39254 40610 41969 1.83 43331 44697 46065 47437 48812 50190 5157152955 54342 55733 1.84 57126 58522 59922 61325 62730 64140 65551 66966 68384 69805 1.85 71230 72657 74087 75521 76957 78397 79839 81285 82734 84186 1.86 85640 87098 88559 90023 91490 92960 94433 95910 97389 98871 1.87 1.88 9.9800356 01844 03335 04830 06327 07827 09331 10837 12346 13859 15374 16893 18414 | 19939 21466 22996 24530260662760629148 1.89 30693 32242 33793 35348 36905 1.90 46311 47890 49471 51055 52642 1.91 1.92 62226 63834 65445 67058 68675 78436 80073 81713 83356 85002 1.93 1.94 94938 96605 98274 99946 01621 1.95 9.9911732 13427 15125 16826 18530 1.96 28815 30539 32266 33995 35728 1.97 46185 47937 49693 51451 53213 1.98 63840 65621 67405 69192 70982 1.99 81779 83588 85401 87216 89034

38465 40028 54232 55825 70294 71917 8665188302 03299 04980 20237 21947 37464 39202 54977 56744 72774 74570 90854 92678

41595 43164 44736 57421 5902060622 73542 75170 76802 89957 91614 93275 76663 08350 T0039 23659 25375 27093 4094342688 44435 58513 60286 62062 76368 78169 79972 94504 96333 98165

the one nearest to the true value whether in excess or defect. This table, and the table of Least Factors, have each been subjected to two complete and in dependent revisions before finally printing off."

2

ALGEBRA .

FACTORS .

123

a²b² = (ab) (a +b) . a³ - b³ = (a — b) (a² +ab + b²) . a³ +b³ = (a +b) (a² — ab +b²) .

And generally, 4

a" -b" = (a − b) ( a" -¹ + a" -2b + ... + b" -1) always .

5

a" -b" = (a +b) ( a¹ -¹ — a” -2b + ...— b" -¹) if n be even.

6

a" + b" = (a +b) ( a " -¹ — a" -2b + ... + b" -¹) if n be odd.

* 7

(x + a) (x +b) = x² + (a +b) x + ab.

8

(x +a) (x + b) ( x + c) = x² + (a + b + c) x² +(be + ca + ab) x +abc. (a + b)² = a² +2ab + b² .

9 10

(a - b)² = a² — 2ab +b².

11

(a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab (a +b) .

12

(a - b)³ = a³ -3a2b + 3ab³ - b³

a³ -b³ - 3ab (a — b).

Generally, (a +b)' = a² ± 7a®b + 21a³b² ± 35a¹b³ + 35a³b¹ ± 21a²b³ + 7ab® ± b² . Newton's Rule for forming the coefficients : Multiply any coefficient by the index of the leading quantity, and divide by the number of terms to that place to obtain the coefficient of the term next following. Thus 21x5 ÷ 3 gives 35 , the following See also (125) . coefficient in the example given above.

7

To square a polynomial : Add to the square of each term twice the product of that term and every term that follows it. Thus, (a + b + c + d)² = a² +2a (b +c + d ) + b² + 2b ( c + d ) +c² + 2cd + d². F

ALGEBRA .

34

13

a* + a²b² + b* = (a² + ab + b²) (a² — ab + b²) .

14

a* + b¹ = (a² +ab √2 + b²) (a² — ab √2 +b²) . 2

15

( x + 1 )² = x² + 1 + 2,

16

(a + b + c)² = a² + b² +c² + 2bc +2ca +2ab.

17

(a + b + c)³ = a³ + b³ + c³ + 3 (b³c + bc² +c²a + ca²

(x + 1 ) ² = x² + 1 + 3(x + 1 ).

+ ab + ab²) +6abc.

Observe that in an algebraical equation the sign of any letter may be changed throughout , and thus a new formula obtained, it being borne in mind that an even power of a negative quantity is positive. sign of c in (16) , we obtain

For example, by changing the

(a + b − c)² = a² + b² +c² - 2bc - 2ca + 2ab.

18

a² + b² - c² +2ab = (a + b)² — c² = (a +b +c) ( a + b − c) by (1 ).

19

a² - b² - c² +2bc = a²- (b − c)² = ( a + b − c) (a − b +c).

20

a³ + b + c³ - 3abc = ( a + b + c) (a² + b² +c² - bc - ca - ab) .

21

be² + b²c + ca² + c²a + ab² +a²b + a³ + b³ + c³ = (a +b + c) ( a² +b² +c²).

22

bc²+b²c + ca² + c²a + ab² + a2b + 3abc = = (a + b + c) (be + ca + ab) .

23

bc² + b²c + ca² +c²a + ab² + a²b + 2abc = (b +c) ( c + a) (a +b)

24

25

be² + be + ca² + c²a +ab² + a²b - 2abc - a³ - b³ —c³ = = (b + c − a) (c + a − b) (a +b− c). be² - b³c + ca² — c²a + ab² — a²b = (b − c) (c — a) (a — b) .

26

26 c² + 2c a² +2a²b² —a* —b¹ — c¹

= (a + b + c) (b + c − a) ( c + a − b) (a + b − c) . 27

x³ + 2x²y + 2xy² +y³ = ( x +y) ( x² + xy + y²) . Generally for the division of (x +y)" — (x² + y") by x² + xy + y²

see (545) .

MULTIPLICATION AND DIVISION.

MULTIPLICATION

AND

35

DIVISION ,

BY THE METHOD OF DETACHED COEFFICIENTS .

j

28

Ex. 1 :

(a*-3a²b² + 2ab³ + b* ) × (a³ — 2ab² - 2b³).

1 +0-3 +2 + 1 1 + 0-2-2 1 + 0-3 +2 + 1 -2-0 +6-4-2 —2—0 + 6-4-2 170-5 + 0 + 7 + 2−6-2 Result

Ex . 2 :

a7-5ab² + 7a³b' + 2ab5-6ab* -2b7.

(x − 5x + 7x³ + 2x² — 6x −2 ) ÷ ( x* −3x² + 2x + 1 ) .

1 +0-3 +2 + 1 )

1 + 0-5 + 0 + 7 + 2−6−2 ( 1 + 0−2−2 -1-0 +3-2-1 0-2-2 + 6 +2-6 +2 + 0-6 +1 + 2

-2 +0 +6-4-2 +2 +0-6 +4 +2 Result

— 2x - 2.

Synthetic Division . Ex. 3 : Employing the last example, the work stands thus,

-0 +3 -2

1 +0−5 + 0 + 7+ 2-6-2 0 +0 +0 + 0 +3 +0-6-6 -2 +0 +4 +4 -1 +0 +2 +2

!

-1

Result

1 + 0-2-2 -2x - 2.

[ See also (248 ) .

Note that, in all operations with detached coefficients, the result must be written out in successive powers of the quantity which stood in its successive powers in the original expression .

ALGEBRA.

36

INDICES . 6 3/3 / a5 ; axa = a + = a³ , or 1 +1 m +n 8% m n = a mn " or " am +". am xa" =

Multiplication :

29

Division :

a ÷ a = a 1 1 1 21 m a " ÷ a'm = a

Involution : Evolution :

(a )¹* := a³ × } = at, (a*)

HIGHEST

30

RULE .

a; Nam-n.

or Va.

21

a = = aa³ × + = ar, or a -n =

pressions

6

= a*, or m -n mn or a "

1 ans

a

a².

= = 1.

COMMON FACTOR .

To find the highest common factor of two ex Divide the one which is of the highest dimension by

the other, rejecting first any factor of either expression which Operate in the same manner is not also a factor of the other. upon the remainder and the divisor, and continue the process until there is no remainder. The last divisor will be the highest common factor required.

EXAMPLE.-To find the H. C. F. of 3x - 10x + 15x + 8 and x - 2x - 6x³ + 4x² + 13x + 6. - 2- 6+ 4 + 13 + 6 1 3 + 0−10 + 0 + 15 + 8 3 −3 + 6 + 18-12-39-18

31

1

‫رم‬

5

3-6-18 + 12 + 39 +18 −3− 4+ 6 + 12 + 5

3

2 ) 6+ 8-12 -24-10

2 ) - 10-12 + 24 +44 +18

3+ 4-6-12- 5 -3-9-9-3

- 5- 6 + 12 + 22 + 9 3

+ 5 + 15 + 15 + 5

5-15-15-5

-15-18 +36 +66 + 27 + 15 +20-30-60-25 2 ) 2+ 6+ 6+ 2 1+ 3+ 3+ 1

Result H. C. F. = x² + 3x² + 3x + 1.

EVOLUTION.

Otherwise . -

32

To form the H. C. F. of two

37

or more

algebraical expressions : Separate the expressions into their simplest factors. The H. C. F. will be the product of the factors common to all the expressions , taken in the lowest powers that occur.

LOWEST COMMON MULTIPLE .

33

The L. C. M. of two quantities is equal to their product divided by the H. C. F. Otherwise . - To form the L. C. M. of two

34

or

more

algebraical expressions : Separate them into their simplest factors. The L. C. M. will be the product of all the factors that occur , taken in the highest powers that occur.

EXAMPLE . -The H. C. F. of a² ( b - x )³c'd and a³ (b − x) ³c¹e is a² ( b - x) ²c¹ ; and the L. C. M. is a (b - x)'c'de.

EVOLUTION.

To extract the Square Root of ― 3aa3a41a +1 . a² 16 2 2 Arranging according to powers of a, and reducing to one denominator, the expression becomes

35

16a² -24a + 41a - 24a¹ +16 16

Detaching the coefficients , the work is as follows · : 16-24 +41-24 +16 ( 4-3 +4 16 8-3 -3

-24 +41 24-9

8-6 +4

Result

32-24 + 16 -32 + 24-16

4a - 3a¹ + 4 = a− 4

√a + 1

38

ALGEBRA.

To extract the Cube Root of

37

" 4 W

8xº - 36x³√y + 66x¹y −63x³y √y + 33x³y² —9x² √✅y + y³.

The terms here contain the successive powers of x and detaching the coefficients, the work will be as follows -: II. I. III.

6-3) -65

12

6-9 + 1

12-18 + 9

y ; therefore,

8-36 +66-63 +33-9 + 1 ( 2−3 +1 -8

-18+ 9

-36 +66-63 + 33-9 + 1 + 36-54 +27

+9

12-36 +27

12-36 +33-9 + 1 -12 +36-33 + 9-1

6-9 + 1 12-36 +33-9 + 1

Result

2x² - 3x√y + y.

EXPLANATION.-The cube root of 8 is 2 , the first term of the result. Place 3 x 26 in the first column I., 3 x 2² = 12 in column II. , and 2³ = 8 in III., changing its sign for subtraction. ---36 ÷ 12 = −3, the second term of the result. Put - 3 in I.; (6-3 ) x ( −3 ) gives 18 +9 for II. ( 12-18 +9) × 3 (changing sign) gives 36-54 +27 for III. Then add . Put twice ( -3), the term last found, in I., and the square of it in II. Add the two last rows in I., and the three last in II. 12 ÷ 12 gives 1 , the third term of the result. Put 1 in col. I. , ( 6−9 + 1 ) × 1 gives 6-9 + 1 for col . II. ( 12-36 +33-9 + 1 ) x 1 gives the same for III. Change the signs , and add, and the work is finished.

The foregoing process is but a slight variation of Horner's rule for solving an equation of any degree .

See (533) .

Transformations frequently required .

c+ d

a+ b 38

If b

= a› then d

39

If

x + y = ar

= [ 68.

a-b

then

and x -y = b

c- d ( x = 1 (a + b). X = 1 (a − b).

40

(x +y) ² + (x − y) ² = 2 (x² +y²) .

41

(x +y) ³— (x −y)² = 4xy.

39

EQUATIONS.

42

(x +y) ² = (x — y) ² +4xy.

43

(x− y)² = (x + y)³ — 4xy.

44

EXAMPLES .

2√a²- b² + √c²x² = _ 3√a²- bª + √√c² — ď² 2√a² -b² -√ c² —x²

3√a -b -

c² — d²²

√2²- 202 √c²- d = 2a -b 3a²-b

....... [38.

9 ( c² — x³) = 4 (c² —ď²) , x = √5c² +4d².

To simplify a compound fraction, as

1 1 + a²+ab + b² a² - ab + b² 1 1 a² + ab + b² a² - ab + b² multiply the numerator and denominator by the L. C. M. of all the smaller denominators. (a²+ab + b²) + (a³ — ab + b²) a²+ b² Result = (a² +ab + b³) — (a³ — ab + b²) ab

QUADRATIC

EQUATIONS .

-b ±√b2-4ac 45

If

ax² + bx + c = 0,

x = 2a

46

If

ax²+ 2bx +c = 0 ;

that is , if the coefficient of x a be

-b ± √ b² - ac an even number ,

X a

47

Method of solution without the formula . Ex.:

Divide by 2,

2-7x +3 = 0. 7 . / x² 2-1 2 x + 3 = 0. 2

40

ALGEBRA.

49 x² x² -3x+ ( 2)²= 18 16' 2 = 25 16 - 3

Complete the square,

X-

Take square root,

x=

5 7 =± 4' 4

7 ±5 = 3

or 2*

48

Rule for " completing the square " of an expression like

x²- x : Add the square of half the coefficient of x.

49

The solution of the foregoing equation, employing formula ( 45) , is 1 -b ± √b2-4ac = 749-24 = 7 + 5 = 3 or x= 4 2 2a 4

THEORY OF QUADRATIC EXPRESSIONS .

If a , ẞ be the roots of the equation ax² + be + c = 0 , then

ax² + bx + c = = a (x — a) (x —B) .

50

b

= a +B =

Sum of roots

51

a 52

Product of roots

aß = 2. αβ a

Condition for the existence of equal roots b2-4ac must vanish.

53

54 The solution of equations in one unknown quantity may sometimes be simplified by changing the quantity sought.

Ex. ( 1) :

2x +

18x + 6 3x - 1 = 14 + 3x + 1 6x +5x - 1

(1) .

6x² + 5x - 1 6 (3x + 1 ) = 14. + 3x + 1 6x² + 5x - 1

Put

y=

6x² + 5x - 1 3x + 1

(2) .

EQUATIONS.

41

6 thus

= 14.

y +

y y - 14y +6 = 0. y having been determined from this quadratic, x is afterwards found from (2) .

Ex. 2 :

55

x² + 1/1 + x + 1/2 = 4 . x 2 (x + 1 ) + (x + 1 ) = 6 .

1 Put x + = y, and solve the quadratic in y. x

56

Ex. 3 :

x 2² + x + 3√2x² 232 2 + x + 2 = 2 +1.

2x² + x + 3√2x² + x + 2 = 2, 2x² + x + 2 + 3√2x * + x + 2 = 4.

Put

√2x² + x + 2 = y, and solve the quadratic y² + 3y = 4 .

57

Ex . 4 :

2 16 = x 3 33 /x" 16 2n 3 x3 = + 3 2n У x3

3/2" +

59100

22

23

A quadratic in

To find Maxima and Minima values by means of a

58

Quadratic Equation . Ex .

Given

y = 3x² + 6x + 7,

to find what value of x will make y a maximum or minimum. Solve the quadratic equation

3x² + 6x + 7 - y = 0. Thus

x=

-3 ± √3y - 12 3

[46.

In order that may be a real quantity, we must have 3y not less than 12 ; therefore 4 is a minimum value of y, and the value of x which makes y a minimum is -1. G

42

ALGEBRA.

SIMULTANEOUS

EQUATIONS .

General solution with two unknown quantities. Given

59

a₁x + b₁y = c₁

" a2x + b₂y = c2

x = c₁b₂-cǝb₁ a₁b₁₂- ab₁'

C₁a₂2 -C₂ a₁ y b₁a - ba₁

General solution with three unknown quantities . 60

Given

a₁x + b₁y + c₁z = d₁ ax + b₂y+ c₂≈ = d₂ 3 ax + by + c¸≈ = ds.

1 2 3 1 2 x = d₁ (b¿cз — bçc₂) + d₂ (b¸ ¢₁ − b₁c ) +d² (b₁c₂ — b₂c₁) a₁ (b₂cз - bc2) + α₂ (b¸ ¢₁ − b₁ ¢3) +α3 (b¸ c₂ — b¸c₁ ) ' and symmetrical forms for y and z .

Methods of solving simultaneous equations between two unknown quantities x and y.

61 I. By substitution .— Find one unknown in terms of the other from one of the two equations, and substitute this value in the remaining equation.

Ex.:

From ( 2) , y = 4.

62

Then solve the resulting equation.

x +5y = 23 ...... (1 ) ? 28 ..... (2) S 7y = 28 Substitute in (1 ) ; thus x + 20 = 23, x = 3.

II. By the method of Multipliers.

Ex.:

3x +5y = 36 ...... (1 ) Į 2x - 3y = 5 ...... 8} (2) S

Eliminate a by multiplying eq . ( 1 ) by 2 , and ( 2 ) by 3 ; thus 6x + 10y = 72, 6x - 9y = 15, 19y = 57, by subtraction, y = 3; .. x = 7, by substitution in eq. (2) .

43

EQUATIONS .

63

III. By changing the quantities sought.

Ex. 1 :

x-y = 2. (1) x² -y² + x + y = 30 .... (2) S ·

Let

x + y = u,

x― y = v.

Substitute these values in ( 1 ) and (2 ) , v= 2

; ..

uv + u = 30 ) 2u + u = 30, u = 10 ; x + y = 10, x -y = 2.

From which

x= 6

and

y = 4.

x + y + 10x- y = 9 22 (1) x +y x- y x²+7y² = 64 ...... (2) i x n +y (1) ; Substitute z for x -y 10 2z+ = 9 ; 2

64

Ex . 2 :

22-9z + 10 = 0. 4/2

5

or 2,

x + y = 2 or 11

2

x- y

7 x = 3y or 34 y..

From which Substitute in (2) ; thus

Ner

z=

From which

and

y=2

x = 6,

6 or

У=

and

2√7.

√7

3x +5y = xy ...... (1) 2x +7y = 3xy ....... (2) Divide each quantity by ay ;

65

Ex . 3 :

3 +

y 2

5 = 1 ...... (3) x

+

У

= 3 ...... (4) x

Multiply (3) by 2, and ( 4) by 3, and by subtraction y is eliminated.

44

ALGEBRA.

66

IV. By substituting y = tx, when the equations are homogeneous in the terms which contain x and y.

Ex . 1 :

52x² +7xy = 5y³ ...... (1) 2

(2) 5x-3y = 17 52x +7tx = 5ť²x²..... (3) 17 ...... (4) 5x-3tx 52 + 7t = 5ť , (3) gives a quadratic equation from which t must be found , and its value substituted in (4) . a is thus determined ; and then ม from y = = tx.

From (1), and , from (2 ),

2x² +xy + 3y² = 16 ...... ( 1 ) ? 4 ...... 385 (2) S 3y-2x

Ex. 2 :

67

From (1) , by putting y = tx, x² ( 2 + t + 3ť²) = 16 ...... (3) ?

from (2),

x (3t - 2) = 4 ...... (4) $ ; x² 2 (92-12t +4) = 16 ; 92-12t + 4 = 2 + t + 3ť²,

squaring,

a quadratic equation for t. t being found from this , equation (4) will determine x ; and finally y = tx.

RATIO AND PROPORTION.

68

If a : b :: c : d ; then ad = be , and 2 = ;

a+b = c+d;

a=b = c

a+ b - c + d c-d

;

a -b

d

a

69

a

=

=

If b

d

= &c.; then

F

= a + c + e + &c . b+ d+f+ &c . '

General theorem .

70

If If // =

= & c . = k say, then

== k = { pa" + qc" + re" + & c. ) 1/1 (pb" +qd" +rf” + &c. S

where p, q, r, &c . are any quantities whatever. in ( 71 ) .

Proved as

45

RATIO AND PROPORTION.

RULE. To verify any equation between such proportional

71

quantities : Substitute for a, c, e , &c. , their equivalents kb, kd, kf, &c. respectively, in the given equation. Ex.- If ab :: c : d, to show that

√a- √b

√a- b =

√c- d

√c - √ď

Put kb for a, and kd for c ; thus √a-b ✓kb- b √b√k- 1 √b = == = √d dk- 1 √c - d ✓kd - d

√b ( √k− 1) = √b. √kb- √b Na - √b = = √kd - d √d ( k - 1 ) √c- √d

also

Identical results being obtained, the proposed equation must be true.

72

If a : b : cd : e &c. , forming a continued proportion,

then

a : c :: a² : b², the duplicate ratio of a : b, 3 a : d :: a³ : b³, the triplicate ratio of a : b , and so on.

Also

✔ab is the subduplicate ratio of a : b , a : b

is the sesquiplicate ratio of a : b . a

73

is made to approach nearer to unity in

The fraction b

value, by adding the same quantity to the numerator and Thus denominator . a+x is.

is nearer to 1 than b

b +x

74 DEF. -The ratio compounded of the ratios a : b and c : d is the ratio ac : bd. 75

If a

b

c : d , and a' : b' :: c' : d' ; then, by compound

ing ratios , aa' : bb' :: cc' : dd .

VARIATION. 76

If a ac and bac, then (a + b ) ∞ c and

ab∞ c.

b

If a xb , then ac oc bd and a α d' с cods' and

gry

78

If a xb, we may assume a = mb , where m is some constant.

46

ALGEBRA.

ARITHMETICAL PROGRESSION.

General form of a series in A. P. 79

a, a + d, a + 2d, a + 3d, ...... a + (n - 1 ) d. a = first term , d = common difference ,

1 = last of n terms, 88888

s = sum of n terms ; then

80

l = a + (n - 1 ) d.

81

n s= = (a + 1) 1/2·

82

n s = { 2a + (n − 1) d} ½2.

PROOF. together.

By writing ( 79) in reversed order, and adding both series

GEOMETRICAL PROGRESSION .

General form of a series in G. P.

arn - 1.

a, ar, ar², ar³,

83 a

first term ,

r = common ratio , I

last of n terms ,

s = sum of n terms ; then

84

1 = ar" -¹ .

85

s= a

zon.

1- n or

a 1- r

If r be less than 1 , and n be infinite, a

86

s= 1ª,,

since

‚” = 0 .

PROOF .- (85) is obtained by multiplying (83) by r, and subtracting one series from the other.

47

PERMUTATIONS AND COMBINATIONS.

HARMONICAL PROGRESSION.

888-8

87

a , b, c, d , &c. are in Harm . Prog. when the reciprocals 1 1 1 &c . are in Arith . Prog. , ' d ' b' c

88

Or when ab

: a - bb - c is the relation subsisting

between any three consecutive terms. ab 89

[87, 80.

nth term of the series =

(n - 1) a- (n - 2) 90 1

Approximate sum of n terms of the Harm . Prog. 2 Ja 1a &c. , when d is small compared with a , a+d' a +2d' a + 3d' (a + d)" — an

= d (a + d)" PROOF.-By taking instead the G.P.

1 a² a + .... + + a+d (a +d)² ' ( a + d)³

a+ b 91

Arithmetic mean between a and b =

2 92

Geometric

do .

= √ab.

93

Harmonic

do.

=

2ab

a+b The three means are in continued proportion.

PERMUTATIONS AND COMBINATIONS .

94 time

The number of permutations of n things taken all at a = n (n - 1 ) (n - 2) ... 3.2.1 = n ! or n (") .

PROOF BY INDUCTION .-Assume the formula to be true for n things . Now take n + 1 things. After each of these the remaining n things may be arranged in n ! ways, making in all nxn ! [ that is (n + 1 ) ! ] permutations of n +1 things ; therefore, &c. See also (233 ) for the mode of proof by Induction .

errata

48

ALGEBRA.

95 The number of permutations of n things taken r at a time is denoted by P (n, r) . P (n , r) = n (n − 1 ) ( n − 2) ... ( n − r + 1 ) = n (r) . PROOF.-BY (94) ; for (n - r) things are left out of each permutation ; therefore P (n, r) = n ! ÷ ( n − r) !. Observe that r

the number of factors.

96 The number of combinations of n things taken r at a time is denoted by C (n , r) .

n(r) n (n − 1 ) (n − 2) ... (n − r + 1) = C (n, r) = 1.2.3 ... " r! n! =

= C (n, n − r) . r ! (n − r) !

For every combination of tions ; therefore

things admits of

C (n, r) is greatest when r = in or 97 as n is even or odd.

98

! permuta

C (n , r) = P ( n , r ) ÷ r !

(n + 1 ) , according

The number of homogeneous products of r dimensions

of n things is denoted by H (n , r) .

H (n, r) =

n (n + 1) ( n + 2) ... ( n + r − 1) = (n +r − 1 )(*) · 1.2 ... r · r!

When r is > n , this reduces to

ata

err

(r + 1) (r + 2) ... (n + r − 1 ) 99 (x - 1 ) ! n PROOF. - H (n, r) is equal to the number of terms in the product of the expansions by the Bin. Th. of the n expressions ( 1 - ax) - , ( 1— bæ) ´¹, (1 - cx) ', &c. Put a = b c = &c. = 1. ( 1 - x) -". ( 128 , 129. )

The number will be the coefficient of x in

PERMUTATIONS AND COMBINATIONS.

49

100 The number of permutations of n things taken all to gether , when a of them are alike , b of them alike , c alike , &c. n!

a ! b ! c ! ... & c .' For, if the a things were all different , they would form a ! permutations where there is now but one. So of b , c, &c.

101

The number of combinations of n things r at a time,

in which any p of them will always be found , is

= = C ( n − p , r — p) . For, if the p things be set on one side , we have to add to them r-p things taken from the remaining n -p things in every possible way.

102

THEOREM :

C (n − 1 , r − 1 ) + C ( n − 1 , r) = C ( n , r) .

PROOF BY INDUCTION ;

or as follows :

letters aside ; there are C (

Put one out of n

-1 , r) combinations of the re

maining n - 1 letters r at a time. To complete the total C (n, r), we must place with the excluded letter all the com binations of the remaining n - 1 letters r - 1 at a time.

103 If there be one set of P things , another of Q things, another of R things , and so on ; the number of combinations formed by taking one out of each set is = PQR ... &c. , the product of the numbers in the several sets .

For one of the P things will form Q combinations with A second of the P things will form Q more the Q things. In all , PQ combinations of two combinations ; and so on. Similarly there will be PQR combinations of three things . things ; and so on .

104

This principle is very important.

On the same principle, if p, q , r, &c . things be taken

out of each set respectively, the number of combinations will be the product of the numbers of the separate combinations ; that is ,

= = C ( Pp) . C ( Qq) . C (Rr) ... &c . H

50

105

ALGEBRA.

The number of combinations of n things taken m at a

time, when p of the n things are alike , q of them alike , r of them alike, &c . , will be the sum of all the combinations of each possible form of m dimensions , and this is equal to the coefficient of a" in the expansion of

• ( 1 + x + x² + ... + x² ) ( 1 + x + x² + ... + x²) ( 1 + x + x² + ... + œ”) ….. . 106 The total number of possible combinations under the same circumstances , when the n things are taken in all ways , 1 , 2 , 3 ... n at a time,

= (p + 1 ) (q + 1) ( r + 1) ... −1 . 107

The number of permutations when they are taken m

at a time in all possible ways will be equal to the product of m ! and the coefficient of am in the expansion of 202

xp

23

x³

... (1 + x + 2 2! + 3 ! +

xa ...

... +

) { 1 + * + 2 ! + 3! +

+ 9

... & c .

SURDS .

3 108 To reduce 2808. Decompose the number into its prime factors by (360) ; thus , 2808 Va15 blocs

109

/2³ . 3º . 13 = 6 ³ / 13,

as b' cˆ = a" b³ c² b ' c³ = a³ b³ c² Vuc².

To bring 5/3 to an entire surd .

5/3 =

5.3 =

1875, 30 15 X yº z¹³. x³ y¹ z} = x³8 y¹³ zx = √ /x™

110

To rationalise fractions having surds in their denominators.

1 7

= √7 7

1 = 3/7

3/49 = 3/49 3/ 7 7 x 19

51

SURDS.

111

3 3 (9 + /80) = = 3 (9+ √80), 81-80 9-80

since

(9 − √ /80) (9+ √ /80 ) = 81-80 , by ( 1) .

1 112 1 + 2√3 − √2

= 1 +2√3 + √2_1 = + 2√3 + √2 ( 1 +2 /3)*-2 11 + 4 /3 /3 + /2 ) ( 11-4 /3) = ( 1 +2√√ 73

1 1 3 = 3* —21° 1 3/3- √2

113

Put 3* = x, 2¹ = y, and take 6 the L. C.M. of the denominators 2 and 3, then

1 = x + x*y +x®y³ + x²y³ +xy* + y³, by (4) ; x--y x -y⁰ 1 therefore

3-2

= 3 + 3¹2¹ + 32¹ + 3¹2¹ + 3 } 2} +2} 31-21

/3 + 4√2. = 33/9 +3 /72 + 6 + 23/648 + 4¾

1 Here the result will be the same as inthe last example

114

$ /3 + √2°

[ See 5.

if the signs of the even terms be changed.

115 A surd cannot be partly rational ; that is , a cannot Proved by squaring. be equal to b ± c. 116

The product of two unlike squares is irrational ; √7 × √3 = √21 , an irrational quantity.

117

The sum or difference of two

unlike

surds

cannot

produce a single surd ; to c.

that is , √a + √b cannot be equal By squaring.

118

n ; then ab and m = n.

If a + √m = b +

Theorems (115 ) to ( 118) are proved indirectly.

119 then

If

√a + √b =

√x + √y,

√ a― √b =

√x- Ny.

By squaring and by (118 ).

52

ALGEBRA.

120

To express in two terms

Let

√7 + 26.

√7 + 2√6 = √x + √y ;

then

x+y = 7 ............... by squaring and by (118 ) ,

and

x—y = √ / 7² — (2√✓ /6) ³ = √49—24 = 5, by ( 119) ; ..

x = 6 and y = 1. Result

6 + 1.

General formula for the same

121

√a ± √b = √ } (a + √a² − b ) ± √ } (a − √a² — b) .

Observe that no simplification is effected unless a² -b is a perfect square .

122

Va + √b.

To simplify

Assume

Va + √b = x + ✔y. 3 c = √ a² - b.

Let

Then a must be found by trial from the cubic equation

4x³ - 3cxa , and

y = x² ― c .

No simplification is effected unless a² -b is a perfect cube .

Ex. 1 :

7+ 5√2 = x + √y. c = ³ /49–50 = - ༤.

4x³ + 3x = 7 ; Result

Ex. 2 : Cubing,

.. x = 1 , y = 2. 1+

93-11 /2 = √x +

2.

y, two different surds.

9√3-11√2 = x√x + 3x√y + 3y√x + y√y ;

...

9 √3 = (x + 3y) √x 11√ /2 = (3x + y) √y ..

x = 3 and y = 2.

;

(118)

BINOMIAL THEOREM.

To simplify

123 Assume

53

✓ (12 +4 /3 +4 /5 + 2 / 15) .

/ 15 ) = √x + √y + √x. √ (12 +4√3 +4√5 + 2√√

Square, and equate corresponding surds.

Result

124

√3 + √4 + √5.

A+ B in the form of two surds , where A

To express

Take q and B are one or both quadratic surds and n is odd . p", by power, say perfect nth such that q (A2 - B ) may be a n (A+B)² Take s and t the nearest integers to (361 ) . n /q (A — B)º, then and ~ 1

VA+ B =

{√s + t +2p ± √s + t −2p} . 22/q

EXAMPLE :

Here

89 /3 +109 /2.

To reduce

A89 /3,

B = 109/2,

.. p = 1 and q = 1.

A²- B² = 1 ; √q (A + B)² = 9 +ƒ

f being a proper fraction ;

" .. s = 9, t = 1 .

¾q (A — B)³ = 1 −ƒ ) Result

/ ( √9 + 1 + 2 ± √9 + 1−2) = √3 + √2.

BINOMIAL THEOREM.

125

(a +b)" =

n n a* + na" - 'b + " (n— ¹) a " -¹b³ + " ( n − 1) (n − 2) a »-³b³ + &c. 3! 2! 126

General or (r + 1)th term, n (n − 1) (n − 2) ... (n — r (n − r + 1 ) un - rbr r!

n! 127

an-rbr

or

(n − r) ! r ! if n be a positive integer. If b be negative, the signs of the even terms will be changed .

54

ALGEBRA.

" If n be negative the expansion reduces to 128

(a + b) " =

a - n -2 z²__n (n + 1) (n + 2) a¯ * -³b³ + &c. n (n + 1) a a¯” — na¯n - ¹b+ n 3! 2! 129

General term , ( −1) , n (n + 1) (n + 2) ... ( n +r − 1 ) a - " - rb " . r!

[ See 98 .

Euler's proof - Let the expansion of ( 1 + x) ", as in ( 125) , be called f(n) . Then it may be proved by Induction that the .... equation (1 ) f (m )Xf ( n ) = f (m + n) is true when m and n are integers, and therefore universally true ; because the form of an algebraical product is not altered by changing the letters involved into fractional or negative quantities . Hence f(m + n +p + &c . ) = ƒ (m) ׃ (n ) ׃ (p) , &c. Put m = n = p = &c . to k terms , each equal

, and the

theorem is proved for a fractional index. Again, put - n for m in (1 ) ; thus, whatever n may be , f( -n ) ׃ ( n ) = ƒ ( 0 ) = 1 ,

which proves the theorem for a negative index.

130

For the greatest term in the expansion of ( a + b ) ”, take

r = the integral part of (n + 1 ) b or (n - 1 ) b a-b a+b according as n is positive or negative. But if b be greater than a , and n negative or fractional , the terms increase without limit .

EXAMPLES.

Required the 40th term of ( 1 – 2r) “ 2x n = 42 . a= 1; b = 3 By (127) , the term will be 30 42! ( -20 ) 39 30 = 42.41.40 ( 2r by ( 96). (2x)" 1.2.3 3 3 3 ! 39 ! Here r = 39 ;

55

BINOMIAL THEOREM.

Required the 31st term of (a - x) -*. Here r = 30 ; b = −x ; n = -4. − By (129) , the term is 31.32.33 30 (-1 ) 4.5.6 ... 30.31.32.33 -34 ( −x )0 = 1.2.3 ... 30 1.2.3 a34 by (98).

1 131

when x = 14.

Required the greatest term in the expansion of

(1 + x)* 1 = (1 + x)

.

Here n = 6, a = 1, b = x in the formula

(1 +x) (n - 1 ) b = 5x1 = 23} ; 1-1 a -b therefore r = 23, by ( 130) , and the greatest term 5.6.7 ... 27/1428 = = ( −1)23 5 1.2.3 ... 23 (14)**

132

24.25.26.27/1428 1.2.3.4 (14)**.

Find the first negative term in the expansion of ( 2a + 3b) .

We must take r the first integer which makes n − r + 1 negative ; there fore r > n + 1 = 1 + 1 = 63 ; therefore r = 7. The term will be 17 14 11 8 3 v . j . j . 1 ( − }) (2a) -* (3b)' by (126) y . 4. 7! 17.14.11.8 5.2.1 7 7! (2a)**

133

Required the coefficient of a" in the expansion of (2 +3 ) .

2 -2 = (2 +3x) (2-3 ) ² = (2 +3 m ) ² (1 - 3 x20 ) -² 2 (2-3x)²

(2 + 3 )

2.3 3x 9 = ( 1 + 3x + 2 x² ) { 1 + 2 2 (3 ) + 1.2 (320)+ ...... 33 3x 34 ...... +33 (3x ) * +·3+ ( 3x) + 35 ( 32):)" + &c .... ..... } , 2 the three terms last written being those which produce a after multiplying by the factor (1 + 3x + 2x² ) ; for we have 33 34 9 3x) + 1x35 a² × 33 ( 3x) * + 3.x × 34 (327)*, (37 2 2 4 giving for the coefficient of x in the result 32 33 297

34

4 7 ( 3 ) " + 102 ( 3 ) " + 35 ( 3 ) ” = 306 ( 3 ) ”. The coefficient of a" will in like manner be 9n (3) ” -². UNI Clos

56

ALGEBRA.

134

To write the coefficient of

12 +1 ³ +1 in the expansion of (x² ( a² - 11 )2 + 0.

The general term is (2n + 1) ! (2n + 1 ) ! ‚4n - 4r. x² (2n - r+ 1) • 1 = (2n + 1 - r) ! r !" (2n + 1 − r) ! r !

1 Equate 4n - 4r +2 to 3m + 1 , thus

r=

4n - 3m + 1 4

Substitute this value ofr in the general term ; the required coefficient becomes (2n + 1) ! [ ‡ (4n + 3m + 3 ) ] ! [ ‡ ( 4n −3m + 1 ) ] !' The value of r shows that there is no term in x8m + 1 unless integer.

4n - 3m + 1 is an 4

135 An approximate value of ( 1 + x) " , when a is small , is 1 + nx, by ( 125) , neglecting a and higher powers of a.

136 Ex.- An approximation to 999 by Bin. Th. ( 125 ) is obtained from the first two or three terms of the expansion of = 299 300 nearly. (1000-1 ) = 10-1.1000-10-30 9388

MULTINOMIAL

THEOREM.

The general term in the expansion of ( a + bx + ca² + &c .) " is

137 where

n (n − 1) (n − 2) ... (p + 1 ) aº b²c" d³ ... X9+ 2r+ 3s +... , g ! r ! s ! ...

p + q + r + s + & c . = n,

and the number of terms p, q, r, &c . corresponds to the number of terms in the given multinomial . p is integral, fractional , or negative, according as n is one or the other. If n be an integer, ( 137) may be written n! 138

·aº bª c" d¸... x²+ 2r+ 38¸ ds p! q ! r ! s! [ Deduced from the Bin . Theor.

MULTINOMIAL THEOREM.

57

Ex . 1. — To write the coefficient of a³bc" in the expansion of (a + b + c + d)¹º. Here put n == 10, x = 1 , p = 3 , q = 1 , r = 5 , s = 0 in ( 138 ) . Result

Ex . 2.

10 ! = 7.8.9.10 . 3! 5!

To obtain the coefficient of a8 in the expansion of ( 1—2x + 3x² -4x³) ¹.

Here, comparing with ( 137 ) , we have a = 1 , b = −2, c = 3 , d = — 4, p + q + r + 8 = 4,

q + 2r + 388, 2270

1304

0220

1000

The numbers 1 , 0, 1 , 2 are particular values of p, q, r, s respectively, which satisfy the two equations given above. 0, 2, 0, 2 are another set of values which also satisfy those equations ; andthe four rows of numbers constitute all the solutions . In forming these rows always try the highest possible numbers on the right first. Now substitute each set of values of p, q, r, s in formula ( 138 ) succes sively, as under : 4! 2 1¹ ( − 2) º 3¹ ( — 4) ² = 576

4! 1° (-2 ) 3º ( -4) = 384 2 ! 2! 4! 2; 1º ( −2 ) ' 3º ( —4) ¹ = 864

4! = 4 1º ( − 2 ) ° 3 ′ ( — 4 ) ° = — 81 4! Result

1905

1024 124624722

Ex. 3.-Required the coefficient of a in ( 1 + 2x -4x² - 2a³) -¹ . Here a = 1, b = 2, c = —-4, d = −2 , n = — ; and the two equations are 1 29 p +q + r+ 8 = 4,

q + 2r + 3s = 1000

0220

1111

1

I

58

ALGEBRA.

Employing formula ( 137) , the remainder of the work stands as follows :

(− 1 ) ( − 3) 1 − ¹³ 2¹ ( − 4) ' ( —2 ) ' — — 3

(-

) ( -3 )

-

( - 4) ( -2) =

6 1

5 2! ( 음) ) ( -32 ) 12' ( —4) ' ( −2) " = -3 () ( - ) (-1 5 7 2' 4 4! ( -1 ) ( -3 ) ( -3 ) ( - ) ( - 4) ( 2)" =

Result

139

15 35 8

223

The number of terms in the expansion of the multi

nomial (a + b + c + to n terms ) " is the same as the number of homogeneous products of things of ↑ dimensions . See (97) and (98) .

The greatest coefficient in the expansion of ( a + b + c + to m terms) ", n being an integer , is

1

n! 140

where qm + k = n.

m

(q !)

(g + 1) (*)'

PROOF. By making the denominator in ( 138 ) as small as possible. The notation is explained in (96 ) .

!

LOGARITHMS .

142

loga Na signifies that a

= N, or

DEF. — The logarithm of a number is the power to which the base must be raised to produce that number.

143

log, a = 1 ,

log 1 = 0.

1 144

log MN = log M + log N.

M = log M - log N.

log N

log (M)" = n log M.

log M.

log /M = n

[142

EXPONENTIAL THEOREM.

59

log,a 145

log, a = log, b

That is - The logarithm of a number to any base is equal to the logarithm of the number divided by the logarithm of the base, the two last named logarithms being taken to any the same base at pleasure.

PROOF . -Let log, a := x and log, b = y ; then a = c ", b = c". Eliminate c. x x that is, log, a = c = a* = b ; .. a = b" , Q. e. d. y 1 146

Put ca in ( 145 ) .

log, a =

loga b

log, N 147

N= log10 N

by ( 145 ) .

log, 10 1

= ' 43429448 ...

148

log, 10 is called the modulus of the common system of logarithms ; that is, the factor which will convert logarithms of numbers calculated to the base e into the corresponding logarithms to See (154) . the base 10.

EXPONENTIAL

THEOREM .

C333

c²x² 149

a* = 1 + cx +

where

c = (a − 1 ) -

+ & c. ,

+ 2!

(a − 1 ) ² +

3! ( a − 1 )³- &c .

PROOF. a* = { 1+ (a − 1 ) } . Expand this by Binomial Theorem , and collect the coefficients of a ; thus c is obtained . Assume C2, C3, &c . , as the coefficients of the succeeding powers of a, and with this assumption write out the expansions of a , a", and a* + . Form the product of the first two series, which product must be equivalent to the third . Therefore equate the coefficient of x in this product with that in the expansion of a . In the identity so obtained, equate the coefficients of the successive powers of y to determine C2, C3, &c.

ALGEBRA .

60

Let e be that value of a which makes c = 1 , then

Xp2 e² = 1 + x +

150

ე3 +

+ & c.

2!

8!

1

151

e = 1+ 1 +

1

+ &c.

+ 2!

3!

= 2.718281828 ...

[ See (295).

PROOF.-By making a = 1 in (150 ) .

152

By making a = 1 in ( 149 ) and a = c in ( 150 ) , we obtain a = e ; that is , c = log, a.

Therefore by (149)

1 154

log, a = (a - 1 ) -

155

log (1 + a) =

(a - 1 ) ² +

X -----

( a − 1 ) ³- &c .

4,3

Xt

3

4

+ &c .

+ 2

X3 log ( 1 - x) :

-x- =

156

- &c .

1

2

= 2 {} x +

+ 3

1 -x

[154

4

Xe

X3

1 +x

... log

157

3

+ &c. } 5

1

m Put

for

in ( 157 ) ; thus ,

m+1 3

1 158

log m = 2 ) {

+ 3( \m

m

5

1) + 5 ( \ m +1) + &c. {} ,

1

Put

for x a in ( 157) ; thus , 2n + 1

159

log ( n + 1 ) —log n 1 1 1 =2 + + 2 { 5 (2n + 1 )5 +&c. }. 3 (2n + 1 ) 2n + 1

61

CONTINUED FRACTIONS.

CONTINUED FRACTIONS AND CONVERGENTS .

314159 160

Proceed as

To find convergents to 3.14159 = 100000

in the rule for H. C. F. 7

100000 99113

314159 300000

3

14159

15

The continued fraction is

3+1 1

887 854

1

15+ & c .

as it is more niently written ,

5289

33 29

or,

4435

44

4

7+ 1

887

854 66

4

25

1

conve

1

& c.

3+ 7+ 15+

194 165 29 28

7

1

The convergents are formed as follows : 3 3

7 22

1'7 '

15

1

25

1

7

4

333

355

9208

9563

76149

314159

106'

113 '

2931'

3044'

24239 '

100000'

— 161 RULE. Write the quotients in a row, and the first two convergents at sight (in the example 3 and 3 + 4 ) . Multiply the numerator of any convergent by the next quotient , and add the previous numerator . The result is the numerator of the next convergent . Proceed in the same way to determine the denominator . The last convergent should be the original fraction in its lowest terms .

162

Formula for forming the convergents .

If Pn -2, Pn -1 Pn are any consecutive convergents , and In-2 In-1 In -1 , an the corresponding quotients ; then an- 2, = Pnan Pn - 1 + Pn - 25

In = an In - 1 + In - 2 •

62

ALGEBRA.

The nth convergent is therefore

- = F„ Pn = anPn-1 +Pn -2 In an qn - 1 + gn - 2

The true value of the continued fraction will be expressed by

F = a'n Pn- 1 + Pn- 2¸ a'n qn - 1+ qn - 2

163

in which a', is the complete quotient or value of the continued fraction commencing with a,.

164

Pn¶n - 1 - Pn - 1 ¶n =

1

alternately , by (162 ) .

The convergents are alternately greater and less than the original fraction, and are always in their lowest terms.

165 The difference between F,n and the true value of the continued fraction is 1

1

and

< In In + 1

> In (In + In+ 1)

and this difference therefore diminishes as n increases .

Pn PROOF . - By taking the difference , Pаp₂ +1 + pu In a²qn + 1 + qn

( 163 )

Also F is nearer the true value than any other fraction with a less denominator.

n 166 FF +1 is greater or less than F² according as F greater or less than Fn+ 1.

General Theory of Continued Fractions. 167

First class of continued fraction .

b F= b₁ a₁+ a +

bз a3 + & c .

Second class of continued fraction.

b₁

b3

br

V=

01 a₁

a₁, b₁ , &c. are taken as positive quantities .

-

az

az- & c .

is

CONTINUED FRACTIONS.

63

bba , &c. are termed components of the continued frac A2 a1 tion. If the components be infinite in number , the continued

fraction is said to be infinite . Let the successive convergents be denoted by

Pi

b₁ =

qi

168

; a

P2 = b₁ а₁+ q2

b2

а₂

;

bbs ; and so on . P3 = b₁ a₁+ α₂+ Az 93

The law of formation of the convergents is

For F,

For V,

Pn = аnPn - 1 + bn Pn- 2

P₂ n = аn Pn -1bn Pn - 2 SP

2 Yn = an In - 1 + bn 4 n - 2

( qn = an In - 1 − bn ¶n − 2 [ Proved by Induction.

The relation between the successive differences of the convergents is, by ( 168) ,

169

Pn+1 _ Pn 二千 bn + 1 ¶n - 1 ( PnPn - 1) qn + 1 In In +1 In In-1

Take the ―

170

171

sign for F, and the + for V.

... Pn¶n- 1 — Pn - 1 ¶n = ( −1 ) π - ¹b¸ b¸b¸ …… . bŋ.

(168)

The odd convergents for F, P₁, Ps , &c. , continually 93 91

decrease, and the even convergents , P2, P , &c., continually 94 92 increase . (167) Every odd convergent is greater , and every even con vergent is less , than all following convergents. ( 169)

172

DEF. - If the difference between consecutive conver

gents diminishes without limit , the infinite continued fraction is said to be definite. If the same difference tends to a fixed value greater than zero , the infinite continued fraction is in definite ; the odd convergents tending to one value , and the even convergents to another.

ALGEBRA.

64

173

F is definite if the ratio of every quotient to the next

component is greater than a fixed quantity. PROOF.-Apply ( 169 ) successively. 174 F is incommensurable when the components are all proper fractions and infinite in number . PROOF. -Indirectly, and by ( 168 ) . 175

If a be never less than b + 1 , the convergents of

are

all positive proper fractions , increasing in magnitude, p„ and In also increasing with n. By ( 167 ) and ( 168) . 176

If, in this case , V be infinite, it is also definite , being = 1 , if a always = b + 1 while b is less than 1 , ( 175 ) ; and

being less than 1 , if a is ever greater than b + 1 .

By ( 180) .

177 V is incommensurable when it is less than 1 , and the components are all proper fractions and infinite in number . 180

b, + 1

If in the continued fraction V ( 167 ) , we have a,

always ; then, by ( 168) ,

1

n = P P₂ = b₁ + b₁ b₂ + b,1 b₂ b + ... to n terms , and q = Pa + 1 .

181 If, in the continued fraction F, a, and b, are constant and equal, say, to a and b respectively ; then Pn and In are

H respectively equal to the coefficients of h

" -1 in the expansions

a + bx

and

of

1 - ax -bx²*

bx²

1 -ax

PROOF . —Pn and In are the nth terms of two recurring series. and (251 ).

182

See ( 168 )

To convert a Series into a Continued Fraction. n

x

1 The series

+ ... +

+

+ u

X2

u₁

U₂

Un

is equal to a continued fraction V ( 167 ) , with n + 1 com ponents ; the first , second , and n + 1th components being

1

u² x

un -1 X

u₂ + ux

un + un -jx

" u

[ Proved by Induction .

CONTINUED FRACTIONS.

183

65

The series

1

X +

Xn

x² + ... +

+ vv1

V V 1 Vz ... V n

vv₁v2

is equal to a continued fraction V ( 167) , with n + 1 compo nents , the first, second , and n + 1th components being 1

VX

Vn - 1 X

v'

v₁ + x'

vn + x

[Proved by Induction.

184 The sign of a may be changed in either of the state ments in (182) or ( 183) . 185 Also , if any of these series are convergent and infinite , the continued fractions become infinite .

186

To find the value of a continued fraction with recurring quotients .

Let the continued fraction be bn+m by where y = bn+1 x = b₁ ... ... an+i+ + an +y a₁ + + an+ m +y

so that there are m recurring quotients . Form the nth con vergent for æ, and the mth for y. Then , by substituting the complete quotients an +y for an , and an + m + y for an + m in ( 168) , two equations are obtained of the forms

Ay + B x=

and

Cy + D

Ey +F y= Gy +H'

from which, by eliminating y, a quadratic equation for de termining a is obtained .

bu

b₁

187

If a₁+

+ an+

be a continued fraction, and

Pn

Pi . 91

qn K

66

ALGEBRA.

the corresponding first n convergents ; then 1-1 , developed In by ( 168), produces the continued fraction 1

b₁

b3

bn -1

b2

an + an- 1 + an - 2 + ... + a₂ + a₁ the quotients being the same but in reversed order .

INDETERMINATE

188

Given

EQUATIONS .

ax + by = c

free from fractions , and a, ẞ integral values of a and Y wnich satisfy the equation , the complete integral solution is given by

x = a - bt

1 y = B + at where t is any integer.

EXAMPLE .

Given

5x +3y = 112. Then x = 20, y 4 are values ; x = 20-3t

y=

4 + 5t )

The values of x and y may be exhibited as under : 0 1 2 t = -2 --1 3 4 8 x = 26 23 20 17 14 11

5 5

6 2

7 -1

y = -6 -1 4 9 14 19 24 29 34 39 For solutions in positive integers t must lie between 20 = 63 and — that is, t must be 0, 1 , 2 , 3, 4 , 5 , or 6, giving 7 positive integral solutions.

189

If the equation be

ax -by = c

the solutions are given by

x = a + bt

y = B+ at.

INDETERMINATE EQUATIONS.

EXAMPLE : Here a

67

4x -3y = 19. 10, y = 7 satisfy the equation ;

x = 10 + 3t 7+4t furnish all the solutions. y= The simultaneous values of t, x, and y will be as follows : : 3 -2 -1 1 0 -4 2 3 t = -5 1 4 -2 7 10 13 16 19 x = -5

y = -13

9

-1

-5

3

7

11

15

19

The number of positive integral solutions is infinite, and the least positive integral values of x and y are given by the limiting value of t, viz ., 10 7 and t > t> 4 3 that is, t must be --- 1 , 0, 1 , 2, 3, or greater.

190 If two values , a and ẞ, cannot readily be found by inspection, as, for example, in the equation 17x + 13y = 14900 ,

divide by the least coefficient, and equate the remaining frac tions to t, an integer ; thus y +a+

4x 2 = 1146+ 13 13

4x -2 = 13t. Repeat the process ; thus

x

Put

2 = 3t + " 4 4

.. t + 2 = 4u. W == 1 ,

t = 2, 13t + 2 x = = 7 = a; 4 and

y + x + t = 1146, by ( 1 ) , y = 1146-7-2 = 1137 = B. The general solution will be x=

7-13t , y = 1137 + 17t,

Or, changing the sign of t for convenience, x= 7 + 13t, y = 1137-17t.

( 1) ;

68

ALGEBRA.

Here the number of solutions in positive integers is equal to the number of - 7 and 1137 ; integers lying between 17 13

- 7 and 661 ; that is, 67. 13

or

Otherwise . -Two values of x and y may be found in 191 the following manner :

17 [ By ( 160) .

Find the nearest converging fraction to 13

This is

4 3'

By ( 164) we have 17 x3-13 x 4 = −1 .

Multiply by 14900, and change the signs ; 17 ( −44700 ) +13 ( 59600 ) = 14900 ;

which shews that we may take

S a = −44700 B = 59600

and the general solution may be written x = -44700 + 13t, y= 59600-17t. This method has the disadvantage of producing high values of a aud ß.

192 The values of a and y, in positive integers , which satisfy the equation ax + by c, form two Arithmetic Pro gressions, of which b and a are respectively the common differences . See examples ( 188 ) and ( 189 ) .

193

Abbreviation of the method in (169 ) .

EXAMPLE :

11x - 18y = 63.

Put x = 9z, and divide by 9 ; then proceed as before.

194

To obtain integral solutions of ax + by + cz = d.

Write the equation thus

ax + by = d - cz. Put successive integers for z, and solve for x, y in each case.

69

REDUCTION OF A QUADRATIC SURD.

TO

REDUCE

A

QUADRATIC

CONTINUED

195

SURD

TO

A

FRACTION .

EXAMPLE : 4 √29 =

5+ √/29-5 =

5+

√29 +5' 5

√29 + 5 = 4

-3 2+ √29 = 4 .

2+

√29 +3 = 5

1+

✓ /29-2 = 5

1+

√29 +2 = 5

1+ √29-3 = 5

]+

/29-5 = √ 4

2+

/29 +3 2+

=

4

29 +3 '

5

√29 + 2' 4

√29 + 3' 1

√29 + 5 ' 4

√29 + 5 = 10 + √29 − 5 = 10 + √29 +5° The quotients 5 , 2 , 1 , 1 , 2 , 10 are the greatest integers contained in the quantities in the first column. The quotients now recur, and the surd 29 is equivalent to the continued fraction 1 1 1 1 1 1 1 1 1

1+ 2+

1+

5+ 2+

1'

11 2 '

16 3

2+

1+ 1+ 2 + &c .

29 , formed as in ( 160 ) , will be

The convergents to 5

10+

27 5'

70

727

1524

13 '

135'

283 '

2251

3775

418 '

701 '

9801 1820'

Note that the last quotient 10 is the greatest and twice the first, that the second is the first of the recurring ones, and that the recurring quotients, excluding the last , consist of pairs of equal terms , quotients equi- distant from the first and

196

last being equal . -210) .

These properties are universal.

(See 204

To form high convergents rapidly.

197

Suppose m the number of recurring quotients , or any

ALGEBRA.

70

multiple of that number, and let the mth convergent to Q be represented by Fm ; then the 2mth convergent is given by the

Q formula

F2m =

{F +

by ( 203 ) and ( 210 ) .

m 198 For example, in approximating to 29 above, there are five recurring quotients. Take m = 2x5 = 10 ; therefore, by 297 F20 = 1 " 2 { F10 + F10 } F10 = 9801 the 10th convergent. 1820'

Therefore

1820 9801 F2 = { 1820 + 29 × 9801

the 20th convergent to convergents is saved .

192119201 35675640

29 ; and the labour of calculating the intervening

GENERAL

199

=

THEORY .

The process of ( 174) may be exhibited as follows : r2

√ Q + C₁ = a₁₂ +

r1

√Q + cz

√Q + c2

13 = a₂+

12

√Q + c3

...

...

rn + 1

√Q + cn = a₂ +

In

√Q + en+i 1

200

1

1

Then √ Q = a₁ + a₂+ as + as + &c.

The quotients a1, a2, tions on the left.

3 , &c . are the integral parts of the frac

REDUCTION OF A QUADRATIC SURD.

201

71

The equations connecting the remaining quantities are

G₁ = 0

r₁ = 1

C₂ = a₁r₁ — C1

r2 =

Q-c

r1

2 Q - cs C3

13 =

a₂r2 - C2

r2

Cnan - 1n - 1 - On - 1

The nth convergent to

rn

Q - on rn- 1

Q will be

Pn = anPn- 1 +Pn-2 In An In-1 + In-2

202

The true value of

[By Induction .

Q is what this becomes when we

substitute for a, the complete complete quotient quotient is only the integral part .

+ Vn

, of which a an

This gives

( √Q + Cn) Pn- 1 + VnPn-2¸ 203

√Q = =

~ Q + Cn) In -1 + rn In− 2

By the relations ( 199 ) to ( 203 ) the following theorems are ―― demonstrated : 204

All the quantities a , r, and c are positive integers .

205

The greatest c is C2, and C2 = a₁.

206

No a or

207

If

208

For all values of n greater than 1 , a - c, is < rn .

209

The number of quotients cannot be greater than 2a1.

can be greater than 2ɑ .

n = 1 , then c₁ = a₁.

The last quotient is 2a ,

and after that the terms repeat . √ Cy Q + , and The first complete quotient that is repeated is T2 da, 72, C₂ commence each cycle of repeated terms .

72

ALGEBRA.

210

Let am, I'm Cm be the last terms of the first cycle ; then

am- 19 7m - 19 Cm -1 are respectively equal to a2, 72, C2 ; Am- 25 7m− 29 Cm-2 are equal to 3, 73, C3, and so on. [ By ( 187 ) .

EQUATIONS .

Special Cases in the Solution of Simultaneous Equations. 211

First, with two unknown quantities . =

a₁x +by

1)

aşx + by = c₂S

x = cab₂ --cab₁ abɔ— a‚b₁'

C₁₂ -Caα1

y b₁α - ba₁

If the denominators vanish , we have

a1 =

A2

b₁ " and x = 0, y = ∞ ; ba

unless at the same time the numerators vanish , for then

a1 = b₁ = C1 ; C2 b₂ aa

0

0 x=

y =

0

0

and the equations are not independent, one being produced by multiplying the other by some constant . 212 Next, with three unknown quantities . the equations .

See ( 60 ) for

If d₁ , da, d

all vanish , divide each equation by z, and we -x and " , two have three equations for finding the two ratios 2 only of which equations are necessary, any one being dedu cible from the other two if the three be consistent .

213

To solve simultaneous equations by Indeterminate Multipliers.

Ex.

Take the equations. x + 2y + 3x + 5y +

3z + 4w = 27 , 72 % +

1 = 48 ,

5x + 8y + 10z — 2w = 65, 7x + 6y +

5z + 4w = 53 .

1

73

MISCELLANEOUS EQUATIONS.

Multiply the first by A, the second by B, the third by C, leaving one equation unmultiplied ; and then add the results . Thus

(A + 3B + 5C + 7) x + ( 2A + 5B + 80 + 6) y + (3A + 7B + 10C + 5 ) ≈ + (4A + B − 2C + 4) w = 27A + 48B + 65C + 53 .

To determine either of the unknowns , for instance x, equate the coefficients of the other three separately to zero , and from the three equations find A, B, C. Then 274 +48B + 65C + 53

x= A + 3B + 5C + 7

MISCELLANEOUS EQUATIONS AND SOLUTIONS.

214

x ± 1 = 0.

Divide by a³, and throw into factors , by (2) or (3) .

See also

(480) .

215

x³ - 7x - 6 = 0.

x x= = -1 is a root , by inspection ; therefore +1 is a factor . Divide by x + 1, and solve the resulting quadratic .

216

a³ + 16x = 455 . x* + 16a² = 455x = 65 × 7x, 2 2 65x2 æ¹ + 65a² + ( 65 )² = 49x²‍+65 × 7x + ( 65 2 ) ”, 65 ²+

65

= 7x + 2'

2

a² = 7x; RULE .

..

x = 7.

Divide the absolute term (here 455 ) into two

factors , if possible , such that one of them , minus the square See (483) for general of the other, equals the coefficient of r. solution of a cubic equation . L

74

ALGEBRA.

217

x* —y* = 14560 ,

Put

x = x+v

and

x - y = 8. y = x - v.

Eliminate v, and obtain a cubic in z , which solve as in (216) .

218

x³ —y³ = 3093 ,

x - y = 3.

Divide the first equation by the second , and subtract from the result the fourth power of x - y . Eliminate (x² + y²) , and obtain a quadratic in xy.

219

On forming Symmetrical Expressions.

Take, for example, the equation (y - c) (z - b) = a². To form the remaining equations symmetrical with this , write the corresponding letters in vertical columns, observing the circular order in which a is followed by b, b by c , and c by a. So with x, y, and z . Thus the equations become (y— c) (≈ —b) = a²,

(≈— a) (x− c) = b², (x —b) (y — a) = c². To solve these equations, substitute x = b + c + x',

y = c + a + y' ,

z = a +b+z ;

and, multiplying out, and eliminating y and z , we obtain

x = be (b +c) — a (b² + c²) , bc - ca -ab and therefore, by symmetry, the values of y and z, by the rule just given .

220

..

y² + z² + yz = a²

(1),

≈² + x² + zx = b²

(2) ,

x² + y² + xy = c²

(3) ;

3 (yz + z + xy) ² = 20°c² + 2c²a² + 2a²b² —a * —b* — c* ...... (4).

75

IMAGINARY EXPRESSIONS.

Now add (1) , ( 2) , and ( 3 ) , and we obtain 2 (x + y + z)² − 3 (yz + zx + xy ) = a² + b² +c² ...... (5) . From (4) and ( 5 ) , ( x + y + z) is obtained , and then ( 1 ) , (2 ) , and (3) are readily solved .

x² ―yz = a²

...... ( 1 ),

— zx = b² y³ -

(2) ,

221

≈² —xy = c²

.....

(3).

Multiply (2) by (3) , and subtract the square of ( 1) . Result

x (3xyz — x³ — y³ — z³) = b²c² — a¹ , x =

b2c2 -a¹

y c²a² - b¹

2 = d ……………..

a²b²-c¹

(4) .

Obtain λ² by proportion as a fraction with numerator

= x² — yz = a³.

222

x = cy + bz

(1) ,

y = az + cx

(2) ,

z = bx + ay

(3) .

Eliminate a between (2) and (3) , and substitute the value of æ from equation ( 1) . Result

2° y² = = 1 - a2* 1 - c2 1 - b2

IMAGINARY

223

EXPRESSIONS .

―――― The following are conventions :

―― That ( -a ) is equivalent to a√(-1 ) ; that a√ ( · vanishes when a vanishes ; that the symbol a (-1 ) is sub ject to the ordinary rules of Algebra. (-1 ) is denoted by i.

76

ALGEBRA.

224

If a + iß = y + id ; then a = y and ß = 8.

225 duct

a + iẞ and a - iẞ are conjugate expressions ; their pro a² + ß³.

226 The sum and product of two conjugate expressions are both real, but their difference is imaginary.

227

The modulus is + √a² +ẞ².

228

If the modulus vanishes , a and ẞ must vanish .

229 If two imaginary expressions are equal, their moduli are equal, by (224) .

The modulus of the product of two imaginary expres

230

sions is equal to the product of their moduli . Also the modulus 231 quotient of their moduli .

of the quotient is

equal to the

METHOD OF INDETERMINATE COEFFICIENTS .

232

If A + Bx + Ce² + ... = A′ + B′x + C′x² + ... be an equa

tion which holds for all values of a, the coefficients A, B, &c . not involving x, then A = ', B = B', C = C' , & c.; that is , the coefficients of like powers of a must be equal. Proved by putting = 0, and dividing by a alternately. See ( 234) for an example.

233

METHOD OF PROOF BY INDUCTION .

Ex.-To prove that 1 +2 +3 +... + n² = n (n + 1 ) ( 2n + 1 ) 6

Assume

1 +2 + 3 + ... + n² = n (n + 1 ) (2n + 1 ) ; 6

-77

PARTIAL FRACTIONS.

..

1 + 2² + 3² + ... + n² + (n + 1 ) ² = n (n + 1 ) ( 2n + 1 ) + (n + 1 )' 6 = n (n + 1 ) ( 2n + 1 ) + 6 ( n + 1 ) ² _ ( n + 1 ) { n ( 2n + 1 ) +6 ( n + 1 ) } 6 6

= (n + 1 ) (n + 2 ) (2n +3) = n' (n' + 1 ) (2n' + 1 ) " 6 6 where n' is written for n + 1 ; ..

1 + 2 + 3 + ... + n ^ =¸ n' (n' + 1 ) (2n' + 1) 6

It is thus proved that if the formula be true for n it is also true for n +1. But the formula is true when n = 2 or 3 , as may be shewn by actual trial ; therefore it is true when n = 4 ; therefore also when n = 5 , and so on ; therefore universally true.

234 Ex. - The same theorem proved by the method of In determinate coefficients .

Assume 1 +2 +3 +... + n²

= A + Bn

+ Cn²

+ Dis

+ &c.;

.. 1 + 2² + 3² + ... + n² + ( n + 1 ) ² = A + B ( n + 1 ) + C (n + 1)² + D (n + 1 }³ + &c .; therefore, by subtraction, n² + 2n + 1 = B + C (2n + 1 ) + D (3n² + 3n + 1 ) , writing no terms in this equation which contain higher powers of n than the highest which occurs on the left-hand side, for the coefficients of such terms may be shewn to be separately equal to zero. Now equate the coefficients of like powers of n ; thus

3D == 1 ,

...

20 + 3D = 2,

D=

1 ; 3

C =

1 " 2

and

A = 0;

1

B + C + D = 1,

..

B=

;

therefore the sum of the series is equal to n n² n³ n (n + 1 ) (2n + 1 ) = + + 6 3 2 6

PARTIAL

FRACTIONS .

In the resolution of a fraction into partial fractions four cases present themselves , which are illustrated in the follow ing examples .

78

ALGEBRA.

235 First.- When there are no repeated factors in the de nominator of the given fraction.

Ex.—

3x - 2 To resolve

into partial fractions . (x - 1 ) (x - 2) (x - 3 )

3α - 2

Assume

= (x - 1) (x - 2) (x - 3)

B A C + = 3; x- 1 x- 2 + x-

3x-2A (x - 2) (x - 3) + B (x - 3 ) (x - 1 ) +0 (x - 1 ) (x - 2) . Since A, B, and C do not contain x, and this equation is true for all values of x, put x = 1 ; then 1 3-2A (1-2) ( 1-3) , from which A = 2' Similarly, if a be put = 2, we have

6—2 = B (2−3) ( 2-1 ) ; and, putting a = 3,

..

9-2 = 0 (3−1 ) (3—2) ;

..

1

3x - 2 =

Hence (x- 1) (x- 2) ( x - 3)

236

2 (x - 1 )

B = -4 ;

0=

7 2

7 4 · + x - 2 2 (x- 3)

Secondly.- When there is a repeated factor.

Ex. -Resolve into partial fractions

7x³ - 10x² + 6x X--1 )³ (x + 2 )*

A B 7x³ - 10x² + 6x = с D + + + (x1 )¹ x--1 x+ 2 )* (2-1 (x− 1 ) ( x + 2)

Assume

These forms are necessary and sufficient. Multiplying up , we have - 1)³ 7x³- 10x² + 6x = A ( x + 2 ) + B (x− 1 ) ( x + 2 ) + C ( x − 1 ) ³ ( x + 2 ) + D (x − ......... (1) . Make x = 1 ; .. 7-10 +6A ( 1 + 2) ; ... A = 1.

Substitute this value of A in ( 1) ; thus 7x³ - 10x² + 5x -− 2 = B ( x − 1 ) ( ≈ + 2 ) + C ( x − 1 ) ² ( x + 2 ) + D ( x − 1 ) ³. Divide by x - 1 ; thus 7x -3x +2B (x + 2) + C (x - 1 ) (x + 2) + D (x - 1 ) ......... ( 2) . Make

1 again,

7-3 + 2 = B (1 + 2) ;

..

B = 2.

Substitute this value of B in ( 2) , and we have 7x² - 5x - 2 = C (x - 1 ) ( x + 2 ) + D (x − 1 ) ². Divide by x - 1, Put

7x+ 2 = C (x + 2) + D (x − 1 ) ...... ……….

1 a third time,

7 +2 = C (1 + 2) ;

.. C = 3 .

..... (3) .

PARTIAL FRACTIONS.

79

Lastly, make x = −2 in (3) , ..

-14+2 = D ( -2-1 ) ; Result

D = 4.

2 1 3 (2-1 ) ² + (2-1 -- 1 + x+ ); + x = +422 (x−1)²

237 Thirdly.- When there is a quadratic factor of imaginary roots not repeated . 1

Ex. -Resolve

into partial fractions. (x² + 1 ) ( x² + x + 1 )

Here we must assume Ax + B Ca + D 1 = + 2² +x + 1 23+ 1 (x² + 1 ) (x² + x + 1 ) x² + 1 and x² +a + 1 have no real factors, and are therefore retained as denominators. The requisite form of the numerators is seen by adding a с together two simple fractions, such as + x+ b x+d' Multiplying up, we have the equation 1 = (Ax + B) (x² + x + 1 ) + ( Cx + D ) ( x² + 1 ) .......……………….. ( 1 ) . .. x² = -1 . 2²+ 1 = 0 ;

Let

Substitute this value of a² in ( 1 ) repeatedly ; thus 1 = (A + B) x = Ax² + Bx = -A + Bx ; Bx-A - 1 = 0.

or

Equate coefficients to zero ;

Again, let

..

B = 0, A = -1.

x²+x + 1 = 0 ; 2 = -x - 1.

Substitute this value of a repeatedly in ( 1 ) ; thus 1 = (Cx + D) ( −x) = -Ca³ - Dx = Сx + C − Dx ; or (C - D) x + C - 1 = 0. Equate coefficients to zero ; thus C = 1 , D = 1. 1 = x+ 1 Hence (x² + 1 ) ( x² + x + 1 ) x² +x + 1

238

æ 2 +1

Fourthly.-When there is a repeated quadratic factor

of imaginary roots. 40x -103

Ex.-Resolve

into partial fractions . (x + 1 ) ² ( x² - 4x + 8)³

80

ALGEBRA.

Assume 40x - 103 = (x + 1 ) ³ (x² -4x + 8 ) ³

Ex+F Ax + B Cx + D + + (x² −4x + 8) ³¹ ( x² - 4x + 8 ) ²¯¯ x² -4x + 8 H

G

+ (x+ 1)² + x+ 1 ' 40x − 103 = { (Ax + B) + (Cx + D) (x² −4x + 8) + (Ex + F) (x² − 4x + 8 )² } (x + 1 ) ² + { G + H (x + 1 ) } ( x² − 4x + 8 )³ ......

...... (1) .

In the first place, to determine A and B, equate a² -4x + 8 to zero ; thus x² = 4x - 8. Substitute this value of a repeatedly in ( 1 ) , as in the previous example , until the first power of a alone remains. The resulting equation is 40x103

(17A + 6B) x - 48A - 7B.

Equating coefficients, we obtain two equations

17A + 6B = 40 484 +7B = 103 }

from which

A= 2 B = 1.

Next, to determine C and D, substitute these values of A and B in ( 1 ) ; the equation will then be divisible by x² - 4x + 8 . Divide, and the resulting equation is 0 = 2x + 13 + { Cx + D + ( Ex + F ) ( x² −4x + 8) } (x + 1 ) ³ + { G + H (x + 1 ) } (œ² —4x + 8 ) ² ......

......... (2) .

Equate ² - 4x + 8 again to zero, and proceed exactly as before, when finding A and B. Next, to determine E and F, substitute the values of C and D, last found in equation (2) ; divide, and proceed as before. Lastly, G and H are determined by equating +1 to zero successively, as in Example 2.

CONVERGENCY AND DIVERGENCY OF SERIES .

Let a₁ +a₂ + a + &c . be a series , and an, an +1 any two 239 consecutive terms . The following tests of convergency may The series will converge , if, after any fixed term (i ) The terms decrease and are alternately positive and negative . An (ii. ) Or if is always greater than some quantity An+1

be applied .

greater than unity.

81

SERIES.

an is never less than the corresponding ratio

(iii. ) Or if

An+ 1 in a known converging series . nan ― n ) is always greater than some quan (iv. ) Or if ( An + 1 [By 244 and iii. tity greater than unity.

nan (v.) Or if if ( An + 1

-n -1 ) log n

is

always

greater than

some quantity greater than unity.

240

The conditions of divergency are obviously the converse

of rules (i . ) to (v.) .

an +1 is 241

The

series

a₁ + a

+ ax² + &c .

if

converges ,

An

1 always less than some quantity p, and a less than

p [ By 239 (ii. )

242

To make the sum of the last series less than an assigned p

quantity p , make a less than

k being the greatest co

P + k' efficient .

General Theorem . 243 If Φ (a) be positive for all positive integral values of a, and continually diminish as a increases , and if m be any posi tive integer, then the two series (1) +

(2) +4 (3 ) +6 (4) + ......

$ (1) + mp (m) + m²4 ( m²) + m³4 ( m³) + ...... are either both convergent or divergent.

244 series

Application of this theorem.

To ascertain whether the

1 1 1 + 1/2 + 3 + 0+30 + 2p 4p 1P

is divergent or convergent when p is greater than unity. M

ALGEBRA.

82

Taking m2, the second series in ( 243) becomes

4

2 1+

+ 2p

8

+ 4P

+ & c ....... 8P

a geometrical progression which converges ; given series converges .

therefore the

1 245

The series of which

is the general term is n (log n)r convergent if p be greater than unity , and divergent if P be

[ By ( 243) , (244).

not greater than unity.

246

The series of which the general term is 1 ηλ (η ) λ * (n ) ...... X " ( n) { \ r + ¹ ( n ) } p '

where (n) signifies log n , λ² (n ) signifies log { log (n ) } , and so on, is convergent if p be greater than unity, and divergent if p be not greater than unity . [ By Induction, and by (243) .

247

The series a₁ + a₂ + &c . is convergent if nan log (n) log² (n) ...... log" (n ) { logr+1 (n) } r

is always finite for a value of p greater than unity ; log² (n ) here signifying log (log n) , and so on. [ See Todhunter's Algebra, or Boole's Finite Differences.

EXPANSION OF A FRACTION .

4x - 10x2 248

A fractional expression such as

may 1-6x + 11x² - 6x³

be expanded in ascending powers of a in three different ways . First, by dividing the numerator by the denominator in the ordinary way, or by Synthetic Division , as shewn in (28) . Secondly, by the method of Indeterminate Coefficients (232 ) . Thirdly, by Partial Fractions and the Binomial Theorem.

83

SERIES.

To expand by the method of Indeterminate Coefficients , proceed as follows : 4x - 10x2 = A + Bx + Ca² + Dx³ + Ex + &c. 1-6x + 11x² - 6x³ 4x - 10x2 = A + Bж+ Cx² + Dx³ + Ex + Fæ³ + ... -6Аx- 6Bx²- 6Сx³- 6Dx¹- 6Ex -...

Assume ..

+11 Ax² + 11 Bx³ +11 Ca¹ + 11Dx² + ... - 6Ax³- 6B - 6Сx³ -... Equate coefficients of like powers of X, thus A= 0, 4, B- 6A = C- 6B + 114 = —-10 ,

D - 6C +11B- 6A = E - 6D +110- 6B = F- 6E + 11D– 6C = ... ... ...

0, 0, 0,

.. .. .. .. ..

4; B= C = 14 ; D = 40 ; E = 110 ; F = 304 ;

The formation of the same coefficients by synthetic division is now exhibited, in order that the connexion between the two processes may be clearly seen. The division of 4-10x² by 1—6x + 11æ² - 6x is as follows :

+6 -11

+ 6

0 +4-10 24 +84 +240 + 660 -44-154-440-1210 + 24+ 84+ 240 + 660 0 +4 +14 +40 + 110 +304 + A B C D E DE F

If we stop at the term 110x , then the undivided remainder will be 304-970 + 6607 , and the complete result will be 304-970x + 660x7 3 4x + 14x² + 40x³ + 110x¹ + 1-6x + 11² — 6x³ 249

Here the concluding fraction may be regarded as the

sum to infinity after four terms of the series , just as the original expression is considered to be the sum to infinity of the whole series .

250

If the general term be required, the method of ex

pansion by partial fractions must be adopted .

See (257 ) ,

where the general term of the foregoing series is obtained .

ALGEBRA.

84

RECURRING

а +a¸x+a²x² +a

SERIES .

+ &c . is a recurring series if the co

efficients are connected by the relation 251

an = p₁an - 1 + P₂ªn - 2 + ... + Pmªn - m²

The Scale of Relation is 252

1 1 -р₁x - pæ -р²x² —...— Pmxm.

The sum of n terms of the series is equal to 253

[The first m terms

-p₁x (first m - 1 terms + the last term) -p2x² (first m —- 2 terms + the last 2 terms) -Pa³ (first m - 3 terms + the last 3 terms) .. ... ... ... ... -Pm-1-1 (first term + the last m - 1 terms) m -P (the last m terms) ] ÷ [ 1 - p₁x — P²x² —...— Pmx ”] . 254 If the series converges , and the sum to infinity is re quired, omit all " the last terms " from the formula .

255

EXAMPLE. - Required the Scale of Relation, the general

term, and the apparent sum to infinity, of the series. 4x + 14x² + 40x³ + 110x¹ + 304x³ − 854xº + .... …….. Observe that six arbitrary terms given are sufficient to determine a Scale of Relation of the form 1 - px - qa² - ra³, involving three constants p, q, r, for, by (251 ) , we can write three equations to determine these constants ; The solution gives namely, 40p + 14q + 4r' 110 p = 6, q = -11 , r = 6. 110p + 40q + 14r . 304 854304p + 110g +40r ) Hence the Scale of Relation is 1-6x + 11x² - 6x³ . The sum of the series without limit will be found from (254) , by putting P₁ = 6, p₁ = —11 , p¸ = 6, m = 3. The first three terms = 4x + 14x² + 40x³ -24x² -84x³ -6xthe first two terms = = +44x³ + 11xx the first term

4x - 10x²

RECURRING SERIES.

85

4x - 10x² S = 1-6x +112-63 the meaning of which is that, if this fraction be expanded in ascending powers of X, the first six terms will be those given in the question .

256 To obtain more terms of the series, we may use the Scale of Relation ; thus the 7th term will be (6 x 854-11 x 304 +6 x 110) x

257

2440x7.

To find the general term, S must be decomposed into

partial fractions ; thus , by the method of ( 235 ) ,

4. - 10.2 2 - 3 1 = + 1-2x 1 -x 1-3x 1-6x + 11x² - 6x³ By the Binomial Theorem ( 128) , 1 = 1 + 3x + 3²x² + ...... + 3"x", 1-3x 2 = 1-2x

2 + 2x + 2x² + ....

3 = -3-3x - 3x² 1-x

+ 2 +¹x", -3x².

Hence the general term involving a" is (3 + 2 +1-3) æ". the " last terms " required in (253 ) , and write formula can we this And by so obtain the sum of any finite number of terms of the given series . Also, by the same formula we can calculate the successive terms at the beginning of the series. In the present case this mode will be more expeditious than that of employing the Scale of Relation .

258

If, in decomposing S into partial fractions for the sake of obtaining the general term, a quadratic factor with ima ginary roots should occur as a denominator, the same method must be pursued , for the imaginary quantities will disappear in the final result. In this case, however, it is more con venient to employ a general formula. Suppose the fraction which gives rise to the imaginary roots to be

L+ Mx

L+ Mx =

a + bx + x²

(p - x) (q - x) '

p and q being the imaginary roots of a + bx +a² = 0. Suppose

p = a + iß, q = a-iß, where i = √1.

86

ALGEBRA.

If, now, the above fraction be resolved into two partial fractions in the ordinary way, and if these fractions be ex panded separately by the Binomial Theorem, and that part of the general term furnished by these two expansions written out , still retaining p and q, and if the imaginary values of p and q be then substituted , it will be found that the factor will disappear, and that the result may be enunciated as follows .

259

The coefficient of a" -1 in the expansion of

L+ Mx

ta

a err

(a² +B²) —2ax +x²

will be L

-1 n- 3 { na" -¹B - C ( n , 3) an − ³ ß³ + C ( n , 5 ) a” -5 ß³ — ….. }

B (a² + B ) M

n-2 n-4 { (n − 1) a" -² ß — C (n − 1 , 3) a” -¹ ß³

+ B(a² + B²) -1

+ C (n − 1 , 5 ) a” -6 ß³ — ...} ·

260 With the aid of the known expansion of sin no in Trigonometry, this formula for the nth term may be reduced to '(L + Ma ) ² + M² ß² · sin (no - 4) , B² (a² + B²)"

in which

0 = tan -1B a

-1 tan -2

MB L+Ma

If n be not greater than 100 , sin ( n0 − q ) may be obtained from the tables correct to about six places of decimals , and accordingly the nth term of the expansion may be found with corresponding accuracy. As an example, the 100th term in 1 +x is readily found by this method the expansion of 5-2x + x² 41824 .99 to be . 1041

To determine whether a given Series is recurring or not. 261 If certain first terms only of the series be given, a scale of relation may be found which shall produce a recurring

RECURRING SERIES.

87

series whose first terms are those given. The method is exemplified in (255) . The number of unknown coefficients P, q, r, &c . to be assumed for the scale of relation must be equal to half the number of the given terms of the series , if that number be even. If the number of given terms be odd , it may be made even by prefixing zero for the first term of the series . 262 Since this method may, however, produce zero values for one or more of the last coefficients in the scale of relation , it may be advisable in practice to determine a scale from the first two terms of the series , and if that scale does not produce the following terms , we may try a scale determined from the first four terms , and so on until the true scale is arrived at . If an indefinite number of terms of the series be given, we may find whether it is recurring or not by a rule of Lagrange's . 263

Let the series be S = A + Bx + Ca² + Dx³ + &c .

Divide unity by S as far as two terms of the quotient, which will be of the form p + qe, and write the remainder in the form S'a², S' being another indefinite series of the same form as S. Next, divide S by S' as far as two terms of the quotient , and write the remainder in the form S"x². " by S", and proceed as before, and repeat Again, divide S this process until there is no remainder after one of the divisions. The series will then be proved to be a recurring series, and the order of the series , that is, the degree of the scale of relation , will be the same as the number of divisions which have been effected in the process .

EXAMPLE. To determine whether the series 1, 3, 6, 10, 15 , 21 , 28, 36, is recurring or not. Introducing x, we may write S = 1 + 3x + 6x² + 10x³ + 15x* + 21x + 28x + 36x + 45x³ .... Then we shall have S = 1-3x + ... with a remainder 1

45,

3x² + 8x + 15x* + 24x³ + 35x® + &c .

Therefore

S3 + 8x + 15x² + 24x³ + 35x¹ + &c . , S = 1 x + S' 3 9

88

ALGEBRA.

with a remainder

9 (x² + 3x³ + 6x¹ + 10x³ + &c. ... ) .

Therefore we may take S" = 1 + 3x + 6x² + 10x³ + &c.

Lastly

S' -= 3-S"

without any remainder.

Consequently the series is a recurring series of the third order. It is, in 1 fact, the expansion of 1-3x + 3x² - 23

SUMMATION OF SERIES BY THE

METHOD

OF

DIFFERENCES .

RULE . - Form successive series ofdifferences until a series Let a, b, c, d, &c. be the first of equal differences is obtained. terms of the several series ; then the nth term of the given series is 264

265 a + (n − 1 ) b +

(n - 1 ) (n - 2) n − 1 ) (n - 2) ( n − 3) d+ c+ 1.2 1.2.3

The sum of n terms

266

= na +

n (n - 1 ) n (n − 1 ) (n - 2) c + & c. b+ 1.2 1.2.3

Proved by Induction.

EXAMPLE :

a ... 1+ 5 + 15 + 35 +70 +126 + ... b ... 4 +10 +20 + 35 +56 +... c ... 6 +10 + 15 + 21+ ... d ... 4+ 5+ 6 + ... e ... 1 + 1 + ...

The 100th term of the first series = 1 + 99.4 +

99.98 99.98.97 4 99.98.97.96 6+ + 1.2 1.2.3 1.2.3.4

The sum of 100 terms

= 100+

100.99 100.99.98 100.99.98.97 100.99.98.97.96 4+ 6+ 4+ 1.2 1.2.3.4 1.2.3 1.2.3.4.5

89

FACTORIAL SERIES.

267 To interpolate a term between two terms of a series by the method of differences . Ex.-Given log 71 , log 72, log 73, log 74, it is required to find log 72:54. Form the series of differences from the given logarithms, as in (266) ,

log 71 a ... 1.8512583 b ... ⚫0060742 C ... -.0000838 d ... - 0000022

log 72 log 73 1.8633229 1.8573325 ·0059088 0059904 -'0000816 considered to vanish.

log 74 1.8692317

Log 72.54 must be regarded as an interpolated term , the number of its place being 2.54. Therefore put 2:54 for n in formula ( 265 ) . Result 1.8605777. log 72.54

DIRECT

268

Ex.:

•

FACTORIAL

SERIES .

5.7.9 + 7.9.11 + 9.11.13 + 11.13.15 + ... d

common difference of factors ,

m = number of factors in each term , n = number of terms ,

α = first factor of first term -d.

nth term == ( a + nd) ( a + n + 1d) ...... (a + n + m - 1d) . 269

To find the sum of n terms .

RULE .

From the last term with the next highest factor take

thefirst term with the next lowest factor , and divide by (m + 1 ) d.

PROOF.- By Induction . Thus the sum of 4 terms of the above series will be, putting d = 2 , m = 3, 11.13.15.17-3.5.7.9 n = 4, a = 3,

S= (3 + 1 ) 2

Proved either by Induction, or by the method of Indeterminate Coefficients.

N

90

ALGEBRA.

대 INVERSE

1

1

270

FACTORIAL

Ex.:

1

1 +

+ 5.7.9

SERIES .

7.9.11

+ 9.11.13

+ ... 11.13.15

Defining d, m, n, a as before, the 1

nth term = (a + nd) (a +n + 1d) ... (a + n + m − 1d)' 271

To find the sum of n terms .

RULE. - From the first

term wanting its last factor take the last term wanting its first factor, and divide by (m - 1) d .

Thus the sum of 4 terms of the above series will be, putting d = 2, m = 3, 1 1 5.7 13.15 n = 4, a = 3, (3-1) 2 PROOF. -By Induction , or by decomposing the terms, as in the following example. 272 Ex.: To sum the same series by decomposing the terms into partial fractions . Take the general term in the simple form 2 (r − 2) j' ( +2) Resolve this into the three fractions 1 1 - 1 + 4r 8 (r +2) by (235) . 8 (r- 2)

Substitute 7 , 9, 11 , &c . successively for r, and the given series has for its equivalent the three series 1 1 1 1 1 1 1 ( + + + + + 5 7 8 11 13 2n +3 1 2 - 2 2 - 2 2 2 + 9 7 13 8 11 2n +3 2n +55 ;} +

1 81 {

1 1 1 1 1 1 ? + + + ****** + + + 13 11 9 2n +3 2n +5 2n +7

and the sum of n terms is seen, by inspection , to be 1 ( 1 - 1 ---1 1 1 = 161 -+ 2n +5 45.7 85 7 2n +75 (2n + 5) (2x (2n + 7) } , + 7) 1 taking , a result obtained at once by the rule in (271 ) for the first 5.7.9 1 term , and for the nth or last term . (2n + 3) (2n + 5) (2n + 7)

91

FACTORIAL SERIES.

273

Analogous series may be reduced to the types in ( 268 )

and ( 270) , or else the terms may be decomposed in the manner shewn in ( 272) .

Ex.:

10 4 1 7 + + + + ...... 4.5.6 3.4.5 1.2.3 2.3.4

has for its general term 3n -2 == n (n + 1 ) (n + 2)

1 5 + n +1 n

4 n + 2 by (235) ,

and we may proceed as in (272) to find the sum of n terms. The method of ( 272) includes the method known as " Summation by Subtraction," but it has the advantage of being more general and easier of application to complex series.

COMPOSITE

274

FACTORIAL

SERIES .

If the two series

5.6 5.6.7 5.6.7.8 20 + x² + ... (l− x )- = 1+ 5 + 1.2 x²+ 1.2.3 ³ 1.2.3.4 3.4.5.6 3.4 3.4.5 -3 x²+ ·20 +... ·203+ (1 − x) -³ = 1 + 3x + 1.2.3.4 1.2 1.2.3 be multiplied together, and the coefficient of a¹ in the product 8 be equated to the coefficient of a¹ in the expansion of ( 1 − x) -³ , we obtain as the result the sum of the composite series 5.6.7.8x1.2 +4.5.6.7x2.3 + 3.4.5.6 × 3.4 4! 2.11 ! + 2.3.4.5x4.5 + 1.2.3.4x5.6 = 7! 4!

275

Generally , if the given series be P₁Q₁ + P₂Q₂ + ... + Pn - 1Qn - 1

where and

Qr = r (r + 1 ) (r + 2) ...· ( r + q − 1 ) , P‚ = (n − r) (n − r + 1 ) ... (n − r + p − 1 ) ;

the sum of n- 1 terms will be

p! q! (p + q + 1 ) ! *

(n + p + g − 1 ) ! (n - 2) !

(1) ,

92

ALGEBRA.

MISCELLANEOUS

276

SERIES .

Sum of the powers of the terms of an Arithmetical Progression.

1 + 2 +3 + ... + n =

n (n + 1 ) 2

= S₁

1 + 2²+ 3² + ... + n² =

n ( n + 1 ) ( 2n + 1 ) 6

= S2

n ( n + 1 ) 2² 1 + 2 + 3 + ...+ n³ = {n(n + 2

= S3

— S.. 1 + 2' +3′ + ... + n ' = " (n + 1) (2n + 1) (3n² +3n − 1 ) = 30 [By the method of Indeterminate Coefficients (234) . A general formula for the sum of the 7th powers of 1.2.3 ... n , obtained in the same way is

1 ST =

-- 1 n”+¹ + { n” + A¸n ” -¹ + … .. A¸ - 1n,

+

r+ 1 where A , A , &c. , are determined by putting p = 1 , 2 , 3, &c . successively in the equation

1 2 (p + 1 ) !

A A₁ = + + + = (p +2 ) ! r (r − 1) (p − 1) ! r (p) !

Ap 'r (r − 1) ... (r − p + 1 )'

277

a™ + (a + d)™ + (a + 2d) ™ + ... + (a + nd)m m-2 = ( n + 1 ) a™ + S¸ma" -¹d + S¸C (m, 2 ) a™ -² d²

+ S¸C (m , 3 ) am - ³d³ + &c . PROOF.- By Binomial Theorem and (276) .

278

Summation of a series partly Arithmetical and partly Geometrical .

EXAMPLE . terms.

To find the sum of the series 1 + 3x + 5x² + to n

93

MISCELLANEOUS SERIES.

Let

8 = 1 + 3x + 5x² + 7x³ + ... + ( 2n − 1 ) x²-¹ , sx = x + 3x² + 5x + ... + ( 2n − 3) x" ¹ + ( 2n − 1 ) x”,

.. by subtraction, s (1 − x) = 1 + 2x + 2x² + 2x³ + ... + 2x² - ¹— ( 2n - 1 ) x” 1 - x -1 = 1 + 2x — ( 2n − 1 ) x”, 1--x s = 1- (2n - 1 ) x" + 2x ( 1 − x² - ¹) 1 -x (1 - x)'

279

A general formula for the sum of n terms of a + (a +d) r + (a + 2d) r² + (a + 3d) µ³ + &c .

is

S=

a− (a +n − 1d) r² + dr (1 — p²-¹) 2 l-r ( 1 —r)² Obtained as in ( 278) .

RULE.-Multiply by the ratio and subtract the resulting series.

280

1 -x = 1 + x + x² + x³ + ... + x² - ¹ +

281

= 1 + 2x + 3x² +4a³ + ... - x)²

trở

Xn 1−x

nước t

t (n + 1 )

... ( 1 − x)²

282

(n − 1 ) x + (n − 2) x² + (n − 3) x³ + ... + 2x² - ² + x² - ¹ = (n − 1)

−n + 1

By (253).

(1 - x)²

283

1 + n + n (n − 1 ) + 2!

n (n - 1 ) (n - 2) + & c. = 2", 3!

1 −n + n (n − 1 ) 2!

n (n − 1 ) (n − 2) + & c. = 0. 3! By making a bin (125). X=1

evrata

94

ALGEBRA.

284

The series n- 3

1

+

(n − 4) (n − 5) ___ (n − 5 ) ( n − 6 ) ( n − 7 ) + ... —5 4! 3!

...· + (− 1 ) r - 1 (n — r — 1 ) ( n — r — 2 ) ... ( n − 2r + 1 ) r! 22

consists of

n- 1 or

terms , and the sum is given by

2 3

S =

if n be of the form 6m + 3 , n

S =

0 if n be of the form 6m + 1 , 1

if n be of the form 6m, ||

|

S N

2

S =

if n be of the form 6m + 2.

n PROOF. - By ( 545 ) , putting p = x + y , q = xy, and applying (546) .

285

The series

n (n − 1 ) (n- 2)" n” n " —n -n (n (n − 1) "+ +? 2!

_n (n − 1 ) (n − 2 ) ( n − 3) ' + &c ... 3! takes the values

0,

according as r is

< n,

n !,

in ( n + 1 ) !

= n + 1. n , or =

PROOF. By expanding (e - 1 ) ", in two ways : first, bythe Exponential Theorem and Multinomial ; secondly, by the Bin. Th., and each term of the expansion by the Exponential . Equate the coefficients of a" in the two results . Other results are obtained by putting r = n +2 , n + 3, &c . The series (285 ) , when divided by r !, is , in fact , equal to the coefficient of a" in the expansion of 22

n

23 ...

3! + { (x+ 2 2! +3

}

POLYGONAL NUMBERS.

95

286 By exactly the same process we may deduce from the function { e -e-* )" the result that the series

n²—n (n — 2) " + n (n − 1) (n − 4)” — &c. 2! takes the values 0 or 2" .n !, according as r is < n or = n ; this series, divided by r !, being equal to the coefficient of ar in the expansion of n ენ 2n + +3 3 " x+ {x •{ +3! 5!

POLYGONAL

NUMBERS .

The nth term of the 7th order of polygonal numbers is equal to the sum of n terms of an Arith . Prog . whose first term is unity and common difference r - 2 ; that is 287

= = ½ { 2+ (n − 1 ) ( r − 2 ) } = n + ³n (n − 1) (r − 2 ) . 288

The sum of n terms

= n (n + 1) + n ( n − 1 ) (n +1) (r − 2) ¸ 2 6 By resolving into two series.

Order.

nth term .

1230

1 n

56

1 2

in ( n + 1 ) n²

5 6

...

T

1 1 1 1 1 1

In (3n - 1 ) (2n - 1 ) n ...

...

n + n (n − 1) (r − 2)

...

1 2 3 4 5 6

1 1 1 1 1 7 6 5 3 4 6 10 15 21 28 9 16 25 36 49 12 22 35 51 70 15 28 45 66 91 ... ... ... ... ... ...

1 , r, 3 + 3 ( r − 2 ) , 4 + 6 (r − 2) , 5 + 10 (r − 2 ) , 6 + 15 (r - 2), &c.

96

ALGEBRA.

In practice to form , for instance, the 6th order of poly gonal numbers -write the first three terms by the formula , and form the rest by the method of differences.

Ex.:

1

6 5

15

9 4

[r- 2 = 4 ]

28

13

45

17 4

4

66

21 4

FIGURATE

25 4

91

120 ... 29 ...

4 ...

NUMBERS .

289 The nth term of any order is the sum of n terms of the preceding order. The nth term of the 7th order is

n (n + 1). ... (n +r − 2) = – H ( n, r − 1) . ( -1 )!

290

[By 98.

The sum of n terms is n ( n + 1 ) ... ( n + r − 1 ) = H (n , r) . r!

Order.

nth term .

Figurate Numbers .

1

1, 1 ,

1,

1,

ཇ.

1

1

2

1, 2,

3,

4,

5,

6

n

3

1, 3,

6, 10 ,

15 ,

21

n (n + 1 ) 1.2

4

1 , 4 , 10 , 20 ,

35 ,

56

n (n + 1 ) (n + 2 ) 1.2.3

5

1 , 5 , 15 , 35,

70, 126

6

1 , 6 , 21 , 56, 126, 252 n (n + 1 ) (n + 2 ) ( n + 3) (n + 4) 1.2.3.4.5

n (n + 1 ) ( n + 2) ( n + 3) 1.2.3.4

97

HYPERGEOMETRICAL SERIES.

HYPERGEOMETRICAL

SERIES .

α.β 291 1+ x + a (a + 1 ) B (B + 1 ) x2 1.Y 1.2.y(y + 1 ) a (a + 1 ) (a + 2) B (B + 1 ) (B + 2)

x² + & c .

+ 1.2.3.y (y + 1 ) (y + 2) is convergent if a is < 1 ,

(239 ii. )

and divergent if a is > 1 ;

and if a = 1 , the series is convergent if y -a -ẞ is positive ,

divergent if y a -a -ẞ is negative ,

(239 iv.)

and divergent if y- a -ẞ is zero .

(239 v.)

Let the hypergeometrical series ( 291 ) be denoted by F(a, B, y) ; then, the series being convergent, it is shewn by induct ion that

292

F(a, B + 1 , y + 1) = 1 F (a, ẞ, y) 1 - k₁

concluding with

1 - k₂ I- &c. ...

1 - kar-1

I — kąrz r

where k₁, kg, kz, &c . , with z2 , are given by the formulæ (a + r − 1 ) (y + r − 1 − ß) x kyr-1 = (y +2r - 2) (y +2r − 1 )

Kar

= (B +r) (y + r − a) x (y+2r − 1) (y+2r)

22r

= F (a + r, B + r + 1 , y + 2r + 1 ) F (a + r, B + r , y + 2r)

The continued fraction may be concluded at any point with . When is infinite , % , - 1 and the continued = fraction is infinite .

98

293

ALGEBRA.

Let

x² a4 x6 + 1 f 1 + f (n (x)) = = 1 1 . Y + 1.2.y (y + 1 ) + 1.2.3.y (y + 1 ) (y +2) +&c. x²

the result of substituting

for a in (291 ) , and making αβ

β Ba = ∞ ..

Then , by last , or independently by induction , f(y + 1 ) = 1 P₁ P2 P 1+& f (y) 1+ 1+ 1 + ... + 1 + &c c. a²

with P p₁ m= (y + m − 1) (y+ m)

294

In this result put y =

and

½ 2 for a, and we obtain by

Exp . Th . ( 150) , ey —p¯y e" + e˜¯ÿ

y y y² =y the 7th component being 2r- 1 1+ 3+ 5 + &c .

Or the continued fraction may be formed by ordinary division of one series by the other. m 295

e " is incommensurable, m and n being integers . m the last and (174) , by putting a = ก

From

INTEREST .

If ľ be the Interest on £ 1 for 1 year, n

the number of years,

P

the Principal,

A

the amount in n years .

Then

A = P (1 +nr) .

296

At Simple Interest

297

At Compound Interest A = P ( 1 + r) ".

By ( 84) .

99

INTEREST AND ANNUITIES.

298

But if the payments of Interest be made I times a year ... ... }

nq

A = P ( 1 + 5 )" .

If A be an amount due in n years' time , and P the present Then worth of A. A P= By (296 ) . 299 At Simple Interest 1 + nr A

300

At Compound Interest

P=

By (297) . (1 +r ) *

301

Discount

A — P.

ANNUITIES .

302

The amount of an Annu ity of £ 1 in n years ,

= n + n (n − 1) ጥ. 2

By (82).

at Simple Interest ... 303

Present value of same

− 1) r¸ = n +¹n (n 1+ nr

304

Amount at Compound ...

= (1 +r ) *−1 (1 +r ) — 1 °

Present worth of same

= 1- ( 1 + r) -n ( 1 + r) −1

Amount when the pay. ) Interest ments of

By (85) .

By (300) .

(1 + 7)” - 1 By (298) .

are made q times per ... annum

-1 ' −1 (1+2) -ny − (1+

Present value of same

‫ام‬ ‫ر‬

305

By (299) .

( +

)-1

ALGEBRA.

100

306

Amount when the pay ments of the Annuity

(1 + r ) " − 1 1

are made m times per annum ... ... ...

m { (1 + r) = - 1 } 1− (1 +r) ¯”

Present value of same

m { ( 1 + r) " − 1 }

307

Amount when the

In

1+ (

terest is paid q times and the Annuity m times per annum ....

) -1

m m {( 1+ 2) = -1 } -nq

- (1 + 2) Present value of same

= m m {( 1 ++) =-1 }

PROBABILITIES .

309 If all the ways in which an event can happen be m in number, all being equally likely to occur, and if in n of these m ways the event would happen under certain restrictive conditions ; then the probability of the restricted event hap pening is equal to n ÷ m. Thus , if the letters of the alphabet be chosen at random , any letter being equally likely to be taken, the probability of 5 a vowel being selected is equal to . The number of un restricted cases here is 26 , and the number of restricted ones 5. 310

If, however, all the m events are not equally probable ,

they may be divided into groups of equally probable cases . The probability of the restricted event happening in each group separately must be calculated , and the sum of these probabilities will be the total probability of the restricted event happening at all.

PROBABILITIES.

101

EXAMPLE.-There are three bags A, B, and C. A contains 2 white and 3 black balls. B 4 3 99 "" 99 5 4 C 29 "" 99 A bag is taken at random and a ball drawn from it. bability of the ball being white.

Required the pro

Here the probability of the bag A being chosen , and the subsequent probability of a white ball being drawn = 3. Therefore the probability of a white ball being drawn from A 2 2 = 13X = 5 15 .

Similarly the probability of a white ball being drawn from B 13 3 1! = = 3x 7 And the probability of a white ball being drawn from C 1 4 4 = X = 3 9 27' Therefore the total probability of a white ball being drawn 4 = 401 1 2 = + 945 27 7 15

If a be the number of ways in which an event can happen , and b the number of ways in which it can fail , then the

311

Probability of the event happening

312

Probability of the event failing

a +b b

= a+b

Certainty

=

Thus

1.

If p, p' be the respective probabilities of two independent events , then

313

Probability of both happening

= pp'.

314

""

of not both happening =: 1 -pp'.

315

99

of one happening and one failing = p +p' - 2pp'.

316

""

of both failing = (1 — p) (1 — p') .

1 ALGEBRA.

102

If the probability of an event happening in one trial be P, and the probability of its failing 7, then

317

Probability of the event happening

times in n trials.

= C (n, r) p” q” -r. 318

Probability of the event failing r times in n trials = C (n, r) p” -rqr.

[ By induction.

319 Probability of the event happening at least r times in n trials the sum of the first n -r + 1 terms in the expansion of (p +q)". 320 Probability of the event failing at least r times in n trials = the sum of the last n - r + 1 terms in the same ex pansion.

321 The number of trials in which the probability of the same event happening amounts to p'

= log (1 - p') log ( 1 - p ) From the equation ( 1 - p)

322

= 1 - p'.

DEFINITION . -When a sum of money is to be received if

a certain event happens, that sum multiplied into the proba bility of the event is termed the expectation . EXAMPLE . If three coins be taken at random from a bag containing one sovereign, four half-crowns, and five shillings , the expectation will be the sum of the expectations founded upon each way of drawing three coins . But this is also equal to the average value of three coins out of the ten ; that is, ths of 35 shillings , or 10s. 6d.

323 The probability that , after r chance selections of the numbers 0 , 1 , 2 , 3 ... n, the sum of the numbers drawn will be s, is equal to the coefficient of as in the expansion of (æº + a¹ + æ² + ... + a") " ÷ (n + 1 ) ”.

PROBABILITIES.

103

324 The probability of the existence of a certain cause of an observed event out of several known causes , one of which must have produced the event , is proportional to the a priori probability of the cause existing multiplied by the probability of the event happening from it if it does exist. Thus, if the a priori probabilities of the causes be P₁ , P₂ ... &c. , and the corresponding probabilities of the event hap pening from those causes Q1 , Q2 ... &c . , then the probability of the 7th cause having produced the event is

PrQr Σ (PQ)*

325

If P₁ , P ... &c . be the a priori probabilities of a second

event happening from the same causes respectively , then , after the first event has happened , the probability of the

second happening is

(PQP) Σ (PQ )

P.Qr Pr . which is For this is the sum of such probabilities as Σ (PQ)' the probability of the 7th cause existing multiplied by the probability of the second event happening from it.

Ex. 1.- Suppose there are 4 vases containing each 5 white and 6 black balls, 2 vases containing each 3 white and 5 black balls, 2 white and 1 black ball . and 1 vase containing A white ball has been drawn, and the probability that it came from the group . of 2 vases is required .

Here

4 P₁ = 7 Q₁ = 5 11'

P₁ = 2 / 7'

P3 P₁ = 1/1/1 7

3 Q₂2 =

•

Q3 =

2 3

Therefore, by (324) , the probability required is 2.3 7.8 99 = 1.2 2.3 4.5 427 + + 7.3 7.8 7.11

Ex. 2. After the white ball has been drawn and replaced , a ball is drawn again ; required the probability of the ball being black.

ALGEBRA.

104

Here

P₁ =

6 11'

P₂:=

5 8'

P; = 1/1/0

The probability, by (325) , will be 1.2.1 4.5.6 2.3.5 + + 7.3.3 7.11.11 7.8.8 58639 = 2.3 4.5 112728 1.2 + + 7.3 7.11 7.8 If the probability of the second ball being white is required, Q₁Q, Q must be employed instead of P¦ P'P'.

326 The probability of one event at least happening out of a number of events whose respective probabilities are a , b, c,

& c. , is

P₁ − P₂+P¸3 −P₁4 + &c.

where

P₁ is the probability of 1 event happening, P2 2 "" 99 ""

and so on.

For, by (316) , the probability is

1- (1 - a) (1 - b ) ( 1 — c) ·... = Σa - Σab + Σabc- , ...

327 The probability of the occurrence of r assigned events and no more out of n events is QrQr+1+ Qr+ 2 − Qr+s + &c., where Q, is the probability of the r assigned events ; Q +1 the probability of r + 1 events including the r assigned events . For if a, b, c ... be the probabilities of the r events , and a', b', c' ... the probabilities of the excluded events, the re quired probability will be

abc ... (1 — a') ( 1 — b') ( 1 — c') ... = abc ... ( 1 - Σa' + Za'b' - Za'b'c' + ... ) .

328 is

The probability of any r events happening and no more EQ,

EQr+1 + ΣQr +2'- &c .

NOTE.- If a = b = c = &c. , then ΣQ, = C (n , r ) Qr, &c.

INEQUALITIES.

105

INEQUALITIES .

330

a₁ + a2 + ... + an lies between the greatest and least of + b₁+ b₂+ ... + bn

the fractions a

an

a2

" ...

, the denominators being all of

the same sign . PROOF. -Let k be the greatest of the fractions, and ar any other ; then br a, √ab.

331

a₁ + a₂+ ... +an > n

332

a₁a ... an

Arithmetic mean > Geometric mean .

or,

PROOF. - Substitute both for the greatest and least factors their Arith metic mean . The product is thus increased in value. Repeat the process indefinitely. The limiting value of the G. M. is the A. M. of the quantities.

m m b \m a +b " >(a+b)" , 2

333

excepting when m is a positive proper fraction. ከ a" + b " = ( " + ¹ ) " { (1 + x) " + (1 − x) " } ,

PROOF :

where x = a-b a+ b

334

Employ Bin. Th .

m m m m an)" , atait ... ta > (a +a + ... +a. n

excepting when m is a positive proper fraction . P

106

ALGEBRA.

Otherwise .-The Arithmetic mean of the mth powers is greater than the mth power of the Arithmetic mean, excepting when m is a positive proper fraction . PROOF . Similar to (332 ) . left side, employing (333) .

336 then

Substitute for the greatest and least on the

If a and m are positive, and a and me less than unity ; (1 + x) -

> 1 - mx.

(125 , 240 )

337 If x, m , and n are positive , and n greater than m ; then , by taking a small enough, we can make 1 + nữ > (1 + r )” . For a may be diminished until 1 + nx is > ( 1 − mx) -1 , and this is (1 + x)", by last.

338

If a be positive ,

(150)

log ( 1 + x) < x.

If æ be positive and > 1 ,

X2 x 2² . log (1 + x) > w−

If a be positive and < 1 ,

log log

( 155 , 240)

1 > x.

(156)

1 -x

339

When n becomes infinite in the two expressions

1.3.5 ... (2n - 1 ) 2.4.6 ... 2n

and

the first vanishes , the second product lies between and 1 .

3.5.7 ... (2n + 1) " 2.4.6 ... 2n becomes infinite, and their

Shewn by adding 1 to each factor (see 73 ) , and multi plying the result by the original fraction .

340

If m be > n , and n > a , m

m+a \m-- -a

is < ( n +a)". n- a

SCALES OF NOTATION.

107

If a, b be positive quantities ,

341

a+ b\ a + b a b' is > ( a + b)* + *.

a+ b+ c aa bb c ° > ( a + b + c) +++ o a²b³

Similarly

These and similar theorems may be proved by taking loga rithms of each side, and employing the Expon . Th . (158) , &c .

SCALES OF NOTATION.

342

If N be a whole number of n + 1 digits , and r the radix

of the scale ,

N = P₂r” + Pn - 1 ?” - ¹ +Pn - 2 ?¹²¬² + ... + P₁r + Po

where Pn , Pn-19 ... Po ar th di gits . e e

343

Similarly a radix-fraction will be expressed by Pi P2 P3 + + + & c. , r 2.2 2.3

where P1 , P2 , & c . are the digits . EXAMPLES :

3426 in the scale of 7

= 3.7° +4.73 + 2.7 + 6 ; 1

1045 in the same scale =

0 72

5 + 74

344 Ex. - To transform 34268 from the scale of 5 to the scale of 11.

RULE . -Divide successively by the new radix.

11 34268 111343 -t 11 | 40-3 1-9

Result 193t, in which t stands for 10.

108

ALGEBRA.

345

t0el from the scale of 12 to that

Ex. - To transform

of 7, e standing for 11 , and t for 10. RULE . -Multiply successively by the new radix.

'toel 7 5 1657 7 6.1931 7 1.0497 7 0.2971

Result

5610 ...

346 Ex.- In what scale does 2t7 represent the number 475 in the scale of ten ?

Solve the equation 2² +10r + 7 = 475. Result r = 13.

347

[178

The sum of the digits of any number divided by r - 1

leaves the same remainder as the number itself divided by r- 1 ; being the radix of the scale. (401 ) 348

The difference between the sums of the digits in the

even and odd places divided by r + 1 leaves the same re mainder as the number itself when divided by r + 1 .

THEORY

OF NUMBERS .

a 349

is in its lowest terms .

If a is prime to b ,

b PROOF.-Let

a = a1 a fraction in lower terms. b B Divide a by a,, remainder a, quotient q₁ , b by b₁, remainder b, quotient q₁;

and so on, as in finding the H. C. F. of a and a₁ , and of b and b₁ ( see 30) . Let an and b, be the highest common factors thus determined .

109

THEORY OF NUMBERS.

Then, because

and so on ; thus

a = a1 2 b b₁

α = a -91a1 = 029 b ba b- q, b₁

(70)

a = a1 = aq = &c. ...... = an b b₁ ba ხა

Therefore a and b are equimultiples of a, and b„ ; that is, a is not prime to b if any fraction exists in lower terms.

a

α 350

then a and b' are

=

If a is prime to b , and b

b

equimultiples of a and b . PROOF.- Let

reduced to its lowest terms be P. 9

Then P = a ' and , b' q

since p is now prime to q, and a prime to b, it follows , by 349 , that

is Չ

a neither greater nor less than b ; that is, it is equal to it.

351

Therefore, &c .

If ab is divisible by c, and a is not ; then b must be.

PROOF .-Let

ab с

q;

...

a = 9 с b

But a is prime to c ; therefore, by last, b is a multiple of c.

352 to c.

If a and b be each of them prime to c, ab is prime [ By (351 ) .

353

If abcd ... is divisible by a prime, one at least of the

factors a, b, c , &c. must also be divisible by it .

Or, if p be prime to all but one of the factors , that factor is divisible by p . (351 ) 354

Therefore, if a" is divisible by p, p cannot be prime to

a; and if p be a prime it must divide a.

355 If a is prime to b, any power of a is prime to any power of b. Also, if a, b , c, &c . are prime to each other, the product of any of their powers is prime to any other product of their powers .

110

ALGEBRA.

356 No expression with integral coefficients , A + Bx + Cx² + ..., can represent primes only.

such

as

PROOF. For it is divisible by r if A = 0 ; and if not, it is divisible by A , when A.

357

The number of primes is infinite.

PROOF. Suppose if possible p to be the greatest prime. Then the pro duct of all primes up to p, plus unity, is either a prime, in which case it would be a greater prime than p, or it must be divisible by a prime ; but no prime up to p divides it, because there is a remainder I in each case. Therefore, if divisible at all, it must be by a prime greater than p. In either case, then, a prime greater than p exists.

If a be prime to b, and the quantities a , 2a, 3a, ... (b - 1 ) a be divided by b, the remainders will be different . 358

PROOF.

Assume ma - nb = m'a — n'b, m and n being less than b, a n —n' = m -m b

359

Then by (350).

A number can be resolved into prime factors in one

way only. 360

[ By (353).

To resolve 5040 into its prime factors .

RULE .-Divide by the prime numbers successively. 2 × 5 5040 2504 2252 2126

763 319 3

Thus

5040 = 2¹.3.5.7.

361 Required the least multiplier of 4704 which will make the product a perfect fourth power.

By (196 ), Then

4704

25.3.72.

25.3¹.7 × 23.3³ . 7² = 2º . 3¹ . 7' — 84*,

the indices 8, 4, 4 being the least multiples of 4 which are not less than 5, 1 , 2 respectively. Thus 28.33.73584 is the multiplier required .

111

THEORY OF NUMBERS.

362 All numbers are of one of the forms 2n or 2n + 1 99

99

2n or 2n - 1

99

99

3n or 3n + 1

99

99

4n or 4n +1 or 4n +2 4n or 4n + 1 or 4n - 2

33

99

5n or 5n + 1 or 5n + 2

99

99

and so on . 363

All square numbers are of the form 5n or 5n ± 1.

PROOF. By squaring the forms ɔ̃n , 5n ± 1 , 5n ± 2 , which comprehend all numbers whatever. 364 All cube numbers are of the form 7n or 7n ±1 . And similarly for other powers .

365 The highest power of a prime p, which is contained in the product m !, is the sum of the integral parts of M

m

P

p2 ,

m 39 p

&c .

m

For there are

m 2 which p² The successive

factors in m ! which p will divide ;

P it will divide a second time ; and so on .

divisions are equivalent to dividing by m p" . p

m

m+ m + p2 ... &c . = p

EXAMPLE . The highest power of 3 which will divide 29 !. Here the factors 3 , 6, 9, 12, 15 , 18 , 21 , 24 , 27 can be divided by 3. Their number is 29 3 = 9 (the integral part) . The factors 9, 18, 27 can be divided a second time. 29 32

Their number is

3 (the integral part) . One factor, 27, is divisible a third time.

29 = 1 (integral part) . 35

9 +3 + 1 == 13 ; that is, 3¹³ is the highest power of 3 which will divide 29 !.

366 The product of any r consecutive integers is divisible by r !.

PROOF :

n (n − 1 ) ... ( n − r + 1 ) is necessarily an integer, by (96) , r!

112

367

ALGEBRA.

If n be a prime, every coefficient in the expansion of By last.

(a + b)", except the first and last , is divisible by n.

368

If n be a prime , the coefficient of every term in the ex

pansion of ( a + b + c ... ) " , except a" , b" , &c. , is divisible by n. PROOF. - By ( 367) .

369

Fermat's Theorem. -If p be a prime , and N prime to

p ; then

NP-1-1 is divisible by p.

NP = (1 + 1 + ...)" = N + Mp.

PROOF : 370

Put ß for ( b + c + ... ) .

By (368 ) .

If p be any number, and if 1 , a , b, c,.... (p - 1 ) be all

the numbers less than, and prime to p ; and if n be their number, and a any one of them ; then a" -1 is divisible by p .

PROOF. -If x, ax, bx ... (p - 1 ) x be divided by p, the remainders will be all different and prime to p [ as in (358) ] ; therefore the remainders will be 1, a, b, c ... (p - 1 ) ; therefore the product x"abc ... (p - 1 ) = abc ... (p - 1) + Mp.

371

Wilson's Theorem.- If p be a prime, and only then,

1+ ( p - 1 ) ! is divisible by p. Put p - 1 for r and n in (285) , and apply Fermat's Theorem to each term .

372 If P be a prime by p.

2n + 1 , then (n ! ) ² + ( - 1 ) " is divisible

PROOF. By multiplying together equi - distant factors of (p - 1 ) ! in Wilson's Theorem, and putting 2n + 1 for p.

373 Let N = abc" ... in prime factors ; the number of in tegers , including 1 , which are less than n and prime to it, is

( 1-4) N ( 1-1 a ) ( 1-4) )... 증 .... -풍) (1PROOF. - The number of integers prime to N contained in a Similarly in b , c , &c .

Take the product of these,

is aº

ap a ·

113

THEORY OF NUMBERS.

Also the number of integers

less than and prime to

(NX MX &c .) is the product of the corresponding numbers for N, M, &c. separately .

374

The number of divisors of N, including 1 and N itself, For it is equal to the number

is = (p + 1 ) (q + 1 ) ( r + 1 ) ... of terms in the product

(1 +a + ... + a³) ( 1 + b + ... + b²) ( 1 + c + ... + c") ... &c .

375 The number of ways of resolving N into two factors is half the number of its divisors (374) . If the number be a square the two equal factors must, in this case, be reckoned as two divisors .

376 If the factors of each pair are to be prime to each other, put p, q, r, &c. each equal to one.

377

The sum of the divisors of N is ap+1-1

ba+1 _1

cr+1-1

b- 1

c- 1

• a -1

PROOF.

...

By the product in (374) , and by (85) .

378 If p be a prime, then the p - 1th power of any number is of the form mp or mp + 1 . By Fermat's Theorem (369) . Ex.-The 12th power of any number is of the form 13m or 13m + 1.

379

To find all the divisors of a number ; for instance , of 504.

I.

II.

504 252 126 63 21 7

2 2 2 3 3 7

1

8 6 3 9 18 7 14 84 168

12 36 28 63

24 72 56 126

21 252

42 504

EXPLANATIO N. - Resolve 504 into its prime factors, placing them in column II.

Q

114

ALGEBRA.

The divisors of 504 are now formed from the numbers in column II., and placed to the right of that column in the following manner : Place the divisor 1 to the right of column II. , and follow this rule Multiply in order all the divisors which are written down by the next number in column II., which has not already been used as a multiplier : place the first new divisor so obtained and all the following products in order to the right of column II.

380 S, the sum of the 7th powers of the first n natural numbers is divisible by 2n + 1 .

PROOF :

x ( x² — 1²) ( x² — 2²) .… . ( x² — n²)

constitutes 2n + 1 factors divisible by 2n + 1 , by (366) . Multiply out, re jecting a, which is to be less than 2n +1. Thus, using (372), x² - S₁x² - ² + S₂22 — ... S. 12² + ( - 1 ) " ( n ) ² = M (2n + 1) . Put 1 , 2, 3 ... (n - 1) in succession for 2, and the solution of the (n - 1) equations is of the form

S, = M (2n + 1) .

THEORY

OF

EQUATIONS .

FACTORS OF AN EQUATION .

Generalform of a rational integral equation ofthe nth degree . 400

Poxn +P₁xn-¹ +P²x²¬² + ... + Рn - 1x +Pn = 0.

The left side summary .

will be designated f(x) in the following

401 If f (x) be divided by -a , the remainder will be ƒ(a) . By assuming f(x) = P (x - a) + R. 402

If a be a root of the equation f (x) = 0 , then ƒ (a) = 0.

403 To compute f(a) numerically ; and the remainder will be f(a) . 404

EXAMPLE.

divide f(x) by x - a, [401

To find the value of 4x - 3x + 12x -x² + 10 when

2

= 2.

4-3 +12 +0 −1 +0 +10 8 +10 +44 +88 +174 + 348 4+5 +22 +44 +87 +174 +358

Thus ƒ (2) = 358.

If a, b, с ... k be the roots of the equati on f (x) = 0 ; then, by (401 ) and (402 ) , 405

f(x) = p。 (x - a) (x − b) (x − c) ... (x− k).

By multiplying out the last equation, and equating coefficients with equation (400), considering po = 1 , the following results are obtained :

116

406

THEORY OF EQUATIONS.

-P₁ =

the sum of all the roots of ƒ (x) .

the sum of the products of the roots taken P₂ = two at a time . {

-P3 = {

the sum of the products of the roots taken three at a time . ...

(−1)″p, = {

(-1)"P

=

... ...

... ...

... ...

the sum of the products of the roots taken r at a time. ... ... ...

product of all the roots.

407 The number of roots of f(x) is equal to the degree of the equation.

408

Imaginary roots must occur in pairs of the form a + ß√ - 1 ,

a - ẞ√ — 1 .

The quadratic factor corresponding to these roots will then have real coefficients ; for it will be

x² —2ax + a² +ß².

[405 , 226

409 If f (x) be of an odd degree , it has at least one real root of the opposite sign to P

Thus 7-10 has at least one positive root. 410 If f (x) be of an even degree, and p, negative , there is at least one positive and one negative root. Thus a - 1 has +1 and -1 for roots.

411 If several terms at the beginning of the equation are of one sign, and all the rest of another, there is one , and only one, positive root. Thus x + 2x + 3x³ + x³ - 5x - 4 = 0 has only one positive root.

412

If all the terms are positive there is no positive root.

413

If all the terms of an even order are of one sign, and

all the rest are of another sign , there is no negative root, 414

Thus 2-2 + x² - x + 1 = 0 has no negative root,

DISCRIMINATION OF ROOTS.

415

117

If all the indices are even , and all the terms of the same

sign, there is no real root ; and if all the indices are odd , and all the terms of the same sign, there is no real root but zero . Thus x +x + 1 = 0 has no real root, and 25 + x³ +x = 0 has no real root but zero . In this last equation there is no absolute term , because such a term would involve the zero power of x, which is even, and by hypothesis is wanting.

DESCARTES' RULE OF SIGNS .

416 In the following theorems every two adjacent terms in f(x) , which have the same signs, count as one " continuation of sign " ; and every two adjacent terms , with different signs , count as one change of sign .

417 f(x), ƒ (x) , multiplied by ( -a) , has an odd number of changes of sign thereby introduced , and one at least. 418 f(x) cannot have more positive roots than changes of sign, or more negative roots than continuations of sign.

419 When all the roots of f (x) are real, the number of positive roots is equal to the number of changes of sign in f(x) ; and the number of negative roots is equal to the number of changes of sign in ƒ ( -x) .

420

Thus, it being known that the roots of the equation x¹ - 10x³ + 35x² — 50x + 24 = 0

are all real ; the number of positive roots will be equal to the number of changes of sign, which is four. Also ƒ( −x ) = x² + 10x³ + 35x² + 50x + 24 = 0, and since there is no change of sign, there is consequently, by the rule, no negative root.

421

If the degree of f (x) exceeds the number of changes of

sign in f(x) and f( -a) together, by μ, there are at least μ imaginary roots . 422 If, between two terms in f(x) of the same sign, there be an odd number of consecutive terms wanting, then there must be at least one more than that number of imaginary roots ; and if the missing terms lie between terms of different

THEORY OF EQUATIONS.

118

sign, there is at least one less than the same number of imaginary roots . Thus, in the cubic equation a³ +4x - 7 = 0 , there must be two imaginary roots. And in the equation 2-1 = 0 there are, for certain, four imaginary roots. 423

If an even number of consecutive terms be wanting in

f (x), there is at least the same number of imaginary roots. Thus the equation 25 +1 = 0 has four terms absent ; and therefore four imaginary roots at least.

THE DERIVED FUNCTIONS OF ƒ(x) .

Rule for forming the derived functions . 424

Multiply each term by the index of x, and reduce the

index by one ; that is, differentiate the function with respect to x. EXAMPLE.-Take f (x) f' (x) f (x) ƒ³ (x) f (x)

x²+ x + x³ — x³ - x - 1 = 5x¹ + 4x³ + 3x² — 2x −1 = = 20x³ + 12x² + 6x −2 = 60x² +24x + 6 = 120x +24

f (x) = 120 ƒ¹ (x), ƒª ( ) , &c. are called the first, second, &c. derived functions off (x).

425

To form the equation whose roots differ from those of

f(x) by a quantity a. Put x = y + a in f(x) , and expand each term by the Binomial Theorem, arranging the results in vertical columns in the fol lowing manner :

f(a +y) = =

(a + y) as +( 5a + (10a³

+ (a + y) * + (a + y) ³— ( a + y ) ²— (a + y) —1 a³ + a³ + a -1 + 3a² - 2a + 4a³ 1) y - 1) y² + 3a + 6a²

+ (10a² + ( 5a + yo

+ + 4a + 1) y*

1 ) y³

TRANSFORMATION OF AN EQUATION.

119

Comparing this result with that seen in (424) , it is seen that

426

f(a +y) = ƒ (a) +f¹ (a) y + £( a) ( * a) y + £ ª( a )y + £ £ ² (a) y *4 ) y + £ (a 3 2

so that the coefficient

generally of y" in the transformed

equation is fr (a) r

427 To form the equation most expeditiously when a has a numerical value, divide f(x) continuously by x - a, and the successive remainders will furnish the coefficients .

EXAMPLE.-To expand f(y +2) when, as in (425 ) , f(x) = x + 2 + x³ - x³ - x - 1 . Divide repeatedly by a - 2, as follows : 1+ 1 + 1 1 1 - 1 2 + 2 + 6 + 14 + 26 +50

2

1 + 3 + 7 +13 + 25 +49 = ƒ (2) + 2 + 10 + 34 + 94

1 + 5 +17 +47 +119 = f¹ (2) +2 +14 + 62 1 + 7 +31 +109 = ƒª (2) 2 12 + 2 +18

2

1 + 9 +49 = f (2) 3 2 + 2 1 + 11 = f* (2) 14

1 = ƒ³(2) 15

That these remainders are the required coefficients is seen by inspecting the form ofthe equation (426) ; for if that equation be di vided by a = y_repeat edly, these remainders are obviously produced when a = 2.

Thus the equation, whose roots are each less by 2 than the roots of the proposed equation, is y +11y +49y + 109y +119y +49 = 0.

428 To make any assigned term vanish in the transformed equation, a must be so determined that the coefficient of that term shall vanish.

EXAMPLE. -In order that there may be no term involving y* in equation (426), we must have f* (a) = 0. Find f '(a) asin (424) ; thus 120a + 24 = 0 ; .. a = --중. The equation in (424) must now be divided repeatedly by + after the manner of (427) , and the resulting equation will be minus its second term.

120

THEORY OF EQUATIONS.

429

Note , that to remove the second term of the equation P1 f(x ) = 0, the requisite value of a is = ; that is , the npo coefficient ofthe second term, with the sign changed , divided by the coefficient of the first term , and by the number expressing the degree of the equation.

430

To transform f(x) into an equation in y so that y =

a given function of x, put x =

(x) ,

¹ (y) , the inverse function ofy.

EXAMPLE . To obtain an equation whose roots are respectively three times y the roots of the equation a³ - 6x + 1 = 0 . Here y = 32 ; therefore x = 3' by +10, or y³ - 54y + 27 = 0. and the equation becomes 27 3

431 To transform ƒ ( x) = 0 into an equation in which the coefficient of the first term shall be unity, and the other coefficients the least possible integers .

EXAMPLE .-Take the equation 288x³ +240x³ -176x - 21 = 0. Divide by the coefficient of the first term, equation becomes 2³ + 25 / - x²2 - 1130 x -18 6 Y Substitute for x, and multiply by k³ ; we k 112 y y³ + 5k 6 y³ — 18

and reduce the fractions ; the 7 = 0. 96

get

77 = 0. 96

Next resolve the denominators into their prime factors, 5k 112 7k³ = 0. y³ + 25.3 2.3 y³. 2.32 Y The smallest value must now be assigned to k, which will suffice to make each coefficient an integer. This is easily seen by inspection to be 22.3 = 12, and the resulting equation is y³ + 10y³ — 88y — 126 = 0 , the roots of which are connected with the roots of the original equation by y = 12x. the relation

EQUAL ROOTS OF AN EQUATION.

By expanding f (x +z) in powers of z by (405 ) , and also by (426), and equating the coefficients of z in the two ex

121

EQUAL ROOTS.

pansions , it is proved that

f(x) 432

+

' (x) = f

(x− a)

f(x) f(x) + & c. , + (x—b) (x−c)

from which result it appears that , if the roots a , b, c , &c . are all unequal, ƒ (a) and f (x) can have no common measure in volving x.. If, however, there are r roots each equal to a, s roots equal to b , t roots equal to c, & c . , so that

f(x) = p (x — a) ” (x — b) ³ ( x — c) ' ... then

433

“ (x) = f(x) + sf(x) + if(x) + & c .; ƒ (x—b) x -b x -a

and the greatest common measure of ƒ (x) and ƒ' (x) will be 444

(x− a)r−¹ (x — b) ³ - ¹ (x —c) '—¹…… .

When a a, f (x) , ƒ ′ (x) , ... fr-1(a ) all vanish . when ab, &c.

Similarly

Practical method offinding the equal roots . 445

X x X x x ... X , where Let f(x) = x, x X product of all the factors like (x - a ) , X= "" "" (x- a)³,

X; =

""

""

(x − a)³.

Find the greatest common measure of f (x) and ƒ ′ (x) = F₁ (x) say, "" 99 F (x) and F (x) = F₂ (x) , "" 33 F (x) and F (x) = F3 (i ) , ...

...

...

...

Lastly, the greatest common measure of F -1(x ) and F -1 (x) = F (x ) = 1 . Next perform the divisions

f (x) ÷ F (x) = F (x) ÷ F, (x) = ...

And, finally,

...

(x) say, ½ (x) ,

...

Fm - 1 (x) + 1 = P (x). $1 (x) ÷ $₂2 (x) = X₁ , P (x) +3 (x) = X , ... Pm-1 (x) + F - 1 (x) =

...

m (x) = Xm - 1 , (x) = X · R

[T. 82.

122

THEORY OF EQUATIONS.

The solution of the equations X₁ = 0, X, = 0, &c. will furnish all the roots off(x) ; those which occur twice being found from X,; those which occur three times each, from X,; and so on. 446 If f(x) has all its coefficients commensurable, X₁ , X2, X3, &c. have likewise their coefficients commensurable . Hence, if only one root be repeated r times , that root must be commensurable. 447

In all the following theorems , unless otherwise stated ,

f(x ) is understood to have unity for the coefficient of its first term .

LIMITS OF THE ROOTS .

448 If the greatest negative coefficients in f(x) and ƒ ( -x) be p and q respectively, then p + 1 and - (q + 1 ) are limits of the roots.

449

If a" -" and

-8 " -s are the highest negative terms in f(x)

and f ( -a) respectively, ( 1 + √p) and − ( 1 + √g) are limits of the roots .

450

then

If k be a superior limit to the positive roots of f (1 ) , 1 will be an inferior limit to the positive roots of ƒ (x) . k

451

If each negative coefficient be divided by the sum of all the preceding positive coefficients , the greatest of the fractions so formed + unity will be a superior limit to the positive roots .

452

Newton's method .— Put x = h +y in f(x) ; then , by (426 ) ,

y² fn (k ) = 0 . f(h +y) = f(h) + yf ' (h ) + 2 ƒ² (h) + ... + * n Take h so that f ( h) , ƒ ′ (h) , ƒ² (h) ...f" (h) are all positive ; then h is a superior limit to the positive roots .

According as ƒ (a) and ƒ (b) have the same or different 453 signs , the number of roots intermediate between a and b is even or odd.

123

INTEGRAL ROOTS.

454

' (x) Rolle's Theorem .-One real root of the equation f

lies between every two adjacent real roots of ƒ(x). COR. 1.-f(x) cannot have more than one root greater ' (x) ; or more than one less than than the greatest root in f the least root in ƒ' (x) .

455

456 COR. 2.- If ƒ (x) has m real roots , fr (x) has at least m -r real roots. 457 COR. 3.- If f (x) has μ imaginary roots , f (x) has also μ at least . 458 COR . 4. - If a, B, y ... x be the roots of f ' ( ) ; then the number of changes of sign in the series of terms

f( ∞ ) , ƒ(a) , ƒ (ß) , ƒ (y) ... ƒ ( —∞ ) is equal to the number of roots of f (x) .

NEWTON'S METHOD OF DIVISORS .

459

To discover the integral roots of an equation .

EXAMPLE.-To ascertain if 5 be a root of x -6x +86x2-176x + 105 = 0. If 5 be a root it will divide 105. next coefficient. Result, -155 .

5 ) 105 21 -176

Add the quotient to the

If 5 be a root it will divide -155 . the next coefficient ; and so on.

Add the quotient to

If the number tried be a root, the divisions will be effectible to the end, and the last quotient will be -1, or -po, if po be not unity .

5 ) - 155 -31 86

5 ) 55 11 -6 5

460 In employing this method , limits of the roots may first be found , and divisors chosen between those limits . 461

Also , to lessen the number of trial divisors , take any

integer m ; then any divisor a of the last term can be rejected if a - m does not divide f (m). In practice take m = + 1 and -1. To find whether any of the roots determined as above are repeated, divide f(x) by the factors corresponding to them, and then apply the method of divisors to the resulting equation.

124

THEORY OF EQUATIONS.

EXAMPLE.

Take the equation x +2x - 17x - 26x + 88x² + 72x - 144 = 0.

Putting

= 1 , we find ƒ ( 1 ) = −24 . The divisors of 144 are 1 , 2, 3, 4, 6, 8, 9, 12, 16, 24, &c. The values of a - m (since m = 1 ) are therefore 0,

1,

2,

3,

5,

7,

8,

11 ,

15,

23,

&c.

Of these last numbers only 1 , 2 , 3, and 8 will divide 24 . Hence 2 , 3, 4 , and 9 are the only divisors of 144 which it is of use to try. The only integral roots of the equation will be found to be ± 2 and ± 3 .

If f(x) and F (X) have common roots , they are con tained in the greatest common measure of ƒ(a) and F (X) .

462

463

If f (x) has for its roots a , p (a) , b, p (b) amongst others ;

then the equations f(x) = 0 and f roots a and b.

(x )

= 0 have the common

464 But, if all the roots occur in pairs in this way, these equations coincide . For example, suppose that each pair of roots, a and b, satisfies the equation a + b = 2r. We may then assume a - b = 2z . Therefore f (z + r) = 0. This equation involves only even powers of z , and may be solved for z².

465 Otherwise : Let ab z ; then f (e) is divisible by ( -a) ( —b ) = x² − 2rx + z. Perform the division until a remainder is obtained of the form Pr + Q, where P and Q only involve z. The equations P = 0 , Q = 0 determine z , by (462 ) ; and a and b are found from a + b = 2r, ab = z.

RECIPROCAL

EQUATIONS .

A reciprocal equation has its roots in pairs of the form 466 1 a, ; also the relation between the coefficients is a Pn-r• PrP -r, or else Pr = -Pn -r

467 A reciprocal equation of an even degree, with its last term positive, may be made to depend upon the solution of an equation of half the same degree .

BINOMIAL EQUATIONS.

EXAMPLE :

468

125

4x - 24x5 + 57x¹ −73x³ + 57x² - 24x + 4 = 0

is a reciprocal equation of an even degree, with its last term positive . Any reciprocal equation which is not of this form may be reduced to it by dividing by x + 1 if the last term be positive ; and, if the last term be negative, by dividing by x - 1 or x² -1 , so as to bring the equation to an even degree. Then proceed in the following manner : 469

First bring together equidistant terms , and divide the equation by a ; thus

4 (2² + 1 ) — 24( π² +

1 ) + 57(x +

1 ) -73 = 0.

1 y, and by making repeated use of the

By putting a + x 1

relation a² +

= a2

x + 1 ) -2 , the equation is reduced to (

a cubic in y, the degree being one-half that of the original equation .

1

1

" and Pm for

Put p for x + x 470

+

•

X,m

The relation between the successive factors of the form

P may be expressed by the equation

PmPPm-1 -Pm-2 · 471

The equation for Pm , in terms of p , is

Pm = p - mpm -2 +

m (m - 3) pm —4— ….. 1.2

+ (- 1)"

m (m— r — 1) ... (m− 2r + 1 ) pm -2r + ... r

By (545) , putting q = 1.

BINOMIAL EQUATIONS .

m If a be a root of a" -1 = 0 , then a" is likewise a root where m is any positive or negative integer.

472

473

If a be a root of

" + 1 = 0 , then a2m + 1 is likewise a root .

126

THEORY OF EQUATIONS.

474 If m and n be prime to each other, have no common root but unity.

" -1 and æ” -1

Take pm - qn = 1 for an indirect proof. 475 If n be a prime number , and if a be a root of a" -1 = 0 , the other roots are a, a , a³ ... a". These are all roots, by (472) . Prove, by (474) , that no two can be equal. 476 If n be not a prime number, other roots besides these may exist . The successive powers , however, of some root will furnish all the rest.

= has the index n = mpq ; m , p, q being 477` If a" -10 prime factors ; then the roots are the terms of the product (1 + a + a² + ... + am - ¹) ( 1 +ß + ß² + ... + ß³-¹) × (1 + y + y² + ... +79-¹), where a is a root of am - 1 , x² -1 , ‫در‬ β Y but neither a, ß, nor y = 1. 478

99

x² -1 , Proof as in (475) .

If n = m³, and a be a root of x - 1 = 0 , xm α =: 0 , 99 β

"" am — ẞ = 0 ; Y = will be the terms of the product then the roots of a" -1-0 (1 + a + a²+ ... + am -¹) (1 +ß + ẞ² + ... + ẞm - 1)

× (1 + y + y² + ... + ym -¹) .

479 " + 1 = 0 may be treated as a reciprocal equation , and depressed in degree after the manner of (468) . 480

The complete solution of the equation xn - 1 = 0

is obtained by De Moivre's Theorem. The n different roots are given by the formula 2rT

2rT Xx

± √ - 1 sin

COS

n in which

(757)

n

must have the successive values 0 , 1 , 2 , 3 , &c. , n n 1 concluding with " if n be even ; and with 2 if n be odd . 2 2

127

CUBIC EQUATIONS.

481

Similarly the n roots of the equation

x² + 1 = 0 are given by the formula

1 sin (2r+1) π x = COS (2r + 1) π ± √ n n 2

n

if

r taking the successive values 0, 1 , 2, 3, &c. , up to 2 n- 3 n be even ; and up to

if n be odd . 2

482

The number of different values of the product 1 1 Am Bm

is equal to the least common multiple of m and n, when m and n are integers .

CUBIC EQUATIONS .

483

To solve the general cubic equation x³ +px² + qx + r = 0 .

Remove the term pa² by the method of (429) . formed equation be x²+qx + r = 0.

Let the trans

Cardan's method . The complete theoretical solution of this equation by Cardan's method is as follows : Put x = y + z (i.)

484

y³ +z³ + (3yz + q) (y + z) + r = 0 .

Put

L 3z* ute this valu of y, and solv the resul ting quadratic e e 3yz + q = 0 ;

..

y=

Substit

; and we ihn y . The roots are equal to ys and 23 respect ively ave, by (i.), 485

3 x=

= { − 3

1/2 − √ } + { }* } }} + { − + √ } + {

THEORY OF EQUATIONS .

128

The cubic must have one real root at least, by (409) . 43 - r + Let m be one of the three values of " and n one 2 { √ = + 27 1/3 ) }} - r + q of the three values of { 2 27 486

487

Let 1 , a, a' be the three cube roots of unity, so that 1 - 1 1 1 α= I √−3. + ¹ √—3, and a² = 2 2 2 2 Then, since

[472

m³ m2/1 , the roots of the cubic will be m + n, am + a³n, a³m + an.

Now, if in the expansion of p ± 2

4.

+ ‹) 27S

}

by the Binomial Theorem, we put μ the sum of the odd terms, and the sum of the even terms ; then we shall have m = μ + v , and nμ - v ; = - v -1 ; or else m = μ + v√1, and nμ according as

4

+ q³ is real or imaginary. 27

By substituting these expressions for m and n in (487) , it appears that 2.2 Å 488 (i. ) If 4 + 27 be positive, the roots of the cubic will be

2µ,

−µ + v√ − 3,

−µ - v√ − 3.

(ii .) If 2 + 27 2 be negative , the roots will be µ + v√3, −µ - v√3. 2µ,

+ 2 = 0, the roots are 27 2m, —m ; m, since m is now equal to μ.

(iii.) If

4

489

The Trigonometrical method.

The equation

x¹³ + qx + r = 0 may be solved in the following manner, by Trigonometry, 2 q³ is negative . + 4 27 Assume x n cos a . Divide the equation by n³ ; thus

when

r cos³ a + q COS a + =: 0. n³ n²

cos 3a

3 But

Cos³ α ―――――――

COS a 4

= 0. 4

By (657)

129

BIQUADRATIC EQUATIONS.

Equate coefficients in the two equations ; the result is 3 3 cos 3a = —·4r n = " (197 3 )³3 ( 4.q a must now be found with the aid of the Trigonometrical tables. 490

The roots of the cubic will be

n cos a,

n cos (

n cos (

+ a),

-a) .

2.2

491

Observe that, according as

is positive or nega

+ 4

27

tive, Cardan's method or the Trigonometrical will be practi cable. In the former case , there will be one real and two imaginary roots ; in the latter case, three real roots.

BIQUADRATIC EQUATIONS .

492

Descartes' Solution.-To solve the equation x² + qx² + rx + s = 0

(i.)

the term in ³ having been removed by the method of (429) .

Assume

(x² +ex +ƒ) (x² — ex +g) = 0 ............... (ii .)

Multiply out , and equate coefficients with (i .) ; and the fol lowing equations for determining f, g, and e are obtained g+f = q+ e³,

g −f= 2,

gf = s ......... (iii .)

493 e®+2qe¹ + (q² - 4s) e² —r² = 0.

(iv.)

494 The cubic in e' is reducible by Cardan's method, when the biquadratic has two real and two imaginary roots. For proof, take a ± iß and - a ± y as the roots of (i. ), since their sum must be zero. Form the sum of each pair for the values of e [see (ii.) ] , and apply the rules in (488) to the cubic in e . Ifthe biquadratic has all its roots real, or all imaginary, the cubic will have all its roots real. Take aiẞ and -aiy for four imaginary roots of (i . ) , and form the values of e as before .

495 will be

If a², ², y² be the roots ofthe cubic in e , the roots of the biquadratic −1 (a +ß + y) , 1 (a + ß − y) , 1 (ẞ + y − a) , 1 ( y + a − ß) . S

130

THEORY OF EQUATIONS.

For proof, take w, x, y, z for the roots of the biquadratic ; then , by (ii . ) , the sum of each pair must give a value of e. Hence, we have only to solve the symmetrical equations w + x = a, y + z = a, z + x = B, w + y = -ẞ, w + z == Y. x + y = Y,

496

Ferrari's solution .—To the left member of the equation x¹+px³ +qx² + rx + s = 0,

b2 add the quantity ax² + bx +

and assume the result 4a'

= (x² + ½ x + m).

497 Expanding and equating coefficients , the cubic equation for determining m is obtained

following

8m³ —4qm² + (2pr − 8s ) m+ 4qs — p²s — r = 0 . Then a is given by the two quadratics

2ax +b x² + ½ x + m = ±

2 va

498 The cubic in m is reducible by Cardan's method when the biquadratic has two real and two imaginary roots. Assume a, 6, y, & for the roots of the biquadratic ; then aß and yo are the respective products of roots of the two quadratics above. From this find m in terms of aßyd.

499

Euler's solution . - Remove the term in a³ ; then we

have x*+qx² + rx + s = 0 . 500 Assume x = y + z + u , and it may be shewn that y³, z² , and u are the roots of the equation

1 + 2/2f² +

501

q -4s - 2.2 t = 0. 16 64

The six values of y , z , and u, thence obtained , are

r restricted by the relation

yzu = — 8

Thus xy +z +u will take four different values .

COMMENSURABLE ROOTS.

131

COMMENSURABLE ROOTS .

502

To find the commensurable roots of an equation.

First transform it by putting a =

into an equation of k

n-2 x" +P₁x²¬¹ +P²x²¬² + ... +P₂ = 0,

the form = = having Po

1 , and the remaining coefficients integers .

(431 )

503 This equation cannot have a rational fractional root, and the integral roots may be found by Newton's method of Divisors (459). These roots , divided each by k, will furnish the commen surable roots of the original equation.

504

EXAMPLE.-To find the commensurable roots of the equation 81x5-207x - 9x³ + 89x² + 2x − 8 = 0.

Dividing by 81, and proceeding as in (431 ) , we find the requisite substitu tion to be x= Y The transformed equation is

y -23y -9y + 801y + 162y - 5832 = 0. The roots all lie between 24 and -34, ,by (451 ) . The method of divisors gives the integral roots 4, and 3. 6, Therefore, dividing each by 9, we find the commensurable roots of the original equation to be ,, and . 505 To obtain the remaining roots ; diminish the transformed equation by the roots 6, -4, and 3, in the following manner (see 427) : — 6

-4

3

1-23 9 + 801 + 162-5832 6-102-666 + 810 + 5832

1-17-111 + 135 + 972 — 4+ 84 + 108-972 1-21-27 + 243 3-54-243 1-18-81

The depressed equation is therefore - 18y81 = 0. y -

The roots of which are 9 (1+ √2) and 9 ( 1-2) ; and, consequently, the incommensurable roots of the proposed equation are 1+ √2 and 1 - √2.

THEORY OF EQUATIONS.

132

INCOMMENSURABLE ROOTS .

506 Sturm's Theorem .—If ƒ (a) , freed from equal roots , be divided by f' (x) , and the last divisor by the last remainder, changing the sign of each remainder before dividing by it , until a remainder independent of a is obtained , or else a re mainder which cannot change its sign ; then f (x) , ƒ ' (x) , and the successive remainders constitute Sturm's functions , and are denoted by

f(x) , f (x) , ƒ : (x) , &c ....... ƒm (x) .

The operation may be exhibited as follows : f (x) = q₁f (x) —ƒ½ (x) , f (x) = ¶½f2 (x) —ƒ3 (x), ƒ½ f2 (x) = q3f3 (x) —ƒ , (x) , ... ... ... ... ƒm-2 (x) = 9m- 1ƒm-1 (x) — ƒm (x) . 507 NOTE .-Any constant factor of a remainder may .be rejected , and the quotient may be set down for the corres ponding function .

508

An inspection of the foregoing equations shews

(1 ) That fm (2) cannot be zero ; for , if it were , f (x) and fi (a) would have a common factor, and therefore f (x) would have equal roots , by (432 ) . (2) Two consecutive functions , after the first, cannot vanish together ; for this would make fm (x) zero . (3) When any function , after the first , vanishes , the two adjacent ones have contrary signs . 509 If, as x increases, f (x) passes through the value zero , Sturm's functions lose one change of sign. For, before f(x) takes the value zero, ƒ (x) and ƒ, (x) have contrary signs, and afterwards they have the same sign ; as may be shewn by making h small, and changing its sign in the expansion off (x + h) , by (426) . 510 If any other of Sturm's functions vanishes, there is neither loss nor gain in the number of changes of sign .

This will appear on inspecting the equations. RESULT. - The number of roots off(x) between a and b is 511 equal to the difference in the number of changes of sign in Sturm's functions , when xa and when x = b .

133

INCOMMENSURABLE ROOTS.

512 COR.-The total number of roots of ƒ (a) will be found by taking a = + ∞ and b = ∞ ; the sign of each function will then be the same as that of its first term .

When the number of functions exceeds the degree of ƒ (x) by unity, the two following theorems hold :— 513 If the first terms in all the functions, after the first, are positive ; all the roots off(x) are real . 514 If the first terms are not all positive ; then , for every change of sign, there will be a pair of imaginary roots . + ∞ and - co , and examine the number of For the proof put (416) . changes of sign in each case, applying Descartes' rule. (x) has no factor in common with ƒ (x) , and if 4 (x) 515 If and f ( ) take the same sign when f(x) = 0 ; then the rest of (a), instead Sturm's functions may be found from ƒ (a) and of f ( ). For the reasoning in (509) and (510) will apply to the new functions . 516

If Sturm's functions be formed without first removing

equal roots from ƒ (a) , the theorem will still give the number of distinct roots , without repetitions , between assigned limits . For iff (x) and f (x) be divided by their highest common factor ( see 444) , and ifthe quotients be used instead of ƒ (a) and ƒ, (e) to form Sturm's func tions ; then, by ( 515 ) , the theorem will apply to the new set of functions, which will differ only from those formed from f (x) and fi (2) by the absence of the same factor in every term of the series.

517

EXAMPLE. -To find the position of the roots of the equation æ³ −4x³ + x² + 6x + 2 = 0 .

+++++

+++++

11 ++ +

+1 1++

1 + 1 +

++ | ++

+ +1

33

Sturm's functions, formed according to ƒ ( x) = x² - 4a³ + x² + 6x + 2 f (x) = 2x - 6x + x+ 3 the rule given above, are here calculated . 5x -10x - 7 The first terms of the functions are all f (x) X- 1 positive ; therefore there is no imaginary f (x) = root. 12 f(x) = The changes of sign in the func = -2-10 2 3 1 4 tions , as a passes through integral f (x) = values, are exhi bited in the adjoin f (x) = ing table. There " f (x) = + are two changes of f (x) = sign lost while f (x) = passes from -1 to 0, and two more chan No. of ges lost while x passes 0 0 4 2 4 2 2 of sign from 2 to 3. There

134

THEORY OF EQUATIONS.

are therefore two roots lying between 0 and - -1; and two roots also between 2 and 3. These roots are all incommensurable, by (503) .

518

Fourier's Theorem.- Fourier's functions are the fol

lowing quantities

f(x) ,

f ' (x) ,

ƒ “' (x) …………..ƒ” (x) .

519

of Fourier's functions .

Properties

As

a increases ,

Fourier's functions lose one change of sign for each root of the equation f(x) = 0 , through which a passes , and r changes of sign for r repeated roots . 520 If any of the other functions vanish, an even number of changes of sign is lost.

521 RESULTS . -The number of real roots of f(x) between a and ẞ cannot be more than the difference between the number of changes of sign in Fourier's functions when x = a , and the number of changes when x = B. 522

When that difference is odd, the number of intermediate

roots is odd, and therefore one at least . 523 When the same difference is even , the number of inter mediate roots is either even or zero .

524 Descartes ' rule of signs follows from the above for the signs of Fourier's functions , when a = 0 are the signs of the terms in f (x) ; and when a = ∞ , Fourier's functions are all positive.

525

Lagrange's method of approximating to the incom mensurable roots of an equation.

Let a be the greatest integer less than an incommen Take surable root of f(x) . Diminish the roots of ƒ (a) by a. the reciprocal of the resulting equation .

Let b be the greatest

integer less than a positive root of this equation. Diminish the roots of this equation by b , and proceed as before . 526 Let a, b , c, &c . be the quantities thus determined ; then, an approximation to the incommensurable root of f (x) will be 1 1 the continued fraction x= a + b+ c +

135

INCOMMENSURABLE ROOTS.

527 Newton's method ofapproximation .- If c, be a quantity a little less than one of the roots of the equation f(x) = 0 , so that f(c + h) = 0 ; then c, is a first approximation to the value of the root . Also because h2 S ƒ" (c₁) + &c....... ( 426) , f(c₁ + h) = f(c₁ ) + hƒ' (c₁) + 2 and h is but small, a second approximation to the root will be

C1 - ƒ(c₁) = Cq . f ' (c₁) T

In the same way a third approximation may be obtained from C₂, and so on.

To ensure 528 Fourier's limitation of Newton's method. that C1 , C2 , C3 , &c . shall successively increase up to the value cth without passing beyond it, it is necessary for all values of a between c, and c₁ + h.

(i.) That f(x) and f' (x) should have contrary signs. (ii.) That f(x) and f" (x) should have the same sign.

P 17.

S N

R Q

S FIG. 1.

FIG. 2.

A proof may be obtained from the figure. Draw the curve y = f(x). Let OX be a root of the equation , and ON = c₁ ; draw the successive ordinates and tangents NP, PQ, QR, &c . Then OQ = C₂, OS = c,, and so on. Fig . ( 2) represents c₂ > OX, and the subsequent approxi mations decreasing towards the root .

e

530 Newton's Rule for Limits of the Roots. - Let the co efficients of f (x) be respectively divided by the Binomial coefficients, and let ao, a , a, ... an be the quotients , so that

ƒ(x) = α。x² + na₁xn − 1 +

n (n - 1 ) a²x² - 2 + ... + na₂ -1x +an • 1.2

THEORY OF EQUATIONS .

136

Let A,, A., Ag ... A, be formed by the law A, = a -a - 1a, +1 Write the first series of quantities over the second , in the fol lowing manner :

aos

a1

Ag

as ...... an- 1,

an

Ao,

A1 ,

A2,

A3 ..... An- 19

An .

Whenever two adjacent terms in the first series have the same sign, and the two corresponding terms below them in the second series also the same sign ; let this be called a double permanence. When two adjacent terms above have different signs , and the two below the same sign, let this be known as a variation-permanence .

RULE .- The number of double permanences in the asso ciated series is a superior limit to the number of negative roots off(x). 531

The number of variation-permanences is a superior limit to the number of positive roots. The number of imaginary roots cannot be less than the number of variations of sign in the second series.

532 Sylvester's Theorem. Let f(x +1) be expanded by (426) in powers of a, and let the two series be formed as in Newton's Rule (530) . Let P (1) denote the number of double permanences . Then P (A) ~ P (μ) is either equal to the number of roots of f (x) , or surpasses that number by an even integer . NOTE .— The first series may be multiplied byn , and will then stand thus , ƒ" (^) ,

ƒn-¹ (^) ,

[2ƒ" -2 (X) ,

n- 3 [3 ƒn− ³ (X ) ... | n ƒ (^) .

The second series may be reduced to Gn ( ) ,

where

533

Gn - 1 (A) ,.

Gn- 2 (A) ... G (X) ,

n − r + 1 fr−1 (X) fr+¹ (X) . G, (^) = {ƒ” (\ ) } ² - ? n-r

Horner's Method.- To find the numerical values of the

roots of an equation .

Take , for example , the equation

x* —4x³ + x² + 6x + 2 = 0 , and find limits of the roots by Sturm's Method or otherwise .

INCOMMENSURABLE ROOTS.

137

It has been shewn in (517) that this equation has two incommensurable roots between 2 and 3. The process of calculating the least of these roots is here exhibited . | m = mo | ཤ ༑།

+1 -4 -3 0 -3 4 C,

D, 40 4 44 4 48 4 52 4 D, 560 1 561 1 562 1 563 1 D, 5640 4 5644

5648 4 5652 4 D. 5,656 METHOD .

100 176 276 192 468 208

C₂

67600 561 68161 562 68723 563

C₁

6928600 22576 6951176 22592 6973768 22608

C.

C₁ C

6996376 11 69974 11 69985 11

+6 -6 0 -6

B₁ - 6000 1104 -4896 1872 B₂ - 3024000 68161 -2955839 68723 B -2887116000 27804704 -2859311296 27895072 B - 2831416224 139948 -283001674 139970 Bs -282861704 700 -28285470 700 Bo -28284770 21 -2828456 21

+2 ( 2·414213 0 20000 A₁ -19584 4160000 A -2955839 As 12041610000 - 11437245184 As 604364816 -566003348 As 38361468 -28285470 10075998 As 8485368 1590630 A,

282843 ) 1590630 ( 562372 1414215 28284 ) 176415 169706 2828 ) 6709 5657 282 ) 1052 848 28 ) 204 197 2)

B, -2828435

7 5

69996 7 Root

2.414213562372 .

Diminish the roots by 2 in the manner of ( 427) .

The resulting coefficients are indicated by A , B , C₁, D₁. A₁ By Newton's rule (527) , - f (c).; that is, is an approximation to B₁ f' (c) the remaining part of the root. This gives 3 for the next figure ; 4 will be found to be the correct one. The highest figure must be taken which will not change the sign of A. Diminish the roots by 4. This is accomplished most easily by affixing ciphers to A₁, B₁, C₁ , D₁ , in the manner shewn, and then employing 4 instead of 4. Having obtained A , and observing that its sign is + , retrace the steps, T

138

THEORY OF EQUATIONS.

trying 5 instead of 4. This gives 4, with a minus sign, thereby proving the existence of a root between 24 and 2.5 . The new coefficients are A2, B2, C₂, D. -1 B gives 1 for the next figure of the root . Affix ciphers as before, and diminish the roots by 1, distinguishing the new coefficients as A , B , C3, D3. Note that at every stage of the work A and B must preserve their signs unchanged. If a change of sign takes place it shews that too large a figure has been tried . To abridge the calculation proceed thus :-After a certain number of figures of the root have been obtained (in this example four), instead of adding ciphers cut off one digit from B , two from C , and three from D. This amounts to the same thing as adding the ciphers, and then dividing each number by 10000 . Continue the work with the numbers so reduced , and cut off digits in like manner at each stage until the D and C columns have disappeared. A, and B, now alone remain, and six additional figures of the root are determined correctly by the division of A, by B. To find the other root which lies between 2 and 3, we proceed as follows :-After diminishing the roots by 2, try 6 for the next figure. This gives A , negative ; 7 does the same, but 8 makes 4, positive. That is to say, ƒ (27) is negative, and ƒ (2· 8 ) positive. Therefore a root exists between 27 and 2.8, and its value may be approximated to, in the manner shewn. Throughout this last calculation A will preserve the negative sign. Observe also that the trial number for the next figure of the root given at f(c) will in this case be always each stage of the process by the formula f' (c)' too great, as in the former case it was always too small.

SYMMETRICAL FUNCTIONS OF THE ROOTS OF

AN EQUATION.

NOTATION .

Let a , b , c ... be the roots of the equation

ƒ(x) = 0 . Let sm denote am +b the roots .

+ ... , the sum of the mth powers of

Let smp denote ambr + bmar + amcP + ... permutations of the roots , two at a time.

through all the

Similarly let 8. p. q denote abc + am bed + ... , taking all the permutations of the roots three at a time ; and so on.

SYMMETRICAL FUNCTIONS OF THE ROOTS.

534

139

SUMS OF THE POWERS OF THE ROOTS. Sm +P1$m-1 + P2Sm- 2 + ... + Pm - 18₁ + mp „ = 0 ,

where m is less than n , the degree of ƒ (x) . Obtained by expanding by division each term in the value of f (x ) given at (432), arranging the whole in powers of e, and equating coefficients in the result and in the value of f ' (x) , found by differentiation as in (424 ) . 535

If m be greater than n , the formula will be

Sm + P₁ Sm- 1 + P2 Sm −2 + ... +Pn Sm-n = 0. Obtained by multiplying ƒ (x ) = 0 by a" - ", substituting for a the roots a, b, c, &c. in succession , and adding the results . By these formulæ 81, 82, 83, &c . may be calculated successively.

536

To find the sum of the negative powers of the roots , put

m equal to n - 1, n − 2 , n − 3 , &c . successively in (535) , in order to obtain 8-1 , $ _2 , 8-3 , & c .

537

To calculate s, independently.

RULE :

8,

rx coefficient of x-r in the expansion

of

log f(x) in descending powers of x. Xn Proved by taking ƒ (x) = (x− a) ( x − b ) ( x − c) ... , dividing by a*, and expanding the logarithm of the right side of the equation by (156).

538

SYMMETRICAL FUNCTIONS WHICH ARE NOT POWERS OF THE ROOTS.

These are expressed in terms of the sums of powers of the roots as under , and thence, by ( 534) , in terms of the roots explicitly , Sm , p = $m$p ――― Sm + p , 539

$m, p. q = $$$q— $m+ p $q ― Sm+q$p— $p + q $ m + 2$ m+p + q •

The last equation may be proved by multiplying 8m, p by and expansions of other symmetrical functions may be 8 obtained in a similar way.

540 If P (a) be a rational integral function of a, then the symmetrical function of the roots of f ( ) , denoted by

140

THEORY OF EQUATIONS.

$ (a) + p (l ) + p (c ) + & c . , is equal to the coefficient of a" -1 in ' (e) by ƒ(x) . (a) f the remainder obtained by dividing & ( c) Proved by multiplying the equation (432) by and by theorem (401) .

541 To find the equation whose roots are the squares of the differences of the roots of a given equation . Let F (x) be the given equation , and S, the sum of the 7th powers of its roots . Let f(a ) and s, have the same meaning with regard to the required equation. The coefficients of the required equation can be calculated —— from those of the given one as follows : The coefficients of each equation may be connected with the sums of the powers of its roots by (534) ; and the sums of the powers of the roots of the two equations are connected by the formula

2r (2r - 1) 542

2s,

S2S2r-2 -... + nS2

nS2-2rS₁S2r- 1 +

1.2 RULE . -28, is equal to the formal expansion of (S - S)²r by the Binomial Theorem , with the first and last terms each mul As the tiplied by n, and the indices all changed to suffixes. equi- distant terms are equal we can divide by 2 , and take half the series .

DEMONSTRATION. - Let a, b, c ... be the roots of F (x) . (x) = (x -− a)² + ( x − b)² + ...

Let

(i.)

Expand each term on the right by the Bin. Theor. , and add, substituting S1 , S2, &c. In the result change a into a, b, c ... successively, and add the ʼn equations to obtain the formula, observing that, by (i. ) , $ (a) + q (b) + ... = 28,. If n be the degree of F (a) , then

n (n - 1 ) is the degree By (96) .

of f (x) . 543

The last term of the equation f(x) = 0 is equal to n" F (a) F (B) F ( 7) ...

where a , ẞ , y,

... are the roots of I" (a) .

that

F " (a) F' (b) ... = n" F (a ) F (ß ) ...

Proved by shewing

544 If F (a) has negative or imaginary roots , f(x) must have imaginary roots.

SYMMETRICAL FUNCTIONS OF THE ROOTS.

545

141

The sum of the mth powers of the roots of the quad

ratic equation

x² -px + q = 0.

Sm = pm - mpm -2q+

2 m (m - 3) pm -sq² -... 2

... + ( − 1 ), m (m — r — 1) ... (m− 2r + 1) pm -2rq" + &c. r By (537) expanding the logarithm by (156) .

546 The sum of the mth powers of the roots of æ” —1 = 0 is n if m be a multiple of n, and zero if it be not. By (537 ) ; expanding the logarithm by ( 156 ) .

547

If

$ (x) = α + α¸æ + α₂æ² + &c .

(i . ) ,

then the sum of the selected terms m n amxm + Am+n xm+n + am m + 2n x +2 + & c.

will be s =

1 { a”-m¤ (ax) +ß”-mÞ (Bx) + y” ¬ ™ þ (yx) + &c . }

n where a, ß , y , &c . are the nth roots of unity. For proof, multiply (i.) by a" -", and change a into ae ; so with ẞ, y, &c. , and add the resulting equations .

548 To approximate to the root of an equation by means of the sums of the powers of the roots .

By taking m large enough , the fraction SmS+1 will approx m imate to the value of the numerically greatest root, unless there be a modulus of imaginary roots greater than any real root, in which case the fraction has no limiting value.

549

Similarly the fraction

2 S mm +2 +21 approximates , as m Sm - 15m + 1 ―― Sm

increases , to the greatest product of any pair of roots , real or imaginary ; excepting in the case in which the product of the pair of imaginary roots , though less than the product of the two real roots , is greater than the square of the least of them, for then the fraction has no limiting value .

THEORY OF EQUATIONS.

142

SmSm+3

550

- Sm + 18m + 2 approximates , 2 SmSm+2 - Sm + 1

Similarly the fraction

as m increases , to the sum of the two numerically greatest roots , or to the sum of the two imaginary roots with the greatest modulus .

EXPANSION OF AN IMPLICIT FUNCTION OF x.

Let

yª (Ax² + ) +y³ (B₁x³ + ) + ... + y° ( Sx' + ) = 0 ...... ………….. (1)

be an equation arranged in descending powers of y , the co efficients being functions of a, the highest powers only of a in each coefficient being written .

It is required to obtain y in a series of descending powers of x. First form the fractions a -·b

―――

a

――――

α ―-8

αY

a- ẞ' a

-d

C

-

a- S

.... ―

. (2) .

α1σ

k

Let

a- n

= t be the greatest of these algebraically , or

if several are equal and greater than the rest, let it be the last of such. Then, with the letters corresponding to these equal and greatest fractions , form the equation Auª + ...... + Ku" = 0

. (3 ) .

Each value of u in this equation corresponds to a value of y, commencing with uat . Next select the greatest of the fractions

k -l -

m

k-s

K- μ

K- σ

k -

K-λ

· (4).

Ji -n = t' be the last of the greatest ones .

Let

Form

κ -ν the corresponding equation

Ku" + ... + Nu' = 0

......... .. (5) .

Then each value of u in this equation gives a corresponding value of y, commencing with ua".

143

EXPANSION OF AN IMPLICIT FUNCTION.

Proceed in this way until the last fraction of the series ( 2 ) is reached .

To obtain the second term in the expansion of y , put . ( 6) ,

y = x² ( u +y₁ ) in ( 1 ) ………………………………… .

employing the different values of u, and again of ť and u, ť" and u, &c . in succession ; and in each case this substitution will produce an equation in y and a similar to the original equation in y. Repeat the foregoing process with the new equation in y , ― observing the following additional rule : When all the values of t, t' , t" , &c . have been obtained, the negative ones only must be employed in forming the equations in u. (7) .

552

To obtain y in a series of ascending powers of a.

Arrange equation (1 ) so that

B, Y, &c. may be in as cending order of magnitude, and a , b, c, &c . the lowest powers of a in the respective coefficients . Select t , the greatest of the fractions in (2) , and proceed exactly as before, with the one exception of substituting the word positive for negative in (7) .

553

EXAMPLE. - Take the equation (x + x ) + (3x - 5æ³) y + ( -4x + 7x² + æ³) y² - y = 0.

It is required to expand y in ascending powers of a. 3-0 The fractions (2) are ― 3-2 - 3-1 ; or 1, 1 , and 0-5 0-2' 0-1'

.

The first two being equal and greatest , we have t = 1. 1-0 == ť. The fractions (4) reduce to 2-5 Equation (3) is 1 + 3u4u² = 0, whichgives u = 1 and -- 1, with t = 1. Equation (5) is and from this

-4u² - u³ = 0,

u = 0 and −4³, with ť = . We have now to substitute for y, according to (6 ) , either

x (1 + y₁) , x ( −1 + y₁ ) , æ³y, or x¹³ ( −4³ + y₁) . Put y = ( 1+ y₁ ) , the first of these values, in the original equation , and arrange in ascending powers of y, thus −4œ* + ( − 5œ³ + ) Y₁ + ( −4x³ + ) y} − 10x³y¡ — 5x³y; — æ³y{ = 0. The lowest power only of x a in each coefficient is here written.

144

THEORY OF EQUATIONS.

The fractions (2) now become 4-3 G 4-5 4-3 0-2' 0-1' 0-3' or 1, 1, −1, −1,

4-5 0-4'

4-5 0-5 '

− From these t = 1 , and equation ( 3) becomes —4—5u = 0 ; .. u = - 4. Hence one of the values of y, is, as in ( 6 ) , y₁ = x ( − § + Y₂) . --Therefore y = x { 1 + x ( − + Y₂ ) } = x — §ƒœ³ + ... Thus the first two terms of one of the expansions have been obtained .

DETERMINANTS .

554

Definitions.-The determinant

a1 а2

is equivalent

| b₁ b₂ to a₁b₂- ab₁ , and is called a determinant of the second order. A determinant of the third order is A1 A2 A3 = a₁ (b₂c¸ − b₂c₂) + a₂ (b¸¢₁ — b¸ ¢3) + a3 (b₁c₂ — b₂º¹) . b₁ b₂ bз C1 C2 C3 Another notation is

a, b, c,, or simply (a , b, cs).

The letters are named constituents, and the terms are called elements. The determinant is composed of all the elements obtained by permutations of the suffixes 1 , 2 , 3 . The coefficients of the constituents are determinants of the next lower order, and are termed minors of the original determinant. Thus , the first determinant above is the minor of c, in the second determinant . It is denoted by C. minor of a₁ is denoted by A₁ , and so on.

So the

555 A determinant of the nth order may be written in either of the forms below a1 a₂ ... ar ... an br ... by b₁ b ... by ... ... ... ... b h l½ ... i ... In

or

а11 a12 ... air ... ain а21 A22 ... azr ... azn

... an

...

...

...

ang ... anr ... Ann

In the latter, or double suffix notation, the first suffix indicates the row, and the second the column. The former notation will be adopted in these pages .

145

DETERMINANTS.

A Composite determinant is one in which the number of columns exceeds the number of rows , and it is

a1 az aз b₁ by bз 18881

written as in the annexed example .

Its value is the sum of all the determinants obtained by taking a number of rows in every possible way. A Simple determinant has single terms for its constituents . A Compound determinant has more than one term in some or all of its constituents . See (570) for an example . For the definitions of Symmetrical, Reciprocal, Partial, and Complementary determinants ; see (574) , (575) , and (576) .

General Theory. 556

The number of constituents is n².

The number of elements in the complete determinant is

557

The

first

or

leading

element is

a, b, c, ... In .

•

Any

element may be derived from the first by permutation of the suffixes .

The sign of an element is + or - according as it has been obtained from the diagonal element by an even or odd number of permutations of the suffixes . Hence the following rule for determining the sign of an element . RULE . Take the suffixes in order, and put them back_to their places in the first element. Let m be the whole number ofplaces passed over ; then ( -1 )

will give the sign required.

Ex -To find the sign of the element a , b, c, d, e, of the determinant (a, b, c, d, e ).

Move the suffix 1, three places... .. 2, three places ... 99 99 3, one place ... 99 In all, seven places ; therefore ( −1 ) = -1

... ... ...

as 1 1 1

bз cz d₁ eg 4 3 5 2 2 4 3 5 2 3 4 5

gives the sign required.

558 If two suffixes in any element be transposed , the sign of the element is changed . Half of the elements are plus , and half are minus .

559 The elements are not altered by changing the rows into columns. If two rows or columns are transposed , the sign of the U

146

THEORY OF EQUATIONS.

determinant is changed . sign.

Because each element changes its

If two rows or columns are identical, the determinant vanishes . 560

If all the constituents but one in a row or column

vanish, the determinant becomes the product of that con stituent and a determinant of the next lower order .

A cyclical interchange is effected by n - 1 successive transpositions of adjacent rows or columns , until the top row has been brought to the bottom, or the left column to the Hence right side . 561

A cyclical interchange changes the sign of a determinant of an even order only. The 7th row may be brought to the top by r - 1 cyclical interchanges.

562 If each constituent in a row or column be multiplied by the same factor , the determinant becomes multiplied by it. If each constituent of a row or column is the sum of m terms , the compound determinant becomes the sum of m simple determinants of the same order .

Also , if every constituent of the determinant consists of m terms , the compound determinant is resolvable into the sum of m² simple determinants .

563 To express the minor of the 7th row and a determinant of the n - 1th order. Put all the constituents in the 7th row and

th column as

th column equal

to 0, and then make r - 1 cyclical interchanges in the rows and k- 1 in the columns , and multiply by ( -1 ) ( + ) (n − 1) ¸

[···

564

To express a determinant as a deter

= ( − 1)(r − 1 + k− 1) (n −1) ¸

10000

minant of a higher order .

a 1000

Continue the diagonal with constituents of “ ones ," and fill up with zeros on one side, and with any quantities whatever (a , ẞ, y, & c . ) on the other.

Beah g y s h b f d ng fe

147

DETERMINANTS.

565 The sum of the products of each constituent of a column by the corresponding minor in another given column is zero. And the same is true if we read row ' instead of ' column.'

Thus, referring to the determinant in ( 555 ) ,

Taking the pth and qth columns ,

Taking the a and c rows ,

а„Ÿ+b„B₂+ ... + l„Lq = 0.

a₁C₁ + a₂C₂+ ... + a„n C₁ = 0.

For in each case we have a determinant with two columns identical.

566 In any row or column the sum of the products of each constituent by its minor is the determinant itself. That is , Taking the pth column , aAp +bB + ... + l„L₂ = A.

567

Or taking the c row, c₂C₁ + c₂ C₂ + ... + c₂C₂ = A.

The last equation may be expressed by EcpCp = A. n аp аq ; the

Also, if (a, c ) express the determinant

Cр са (a, c ) will represent the sum of all the determinants of the second order which can be formed by taking any two columns out of the a and c rows . The minor of (a , c ) may be written (4p, C ), and signifies the determinant obtained by suppress Thus cq ing the two rows and two columns of a, and C And a similar notation when three or ▲ = Σ (αp, Cq) (Ap, C ) . more rows and columns are selected .

568

Analysis of a determinant.

RULE.— To resolve into its elements a determinant of the nt order. Express it as the sum of n determinants of the (n - 1)th order by (560) , and repeat the process with each of the new determinants.

EXAMPLE : Ar b₁ C1 dr

Ag bg C3 da

Az bз C3 dz

As = a₁ b₂ b₂3 b₁ - a, | bვ ს, ხ, | + a, | b,4 b,3 b, C4 C3 Ca b₁ C2 C3 Cs C3 C4 C1 C4 dз d d₁ d¸ d¸3 d dą dy dr ds

Again,

- a, | b, ხ, ს, C1 C2 C3 d₁ d₂ ds

C1 bq b₁ br2 b Cg bz ^^ C1 C2 & C3 | = 6 | 95 d, d , | + 4| 95 då | + | 32 | d₁ dą dz

and so on.

In the first series the determinants have alternately plus and minus signs, by the rule for cyclical interchanges ( 561 ) , the order being even.

1

148

THEORY OF EQUATIONS.

569

Synthesis of a determinant .

The process is facilitated by making use of two evident rules . Those constituents which belong to the row and column of a given constituent a, will be designated " a's con stituents ." Also , two pairs of constituents such as ap, ca and aq, cp, forming the corners of a rectangle, will be said to be 66 conjugate" to each other . RULE I.- No constituent will be found in the same term with one of its own constituents. RULE II .— The conjugates of any two constituents a and b will be common to a's and b's constituents. Ex.

To write the following terms in the form of a determinant : abcd +bfyl +ƒ³h² + 1edf+cghp + 1ahr + elpr

-fhpr―ablr- ach² - 1ƒhg — bdƒª — efhl — cedp. The determinant will be of the fourth order ; and since every term must contain four constituents, the constituent 1 is supplied to make up the number in some of the terms. Select any term, as abcd, for the leading diagonal.

Now apply Rule I., a is not found with e, f, f, g, p, 0 ….. ( 1 ) .

c is not found with f,f, l, r, 1 , 0 ... (3) .

b is not found with e, h , h , p , 1,0 ... ( 2 ) .

d is not found with g, h, h, l, r, 0 ... (4) .

Each constituent has 2 (n - 1 ) , that is, 6 constituents belonging to it, since n4. Assuming, therefore, that the above letters are the constituents of a, b, c, and d, and that there are no more, we supply a sixth zero constituent in each case . Now apply Rule II.-The constituents common to b and c- 1, 0 ; to a and b are e, p ; to a and c-f, ƒ ; to b and d- h, h, 0 ; to c and d--l, 0. to a and d- y, 0 ; a e f g The determinant may now be formed . The diagonal P b 1 h being abed ; place e, p, the conjugates of a and b , either as ƒ 0 с r in the diagram or transposed . 0 h l d Then ƒ and f, the conjugates of a and c, may be written. 1 and 0, the conjugates of b and c, must be placed as indicated, because 1 is one of p's constituents, since it is not found in any term with p, and must therefore be in the second row. Similarly the places of g and 0, and of 1 and r, are assigned. In the case of b and d we have h, h, 0 from which to choose the two conjugates, but we see that O is not one of them because that would assign two zero constituents to b, whereas b has but one, which is already placed. By similar reasoning the ambiguity in selecting the conjugates 1, r is removed. The foregoing method is rigid in the case of a complete determinant

DETERMINANTS.

149

having different constituents. It becomes uncertain when the zero con stituents increase in number, and when several constituents are identical. But even then, in the majority of cases, it will soon afford a clue to the required arrangement.

570

PRODUCT OF TWO DETERMINANTS OF THE nth ORDER.

(P) a1 a₂ ... an

a1 az ... an

b₁ b₂ ... b n

βι β ... B

B₁ B₂ ... B₂

l1 l2 ... In

λιλ ... λ

L₁ L2 ... L

(Q)

The values of A₁ , B₁ ... L₁ in the first column of S are an

=

(S) A₁ A₂ ... An

A₁ Ꭺ = а₁ α₁ + α₂ aş + ... + α, an B₁ = a₁ẞ₁ + a₂ẞ₂ + ... + an ẞn

nexed . For the second column write b's in the place of a's. For the third column write c's , and so on.

L₁ = αλιταλ + ... + αλη

For proof substitute the values of A₁ , B₁, &c . in the determinant S, and then resolve S into the sum of a number of determinants by ( 562) , and note the determinants which vanish through having identical columns. RULE. To form the determinant S, which is the product of two determinants P and Q. First connect by plus signs the constituents in the rows of both the determinants P and Q. Now place the first row of P upon each row of Q in turn, and let each two constituents as they touch become products. This is the first column of S. Perform the same operation upon Q with the second row of P to obtain the second column of S ; and again with the third row of P to obtain the third column of S, and so on. 571

If the number of columns , both in P and Q, be n, and

the number of rows r, and if n be >r, then the determinant S, found in the same way from P and Q, is equal to the sum of the C (n, r) products of pairs of determinants obtained by taking any columns out of P, and the corresponding r columns out of Q. But if n be < r the determinant S vanishes . For in that case, in every one of the component determinants, there will be two columns identical.

150

THEORY OF EQUATIONS.

572 The product of the determinants P and Q may be formed in four ways by changing the rows into columns in either or both P and Q.

573

Let the following system of n equations in

r ... Xn

be transformed by substituting the accompanying values of the variables , а₁x²₁ + a₂x² + ... + ª‚x„ = 0 ,

x₁ = α₁§1 + α₂§2 + ... + an§ns

b₁x₁ + b₂x₂ + ... + bnxn = 0,

X₂ = ẞ₁₁ + B₂$2₂ + ... + ẞ„§n ,

1½ x₁ + 1½ x½ + ... + lnxn = 0,

xn = λ₁ &i+ λ &z + ... + λn §n •

The eliminant of the resulting equations in ... En is the determinant S in (570 ) , and is therefore equal to the product of the determinants P and Q. The determinant Q is then termed the modulus of transformation .

574 A Symmetrical determinant is symmetrical about the leading diagonal. If the R's form the 7th row, and the K's the throw ; then RK, throughout a symmetrical deter minant. The square of a determinant is a symmetrical determinant .

575

A Reciprocal determinant has for its constituents the first minors of the original determinant, and is equal to its n - 1th power ; that is ,

A₁ ... An

L₁ ... Ln

576 If

=

n- 1 a1 ... An

PROOF. Multiply both sides. of the equation by the original determinaut ( 555 ) . The con stituents on the left side all

Ն ... In

vanish excepting the diagonal of A's.

Partial and Complementary determinants . rows and the same number of columns be selected

from a determinant, and if the rows be brought to the top , and the columns to the left side , without changing their order, then the elements common to the selected rows and columns form a Partial determinant of the order r, and the elements. not found in any of those rows and columns form the Com plementary determinant, its order being n - r.

DETERMINANTS.

151

Ex.-Let the selected rows from the determinant (a, b, c, d, e,) be the second, third, and fifth ; and the selected columns be the third, fourth, and fifth. The original and the transformed determinants will be and αι Az Aq as Ag b b₁4 bs b₁ ba b₁

b₁

bs

b₁

CA

C5

C1

C₂ d

C3 d

C4 d

bs C5

Cs

G1 d

es

es

es

e1

Ca

d

As

as

Az

a1

e1

Eg

eg

e4

e5

ds3

d

d

d

Աջ da

Ca

The partial determinant of the third order is ( b , c, e ) , and its comple mentary of the second order is (a, d ). The complete altered determinant is plus or minus, according as the permutations of the rows and columns are of the same or of different class . In the example they are of the same class, for there have been four trans positions of rows, and six of columns. Thus ( -1) ¹º = + 1 gives the sign of the altered determinant.

THEOREM .- A partial reciprocal determinant of the 7th order is equal to the product of the r- 1th power of the original determinant, and the complementary of its corres ponding partial determinant . 577

Take the last determinant for an example. Here n = 5, r = 3 ; and by the theorem , B₁ B₁ B₁ = A⁹ a1 aa where B, C, E are the respective minors . Os C₁ C5 d₁ da E,

E,4 E,

PROOF.- Raise the Partial Reciprocal to the original order five without altering its value, by (564) ; and multiply it by A, with the rows and columns changed to correspond as in Ex. (576) ; thus, by ( 570) , we have

B, B B

B₁ B,

C₁ C₁ C₂

C₁

0

0

0

0

1

b ა b₁ bs C3 C4 Cg

b₁

b₂ | =

C1

C2

eg

e1

eq

Az d

a₁

ag

d

da

es

es

Ag as d. d

‫ܛ‬

C₂ E, E, E, E, E, 0 0 0 1 0

▲ 0 0 b₁1 b₁ = A³ | а₁ αq Ο Δ Ο C C d, a . 0 0 A e eq

0 0 0 a ą 0 0 0 d d₂

578 The product of the differences between every pair of n quantities a , ag ... An

(α - a₂) (α - aз) (a₁ — a₁) ... (a₁ — a₂) × (a, —aç) (α¸ — α¸) ... ……. (a₂ — a₂)

1

1

1

a₁

a2

as ...

a²

a² __ a

...

...

1 an

a²

× (as - as) ... (a — an)

× (an-1 - an)

an - ¹ a -¹ an -¹ ... a -1

PROOF.-The determinant vanishes when any two of the quantities are

THEORY OF EQUATIONS.

152

equal . Therefore it is divisible by each of the factors on the left ; therefore by their product. And the quotient is seen to be unity, for both sides of the equation are of the same degree ; viz. , in ( n − 1 ) . The product of the squares of the) = 579 differences of the same n quantities PROOF. - Square the determinant in ( 578 ) , and write s, for the sum of the rth powers of the roots.

580

So

$1 ... Sn-1

81

82 ... Sn

Sn - 1 Sn ... San - 2

With the same meaning for 81 , 82 ... , the same deter

minant taken of an order r , less than n , is equal to the sum of the products of the squares of the differences of r of the n quantities taken in every possible way ; that is, in C (n , r) ways .

Ex.:

80 81 81 83

= (α₁— a₂)² + (α₁ — a3) ² + &c. = Σ ( a₁₂ — a₂)³,

80 81 82 81 82 83 82 83 84

-a₂) ³ (α1 — α3) ³ (αg — α3) ³. = Σ (α₁ —

The next determinant in order

-= Σ (α₂ — α₂) ² (α‚¸ — à¸) ³ (α¸ — à¸) ² ( α‚ — α¸) ³ (a, —a¸) ² (a, —a¸) ³. And so on until the equation (579) is reached. Proved by substituting the values of s₁, 8, ... &c ., and resolving the deter minant into its partial determinants by (571 ) .

581

The quotient of

а¸x™ + a₁x³¬¹ + ... + a, xm - r + ... b。 x² +b₁ x²¬¹ + ... + b, x² - r + ... is given by the formula Lo xm¬n +91 xm¬n−1 + ... + q, xm¬n-r + ...9 where

1

bo

0

br+1

Ել

0

...

ao

Ե bo

()

...

a1

bą b₁ ....

bo

...

a2

Ir =

br br-1 br-2 ... b₁a, Proved by Induction .

153

ELIMINATION.

ELIMINATION .

582

Solution of n linear equations in n variables.

The equations and the values of the variables are arranged below : *Δ §¹‚_x₁ ▲= = A₁§1 + B₁ §2 + ... + L1 §n а₁X₁ + a₂ X₂ + ... + án xn = $1, b₁ x₁ + b₂ x2 + ... + bn Xn = $2,

x₂A = A½§1 + B₂§2 + ... + L2 §n .....

hx₁ + l₂ x₂+ ... + ln xn = {n,

x„▲ = An§1 + Вn§2 + ... + Ln§n

where A is the determinant annexed , and A1, B1, &c. are its first minors . To find the value of one of the unknowns x..

a1 ... an ......... 1 Ն ... In

RULE.-Multiply the equations respectively by the minors of the pth column, and add the results. x, will be equal to the fraction whose numerator is the determinant ▲, with its pth column replaced by 1 , 2 ... En, and whose denominator is A itself.

583

If

₁,

2 ... En and ▲ all vanish, then

₁, X22 ... x, are in

the ratios of the minors of any row of the determinant A. For example , in the ratios

C₁ : C₂ : C3 : ... : Cn .

The eliminant of the given equations is now A = 0.

584 Orthogonal Transformation .

If the two sets of variables in the n equations (582 ) be connected by the relation x₁ + x² + ... + x²n = §; + §²2 + ... + &

.....

(1 ) ,

then the changing from one set of variables to the other, by substituting the values of the E's in terms of the a's in any function of the former, or vice versâ, is called orthogonal transformation . When equation (1 ) is satisfied , two results follow. I. The determinant A = ± 1 . X

154

THEORY OF EQUATIONS.

II. Each of the constituents of ▲ is equal to the corre sponding minor, or else to minus that minor according as ▲ is positive or negative . PROOF.- Substitute the values of ₁, 19 ... En in terms of x1, x2 ... Xn in equation ( 1 ) , and equate coefficients of the squares and products of the new variables. We get the n' equations

+ = 1

AgA₁ + b₂b₁ + = 0

аzα₁ + b₂b₁ + = 0

ana, + bqb , + = 0

AzAg + bgbg + = 0

anas + bხვ + = 0 ....

a² + b²2 + = 1 Agаz + b₂b¸ + = 0 ...

a₁an + b₁b₁₂ + = 0 )

aşa + b,b, + = 0

azan + b3bn + = 0

a² + b²

Also A =

a₁ b₁ ... l₁ aqb₂ ... lg asbs ... lg ...

amb ... In

+ = 1

Form the square of the determinant A by the rule ( 570) , and these equations show that the product is a determinant in which the only con stituents that do not vanish constitute a diagonal of ' ones.' Therefore A1 and A = ± 1 .

Again, solving 2 the first set of equations for a (writing a, as a, a,, &c. ) , the second set for a , the third for a,, and so on, we have, by (582) , the results annexed ; which proves the second proposition.

585

a² + b²3

(a₁A = A₁ + A‚0 + A¸0 + = A, а ▲ = A₂0 + A₂ + A30 + = A₂ а¸▲ = ¸0 + ‚0 + A¸ + = Ag &c. &c.

Theorem .-The n - 2th power of a determinant of the

nth order multiplied by any constituent is equal to the corre sponding minor of the reciprocal determinant. PROOF.- Let p ρ be the reciprocal determinant of A, and ẞ, the minor of B, in p. Write the transformed equations (582 ) for the x's in terms of the E's, and solve them for . Then equate the coefficient of x, in the result with its coefficient in the original value of 2. Thus p , = A (B₁₁ + ... + ß‚æ‚ + ... ) , and = b₁x₁ + ... + bx + ….. ;

::

586

Aß, = Δβ,

pb, = An -1b, by (575) ;

, == A"-2bp. Br .. B

To eliminate a from the two equations ax™ + bxm -¹ + cam - 2 + ... = 0 ax 2-1 a'x" + b'x"-¹ + c'an -2 + ... = 0

......

(1) ,

(2) .

If it is desired that the equation should be homogeneous x in x and y ; put instead of x, and clear of fractions . The y following methods will still be applicable.

UN. ELIMINATION.

155

1. Bezout's Method. -Suppose m > n. RULE . Bring the equations to the same degree by multi `plying (2) by xm-n¸ Then multiply (1 ) by a', and (2) by a, and subtract. Again, multiply ( 1 ) by a'x + b' , and (2) by ( ax + b) , and subtract. Again, multiply ( 1 ) by a'x² + b'x + c' , and (2) by ( ax² + bx + c) , and subtract, and so on until n equations have been obtained. Each will be of the degree m - 1 . Write under these the m - n equations obtained by multi The eliminant of the m equations plying (2) successively by x. is the result required.

1

d₁ d 1,

Ex.

Let the equations be

ax + bx + cx³ + dx² + ex + f = 0, a²x² + b²x² + c'x + d ' = 0.

The five equations obtained by the method , and their eliminant, by (583) , are, writing capital letters for the functions of a, b, c, d, e, f, A, B, C, D, E₁

A₁2* + B¸½³ + C₁²² + D¸æ + E₁ = 0 Aqx¹ + Bgx³ + Сqx² + Dqïx + E₂ = = 0

Ag + Bzi + Czπ² + Dzx + Ez = 0 a²x² +b²x² + c' x² + d' x = 0

and

a´x³ + b²x² + c′x + d′ = 0

= 0.

A, B, C, D,2 E, A, B,3 C, D,3 Es a b c d 0 O

a

b

c

d'

Should the equations be of the same degree, the eliminant will be a sym metrical determinant.

587

II. Sylvester's Dialytic Method.

RULE. - Multiply equation ( 1 ) successively by x, n - 1 times ; and equation (2) m - 1 times ; and eliminate x from the m + n resulting equations.

Ex.-To eliminate x from

aa³ + bx² + cx + d = 0 2 81. px + qx + r = 0

The m +n equations and their eliminant are px² + qx + r = 0 px³ +qx² + rx =0 = 0 px¹ +qx³ + ræ²

ax +bx + cx + d = 0 ax +bx +cx² + da =0

O o p q r O p q r 0 and

p qr 0 0 O a b c d a b c d 0

= 0.

156

THEORY OF EQUATIONS.

588

III. Method of elimination by Symmetrical Functions.

Divide the two equations in (586) respectively by the coefficients of their first terms , thus reducing them to the forms

ƒ (x) = x" + P₁x™ -¹ + ... + Pm = 0 , $ (x) = x² + ¶₁x” -¹ + ... + 9₂ = 0.

RULE.- Let a, b, c ... represent the roots of f(x) . Form This will contain sym (a) $ (b) ¢ (c) ... = 0.

the equation

metrical functions only of the roots a, b , c ... . Express these functions in terms of P1, P2 ... by (538 ) , & c . , and the equation becomes the eliminant. Reason of the rule. -The eliminant is the condition for a common root of the two equations . That root must make one of the factors ( a) , ¢ (b) ... vanish, and therefore it makes their product vanish . 589 The eliminant expressed in terms of the roots a, b , c ... of f (x) , and the roots a, ß, y ... of (x ) , will be

(a− a) (a —ẞ) (a− y) ... (b — a) (b − ß) (b − y) ... &c ., being the product of all possible differences between a root of one equation and a root of another. 590

The eliminant is a homogeneous function of the co

efficients of either equation , being of the nth degree in the coefficients of ƒ ( x) , and of the mth degree in the coefficients of (x) . 591 The sum of the suffixes of p and q in each term of the eliminant mn. Also, if p, q contain z ; if p₂, q2 contain z² ; if P3 , 3 contain z³, and so on, the eliminant will contain zn Proved by the fact that p, is a homogeneous function of r dimensions of the roots a, b, c ...3 by (406) . 592 If the two equations involve x and y, the elimination may be conducted with respect to a ; and y will be contained in the coefficients P1, P2 , 91 , X2 Q2 ··· ·

593

Elimination by the Method of Highest Common Factor.

Let two algebraical equations in æ x and y be represented by A = 0 and B = 0.

ELIMINATION.

157

It is required to eliminate a.

1. Arrange A and B according to descending powers of x a, and, having rejected any factor which is a function of y only, proceed to find the Highest Common Factor of A and B.

The process may be exhibited as follows : C₁A = q₁B + r₁R₁

C1 , C2 , C3 , C4 are the multipliers re quired at each stage in order to avoid

c₂B = q₂R₁ + r₂ R₂ C3 R₁ = q3 R₂ + r3 R3

fractional quotients ; and these must be constants or functions of y only.

c₁R₂ = q₁ R₂ + r₁

91, 92, 93, 94 are the successive quo tients .

T₁R₁, T₂R₂, TзRз , r, are the successive remainders ; 71, 72, 73, 74 being functions of y only.

The process terminates as soon as a remainder is obtained which is a function of y only ; r, is here supposed to be such a remainder. Now, the simplest factors having been taken for C1, C2, C3, C4, we see that

1 is the H. C. F. of

d2 ds

99 99

""

C1

and ri

C1

and 2

C₁₂ C1C2

and rs

The values of x and y, which satisfy simulta neously the equations A=0 and B = 0 , are those obtained by the four pairs of simultaneous equations

""

d2 d

C1C₂Cs and r

""

"" following :

da ds

(1)

The final equation in y,

= 0 and R₁ = 0 ......... .... ( 2 )

which gives all admissible values , is

r₁ = 0 and B = 0 .......

Ta da

71727374 = 0 , 2 d dods

73

= 0 and R₂ = 0 ....

(3)

ds

If it should happen that the remainder r, is zero , simultaneous the equa

= 0 and R₁ = 0 ........ ... ... ( 4) i =

tions ( 1 ) , ( 2) , (3 ) , and (4) reduce to

1.

R₁ r₁ = 0 and B = 0;

2 = 0 and

= 0;

Rз

T3 = and 0 R₂ = 0. ds = 0 R3

158

594

THEORY OF EQUATIONS.

To find infinite values of x or y which satisfy the given

equations . 1

Put x =

Clear of fractions , and make z = 0 .

2 If the two resulting equations in y have any common roots , such roots, together with x = ∞ , satisfy simultaneously the equations proposed . Similarly we may put y = 1214·

PLANE

TRIGONOMETRY .

ANGULAR

600

MEASUREMENT.

The unit of Circular measure is a Radian , and is the

angle at the centre of a circle which subtends an arc equal to the radius . Hence arc Circular measure of an angle = 601 radius 602

Circular measure of two right angles = 3.14159 ... = T .

603

The unit of Centesimal measure is a Grade , and is the

one-hundredth part of a right angle. 604 The unit of Sexagesimal measure is a Degree , and is the one- ninetieth part of a right angle. To change degrees into grades , or circular measure , or vice versa, employ one of the three equations included in

D

G =

=

605

90

100

2R 2 π

where D, G, and C are respectively the numbers of degrees , grades , and radians in the angle considered .

TRIGONOMETRICAL

RATIOS .

B

606 Let OA be fixed , and let the revolving line OP describe a circle round 0. Draw PN always perpen dicular to AA'. Then, in all posi tions of OP, PN = the sine of the angle AOP, OP

A N N

ON = the cosine of the angle AOP,

OP B PN = the tangent of the angle AOP.

ON

PLANE TRIGONOMETRY.

160

607

If P be above the line AA' , sin AOP is positive . If P be below the line AA', sin AOP is negative.

608

If P lies to the right of BB' , cos AOP is positive. If P lies to the left

of BB' , cos AOP is negative .

609 Note, that by the angle AOP is meant the angle through which OP has revolved from OA, its initial position ; and this angle of revolution may have any magnitude. If the revolution takes place in the opposite direction, the angle described is reckoned negative.

The secant of an angle is the reciprocal of its cosine,

610

cos A sec A = 1 .

or

The cosecant of an angle is the reciprocal of its sine,

611

sin A cosec A = 1 .

or

612

The cotangent of an angle is the reciprocal of its tangent, tan A cot A = 1 .

or

Relations between the trigonometrical functions of the same angle.

[I. 47

613

sin2 A +cos2 A = 1 .

614

sec² 4 = 1 +tan² 4 .

615

cosec² A 4 = 1 + cot² 4 .

616

tan A =

sin A

[ 606

cos A

a

If tan A =

√ a²+b²

ī

a a

617

sin A =

A

b

√ a² + b² b

[606

cos A = √ a²+b² 1

tan A 618

"

sin A = ✓1 +tan A

cos A =

[617 √1 +tan² A

TRIGONOMETRICAL RATIOS.

619

The Complement of A is =

620

The Supplement of A is = 180° — A.

621

sin (90° -A) = cos A ,

161

90° —A.

tan (90° -4 ) = cot A ,

P

sec (90° — 4) = cosec A.

622

sin ( 180° - A ) =

sin A ,

X

A cos ( 180° — A) = − -cos A , A) tan ( 180° — 4 ) = —tan A.

In the figure 4Q0X180 ° -A.

[607, 608

623

sin (-4) =

sin A.

624

A) = cos (-4)

cos A. By Fig., and (607) , (608) .

The secant, cosecant, and cotangent of 180° -A , and of -4, will follow the same rule as their reciprocals , the cosine , sine, and tangent . [610-612

625 To reduce any ratio of an angle greater than 90° to the ratio of an angle less than 90°. RULE .-Determine the sign of the ratio by the rules ( 607) , and then substitute for the given angle the acute angle formed by its two bounding lines, produced if necessary. Ex.- To find all the ratios of 660°. B Measuring 300° ( = 660° -360°) round the circle from A to P, we find the acute angle AOP to be 60°, and P lies below AA ' , and to the right of BB' . Therefore - sin 60° = sin 660° = -

A

A'

√3 2

cos 660° =

cos 60° =

B' 12/121219

and from the sine and cosine all the remaining ratios may be found by (610-616 ) .

INVERSE NOTATION . -The angle whose sine is a is denoted -1 by sin- ¹æ. Y

162

PLANE TRIGONOMETRY.

626 All the angles which have a given sine , cosine , or tan gent, are given by the formulæ

sin¹x n + ( − 1 )" 0 .... -1 cos¯¹x = 2nπ ±0

(1 ) ,

tan -¹a =

(3).

nπ + 0 ....

(2) ,

In these formulæ is any angle which has a for its sine, cosine, or tangent respectively, and n is any integer. Cosecx , seca, cota have similar general values, by ( 610-612 ) . These formulæ are verified by taking A, in Fig. 622, for 0, and making n an odd or even integer successively.

B FORMULE

INVOLVING MULTIPLE

TWO

ANGLES ,

AND

ANGLES .

627

sin ( A + B) = sin A cos B + cos A sin B,

628

sin ( A − B) = sin A cos B - cos A sin B,

629

cos ( A + B) = cos A cos B - sin A sin B,

630

cos (A - B) = cos A cos B + sin A sin B.

PROOF . -By ( 700 ) and ( 701 ) , we have sin C = sin A cos B + cos A sin B, and sin C sin (A + B) , by ( 622) .

To obtain sin (A - B) change the sign of B in (627) , and employ ( 623 ) , (624) ,

cos (A + B)

sin

( 90° -A) -B} , by ( 621 ) .

Expand by (628) , and use ( 621 ) , ( 623 ) , (624) . sign of B in (629) .

For cos ( A - B) change the

tan Atan B

631

tan (A + B) = 1 -tan A tan B

632

tan A - tan B tan ( A — B) = 1 +tan A tan B'

633

cot (A + B) =

cot A cot B - 1 cot Acot B 634

cot (4 - B) =

cot 4 cot B + 1 cot B - cot A Obtained from (627-630 ) .

64040

MULTIPLE ANGLES.

163

635

sin 24 = 2 sin A cos 4.

636

cos 24 = cos² A - sin² A ,

637

= 2 cos² 4 - 1 ,

638

= 1-2 sin² A.

639

[627.

Put B = A

[ 629, 613

2 cos² A = 1+ cos 24 .

[637

2 sin² A = 1- cos 24.

[ 638

A

cos A

1

sin

[640

=V

2

1+ cos A 642

COS 14 = V v

A 643

tan

1

[639 2

- cos A

=

1- cos A

sin A

sin A

1+ cos A

=

·

1+ cos A

[ 641 , 642, 613 A

A 2 tan

1- tan²

2 646

cos A =

2 sin A =

[643, 613 A

A

1 + tan²

1 + tan² 2 1

648

cos A =

A 1 + tan A tan

2 = √1 +sin A (45° sin (45°+ 4) = cos ( 2 4)

[641

649

650

cos ( 45 + 4 )

1- sin A vi ·1/2 ) = √ 2

[642

651

A 1 + sin A cos A 1 + sin A = = + 4 1 + sin A tan( 15 + = √1 )= 2) 11 - sin A cos A

-

= sin (45°

2 tan A

652

tan24 =

[ 631. 1- tan² A cot² A - 1

653

cot 24 = 2 cot A

Put B = A

PLANE TRIGONOMETRY.

164

1 + tan A tan (45° + 4 ) =

654

1 - tan A 1 - tan A

tan (45° -4) :=

655

[631 , 632 1 + tan A

656

sin 34 = 3 sin A - 4 sin³ A.

657

cos 344 cos'A - 3 cos A.

658

tan 34 =

3 tan Atan³ A

1-3 tan'A By putting B = 24 in ( 627) , ( 629) , and ( 631 ) . 659

sin (A +B) sin ( A — B) = sin² A — sin² B — cos² A. = cos² B -

660

— sin² B cos (A +B) cos ( A - B) = cos² A cos2 B - sin² A. From ( 627) , &c.

sin

+ cos 4 = ± √1 + sin 4 . 144 + [Proved by squaring. A A − sin A. sin COS 4 = ± √1 14 - c

661

662

1

A =

{ √1 + sin A 4 − √1 − sin 4 } .

663

sin

664

COS 4 = 1 { √1 + sin A + √1— sin 4 } ,

when 4 lies between -45° and + 45°.

+

+

A

rant in which

+ +

1

665 In the accompanying diagram the signs exhibited in each quadrant are the signs to be prefixed to the two surds in A the value of sin according to the quad 2

lies .

For cos 4 change the second sign. PROOF.

By examining the changes of sign in ( 661 ) and (662) by ( 607) .

MULTIPLE ANGLES.

165

666

2 sin A cos B = sin ( A + B) + sin ( A — B) .

667

2 cos A sin B = sin ( A + B) — sin ( A — B) .

668

2 cos A cos B = cos ( A + B) + cos ( A — B) .

669

- cos ( A + B) . 2 sin A sin B = cos (AB)

[627-630 670

sin A + sin B = 2 sin 4 + B COS 2

2

A- B

A+ B 671

A- B

sin

sin Asin B = 2 cos

2 A+ B

672

A- B

COS

cos A + cos B = 2 cos 2

A- B

A+B

673

cos B - cos A = 2 sin

sin 2

A-B A+ B " and B into 4—¹, in (666–669) . Obtained by changing A into 4+ 2 2 It is advantageous to commit the foregoing formula to memory, in words, thus 2 sin cos = sin sum + sin difference, 2 cos sin sin sum - sin difference, 2 cos cos = cos sum + cos difference, cos sum . 2 sin sin cos difference sin first + sin second = sin first sin second cos first + cos second = cos second - cos first =

674

2 sin half sum 2 cos half sum 2 cos half sum 2 sin half sum

cos sin cos sin

half difference, half difference, half difference, half difference.

sin (A + B+ C) = sin A cos B cos C + sin B cos C cos A + sin C cos A cos B - sin A sin B sin C.

675

cos (A + B + C) = cos A cos B cos C-cos A sin B sin C - cos B sin C sin A - cos C sin A sin B.

676

tan (A + B + C )

=

tan A + tan B + tan C - tan A tan B tan C - tan A tan B* 1 - tan B tan C - tan C tan A —

PROOF.- Put B + C for B in (627) , ( 629) , and (631) .

166

PLANE TRIGONOMETRY.

If A +B +C = 180°, sin A + sin B + sin C = 4 cos

COS

sin

B

A

678

sin

cos A + cos B + cos C = 4 sin

C COS

2

1232

4 sin

C cos 2

B

A

sin A + sin B - sin C

32 32 323

12 42T

B

A

677

C

sin

+1 .

C A B sinGeCOS n1. cos A + cos B - cos C = 4 cos 81/2 2 679

tan A + tan B + tan C = tan A tan B tan C.

680

cot

681

sin 24+ sin 2B + sin 2C

4 sin A sin B sin C.

682

cos 24+ cos 2B + cos 2C

- 4 cos A cos B cos C - 1 .

B A C tcot = cot cot 2 2 C

+ cot

+ cot

General formulæ, including the foregoing, obtained by applying (666-673) .

, and n be any integer,

If A + B + C = 683

24 sin nB in nA nC sin 4 sin 2 2 Nπ

= sin ("

nA - n4 ) + sin ( nA

684

4 cos

B ) + sin (

-n

) -sin

nB nC COS cos 2 2 Nπ

= COS (2 TnA ) + cos ( TnB ) + cos (

nC ) + cos 2

If A + B + C = 0 , 4 sin

sin

nA 4 cos

sin

2

2

686

= -sin nA - sin nB- sin nC. 2

nB

nC

COS

COS 2

nC

nB

nA

685

2

cosnA +cosnB + cosnC + 1 .

2

RULE.-If, in formula (683) to (686) , two factors on the left be changed by writing sin for cos, or cos for sin, then, on the right side, change the signs of those terms which do not contain the angles of the altered factors.

167

COMMON ANGLES.

Thus , from (683) , we obtain nA nB nC COS 687 4 sin COS 2 2 2

=

sin (

n4 ) + sin ("

- nB ) + sin ("

— nC) + sin " ,

A Formula for the construction of Tables of sines , co sines, &c. 688 sin (n + 1 ) a - sin na = sin na— sin (n - 1 ) a -k sin na, where a = 10", and k = 2 ( 1 — cos a) = ' 0000000023504 .

689

Formulæ for verifying the tables—

sin A +sin ( 72° + A ) —sin ( 72° — A ) = sin ( 36° + A) — sin ( 36° —A) , cos A +cos ( 72° +4) + cos ( 72° + A) = cos (36° + A ) + cos ( 36° —A) , sin (60° +4) -sin (60° -A ) = sin A.

RATIOS

OF CERTAIN

ANGLES .

1

690

sin 45° = cos 45° =

tan 45° = 1 .

2' 1 691

sin 60° = √3 2

cos 60° =

√3-1 692

sin 15° =

tan 60° = √3.

2' "

2√2

cos 15° = √3 + 1 2√2

tan 15°

2√32 cot 15° = 2 + √35·

693

sin 18° =

√5-1 4

cos 18° =

tan 18° =

√5 + √5 " 2√2

5-2 /5 5

694

sin 54° = √5 + 1 4

√5 - √5 cos 54° =

"

2√2

tan 54° = √ 5+2 √ /5 5 695 By taking the complements of these angles , the same table gives the ratios of 30° , 75°, 72° , and 36°.

PLANE TRIGONOMETRY.

168

696

PROOFS .-sin 15° is obtained from sin (45° -30°) , expanded by (628).

697

sin 18° from the equation sin 2x = cos 3x, where x = 18°.

698

sin 54° from sin 3x = 3 sin x- 4 sin³ x, where x = 18°.

699 And the ratios of various angles may be obtained by taking the sum, difference, or some multiple of the angles in the table, and making use of known formulæ. Thus 15° 30° -18°, 71° = 12° 9 &c., &c.

PROPERTIES

700

THE

TRIANGLE .

B

D

c = a cos B +b cos A. b

a

=

=

701

sin B

sin A 702

OF

sin C

a² = b² +c² -2bc cos A.

PROOF.- By Euc. II. 12 and 13, a = b +c - 2c . AD. b²+ c² - a²

cos A =

703

2bc

a+ b+c

If s =

and ▲ denote the area ABC, 2

704

sin AA4 =

-c) (s - b) 8 9 bc

A COS 08/4/2 = √√s a). V (s— bc [641 , 642, 703, 9, 10, 1 .

tan 11

705

(s - b) (s - c) s (s - a)

2

706

sin A =

√s (s — a) (s — b) (s — c).

[635 , 704

bc

707

708

A=

be sin A = √ s (s —a) (s — b) (s —c) ,

=

[707, 706

√20°c² + 20° a² +2ab -a -b' - c'.

169

PROPERTIES OF TRIANGLES.

The Triangle and Circle. Let

A r = radius of inscribed circle, r. radius of escribed circle touching the side a , R = radius of circumscribing circle. Δ 709 r= S 6

ra

ጥሪ

+

+

[From Fig., A =

B

2

710

E

Ta a sin

sin

1°

Ta

A

COS B

+ r cott

[ By arcot ra =

711

B a cos

C COS 2

2

712

-ran + 2

[By A = S -a

[ From a = r, tan

ra

B + rtan 9 2

A

COS

abc

a

713

R =

=

2 sin A

44

[ By (III. 20) and ( 706)

715

R A sec² 4 = } √ { (b + c) ª sec² 4 + ( b −c ) °cosec² 2}

B

a

[ 702 Distance between the centres of inscribed and circum scribed circles

716

= √R²—2Rr.

[ 936

Radius of circle touching b , c and the inscribed circle

717

r' = r tan² 1 (B + C ) .

Z

ryo [By sin4 = r+r "

PLANE TRIGONOMETRY.

170

SOLUTION

OF TRIANGLES .

Right- angled triangles are solved by the formulæ

B

c² = a² + b² ;

718

a

a = e sin A , b

719

c cos A , A

a = b tan A , &c .

Scalene Triangles . 720

CASE I.- The equation a

b

= sin A

[ 701 sin B

b

a

will determine any one of the four quantities A, B, a, b when the re

721

B

с

maining three are known .

The Ambiguous Case.

B

When, in Case I. , two sides and an acute angle opposite to one of them

e sin A sin C =

a

a

are given, we have, from the figure ,

A

1 D

a Then C and 180 - C are the values of C and C', by (622) . Also

b = ccos Ava -csin A,

because

722

and

= AD + DC .

When an angle B is to be determined from the equation b sin A, sin B = a

b is a small fraction ; the circular measure of B may be approximated a

to by putting sin ( B + C) for sin A, and using theorem ( 796) .

171

SOLUTION OF TRIANGLES.

723

CASE II.- When two sides b , c and the included angle

A are known, the third side a is given by the formula a² = b² +c² -2bc cos A,

[702

when logarithms are not used .

Otherwise, employ the following formula with logarithms , b- c

B- C

=

tan

724

cot th

b +c

2 b- c

sin B - sin C

b+c

sin B +sin C

Obtained from

(701 ) , and then applying

(670) and (671) . B- C having been found from the above equation, and 2

B+C 2

A 2

B +C

725

. we have

being equal to 90°

B=

B- C

B +C

B- C

2

2

C=

+ 2

2

B and C having been determined , a can be found by Case I. 726 If the logarithms of b and c are known, the trouble of taking out log (b- c) and log (b + c) may be avoided by employing the subsidiary angle 0 tan-1. b and the formula с A 727 tan (B - C) tan (0 [655 ( 0-1 ) cot /1,

I Or else the subsidiary angle 728

-1 €, and the formula = cos¹

A tan } (B— C ) = tan² 2/2 cot 2

[ 643

If a be required without calculating the angles B and C, we may use the formula A [ From the figure in 960, by (b +c ) sin 2 drawing a perpendicular α= 729 from B to EC produced . cos (B - C)

730 If a be required in terms of b, c , and A alone , and in a form adapted to logarithmic computation , employ the subsidiary angle 4bc -1 0 = sin- ' ( (b + c)" cos2 4), and the formula

a = (b +c) cos 0.

[702, 637

172

PLANE TRIGONOMETRY.

CASE III - When the three sides are known, the angles may be found without employing logarithms, from the formula b²+ c² —a²

[703

cos A =

731

2bc

732 If logarithms are to be used , take the formulæ for A A A ; (704) and (705) . or tan sin COS 2' 2' 2

QUADRILATERAL INSCRIBED IN A CIRCLE. ¦

— d² a² + b² — c² 733

B

cos B =

2 (ab + cd) From AC a + b² -2ab cos B = c + d + 2cd cos B, by (702) , and B + D = 180°.

ს

C

734

sin B =

2Q

[613, 733

ab + cd®

735

C

Q = √ (s —a)(s—b) (s — c) (s — d)

D = area of ABCD, and

(a + b + c + d) . 8 = Area = ab sin B + cd sin B ; substitute sin B from last.

AC2 = (ac + bd) (ad + be) (ab + cd)

736

[702, 733

Radius of circumscribed circle 1 737

=

[ 713, 734, 736 4Q✓ (ab +cd) (ac+bd) (ad +be).

If AD bisect the side of the triangle ABC in D, 4A tan BDA = 738 b²-c²

739 740

cot BAD

2 cot A + cot B.

AD² = } ( l² + c² + 2bc cos A ) = } ( b² + c² — } a²) .

If AD bisect the angle A of a triangle ABC, b+c B- C Α tan tan BDA = cot 742 2 b- c taAva 2bc A AD = COS 743 b +c

SUBSIDIARY ANGLES.

173

If AD be perpendicular to BC, AD = bc sin A = b2 sin C + c sin B 744 a b +c

ī BD ~ CD =

745

tan B-tan C b³ -c² =α α tan B +tan O'

REGULAR POLYGON AND CIRCLE.

Radius of circumscribing circle = R. Radius of inscribed circle = 1. pol = a. ygo Side of n

Number of sides R=

π

cosec

" n

2

r=

Bla

a

746

R

= n. π cot.- . n

a

Area of Polygon 2π 748

==

na² cot

= nr² tan

= InR² sin n

n

USE OF

. n

SUBSIDIARY ANGLES .

749

To adapt ab to logarithmic computation . b Take 0 = tan- ¹ ; then a +b = a sec² 0. Va

750

-1 For a − b take 0 = tan- ¹ ( ( ); thus

a- ba√2 cos (0 +45°) cos 751

To adapt a cos C ± b sin C to logarithmic computation. -1¹ a ; then Take tanb a cos Cb sin C = √ (a + b ) sin (0+ C). [ By 617 For similar instances of the use of a subsidiary angle, see ( 726) to ( 730) .

752 To solve a quadratic equation by employing a subsidiary angle. If x² -2px +q = 0 be the equation, x = p ( 1 ± √1-3 ) .

[ By 45

174

PLANE TRIGONOMETRY.

L = sin' 0 ; then p 0 0 x = 2p cos 9 and 2p sin' 2 q CASE II. -If q be >p , put = sec0 ; then p² xp (litan 9) , imaginary roots .

CASE I. —If q be < p², put

[639, 640 444

[614

Case III.-If q be negative, put 9 = tan20 ; then p* x = √q cotth 2

and

—vqtan

[644, 645 2

LIMITS OF RATIOS.

I

tan 0

sin =

753

PT

= 1, 0

0 when 0 vanishes .

For ultimately

PN AT = = 1. [601 , 606 0. AP AP

N

0 754

n sin

= 0 when n is infinite.

By putting

n

755

0 for n

in last.

n COS = 1 when n is infinite . 44)

PROOF. —Put ( 1 - sin²

)

, and expand the logarithm by ( 156) .

DE MOIVRE'S THEOREM . 756

(cos a +i sin a) cos ẞ + i sin ẞ) ... &c . = cos (a +B + y + ... ) + i sin ( a + B+ y + ...) ,

where i✔ -1. i= (cos

757

[ Proved by Induction. +i sin 0) " = cos no +i sin no.

PROOF.- By Induction, or by putting a, ß, &c. each = 0 in ( 756) .

Expansion of cos no, &c. , in powers sin 758

cos no

and cos 0.

cos " 0 - C (n , 2 ) cos " -20 sin² 0 + C (n , 4) cos" -40 sin¹0- & c.

759

sin non cos" -10 sin 6 - C (n , 3 ) cos" -30 sin³ 0 + & c .

PROOF. - Expand ( 757) by Bin. Th., and equate real and imaginary parts.

175

TRIGONOMETRICAL SERIES.

n tan 0 - C (n , 3) tan³0 + &c .

760

tan no = 1 - C ( n, 2) tan²0 + C ( n , 4) tan¹0 -&c .

In series (758, 759) , stop at, and exclude, all terms with indices greater than n. Note, n is here an integer. sum of the C (n , r) products of tan a, tan ß, tan y, Lets, &c. to n terms.

761

sin (a +B +y + &c .) = cosa cos B ... ($1−83 + $3− &c .) .

762

cos (a +B+ y + &c .) = cos a cosẞ ... ( 1 − s₂ + s₁− &c . ) .

PROOF. -By equating real and imaginary parts in (756) . 763

tan (a +B +y + &c .) = $1−83 + $5−87+ &c . 1-8₂+ 84-86 + &c.

Expansions of the sine and cosine in powers of the angle.

51

3!

PROOF.-Put

0 for n

eie

768

ei +e - i = 2 cos 0.

cos 0 +i sin 0.

itan 0 =

0+ · + . + &c. 2! 4!

in (757) and n = ∞ , employing ( 754) and ( 755) .

766

770

cos e = 1

- & c.

+

0—

sin 0

02

05

03 764

eio-e - io io e¹º +e - ¿Ð °

Expansion of cos"

e -io = cos 0 - i sin 0.

By ( 150)

eio - e - io = 2i sin 0.

1 +itan 0 e2i0

1 - i tan o

and sin"

in cosines or sines of

multiples of 0.

772

2n -1 cos" 0 = cos no + n cos (n − 2) 0 +C ' (n , 2) cos (n − 4) 0 + C ( n , 3 ) cos (n − 6) 0 + &c .

773

When n is even , 2" -¹ ( −1 )³" sin" 0 = cos n0 - n cos (n − 2) 0 + C (n , 2) cos (n − 4) 0 — C (n , 3) cos (n − 6)

774

+ &c. ,

And when n is odd, n -1 2″-¹ ( −1 ) = sin” 0 = sin n0 — n sin ( n − 2) 0 + C (n, 2) sin (n − 4) 0 — C (n , 3) sin (n − 6) 0 + &c .

176

PLANE TRIGONOMETRY.

Observe that in these series the coefficients are those of the Binomial Theorem, with this exception : If n be even, the last term must be divided by 2. The series are obtained by expanding ( e" + " )" by the Binomial Theorem, collecting the equidistant terms in pairs, and employing (768) and ( 769 ) .

Expansion of cos no and sin no in powers of sin 0.

775

When n is even, n² n² (n²-2²) sin' 0 cos n0 = 1 − 2+ sin² 0 + 4! n² (n² —2²) (n² —4²) sin® 0+ & c . 6!

776

When n is odd ,

(n² — 1 ) ( n² — 3²) - n² -1 sin' cos ne = cos @ { 1 – ~²71 sin0 + 2! 4!

(n² —1 ) ( n² — 3²) ( n² - 5°) sinº 0 + &c. } . 6!

777

When n is even, n² -22 sin non cos 0

sin 0 -

sin³ 0+ 3!

0— sin³ — ( n² — 2²) ( n² — 4³) ( n² — 6²) (n² — 2¹) ( n² —4²) sin ³0 sin²0 +&c .} . 5! 7! 778

When n is odd,

sin n✪ = n sin 0 — n (n² —1) sin³0 + n (n² - 1) (n² - 3²) sin' 3! 5! n (n² — 1 ) ( n² —3²) ( n² — 5²) sin' + &c. 7! PROOF. -By (758 ) , we may assume, when a is an even integer, 1+ 4, sin' 0+ A, sin * 0 +... + A2, sin² + .... cos no Put 0 +x for 0, and in cos no cos nx - sin no sin ne substitute for cos na and sinne their values in powers of na from (764 ) . Each term on the right is of the type A2, ( sin cos x + cos sin x) . Make similar substitutions for cos x and sine in powers of x. Collect the two coefficients of a³ in each term by the multinomial theorem ( 137 ) and equate them all to the coefficient of a on the left . In this equation write cos' for 1 - sin' everywhere, and then equate the coefficients of sin"" to obtain the relation between the successive quantities A , and A2 +2 for the series (775). To obtain the series ( 777) equate the coefficients of x instead of those of æ³ . When n is an odd integer, begin by assuming, by (759) , sin no

A, sin 0 + A , sin³ 0+ &c.

177

TRIGONOMETRICAL SERIES.

779 The expansions of cos no and sin no in powers of cos 0 are obtained by changing 0 into 1-0 in ( 775 ) to ( 778) .

780

Expansion of cos no in descending powers of cos 0.

2 cos no = (2 cos 0) ” — n (2 cos 0)~~² + n (n − 3) (2 cos 0) -4 2!

... + ( − 1) , n (n — r — 1 ) ( n — r — 2) ... (n − 2r + 1) (2 cos 0)n - 2r + r! up to the last positive power of 2 cos 0. PROOF.

By expanding each term of the identity log (1 - zz) + log ( 1-2 ) = log ( 1-2 ( x + 1

)) =)}

by (156) , equating coefficients of z", and substituting from ( 768 ) .

783

sin a +c sin (a +ẞ) +c² sin (a +2B) + &c . to n terms

sin a - c sin (a - B) —c" sin (a + nẞ) + c” + ¹ sin ( a + n − 18} 1-2c cos 8 +c² If c be < 1 and n infinite, this becomes = sin a -c sin (a - B) 1-2c cos B + c²

784

785

cos a +c cos (a +B) + c² cos (a +28) + &c . to n terms

= a similar result , changing sin into cos in the numerator . 786

Similarly when c is < 1 and n infinite.

787 Method of summation . — Substitute for the sines or cosines their exponential values ( 768) . Sum the two resulting geometrical series , and substitute the sines or cosines again for the exponential values by (766) .

C3 788

c sin (a + ß) ++ sin (a (a +28) + 28) + + infinity

= eccos sin (a +c sin ẞ) -sin a.

C3

c² 789

ccos (a +B) +

cos (a + 38) + & c . to

cos (a + 28) +

21 infinity

sin (a + 3ß) + &c . to

3!

= ec cosß cos (a + c sin ẞ) — cos a .

Obtained by the rule in ( 787) . 2 A

178

PLANE TRIGONOMETRY.

790 If, in the series ( 783 ) to ( 789 ) , B be changed into 3+ , the signs of the alternate terms will thereby be changed . Expansion of 0 in powers of tan✪ (Gregory's series) .

tan50

tan³0 0 = tan 0 -

791

5

3 The series converges if tan

- &c .

+ be not > 1 .

PROOF. -By expanding the logarithm of the value of e² in ( 771 ) by (158). Formulæ for the calculation of the value of π by Gregory's series.

792

+ tan 7 = tan -¹11 12+ 4

1 -1 -tan -1 11 = = 0 4 tan -111-08 239 3 -1 1

= 4 tan

794 PROOF.

tan 1 -te

[791

1 + tan -¹

70

99

By employing the formula for tan ( 4 ± B ) , ( 631 ) . To prove that

is incommensurable.

in terms of from ( 764) and ( 765 ) into 62 02 03 Ө ; or this result may a continued fraction, thus tan 0 = 1-3-- 5-7- & c.

795

Convert the value of tan

may be obtained by putting it for y in ( 294) , and by ( 770) . Hence - Ꮎ = 02 89 02 1 3.――― 5-7- & c. tan U π π² for 0, and assume that π, and therefore Put " is commensurable. Let 4 2 m mn mn π3 = m " m and being integers. Then we shall have 1 = 20 3n - 5n - 7n- &c. 4 The continued fraction is incommensurable, by ( 177) . But unity cannot be equal to an incommensurable quantity. Therefore is not commensurable. 796

If

tanan tan y, 1 n where m = 1+ " 797

If

n sin 2a + sin 3a + &c. 3 2 m3 ทา sin 4y a = y - m sin 2y + sin 6y + &c ., 2 3

sinn sin (x + a),

a = n sina +

PROOF - By subs'ituting the exp mential values of the sine or tangent ( 769) and (770) , and then eliminating x. n (a² + b²) Coefficient of a" in the expansion of ear cos ba = cos no, 798 n! where a = r cos 0 and b = r sin 0.

For proof, substitute for cos be from (768) ; expand by (150) ; put a = r cos 0 , b = r sin in the coefficient of a " , and employ (757) .

179

TRIGONOMETRICAL SERIES.

799

When e is < 1 ,

√1-- €² = 1 + 2b cos 0 + 252 cos 20 + 2b³ cos 30 + .... 1 - e cos 0

e

where b = 1+

1 - e²

26 and 2 cos 0 = x + 1, expand the fraction in two x 1+ び series of powers of a by the method of ( 257 ) , and substitute from ( 768 ).

For proof, put e =

800

sin a +sin (a +ß) + sin ( a + 2ẞ) + ... + sin { a + ( n − 1) ß }

sin ( a + " > ¹8) sinß sin B 2 801

cosa + cos (a +ß) + cos ( a +2ß) + ... + cos { a + ( n − 1)ß }

cos ( a +

ηβ 18) sin "B 2 sin B 2

802 If the terms in these series have the signs alternately, change ẞ into ẞ + in the results.

and

β PROOF.-Multiply the series by 2 sin 2 " and apply (669) and (666) . 2π 803

If ẞ =

in (800 ) and ( 801 ) , each series vanishes . n

804

Generally, If ß =

2π " and if ጎ

be an integer not a

multiple of n, the sum of the 7th powers of the sines or cosines in ( 800) or ( 801 ) is zero if r be odd ; and if r be even it is n = 2r C (r, 1) ; by (772) to ( 774) .

General Theorem . -Denoting the sum of the series 805 c + cx + cx² + ... +cx" by F (x) ; then ccosa + c₁ cos (a + b) + ... + c, cos (a + nß ) = } { e¹• F (e¹³) + € ¯¹ •F (e −¹³) } , and

806 csina + c, sin (a +ß ) + ... + c„ sin ( a + nß) = 1; { c¹ªF (e¹³) — e¯¹ªF(e¯iª) } . Proved by substituting for the sines and cosines their exponential values. (766), &c.

180

PLANE TRIGONOMETRY.

Expansion of the sine and cosine in factors.

807

x²” — 2x¹ y” cos no + y²n

-2.ry 2xy cos 0 +y' } { ' -2ry cos (0 + 2n ) + y = { x²2π to the angle successively . to n factors , adding n

..

10)) " . PROOF. By solving the quadraticon the left, weget x = y ( cos no + isinn0 Then values of x are found by ( 757) and ( 626 ) , and thence the factors. For the factors of æ” ± y ” see ( 480 ) .

808

π

-1 sin no = 2" -¹ sin & sin (

+27) ...

sin (

+

as far as n factors of sines . PROOF. - By putting x = y = 1 and 6 = 24 in the last . 809

If n be even ,

sin n

= 2″ -¹ sin & cos $ ( sin²

π

sin2 n

2π - sin2 sin' a² n $) (

) &c.

If n be odd , omit cos and make up n factors , reckoning two factors for each pair of terms in brackets . PROOF. From (808 ) , by collecting equidistant factors in pairs , and applying ( 659) . 810

811

PROOF. -Put 812

2 )... to n factors .

cos no = 2″ 2" -¹ sin ( ☀ +

+

) sin (

+

T for 4 in (808 ). 2n

Also , if n be odd , π ( sin² 2n

cos 2n - 1 cos cos no = 2"-¹

3πT sin² 4) ( sin² 2n sin³ $) …...

813 If n be even, omit cos p . PROOF. As in (809 ) . 814

-1 sin n = 2" -¹

sin n

3π

2π

π

w (n - 1 ) π ... sin

sin n

n

n

PROOF.-Divide (809) by sing , and make o vanish ; then apply (754) . 815

sin 0

0 {1 - (--) } { 1 - (/ -)*

20

20

T )33 -( 20

..... cost = { 1- ( 9) } { 1- ( ) } { 1- (9) } 0 in (809) and ( 812 ) ; divide by (814) and make n PROOF. Put & = n infinite. 816

181

ADDITIONAL FORMULE.

817

e -2 cos 0 +e¯ * 0 = 4 sin ' ( 1+

.... { 1+ (210 ) } { 1+ (110) } 2 0 for 0 in (807), 2 Proved by substituting a = 1 + 2 , y = 1 − 2 and n 2n 2n 0 making n infinite , and reducing one series of factors to 4 sin 2 by putting z= 0. De

Moivre's

}

Property of the

b

and

Circle. - Take P any point , POB = 0 any angle ,

E

B

a 2π

BOC

COD = & c . =

;

美

n

QP = x; 819

OB = r. cos no + p²n

x² -2x

= PB2 PC² PD2 ... to n factors . By (807) and ( 702 ) , since PB² = x² - 2xr cos 0 + r² , &c .

no

820

If x = r,

2p" sin

= PB.PC.PD ... &c. 2 2π

821

9

Cotes's properties .— If 0 = n

xn x" ~

"

PB.PC.PD ... & c.

n 822

+p

Pa . Pb . Pc ... & c.

ADDITIONAL FORMULE.

823

cot A +tan 4 = 2 cosec 24

824

cosec 24 + cot 24 = cot A.

826

cos A

sec A cosec A. sec A1 +tan A tan 음.

cos*.4 - sin 4

827 +4). tan A + sec A = tan (45° +

tan A +tan B = tan A tan B.

828 cot A +cot B 829

sec² A cosec² A

sec² 4+ cosec² 4.

182

PLANE TRIGONOMETRY.

830

If A + B + C = 2 , tan B tan C + tan C tan A + tan A tan B = 1 .

831

If A + B + C = π, cot B cot C + cot C cot A + cot A cot B = 1 .

832

π = sin-11313 + + sin -1. 44 2 5 5

π -1 tan -112 / + tan 1111= 11010· 3 4

In a right-angled triangle ABC, C being the right angle, cos 2B =

833

a² -b2 2 a²+ b² °

2ab

tan 2B = a² —b²°

834

R +r =

(a + b) .

tan } 4 = √ ( +b b). In any triangle,

a-b 835

(A - B) :=

sin

cos

C.

sin

C.

C

a+ b cos } ( A - B) =

с a²-b2

sin A - B

838

B

tan

B

=

sin A +B 837

tan 14 + tan

=

836

A - tan

(a² + b² + c ) = be cos A + ca cos B + ab cos C.

Area of triangle ABC = sin B sin C } a² ={ sin A

be sin A

sin A sin B = 1 (a²— b²) 1 sin ( A — B) `

2abc 839

=

cos

4 cos

B cos

C.

a+ b+ c 840

=

(a + b + c) ² tan

A tan

B tan

C.

With the notation of (709) , 841

r=

(a + b + c) tan

4 tan

Btan

C.

abc 842

A = √rrarore.

2Rr = a+ b+c'

843

a cos A +b cos B +c cos C = 4R sin A sin B sin C.

183

ADDITIONAL FORMULA.

844

R+r =

(a cot A + b cot B÷c cot C) = sum of per

pendiculars on the sides from centre of circumscribing circle . This may also be shown by applying Enc. VI . D. to the circle described on R as diameter and the quadrilateral so formed . 845

rarore

846

B cos

4 cos

C.

r = √ (rorc) + √ (re ¥a) + √ (rab) .

1

1

=

847

r 849

abc cos

rra

• +11/2+ 1/1/10 rc ra

tan 14 = √ rorc

If O be the centre of inscribed circle, 2bc

OA =

cos

4.

a+ b+ c 850

a (b cos C - c cos B) = b² - c².

851

b cos B +c cos C = c cos (B — C) .

852

a cos A +b cos B+c cos C = 2a sin B sin C.

853

cos A + cos B + cos C = 1+

2a sin B sin C

a+ b+c 854

If s =

(a + b + c) ,

1 - cos² a - cos2 b - cos² c +2 cos a cos b cos c = 4 sin s sin (s — a) sin ( s — b) sin (s — c) .

855

-1 + cos² a + cos2 b + cos² c + 2 cos a cos b cos c = 4 cos s cos (s — a) cos (s — b) cos (s — c) . a

856

4 co s 08

b COS

2

COS 2

2

= coss + cos (sa) + cos (s — b ) + cos (s — c) .

a 857

b sin

4 sin

2

с sin 2

2

―sin s + sin (s — a) + sin (s — b ) + sin (s — c) .

1 858

= 6 ( 1+ 6 1 ( 1 + 1/2 + 3

+ ... ) = 8 ( 1 + 1 32 + 3

PROOF.- Equate coefficients of in the expansion of (815) or of cos by ( 765 ) and ( 816 ) .

+... ).

sin 0 by (764) and 0

། 184

PLANE TRIGONOMETRY.

859

Examples ofthe Solutions of Triangles.

Ex. 1 : CASE II . ( 724) .— Two sides of a triangle b, c, being 900 and 700 feet, and the included angle 47° 25' , to find the remaining angles. b- c A B- C 1 = cot = cot 23° 42′ 30″ ; tan b +c 2 2

therefore therefore

log tan L tan

(B - C) = log log cot 4 -log 8 ;

(B- C) = L cot 23° 42' 30 " -3 log 2,

10 being added to each side of the equation. ..

..

L cot 23° 42′ 30 ″ = 10 · 3573942* •9030900 3 log 2 = L tan

(B- C)

and

(B - C) = 15° 53' 19:55 " (B + C ) = 66° 17′ 30″

94543042

B

82° 10' 49.55"

And, by subtraction , C = 50° 24' 10-45". Ex. 2 : CASE III. ( 732 ) .- Given the sides a, b, c = 7, 8, 9 respectively, to find the angles. 2 4.3 A (s — b ) ( s — c) = = tan 14= 12.5 10 • √ √ s (s - a) A L tan 4 = 10 + } ( log 2−1 ) = 9 ·650515 ; therefore 2 therefore 24° 5' 41'43".* A B is found in a similar manner, and C = 180° — A — B. Ex. 3. In a right-angled triangle, given the hypotenuse c = 6953 and a side b = 3, to find the remaining angles. 3 Here cos A = But, since A is nearly a right angle, it cannot be 6953 determined accurately from log cos A. Therefore take 3475 A cos A sin1/4 6953 2 = 2

A therefore

L sin

= 10 + 1 ( log 3475 — log 6953) = 9 ·8193913 ;

therefore

A = 44° 59′ 15·52",* 2

therefore

A = 89° 58′ 31 04" and B = 0 ° 1 ′ 28 · 96 ″ .

* See Chambers's Mathematical Tables for a concise explanation of the method of obtaining these figures .

SPHERICAL

TRIGONOMETRY .

INTRODUCTORY

THEOREMS .

870 Definitions.-Planes through the centre of a sphere intersect the surface in great circles ; other planes intersect Unless otherwise stated, all arcs are it in small circles . measured on great circles . The poles of a great circle are the extremities of the diameter perpendicular to its plane. The sides a, b, c of a spherical triangle are the arcs of great circles BC, CA, AB on a sphere of radius unity ; and the angles A, B, C are the angles between the tangents to the sides at the vertices , or the angles between the planes of the great circles. The centre of the sphere will be denoted by O. The polar triangle of a spherical triangle ABC has for its angular points A', B', C' , the poles of the sides BC, CA, AB of the primitive triangle in the directions of A, B, C respec tively (since each great circle has two poles ) . The sides of A'B'C' are denoted by a' , b' , c'. 871

A'

The sides and angles of the

polar triangle are respectively the supplements of the angles and sides of the primitive triangle ; that is ,

b

b a' + A = b' + B = c' + C = π,

0 = π. . a + A' = b + B' = c + C' =

G

Let BC produced cut the sides A'B', C'A' in G, H. B is the pole of A'C ' , therefore BH = π . Similarly CG = т " 2 2

B

H

B D

a

C'

therefore, by addition, a + GH= π and GHA', because A' is the pole of BC . The polar diagram of a spherical polygon is formed in the same way, and the same relations subsist between the sides and angles of the two figures. 2 B

186

SPHERICAL TRIGONOMETRY.

RULE .- Hence, any equation between the sides and angles of a spherical triangle produces a supplementary equation by changing a into π - A and A into T --a, &c . 872

The centre of the inscribed circle , radius r , is also the

centre of the circumscribed circle, radius R' , of the polar triangle , and + R' = ! π . PROOF. In the last figure, let O be the centre of the inscribed circle of ABC ; then OD, the perpendicular on BC, passes through A ' , the pole of BC. Also, OD = r; therefore OA' -r. Similarly OB' = OC' -r; there fore 0 is the centre of the circumscribed circle of A'B'C', and r + R' = ½ ″.

873 The sine of the arc joining a point on the circumference of a small circle with the pole of a parallel great circle , is equal to the ratio of the circumferences or corresponding arcs of the two circles . For it is equal to the radius of the small circle divided by the radius of the sphere ; that is, by the radius of the great circle.

874

Two sides of a triangle are greater than the third . [ By XI. 20.

875 The sides of a triangle are together less than the cir cumference of a great circle. [By XI. 21. 876 The angles of a triangle are together greater than two right angles . For π- A +

877

B +π - С is < 2m, by (875) and the polar triangle.

If two sides of a triangle are equal , the opposite angles

are equal.

[ By the geometrical proof in (894) .

878 If two angles of a triangle are equal, the opposite sides. are equal. [ By the polar triangle and ( 877) .

879 The greater angle of a triangle has the greater side opposite to it. PROOF. If B be > A, draw the arc BD meeting AC in D, and make ▲ ABD = A , therefore BD = AD ; but BD + DC > BC, therefore AC > BC.

880 The greater side of a triangle has the greater angle opposite to it. [ By the polar triangle and ( 879) .

187

OBLIQUE-ANGLED TRIANGLES.

RIGHT- ANGLED

TRIANGLES .

881 Napier's Rules . In the triangle ABC let C be a right angle, then a , ( π — B) , (¦π − c) , ( -A) , and b , are called the five circular parts. Napier's rules are

Taking any part for middle part,

I. sine ofmiddle part = product of tangents of adjacent parts. II. sine of middle part = product ofcosines of opposite parts. In applying the rules we can take A, B, c instead of their complements , and change sine into cos , or vice versa, for those parts at once.

Thus , taking b for the middle part , sin b tan a cot A = sin B sin c.

Ten equations in all are given by the rules .

P PROOF. From any point P in OA, draw PR perpendicular to OC, and RQ to OB; therefore PRQ is a right angle ; therefore OB is perpendicular to PR and QR, and therefore to FQ. Then prove any formula by proportion from the triangles of the tetrahedron OPQR, which are all right-angled . Otherwise, prove bythe formule for oblique-angled triangles.

OBLIQUE - ANGLED

882

cosa

b

0

R a B

TRIANGLES .

cos b cos c +sin b sin c cos A.

p

D

R

B

9

PROOF. Draw tangents at A to the sides c, b to meet OB, OC in D and E. Express DE by (702) applied to each of tho triangles DAE and DOE, and subtract. π If AB and AC are both > 2 produce them to meet in A', the pole of A, and employ the tri angle A'BC. π pro If AB alone be >

N S

duce BA to meet BC.

a

F

The supplementary formula, by (871 ) , is

883

-cos B cos C +sin B sin C cosa. cos A = -

SPHERICAL TRIGONOMETRY.

188

A =

884

sin

885

A COS 084 = 2

sin (s - b) sin (s sin b sin c

c)

sins sin (s -- a) sin b sin c

15 sin (s -b) sin (s - c)

886

where 8 =

tan 1/4/1 =

(a + b +c).

sins sin (s - a) A = (1 - cos A) . Substitute for cos A from ( 872) , and 2 throw the numerator of the whole expression into factors by (673). Similarly A for cos 2 PROOF.-Sin²

The supplementary formulæ are obtained in a similar way, They are

or by the rule in (871) .

887

a = COS s” .

888

= sin V te

889

= tan 2 1/

890

Let =

cos (S— B) cos ( S − C) ¸ sin B sin C

cos S cos ( S - A sin B sin C --cos S cos ( S — A) cos (S - B) cos ( S - C) where S =

(A + B + C) .

σ = √ sin s sin ( s — 1! ) sin ( s — b ) sin ( s — c)

✓1 + 2 cos a cos b cos c - cos² a -cos² b - cos² c.

Then the supplementary form , by ( 871 ) , is

891

Σ == = √ —cos S cos ( S — A ) cos ( S — B) cos ( S − C) = ½ √ 1 −2 cos A cos B cos C — cos² A - cos² B- cos˚C. 20

892

sin A =

ΩΣ

sin a = sin b sin c

sin B sin C

A

A [ By sin A

893 The following rules will (884 to 892)

2 sin

COS

and (884, 885) , &c.

produce the ten formulæ

A I. Write sin before each factor in the s values of sin

OBLIQUE- ANGLED TRIANGLES.

189

A

COS

tan 4 , sin A, and ▲, A,

in Plane Trigonometry ( 704–

2 707), to obtain the corresponding formula in Spherical Trigo nometry . II. To obtain the supplementary forms of the five results , transpose large and small letters everywhere, and transpose sin and cos everywhere but in the denominators, and write minus before cos S.

sin A

sin B

sin a

sin b

sin C =

894

sin c

PROOF.-By (882 ) . Otherwise , in the figure of 882 , draw PN perpendi cular to BOC, and NR, NS to OB, OC. Prove PRO and PSO right angles by I. 47, and therefore PN = OP sin c sin B = OP sin b sin C.

895

cos b cos C

cot a sin b - cot A sin C.

To remember this formula, take any four consecutive angles and sides (as a , C, b, A) , and , calling the first and fourth the extremes, and the second and third the middle parts, employ the following rule : cot extreme RULE. -Product of cosines of middle parts side X sin middle side - cot extreme angle X sin middle angle. In the formula for cos a (882) substitute a similar value for sin a cos c, and for sin c put sin C sin A PROOF.

896

NAPIER'S FORMULE.

(1 )

tan

sin ( A - B) = sin

(2)

tan

( A + B) =

cos cos (3)

tan

sin (a - b) := sin

(4)

tan

= cos (a + b) : cos

RULE. -

(a - b) cot C b . (a + b) (a —b) cot C 2 (a + b)

( A - B) tan (A + B) (A — B) tan 2. (A + B)

In the value of tan

obtain tan 12/24 (A + B ) .

(A - B ) change sin to cos to To obtain (3 ) and (4) from ( 1 ) and ( 2) ,

transpose sides and angles, and change cot to tan. PROOF. -- In the values of cos A and cos B, by ( 883 ) , put m sin a and Then put m sin b for sin A and sin B, and add the two equations. m = sin A + sin B and transform by (670-672) . sin a sin b '

190

SPHERICAL TRIGONOMETRY.

897

GAUSS'S FORMULE.

sin

( A + B)

cos

(a - b)

(1 )

cos

C

cos

c

sin } ( A − B) = sin (a - b) sinc cos C

(2)

cos

(3)

cos (4)

( A + B) sin C

cos 14 (a + b)

(A - B) sin C

sin

cos

From any of these formulæ the -by the following rule :

c

(a +b) sinc

others

may be

obtained

RULE - Change the sign of the letter B (large or small) on one side of the equation , and write sin for cos and cos for sin on the other side. PROOF.

Take

(A + B) = sin A cos

sin

B + cos

A sin B,

substitute the s values by (884, 885 ) , and reduce.

SPHERICAL TRIANGLE AND CIRCLE .

898

Let r be the radius of the in

scribed circle of ABC ; r, the radius of the escribed circle touching the side a, and R, R.a the radii of the circumscribed

E circles ; then σ (1 )

tan rtan

4 sin (s - a) = sin s

2 sin a (3)

=

sin

sin

B sin C

sin A Σ

= (4)

D 2 cos

A cos

B cos

C

ΩΣ

= cos S + cos ( S - A ) + cos ( S − B) + &c. PROOF. -The first value is found from the right-angled triangle OAF, in which AFs - a . The other values by (884-892).

E

191

SPHERICAL TRIANGLE AND CIRCLE.

899

( 3)

σ tan r₁a = tan 14 sin s =• sin (s - a) 2 sin a = sin 4 cos B cos C sin A (1 )

Σ

=

(4)

2 cos

A sin

B sin ≥C ΩΣ

= —cos S ― cos ( S — A ) + cos ( S − B) + cos ( S − C ') ' PROOF . From the right-angled triangle O'AF ' , in which AF's. NOTE. The first two values of tan ra may be obtained from those of tan r by interchanging s and s -a.

tan 900

cos S

a

( 1 ) tan R =

Σ

cos (S - A ) sin

a

=

(3)

sin A cos ab cos

c

R 2 sina sin b sinc (4)

B

D

Ra -sins +sin (sa) + sin (s - b) + &c. 20

PROOF . The first value from the right-angled triangle OBD, in which ▲ OBD = S - A. The other values by the formulæ ( 887-892).

tana

901

tan Ra =

(1)

-- cos S

sin (3)

= cos (S - A ) Σ

a

=

sin A sin

b sinc

2 sin la cosb cos

c

(4) σ

(5)

PROOF.

sin s — sin (s — a ) + sin (s — b ) + sin (s — c) ¸ 26 From the right-angled triangle O'BD, in which 4 O'BD =

-S.

192

SPHERICAL TRIGONOMETRY.

SPHERICAL

902

AREAS .

area of ABC = ( A + B + C −π) r² = Er²,

where

E = A + B + C -T, the spherical excess.

PROOF. - By adding the three lunes ABDC,

BCEA,

and observing that ABF

we get

A

CAFB,

CDE,

B (4 + π + c π ) 2wr² = 2«r² + 2ABC.

903

E

C

AREA OF SPHERICAL POLYGON.

n being the number of sides , Area = { Interior Angles- ( n - 2) π } r

= { 27 -Exterior Angles } r = { 27 - sides of Polar Diagram}

².

The last value holds for a curvilinear area in the limit. PROOF. By joining the vertices with an interior point, and adding the areas of the spherical triangles so formed .

Cagnoli's Theorem .

904

sin

E = ✓ { sins sin (s — a ) sin (s — b) sin (s — c) } 2 cosa cosb cos c

PROOF . - Expand sin [ ½ ( A + B ) − 12 ( π - C) ] by (628) , and transform by Gauss's equations (897 i ., iii . ) and ( 669, 890) .

Llhuillier's Theorem.

905 tan

E =

{ tans tan

(s — a) tan

( s — b) tan

(s — c) } .

PROOF. -Multiply numerator and denominator of the left side by 2 cos (A + B - C + ) and reduce by ( 667, 668) , then eliminate (A + B) by Gauss's formulæ (897 i., iii. ) Transform by (672, 673) , and substitute from ( 886).

193

POLYHEDRONS.

POLYHEDRONS .

Let the number of faces , solid angles , and edges , of any polyhedron be F, S, and E; then

906

F+ SE + 2 .

PROOF. Project the polyhedron upon an internal sphere. Let m = number of sides, and s = sum of angles of one of the spherical polygons so formed . Then its area = {s— (m − 2) π } r², by (903 ) . Sum this for all the polygons, and equate to 4m².

THE FIVE REGULAR SOLIDS. Let m be the number of sides in each face , n the number of plane angles in each solid angle ; therefore mFnS = 2E.

907

From these equations and (906 ) , find F, S, and n, thus, 1 1 1 1 1 n 1 m 1 1 F = 2 (m + n - ) } S = 2 (m + n - 2 )

in terms of m and

1 1 1 -- 1 = + E m n 2 1 1 1 > + In order that F, S, and E may be positive, we must have 2" n m a relation which admits of five solutions in whole numbers , corresponding to the five regular solids . The values of m, n , F, S, and E for the five regular solids are exhibited in the following table :

20

...

Hexahedron

...

Octahedron

...

...

F

S

E

3

3

4

4

6

4

3

6

8

12

3

4

86

12

:

Tetrahedron

20

m

Icosahedron

908

Ст

:

Dodecahedron ...

5

3

12

20

30

3

5

20

12

30

The sum of all the plane angles of any polyhedron = 2π (S - 2) ;

Or,

Four right angles for every vertex less eight right angles, 20

194

SPHERICAL TRIGONOMETRY.

909 If I be the angle between two adjacent faces of a regular polyhedron , π sin I = cos ÷ sin . n m

PROOF. -Let PQ == a be the edge, and S the centre of a face, T the middle point of PQ, O the centre of the inscribed and circum scribed spheres, ABC the projection of PST upon a concentric sphere. In this spherical triangle, π π π A = 9 and B = C= = PST. 2 n m Also STO = I.

B

A 1 T

m

23

Q. e. d .

P

If r, R be th ra dii of the inscri e

bed

spher

S

HE

Now, by (881 , ii. ), cos Asin B cos BC; π π sin I. COS == sin that is, m n

and circu m

scrib

of a regul p , ar olyhedro n π π tan I cot r = " R = 2 tan } I tan m n

ed

es

910

PROOF . — In the above figure, OS

PT cot

π tan I. m

sin AC

OS = r,

OP = R,

a PT = 2 ;

Also OP = PT cosec AC, and by (881 , i . ) ,

tan BC cot A

cot I cot

π ; therefore , &c. n

and

ELEMENTARY

MISCELLANEOUS

GEOMETRY .

PROPOSITIONS .

920 To find the point in a given line QY, the sum of whose distances from two fixed points S, S' is a minimum .

R Draw SYR at right angles to QY, making YR = YS. Join RS', cutting QY in P. Then P will be the required point. Y PROOF.- For, if D be any other point on the line, SD = DR and SP = PR. But RD + DS' is >RS'; therefore , &c. R is called the reflection of the point S, and SPS' is the path of a ray of light S reflected at the line QY. If S, S' and QY are not in the same plane, make SY, YR equal perpen diculars as before, but the last in the plane of S' and QY. Similarly, the point Q in the given line, the difference of whose distances from the fixed points S and R' is a maximum, is found by a like construction . The minimum sum of distances from S, S' is given by (SP + S'P) = SS2 +4SY. S'Y'. And the maximum difference from S and R' is given by (SQ - R'Q)² = ( SR ) -4SY. R'Y'. Proved by VI . D. , since SRR'S' can be inscribed in a circle.

Hence , to find the 921 shortest distance from P to Q en route of the lines AB, BC,

CD ;

in

P

other

B

words , the path of the ray reflected at the successive surfaces AB, BC, CD. Find P₁ , the reflection of Pat the first surface ; then P , the reflection of P, at the second sur face ; next P,, the reflection of P, at the third surface ; and so on if

P

b

.

P A

a

D

196

ELEMENTARY GEOMETRY.

there be more surfaces . Lastly, join Q with P,, the last reflection, cutting CD in a. Join aP,, cutting BCin b. Join bP , cutting AB in c. Join cP. PcbaQ is the path required. The same construction will give the path when the surfaces are not, as in the case considered , all perpendicular to the same plane.

922 If the straight line d from the vertex of a triangle divide the base into segments p, q, and if h be the distance from the point of section to the foot of the perpendicular from the vertex on the base , then

[II . 12, 13.

b² +c² = p² + q² +2d² + 2h (p − q) . The following cases are important : (i.) When pq,

b² +c² = 2q² +2d² ;

i.e., the sum of the squares of two sides of a triangle is equal to twice the square of half the base, together with twice the square of the bisecting line drawn from the vertex. (ii.) When p2q,

q

P (II. 12 or 13)

2b² +c² = 6q² +3d².

(iii. ) When the triangle is isosceles ,

b² = c² = pq + ď².

923 If O be the centre of an equilateral triangle ABC and P any point in space . Then PA²+PB²+ PC² = 3 (PO² + 0A²) .

A PROOF. - PB + PC = 2PD² + 2BD".

Also

PA' +2PD260D² +3P0³,

and

BO = 20D ;

therefore, &c.

(922, i.) (922, ii.)

B

COR.-Hence , if P be any point on the surface of a sphere, centre O, the sum ofthe squares of And if r , the radius its distances from A, B, C is constant . the same squares is of sum the OA, to of the sphere, be equal equal to 672.

197

MISCELLANEOUS PROPOSITIONS.

D

The sum of the squares of the sides of a quadrilateral is equal to the sum of the squares of the 924

diagonals plus four times the square of the line joining the middle points of the diagonals . (922, i .)

F

925

COR. -The sum of the squares of the sides of a parallelogram is

A

equal to the sum of the squares of the diagonals. 926 In a given line AC, to find a point X whose distance from a point P shall have a given ratio to its distance in a given direction from a line AB.

E Through P draw BPC parallel to the given direction. Produce AP, and make CE in the given ratio to CB. Draw PX parallel to EC, and XY to CB. There are two solutions when CE cuts AP in two points. [ PROOF.- By (VI . 2) .

P

B

F

A 927 To find a point X in AC, whose distance XY from AB parallel to BC shall have a given ratio to its distance XZ from BC parallel to AD.

Y

Draw AE parallel to BC, and having to AD the given ratio. Join BE cutting AO in X, the point required. [ Proved by (VI. 2 ) .

928

B

To find a point X on any

or curved , whose distances XY, XZ, in given direc tions from two given lines AP, AB,

line,

Z

D

straight

Y

shall be in a given ratio. Take P any point in the first line. Draw PB parallel to the direction of XY, and BC parallel to that of XZ, making PB have to BC the given ratio . Join PC, cutting AB in D. Draw DE parallel to CB. Then AE produced cuts the line in X, the point required, and is the locus of such points. [PROOF.-BY (VI. 2) .

P

B

Ꮓ A

ELEMENTARY GEOMETRY.

198

929 To draw a line XY through a given point P so that the segments XP, PY, intercepted by a given circle, shall be in a given ratio. Divide the radius of the circle in that ratio, and, with the parts for sides, construct a triangle PDC upon PC as base. Produce CD to cut the circle in X. Draw XPY and CY.

P X

Then therefore But

PD+DC radius ; PD = DX ; CY = CX ;

therefore PD is parallel to CY (I.5 , 28) ; therefore &c., by (VI. 2).

A

930 From a given point P in the side of a triangle, to draw a line PX which shall divide the area of the tri

P

angle in a given ratio . Divide BC in D in the given ratio, and draw AX parallel to PD. PX will be the line required. ABD : ADC = the given ratio (VI . 1 ) , and APD = XPD (I. 37) ; therefore, &c.

931

B

D

X

To divide the triangle ABC in a given ratio by a line

XY drawn parallel to any given line AE. Make BD to DC in the given ratio. Then make BY a mean proportional to BE and BD, and draw YX parallel to EA.

X

PROOF. -AD divides ABC in the given ratio (VI . 1) . Now

ABE

XBY :: BE : BD,

or

:: ABE : ABD ; XBY

ABD.

B

E Y

P

therefore

(VI. 19 )

932 If the interior and exterior vertical angles at P of the triangle APB be bisected by straight lines which cut the base in C and D, then the circle circumscribing CPD gives the locus of the vertices of all triangles on the base AB whose sides AP, BP are in a constant ratio .

199

MISCELLANEOUS PROPOSITIONS.

PROOF. The 4CPD

E

(APB+ BPE) = a right angle ; therefore P lies on the circumference of the circle, diameter CD (III . 31 ) . Also AP : PB :: AC : CB :: AD : DB

B (VI. 3, and A. ) , a fixed ratio. AD is divided harmonically in B and C ; i.e. , AD : DB :: AO : CB ; 933 or, the whole line is to one extreme part as the other extreme part is to the middle part. If we put a, b, c for the lengths AD, BD, CD, the proportion is expressed algebraically by a : b :: a - c : c - b, which is equivalent to 1 1 = 2 + a b

934 and

AP : BP OA : 00 = 0C : OB AP : BP² = OA : OB, ᎪᏢ - ᎪᏅ : CP : BP- Ᏼ .

Also

(VI. 19) (VI. 3, & B.)

935 If Q be the centre of the inscribed circle of the triangle ABC, and if AQ produced meet the circumscribed circle , radius R, in F ; and if FOG be a diameter, and AD perpendi cular to BC ; then A G (i.) FC=FQ =FB = 2R sin 4. (ii.) ▲ FAD = FAO = } (B — C), and

CAG =

(B + C ) .

PROOF OF ( i .)— ▲ FQC = QOA + QAC. QAB = BCF ; (III. 21) But QAC .. FCFQ. .. FQC = FOQ ;

H

Similarly FB = FQ. Also GCF is a right angle, and FGC = FAC = } A ; (III. 21 )

..

936

FO = 2R sin 4.

If R, r be the radii of the circumscribed and inscribed

circles of the triangle ABC (see last figure) , and O, Q the centres ; then OQ' = R²- 2Rr. PROOF. - Draw QH perpendicular to AC ; then QHr. By the isosceles triangle AOF, OQ³ = R² — AQ.QF (922, iii. ) , and QF = FC ( 935, i .) , and by similar triangles GFC, AQH, AQ : QH :: GF : FC ; therefore AQ.FC = GF. QH 2Rr.

ELEMENTARY GEOMETRY.

200

The problems known as the Tangencies.

937 Given in position any three of the following nine data viz., three points , three straight lines, and three circles , -it is required to describe a circle passing through the given points The following five and touching the given lines or circles. •

principal cases occur. 938

I. Given two points A, B, and the straight line CD.

ANALYSIS.-Let ABX be the required circle, touching CD in X. 4 Therefore

E CX

CA.CB.

(III. 36)

Hence the point X can be found, and the centre of the circle defined by the inter section ofthe perpendicular to CD through X and the perpendicular bisector of AB. There are two solutions.

B Otherwise, by (926 ) , making the ratio one of equality, and DO the given line.

X C

COR.-The point X thus determined is the point in CD at which the distance AB subtends the greatest angle. In the solution of (941 ) Q is a similar point in the circumference CD. (III. 21, & I. 16) 939

II. Given one point A and two straight lines DC, DE.

In the last figure draw AOC perpendicular to DO, the bisector of the angle D, and make OB OA, and this case is solved by Case I. 940 III. Given the point P, the straight line DE, and the circle ACF. ANALYSIS.-Let PEF be the required circle touching the given line in E and the circle in F. Through H, the centre of the given circle, draw AHCD perpendicular to DE. Let K be the centre of the other circle. Join HK, passing through F, the point of contact. Join AF, EF, and AP, cutting the required circle in X. Then LKF ; 4 DHF (I. 27)

2012. I H

CE

P therefore HFA = KFE (the halves of equal D E angles) ; therefore AF, FE are in the same straight line. Then, because AX.AP (III. 36) AF.AE, and AF.AE AC . AD by similar triangles, therefore AX can be found . A circle must then be described through P and X to touch the given line,

201

THE PROBLEMS OF THE TANGENCIES.

by Case I. There are two solutions with exterior contact, as appears from Case I. These are indicated in the diagram . There are two more in which the circle AC lies within the described circle. The construction is quite analogous, C taking the place of A.

941 IV. Given two points A, B and the circle CD. Draw any circle through A, B, cutting the required circle in C, D. Draw AB and DC, and let them meet in P. Draw PQ to touch the given circle. Then, because PC.PDPA . PB = PQ³, (III. 36)

P

Q

and the required circle is to pass through A, B; therefore a circle drawn through A, B, Q must touch PQ, and therefore the circle CD, in Q (III. 37) , and it can be described by Case I. There are two solutions corresponding to the two tangents from P to the circle CD.

942

D

V. Given one point P, and two circles , centres A and B.

E I

H M

o

K

ANALYSIS.-Let PFG be the required circle touching the given ones in F and G. Join the centres QA, QB. Join FG, and produce it to cut the circles in E and H, and the line of centres in O. Then, by the isosceles triangles, the four angles at E, F, G, H are all equal ; therefore AE, BG are parallel, and so are AF, BH ; therefore AO : BO :: AF : BH, and O is a centre of similitude for the two circles. Again, HBK := 2HLK, and FAM = 2FNM ( III. 20) ; therefore FNM = HLK = HGK ( III. 21 ) ; there fore the triangles OFN, OKG are similar ; therefore OF . OG = OK . ON ; therefore, if OP cut the required circle in X, OX.OP = OK . ON. Thus the point X can be found, and the problem is reduced to Case IV. Two circles can be drawn through P and X to touch the given circles. One is the circle PFX. The centre of the other is at the point where EA and HB meet if produced, and this circle touches the given ones in E and H. 943 An analogous construction, employing the internal centre of simili tude O ', determines the circle which passes through P, and touches one given circle externally and the other internally. See (1047-9) . The centres of similitude are the two points which divide the distance between the centres in the ratio of the radii. See ( 1037). 2 D

202

944

ELEMENTARY GEOMETRY.

COR. - The tangents from 0 to all circles which touch

the given circles , either both externally or both internally, are equal. For the square of the tangent is always equal to OK. ON or OL . OM. 945

The solutions for the cases of three given straight lines or three given points are to be found in Euc. IV. , Props . 4 , 5 .

946 In the remaining cases of the tangencies , straight lines and circles alone are given. By drawing a circle concentric with the required one through the centre of the least given circle , the problem can always be made to depend upon one of the preceding cases ; the centre of the least circle becoming one of the given points .

DEFINITION . -A centre of similarity of two plane curves 947 is a point such that, any straight line being drawn through it to cut the curves, the segments of the line intercepted between the point and the curves are in a constant ratio. 948

If AB, AC touch a circle at B and

C, then any straight line AEDF, cutting the circle , is divided harmonically by the circumference and the chord of con tact BC. Proof from

AE. AF AB

and

AB².

BD . DC + AD²,

BD . DC

ED.DF.

α

(III. 36) B (923) (III. 35)

949 If a , ß, y, in the same figure , be the perpendiculars to the sides of ABC from any point E on the circumference of the circle , then

By = a³.

ßd, because BEH is a PROOF. Draw the diameter BHd ; then EB² right angle. Similarly EC = yd. But EB . EC = ad (VI. D.) , therefore &c . 950 If FE be drawn parallel to the base BC of a triangle, and if EB, FC intersect in O, 0, then

A

F

AE : AC : EO : OB :: FO : OC. By VI. 2. Since each ratio = FE : BC. COR.-If AC = n . AE, then BE

(n + 1) OE.

B

E

203

MISCELLANEOUS PROPOSITIONS.

951

The three lines drawn from the angles of a triangle to

the middle points of the opposite sides , intersect in the same point , and divide each other in the ratio of two to one. For, by the last theorem, any one of these lines is divided by each of the others in the ratio of two to one, measuring from the same extremity, and must therefore be intersected by them in the same point. This point will be referred to as the centroid of the triangle.

952 The perpendiculars from the angles upon the opposite sides of a triangle intersect in the same point . Draw BE, CF perpendicular to the sides, and let them intersect in O. Let AO meet BC in D. Circles will circumscribe AEOF and BFEC, by (III . 31 ) ; therefore LFAO = FEO = FCB ; (III. 21)

F therefore

▲ BDA = BFC = a right angle ;

i.e., AO is perpendicular to BC, and therefore the perpendicular from A on BC passes through 0. O is called the orthocentre of the triangle ABC.

B

D

COR. -The perpendiculars on the sides bisect the angles of the triangle DEF, and the point O is therefore the centre of the inscribed circle of that triangle. PROOF .

From ( III. 21) , and the circles circumscribing OEAF and OECD.

953 If the inscribed circle of a triangle ABC touches the sides a, b, c in the points D, E, F; and if the escribed circle to the side a touches a and b, c produced in D' , E ', F " ; and if

s =

(a + b + c) ;

then

BF' = BD ' = CD = s — c, and

AE'

AF' = s ; C

and similarly with respect to the other segments . B PROOF. -The two tangents from anyvertex to either circle being equal, it follows that CD+c = half the perimeter of ABC, which is made up of three pairs of equal segments ; therefore CD = s− t.

D' F

E' Ta

Also AE +AF

AC + CD + AB + BD = 28 ; therefore AE ' = AF' = s.

O'

ELEMENTARY GEOMETRY.

204

The Nine-Point Circle. 954

The Nine-point circle is the circle described through

D, E, F, the feet of the perpendiculars on the sides of the triangle ABC . It also passes through the middle points of the sides of ABC and the middle points of OA , OB, OC ; in all, through nine points . PROOF.-Let the circle cut the sides of ABC again in G, H, K ; and OA, OB, OC in L, M, N. ▲ EMF = EDF (III. 21 ) = 20DF (952, Cor.) ; therefore, since OB is the diameter of the circle cir cumscribing OFBD (III. 31 ) , M is the centre of that circle (III. 20) , and therefore bisects OB. Similarly OC and OA are bisected at N and L. B Again, MGB = MED (III. 22) = OCD, (III. 21) , by the circle circum scribing OECD. Therefore MG is parallel to OC, and therefore bisects BC. Similarly H and K bisect CA and AB.

955

The centre of the

nine-point circle is the middle point of OQ, the line joining the ortho centre and the centre of the circumscribing circle of

F

the triangle

ABC. For the centre of the N. P. circle is the intersection of the perpendicular bisectors of the chords DG, EH, FK, and these perpendiculars bisect OQ in the same point N, by (VI. 2).

956

R N

The centroid of

the triangle ABC also lies on the line OQ and divides it in R so that OR = 2RQ. PROOF. The triangles QHG, OAB are similar, and AB = 2HG ; there fore AO = 2GQ ; therefore OR = 2RQ ; and AR = 2RG ; therefore R is the centroid, and it divides OQ as stated ( 951 ) .

4

205

CONSTRUCTION OF TRIANGLES.

957 Hence the line joining the centres of the circumscribed and nine-point circles is divided harmonically in the ratio of 2 : 1 by the centroid and the orthocentre of the triangle. These two points are therefore centres of similitude of the circumscribed and nine-point circles ; and any line drawn through either of the points is divided by the circumferences in the ratio of 2 : 1. See (1037. ) 958 The lines DE, EF, FD intersect the sides of ABC in the radical axis of the two circles . For, if EF meets BC in P, then by the circle circumscribing BCEF, PE.PF = PC . PB ; therefore ( III . 36) the tangents from P to the circles are equal (985) . 959 The nine-point circle touches the inscribed and escribed circles of the triangle . PROOF.- Let O be the orthocentre, and I, Q the centres of the inscribed and circumscribed circles. Produce AI to bisect the arc BC in T. Bisect AO in L, and join GL, cutting AT in S. The N. P. circle passes through G, D, and L (954) , and D is a right angle. Therefore GL is a diameter, and is therefore R QA (957) . Therefore GL and QA are parallel. But QA = QT, therefore A A GS = GT = CT sin 4 = 2R sin². (935, i.) 2

A

IN

ST2GS cos 0 (0 being the angle GST = GTS) . N being the centre of the N. P. circle, its B radius = NG = 4R ; and r being the radius of the inscribed circle, it is required to shew that NING- r. Now NISN' + SIª —2SN . SI cos 0. Substitute SNR- GS; A SI = TI— ST = 2R sin4-2GS cos 0 ; Also

and GS

S 0 G

D

C

T (702)

2R sin' A, to prove the proposition.

If J be the centre of the escribed circle touching BC, and r, its radius, it is shewn in a similar way that NJ = NG +r。. To construct a triangle from certain data. 960 When amongst the data we have the sum or difference of the two sides AB, AC ; or the sum of the segments of the base made by AG, the bisector of the exterior vertical angle ; or the difference of the segments made by AF, the bisector of

ELEMENTARY GEOMETRY.

206

the interior vertical angle ; the following construction will lead to the solution . Make AE AD AC. Draw DH parallel to AF, and suppose EK drawn parallel to AG to meet the base produced in K ; and complete the figure. Then BE is the sum, and BD is the differ ence of the sides. BK is the sum of the exterior segments of the base, and BH is the difference of the interior seg ments. ▲ BDH = BEC = {A, LADO EAG = (B + C) , B H 4 DCBDFB = (CB).

F

961 When the base and the vertical angle are given ; the locus of the vertex is the circle ABC in figure (935) ; and the locus of the centre of the inscribed circle is the circle , centre F and radius FB. When the ratio of the sides is given, see (932).

To construct a triangle when its form and the distances of its vertices from a point A' are given .

962

ANALYSIS.- Let ABC be the required triangle. On A'B make the triangle ABC ' similar to ABC, so that AB : A'B :: CB CB. The angles ABA' , CBC will also be equal ; therefore AB : BỜ :: AA′ : CƠ , which gives CC , since the ratio AB : BC is known. Hence the point C is found by constructing the triangle A'CC ' . Thus BC is determined, and thence the tri angle ABC from the known angles.

A

B

K 963

P

To find the locus

of a point P, the tan gent from which to a F

given circle, centre A, has a constant ratio to its distance from a given point B. Let AK be the radius of the circle, and pq the On AB take given ratio. AC, a third proportional to AB and AK, and make AD : DB = p² : q². With centre D, and a radius

G

TANGENTS TO TWO CIRCLES.

207

equal to a mean proportional between DB and DC, describe a circle. be the required locus. PROOF. Suppose P to be a point on the required locus. A, B, C, and D.

It will

Join P with

Describe a circle about PBC cutting AP in F, and another about ABF cutting PB in G, and join AG and BF. Then PK² = AP² —AK² = APª —BA . AC (by constr. ) = AP² -PA.AF (III. 36) = AP.PF (II. 2) = GP.PB (III . 36) .

Therefore, by hypothesis , p² : q' = GP.PB : PB² = GP : PB = AD : DB (by constr. ) ; therefore

▲ DPG = PGA (VI. 2) = PFB (III. 22 ) = PCB (III. 21 ) .

Therefore the triangles DPB, DCP are similar ; therefore DP is a mean pro portional to DB and DC. Hence the construction . 964 COR. - If pq the locus becomes the perpendicular bisector of BC, as is otherwise shown in ( 1003) .

965

To find the locus of a point P, the tangents from which

to two given circles shall have a given ratio . Let A, B be the centres, a, b the radii (a>b), and p q the given ratio. Take c, so that cbpq, and describe a circle with centre A and radius AN = √a² -c² . Find the locus of P by the last proposition, so that the tangent from P to this circle may have the given ratio to PB. It will be the re quired locus.

(See also 1036.)

K

P T

a N

A

B

PROOF.-By hypothesis and construction, 22 PK +c² PK2 c² AP²-AN' AP² - a² + c² = = = = = BP BP2 PT + q³ PT v2 COR. -Hence the point can be found on any curve from which the tangents to two circles shall have a given ratio .

966 To find the locus of the point from which the tangents to two given circles are equal.

Since, in ( 965 ) , we have now p = q, and therefore c = b, the construction simplifies to the following : Take AN = √ (a² — b²) , and in AB take AB : AN : AC. The perpen dicular bisector of BC is the required locus. But, if the circles intersect, then their common chord is at once the line required . See Radical Axis (985).

208

ELEMENTARY GEOMETRY.

Collinear and Concurrent systems ofpoints and lines. 967 DEFINITIONS .-Points lying in the same straight line are collinear. Straight lines passing through the same point are concurrent, and the point is called the focus of the pencil of lines . Theorem.-If the sides of the triangle ABC, or the sides produced , be cut by any straight line in the points a , b, c respectively, the line is called a transversal, and the segments of the sides are connected by the equation 968

(Ab

bC) (Ca : aB) (Bc : cA ) = 1 .

Conversely, if this relation holds , the points a , b, c will be collinear. PROOF. Through any vertex A draw AD parallel to the opposite side BC, to meet the transversal in D, then Ab : bC AD : Ca and Bc : cAaB : AD

A

(VI. 4), which proves the theorem. NOTE . In the formula the segments of the sides are estimated positive, independently of d C B direction, the sequence of the letters being pre served the better to assist the memory. A point may be supposed to travel from A over the segments Ab, bC, &c. continuously, until it reaches A again.

969 By the aid of (701 ) the above relation may be put in the form (sin ABb : sin bBC) ( sin CAa : sin a AB) ( sin BCc : sin cCA) = 1

970 If O be any focus in the plane of the triangle ABC, and if AO, BO , CO meet the sides in a, b , c ; then, as before , (Ab : bC) (Ca : aB) ( Bc : cA ) = 1. Conversely, if this relation holds , the lines Aa , Bb, Cc will be concurrent. PROOF.-Bythe trans versal Bb to the triangle A AaC, we have (968)

(Ab : bC) (CB : Ba) × (a0 : 0A ) = 1.

PO

And, by the transversal Cc to the triangle AaB,

B (Be

cA) (AO : 0a) x (aC: CB) = 1 . P Multiply these equations together.

R

COLLINEAR AND CONCURRENT SYSTEMS.

209

971 If bc , ca, ab, in the last figure, be produced to meet the sides of ABC in P, Q, R, then each of the nine lines in the figure will be divided harmonically, and the points P, Q, R will be collinear. PROOF. ( i . ) Take bP a transversal to ABC ; therefore, by (968) ,

therefore, by (970),

(CP : PB) (Bc : cA ) ( Ab : bC) = 1 ; CP : PB Ca : aB.

(ii. ) Take CP a transversal to Abc, therefore (AB : Bc) (cP : Pb) ( bC : CA) = 1 . But, by (970) , taking O for focus to Abc, (AB : Be) (cp : pb) (bC : CA ) = 1 ; therefore cP : Pb = cp : pb. (iii.) Take PC a transversal to AOc, and b a focus to A0c ; therefore, by (968 & 970), (Aa : a0) (OC : Ce ) (cB : BA ) = 1, and (AppO) (OC : Ce) (cB : BA) = 1 ; therefore Aa : a0 Ap : pO. Thus all the lines are divided harmonically.

= (iv.) In the equation of (970) put Ab : b0AQ : QC the harmonic. ratio, and similarly for each ratio, and the result proves that P, Q, R are collinear, by (968) . COR. -If in the same figure qr, rp, pq be joined , the three lines will pass through P, Q, R respectively. PROOF.- Take O as a focus to the triangle abc, and employ (970) and the harmonic division of be to show that the transversal rq cuts bc in P.

If a transversal intersects the sides AB, BC, CD, &c. of any polygon in the points a, b, c, &c. in order, then

972

(Aa : aB) (Bb : bC) ( Cc : cD) (Dd : dE) ... &c . = 1 . PROOF. Divide the polygon into triangles by lines drawn from one of the angles, and, applying ( 968) to each triangle, combine the results .

973 Let any transversal cut the sides of a triangle and their three intersectors AO, BO, CO (see figure of 970) in the points A' , B' , C' , a' , b' , c' , respectively ; then , as before , (A'b' : b'C') ( C'a'

a'B') ( B'é : c'A') = 1 .

PROOF . Each side forms a triangle with its intersector and the trans versal . Take the four remaining lines in succession for transversals to each triangle, applying (968) symmetrically, and combine the twelve equations. 2 E

210

ELEMENTARY GEOMETRY.

974 If the lines joining corresponding vertices of two tri angles ABC, abc are concurrent, the points of intersection . of the pairs of corresponding sides are collinear, and con

versely. PROOF. Let the concurrent lines Take bc, Aa, Bb, Cc meet in 0. ca, ab transversals respectively to the triangles OBC, OCA, OAB, ap plying (968), and the product of the three equations shows that P, R, Q lie on a transversal to ABC.

B

P

Ꭱ

975 Hence it follows that, if the lines joining each pair of corresponding vertices of any two rectilineal figures are con current, the pairs of corresponding sides intersect in points which are collinear. The figures in this case are said to be in perspective , or in homology, with each other. The point of concurrence and the line of collinearity are called respectively the centre and axis of perspective or homology. See ( 1083 ) . 976 Theorem . When three perpendiculars to the sides of a triangle ABC, intersecting them in the points a, b , c respec tively, are concurrent, the following relation is satisfied ; and conversely, if the relation be satisfied , the perpendiculars are concurrent . Ab² - bC² + Ca² — a B² + Bc² — cA² = 0 . PROOF. If the perpendiculars meet in O, then Ab³ - bC² = A0²— OC³, &c. (I. 47) . EXAMPLES. By the application of this theorem, the concurrence of the three perpendiculars is readily established in the following cases : (1 ) When the perpendiculars bisect the sides of the triangle (2 ) When they pass through the vertices. (By employing I. 47. ) (3) The three radii of the escribed circles of a triangle at the points of contact between the vertices are concurrent. So also are the radius of the inscribed circle at the point of contact with one side, and the radii of the two escribed circles of the remaining sides at the points of contact beyond the included angle. In these cases employ the values of the segments given in ( 953 ) . (4) The perpendiculars equidistant from the vertices with three con current perpendiculars are also concurrent. (5) When the three perpendiculars from the vertices of one triangle upon the sides of the other are concurrent, then the perpendiculars from the vertices of the second triangle upon the sides of the first are also concurrent. PROOF.- If A, B, C and A', B, C' are corresponding vertices of the tri angles, join AB , AC′, BC′, BA', CA' , C'B ' , and apply the theorem in conjunc tion with (I. 47 ) .

211

TRIANGLES CIRCUMSCRIBING A TRIANGLE.

Triangles ofconstant species circumscribed to a triangle. 977 Let ABC be any triangle, and O any point ; and let circles circumscribe AOB, BOC, COA. The circumferences will be the loci of the vertices of a triangle of constant form whose sides pass through the points A, B, C.

PROOF.- Draw any line bAc from circle to circle, and produce bC, cB to meet in a. The angles AOB, COA are supplements of the angles c and b ( III. 22 ) ; therefore BOC is the supplement of a ( I. 32 ) ; there fore a lies on the circle OBC. Also, the angles at O being constant, the angles a , b, c are constant.

B

978 The triangle abc is a maximum when its sides are per pendicular to ОA , OB, OC. PROOF. -The triangle is greatest when its sides are greatest . But the sides vary as Oa, Ob, Oc, which are greatest when they are diameters of the circles ; therefore &c., by (III. 31 ) . 979 To construct a triangle of given species and of given limited magnitude which shall have its sides passing through three given points A, B, C. Determine O by describing circles on the sides of ABC to contain angles equal to the supplements of the angles of the specified triangle. Construct the figure abc0 independently from the known sides of abc, and the now known angles ObCOAC, OaC = OBC, &c. Thus the lengths Oa, Ob, Oc are found, and therefore the points a, b, c, on the circles, can be determined . The demonstrations of the following propositions will now be obvious.

Triangles of constant species inscribed to a triangle. 980 Let abc , in the last figure , be a fixed triangle, and O any point . Take any point A on be, and let the circles cir cumscribing OAc , OAb cut the other sides in B, C.

Then

ABC will be a triangle of constant form, and its angles will have the values A Oba + Oca, &c . ( III . 21.) 981

The triangle ABC will evidently be a minimum when OA , OB, OC are drawn perpendicular to the sides of abc .

982 To construct a triangle of given form and of given limited magnitude having its vertices upon three fixed lines bc, ca, ab.

212

ELEMENTARY GEOMETRY.

Construct the figure ABCO independently from the known sides of ABC and the angles at O, which are equal to the supplements of the given angles a, b, c. Thus the angles OAC, &c are found, and therefore the angles ObC, &c., equal to them ( III. 21 ) , are known. From these last angles the point ( can be determined, and the lengths OA, OB, OC being known from the inde pendent figure, the points A, B, C can be found. Observe that, wherever the point O may be taken, the angles AOB, BOC COA are in all cases either the supplements of, or equal to, the angles c, a , b respectively ; while the angles aOb , bОc, coa are in all cases equal to C ± c, A ± a, B ± b . 983 NOTE .- In general problems , like the foregoing, which admit of different cases , it is advisable to choose for reference a standard figure which has all its elements of the same affec tion or sign. In adapting the figure to other cases , all that is necessary is to follow the same construction , letter for letter , observing the convention respecting positive and negative , which applies both to the lengths of lines and to the magnitudes of angles , as explained in ( 607-609) .

Radical Axis . 984 DEFINITION .-The radical axis of two circles is that perpendicular to the line of centres which divides the dis tance between the centres into segments , the difference of whose squares is equal to the difference of the squares of the radii . Thus , A, B being the centres , a , b the radii , and IP the the radical axis ,

AI³ —BI² — a² — b².

P

Τ

K

Τ

Y

K

Y

T T a

A

985

B

B

It follows that, if the circles intersect , the radical axis

is their common chord ; and that , if they do not intersect , the radical axis cuts the line of centres in a point the tangents from which to the circles are equal (I. 47) .

To draw the axis in this case , see (966 ) .

213

RADICAL AXIS.

Otherwise let the two circles cut the line of centres in C, D and C′, D' respectively. Describe any circle through C and D, and another through Cand D ' , intersecting the former in E and F. Their common chord EF will cut the central axis in the required point I.

PROOF.-IC.ID = IE . IF = IC . ID ( III . 36) ; therefore the tangents from I to the circles are equal.

986

Theorem .-The difference of the squares of tangents

from any point P to two circles is equal to twice the rectangle under the distance between their centres and the distance of the point from their radical axis , or PK -PT = 2AB . PN.

P

K

B

Y'

Bh

PROOF. - (a² — b³) = ( AQ³ — BQ²) — (AIª— BI²) , ' PK' -- PT² = (AP²- BP²) — by (I. 47) & ( 984) . Bisect AB in C, and substitute for each difference of squares, by ( II. 12 ) .

987

COR. 1. - If P be on the circle whose centre is B, then PK2 = 2AB.PN.

988 COR . 2. -If two chords be drawn through P to cut the circles in X, X' , Y, Y' ; then , by (III . 36) , PX.PX ' - PY.PY' = 2AB.PN.

989 If a variable circle intersect two given circles at con stant angles a and ẞ, it will intersect their radical axis at a constant angle ; and its radius will bear a constant ratio to the distance of its centre from the radical axis . Or PN : PX

a cos a - b cos ẞ : AB.

ELEMENTARY GEOMETRY.

214

PROOF. In the same figure, if P be the centre of the variable circle, and if PX = PY be its radius ; then, by ( 988) , PX (XX -YY') = 2AB . PN. XX 2a cos a and YY' = 2b cos ß ; ' But therefore a cos a - b cos ß : AB, PN: PX which is a constant ratio if the angles a, ẞ are constant. 990 Also PX : PN = the cosine of the angle at which the circle of radius PX cuts the radical axis . This angle is therefore constant. 991 COR.-A circle which touches two fixed circles has its radius in a constant ratio to the distance of its centre from their radical axis . This follows from the proposition by making a = ß = 0 or 2″ . If P be on the radical axis ; then (see Figs . 1 and 2 of 984) 992

(i .) The tangents from P to the two circles are equal , PK

or 993

(i . )

The rectangles

PT.

under the segments

through P are equal, or PX . PX' = PY . PY'. 994

(986) of chords (988)

(iii. ) Therefore the four points X, X ' , Y, Y' are con

cyclic (III . 36) ; and , conversely, if they are concyclic , the ' intersect in the radical axis . chords XX ' , YY 995

DEFINITION. -Points which lie on the circumference of

a circle are termed concyclic . 996

(iv .) If P be the centre , and if PX = PY be the radius

of a circle intersecting the two circles in the figure at angles a and ẞ ; then, by ( 993 ) , XX' = YY' , or a cos a = b cos ß ; that is , The cosines of the angles of intersection are inversely as the radii of the fixed circles. 997 The radical axes of three circles (Fig . 1046) , taken two and two together, intersect at a point called their radical centre. PROOF.- Let A, B, C b: t'e centres, a , b, c the radii, and X, Y, Z the points in which the radical axes cut BC, CA , AB. Write the equation of the defini tion (984) for each pair of circles. Add the results, and apply ( 976). 998

A circle whose centre is the radical centre of three

other circles intersects them in angles whose cosines inversely as their radii ( 996 ) .

are

INVERSION.

215

Hence, if this fourth circle cuts one of the others or thogonally, it cuts them all orthogonally. 999 The circle which intersects at angles a , ẞ, y three fixed circles , whose centres are A, B, C and radii a , b , c , has its centre at distances from the radical axes of the fixed circles proportional to

b cos B - c cos y y, BC

― c cos y a cos a " CA

a cos a- b cos ẞ B AB

And therefore the locus of its centre will be a straight line passing through the radical centre and inclined to the three radical axes at angles whose sines are proportional to these fractions . PROOF . The result is obtained immediately by writing out equation (989) for each pair of fixed circles.

The Method of Inversion. 1000 DEFINITIONS . Any two points P, P' , situated on a diameter of a fixed circle whose centre is 0 and radius k, so that OP . OP ' = k² , are R P' called inverse points with re spect to the circle , and either point is said to be the inverse of the other. The circle and its centre are called the circle

P 0

D

D'

and centre of inversion , and k the constant of inversion .

1001

If every point of a plane figure be inverted

with

respect to a circle, or every point of a figure in space with respect to a sphere, the resulting figure is called the inverse or image of the original one. Since OP : k : OP' , therefore

1002

1003

OP : OP' = OP² : k² = k² : OP²².

Let D, D ' , in the same figure , be a pair of inverse

points on the diameter OD . In the perpendicular bisector of DD , take any point Q as the centre of a circle passing through D, D' , cutting the circle of inversion in R, and any straight line through O in the points P, P. Then, by (III . 36) , OP.OP Hence OD . OD OR² ( 1000 ) .

216

ELEMENTARY GEOMETRY.

1004 (i . ) P, P ' are inverse points ; and, conversely, any two pairs of inverse points lie on a circle. 1005 (ii .) The circle cuts orthogonally the circle of inver sion ( III . 37 ) ; and , conversely, every circle cutting another orthogonally intersects each of its diameters in a pair of inverse points . 1006 (iii . ) The line IQ is the locus of a point the tangent from which to a given circle is equal to its distance from a given point D. 1007 DEF . -The line IQ is called the axis of reflexion for the two inverse points D, D ' , because there is another circle of inversion, the reflexion of the former, to the right of IQ , having also D, D' for inverse points .

1008

The straight lines drawn from any point P, within

or without a circle ( Figs . 1 and 2) , to the extremities of any chord AB passing through the inverse point Q, make equal angles with the diameter through PQ. Also , the four points O, A, B, P are concyclic , and QA . QB = QO.QP.

A

(1)

(2) B

P

PO

B

PROOF.- In either figure OP : OA : OQ and OP : OB : 0Q (1000 ), therefore, by similar triangles, ≤OPA = OAB and OPB = OBA in figure ( 1 ) and the supplement of it in figure (2) . But OAB = OBA (I. 5) , there fore, &c . Also, because ▲ OPA = OBA , the four points O. A, B, P lie on a circle in each case ( III. 21 ) , and therefore QA . QB = QO . QP ( III. 35, 36) .

1009 The inverse of a circle is a circle , and the centre of inversion is the centre of similitude of the two figures . See also (1037) . PROOF.In the figure of ( 1043) , let O be the point where the common tangent RT of the two circles, centres A and B, cuts the central axis, and let any other line through O cut the circles in P, Q P, Q. Then, in the demon stration of (942) , it is shown that OP . OQ' = OQ . OP = k² , a constant quantity. Therefore either circle is the inverse of the other, k being the radius of the circle of inversion.

217

INVERSION.

1010 To make the inversions of two given circles equal circles.

RULE . - Take the centre of inversion so that the squares of the tangents from it to the given circles may be proportional to their radii (965) . PROOF.- (Fig. 1043) AT : BR = OT : OR = OT : k³, since OT : k : OR. Therefore OT² : AT = k²³ : BR, therefore BR remains constant if OT ∞ AT. 1011 Hence three circles may be inverted into equal circles , for the required centre of inversion is the intersection of two circles that can be drawn by (965) .

The inverse of a straight line is a circle passing 1012 through the centre of inversion.

PROOF. -Draw OQ perpendicular to the line, and take P any other point on it. Let Q, P ' be the inverse points. Then OP . OP = 02.00 ; therefore, by similar triangles, 4OPQOQP, a right angle ; and OQ' is constant, therefore the locus of P is the circle whose diameter is OQ.

P

1013 EXAMPLE.-The inversion ofa poly gon produces a figure bounded by circular arcs which intersect in angles equal to the corresponding angles of the polygon, the complete circles intersecting in the centre of inversion.

1014 If the extremities of a straight line P'Q ' in the last figure are the inversions of the extremities of PQ, then PQ : P'Q ' = √ (OP.OQ) : √ (OP ' . OQ ') .

OQ

PROOF. OP.

1015

By similar triangles, PQ : PQ Compound these ratios.

OP : OQ and PQ : PQ =

From the above it follows that any homogeneous

equation between the lengths of lines joining pairs of points in space, such as PQ.RS. TU := PR.QT. SU, the same points appearing on both sides of the equation , will be true for the figure obtained by joining the corresponding pairs of inverse points . For the ratio of each side of the equation to the corresponding side of the equation for the inverted points will be the same, namely, (OP.OQ.OR ...) : √ (OP . OQ . OR ...). 2 F

218

ELEMENTARY GEOMETRY.

Pole and Polar. 1016 DEFINITION .- The polar of any point P with respect to a circle is the perpendicular to the diameter OP (Fig. 1012) drawn through the inverse point P. 1017 It follows that the polar of a point exterior to the circle is the chord of contact of the tangents fron the point ; that is, the line joining their points of contact. 1018 Also , PQ ' is the polar of P with respect to the circle , centre 0, and PQ is the polar of Q. In other words , any point P lying on the polar of a point Q' , has its own polar always passing through Q' .

1019 The line joining any two points P, p is the polar of Q' , the point of intersection of their polars . PROOF.-- The point Q' lies on both the lines PQ' , p'Q', and therefore has its polar passing through the pole of each line, by the last theorem. 1020

The polars of any two points P, p, and the line joining

the points form a self- reciprocal triangle with respect to the circle , the three vertices being the poles of the opposite sides . The centre O of the circle is evidently the orthocentre of the triangle ( 952 ) . The circle and its centre are called the polar circle and polar centre of the triangle. If the radii of the polar and circumscribed circles of a triangle ABC be r and R, then 24R2 cos A cos B cos C. PROOF.- In Fig. ( 952) , O is the centre of the polar circle, and the circles described round ABC, BOC, COA, AOB are all equal ; because the angle = OB.OC (VI. C ) BOC is the supplement of A ; &c. Therefore 2R . OD and = OA . OD = OA . OB . OC ÷ 2R. Also, OA = 2R cos A by a diameter through B, and ( III . 21 ) .

Coaxal Circles.

DEFINITION.-A system of circles having a common 1021 line of centres called the central axis, and a common radical axis, is termed a coaxal system . 1022

If O be the variable centre of one of the circles , and

COAXAL CIRCLES.

219

OK its radius, the whole system is included in the equation

012- OK² = ± 8°, where & is a constant length.

R

R

K

K

R

R

1023

O D

D

I

In the first species (Fig. 1 ) , OI -OK² = 8²,

and is the length of the tangent from I to any circle of the system (985) . Let a circle, centre I and radius 8, cut the central axis in D, D '.

When O is at D or D' , the circle whose

radius is OK vanishes .

When O is at an infinite distance,

the circle developes into the radical axis itself and into a line at infinity . The points D, D' are called the limiting points.

1024

In the second species (Fig. 2) , OR² - O12 = 8⁹,

and

is half the chord RR common to all the circles of

These circles vary between the circle with the system. centre I and radius 8, and the circle with its centre at infinity as described above . The points R, R' are the common points of all circles of this system. The two systems are therefore distinguished as the limiting points species and the common points species of coaxal circles.

ELEMENTARY GEOMETRY.

220

1025 There is a conjugate system of circles having R, R' for limiting points , and D, D ' for common points , and the circles of one species intersect all the circles of the conjugate system of the other species orthogonally ( 1005 ) . Thus , in figures ( 1 ) and ( 2) , Q is the centre of a circle of the opposite species intersecting the other circles or

thogonally. 1026

In the first species of coaxal circles , the limiting

points D, D' are inverse points for every circle of the system, the radical axis being the axis of reflexion for the system. OI² — ♪² = OK³, PROOF .-(Fig. 1) OD . OD = OK³, therefore (II. 13)

' are inverse points (1000 ). therefore D, D 1027 Also , the points in which any circle of the system cuts the central axis are inverse points for the circle whose centre is I and radius 8. [ PROOF. Similar to the last. 1028 PROBLEM. - Given two circles of a coaxal system, to describe a circle of the same system- (i . ) to pass through a given point ; or (ii . ) to touch a given circle ; or (iii . ) to cut a given circle orthogonally. 1029 I. If the system be of the common points species, then, since the required circle always passes through two known points, the first and second cases fall under the Tangencies. See (941 ) .

1030 To solve the third case, describe a circle through the given common points, and through the inverse of either of them with respect to the given circle, which will then be cut orthogonally, by ( 1005 ) . 1031 II. If the system be of the limiting points species, the problem is solved in each case by the aid of a circle of the conjugate system. Such a circle always passes through the known limiting points, and may be called a conjugate circle of the limiting points system. Thus, 1032 To solve case (i. ) -Draw a conjugate circle through the given point, and the tangent to it at that point will be the radius of the required circle. 1033 To solve case (ii .) -Draw a conjugate circle through the inverse of either limiting point with respect to the given circle, which will thus be cut orthogonally, and the tangent to the cutting circle at either point of intersection will be the radius of the required circle. 1034 To solve case (iii. ) —Draw a conjugate circle to touch the given one, and the common tangent of the two will be the radius of the required circle. 1035 Thus, according as we wish to make a circle of the system touch, or cut orthogonally, the given circle, we must draw a conjugate circle to cut orthogonally, or touch it.

CENTRES AND AXES OF SIMILITUDE.

1036

221

If three circles be coaxal , the squares of the tangents

drawn to any two of them from a point on the third are in the ratio of the distances of the centre of the third circle from the centres of the other two. PROOF. - Let A, B, C be the centres of the circles ; PK, PT the tangents from a point P on the circle, centre C, to the other two ; PN the perpen dicular on the radical axis. By ( 986) , therefore

PK = 2AC .PN and PT = 2BC . PN, PK' : PT2 = AC : BC.

Centres and axes of similitude. 1037

DEFINITIONS . -

Let 00 ′ be the centres of similitude

(Def. 947) of the two circles in the figure below, and let any line through O cut the circles in P, Q, P, Q. Then the constant ratio OP : OP = OQ : 0Q is called the ratio of similitude of the two figures ; and the constant product OP.OQ = OQ . OP is called the product of anti- similitude. See ( 942 ) , ( 1009 ) , and ( 1043 ) . The corresponding points P, P' or Q, Q on the same straight line through O are termed homologous , and P, Q ' or Q, Pare termed anti-homologous. 1038 Let any other line Opqp'q be drawn through 0. Then, if any two points P, p on the one figure be joined , and if P ' , p' , homologous to P, p on the other figure , be also joined , the lines so formed are termed homologous. But if the points which are joined on the second figure are anti-homologous to those on the first , the two lines are termed anti-homologous. Thus, Pq, Qp' are anti-homologous lines . 1039

The circle whose centre is

0, and whose radius is

equal to the square root of the product of anti- similitude, is called the circle of anti-similitude.

1040 The four pairs of homologous chords Pp and P'p', Qq and Q'q' , Pq and P'q , Qp and Q'p' of the two circles in the figure are parallel.

And in all similar and similarly situated

figures homologous lines are parallel . PROOF.- BY (VI. 2) and the definition (947) .

ELEMENTARY GEOMETRY.

222

M

P

R

S

P B

1041

The four pairs of anti-homologous chords , Pp and

Q'q, Qq and P'p' , Pq and Qp', Qp and P'q ', of the two circles meet on their radical axis . PROOF. OP . OQ = Op . Oq' = k² , where k is the constant of inversion ; therefore P, p, Q ', ' are concyclic ; therefore Pp and Q'g' meet on the radical axis. Similarly for any other pair of anti-homologous chords. ――――― 1042 From this and the preceding proposition it COR . follows that the tangents at homologous points are parallel ; and that the tangents at anti-homologous points meet on the radical axis . For these tangents are the limiting positions of homologous or anti-homologous chords . (1160)

1043 Let C, D be the inverse points of O with respect to two circles , centres A and B ; then the constant product of anti-similitude OP.OQ

or

OQ . OP ' = OA . OD

or

R

P

P C/A

K

D B

OB . OC.

CENTRES AND AXES OF SIMILITUDE.

PROOF.

therefore

and also and

By similar right-angled triangles, OA : OT : OC and OB : OR : OD ; OA . OD OB . OC .

(1 ) ,

OA . OC = OT² = OP . OQ, OB . OD

OR

223

(III. 36)

OP . OQ ;

therefore

OA . OB . OC.OD = OP . OQ . OP ′ . OQ, therefore &c., by (1) .

1044 The foregoing definitions and properties ( 1037 to 1043 ) , which have respect to the external centre of similitude 0, hold good for the internal centre of similitude O ', with the usual convention of positive and negative for measured from O ' upon lines passing through it .

distances.

1045 Two circles will subtend equal angles at any point on the circumference of the circle whose diameter is OO, where 0,0 ' are the centres of similitude (Fig. 1043 ) . This circle is also coaxal with the given circles, and has been called the. circle of similitude . PROOF. -Let A, B be the centres, a, b the radii, and K any point on the circle, diameter 00' . Then, by (932) , KA : KB = AO : B0 = AO′ : BO′ = a : b, by the definition (943) ; therefore a : KA = b : KB ;

that is, the sines of the halves of the angles in question are equal, which proves the first part. Also , because the tangents from K are in the constant ratio of the radii a, b , this circle is coaxal with the given ones, by (1036 , 934) .

1046 The six centres of similitude P, p , Q, q, R, r of three circles lie three and three on four straight lines PQR, Pqr, Qpr, Rpq, called axes of similitude.

A PROOF. - Taking any three of the sets of points named, say P, q, r, they are shewn at once to be col linear by the transversal theorem (968 ) applied to the triangle ABC. For the segments of its sides made by the points P, q, r are in the ratios of the radii of the circles.

1 Y

B

P

R

224

ELEMENTARY GEOMETRY.

1047 From the investigation in ( 942) , it appears that one circle touches two others in a pair of anti -homologous points , and that the following rule obtains : RULE. -The right line joining the points of contact passes through the external or internal centre of similitude of the two circles according as the contacts are of the same or of different kinds. 1048 DEFINITION.-Contact of curves is either internal or external ac cording as the curvatures at the point of contact are in the same or opposite directions .

1049

Gergonne's method of describing the circles which touch

three given circles. Take Pqr, one of the four axes of similitude, and find its poles a, ß, y with respect to the given circles, centres A, B, C ( 1016 ) . From O, the radical centre, draw lines through a, ß, y, cutting the circles in a, a, b, b', c, c'. Then a, b, c and a, b , c will be the points of contact of two of the required circles. PROOF.-Analysis.-Let the circles E, F touch the circles A, B, C in a, b, c, a ', b', c'. Let bc, b'e meet in A P ; ca, c'a' in q; and ab, a'b' in r. E Regarding E and F as touched by A, B, C in turn, Rule ( 1047) shews that aa', bb', cc' meet in O, the centre •B of similitude of E and F ; and ( 1041 ) shews that P, q, and r lie on the F radical axis of E and F. .C Regarding B and C, or C and A, or A and B, as touched by E and F in turn, Rule ( 1047) shews that P, q, r are the centres of similitude of B and p C, C and A, A and B respectively ; and ( 1041 ) shows that O is on the radical axis of each pair, and is therefore the radical centre of A, B, and C. Again, because the tangents to E and F, atthe anti-homologous points a, a' , meet on Pqr, the radical axis of E and F (1042 ) ; therefore the point of meeting is the pole of aa' with respect to the circle A ( 1017) . Therefore au passes through the pole of the line Pqr (1018) . Similarly, b ' and ce pass through the poles of the same line Pqr with respect to B and C. Hence the construction . 1050 In the given configuration of the circles A, B, C, the demonstration shews that each of the three internal axes of similitude Pqr, Qrp, Rpq (Fig . 1046 ) is a radical axis and common chord of two of the eight osculating circles which can be drawn. The external axis of similitude PQR is the

225

ANHARMONIC RATIO.

radical axis of the two remaining circles which touch A , B, and C either all externally or all internally. 1051 The radical centre O of the three given circles is also the common internal centre of similitude of the four pairs of osculating circles . Therefore the central axis of each pair passes through 0, and is perpendicular to the radical axis . Thus, in the figure, EF passes through 0, and is perpen dicular to Pqr . Anharmonic Ratio. 1052

DEFINITION.-Let a pencil of

L

four lines through a point O be cut by a transversal in the points A, B, C, D. The anharmonic ratio of the pencil is any one of the three frac tions

Ho R

AB.CD

AB.CD

or

or AD . BC

AD . BC

AC.BD

AC . BD

1053 The relation between these three different ratios is obtained from the equation AB.CD+ AD . BC = AC . BD. Denoting the terms on the left side by p and q, the three anharmonic ratios may be expressed by pq, pp + q, q : p + q. The ratios are therefore mutually dependent. Hence, if the identity merely of the anharmonic ratio in any two systems is to be established, it is immaterial which of the three ratios is selected .

1054 In future, when the ratio of an anharmonic pencil { 0 , ABCD} is mentioned, the form AB . CD : AD . BC will be the one intended, whatever the actual order of the points A, B, C, D may be. For, it should be observed that, by making the line OD revolve about O, the ratio takes in turn each of the forms given above. This ratio is shortly expressed by the notation {0, ABCD}, or simply { ABCD} . 1055 If the transversal be drawn parallel to one of the lines, for instance OD, the two factors containing D become infinite, and their ratio becomes unity. They may therefore be omitted . The anharmonic ratio then reduces to AB : BC. Thus, when D is at infinity, we may write { 0, ABC∞} = AB : BC. 1056

The anharmonic ratio AB.CD

sin AOB sin COD

= AD . BC

sin AOD sin BOC'

2 G

I

ELEMENTARY GEOMETRY.

226

and its value is therefore the same for all transversals of the pencil . PROOF.- Draw OR parallel to the transversal, and let p be the perpendi cular from A upon OR. Multiply each factor in the fraction by p. Then substitute p.AB = OA . OB sin AOB, &c. ( 707) . 1057 The anharmonic ratio ( 1056) becomes harmonic when its value is unity. See (933) . The harmonic relation there defined may also be stated thus : four points divide a line harmonically when the product of the extreme segments is equal to the product of the whole line and the middle segment.

Homographic Systems of Points. 1058

DEFINITION . — If x , a , b , c be the distances of one

variable point and three fixed points on a straight line from a point O on the same ; and if a' , a', b' , c' be the distances of similar points on another line through 0 ; then the variable points on the two lines will form two homographic systems when they are connected by the anharmonic relation (x - a) (b − c) = (x ' —a') (b ' - c') (x− c) (a −b) (x ' — c') (a' — b')'

1059

Expanding, and writing A, B, C, D for the constant coeffi cients , the equation becomes

1060

Axx + Bx + Cx + D = 0.

From which

Cx + D 1061

Bx+ D

and

x

x = Ax +C®

Ax + B '

THEOREM .-Any four arbitrary points 2 , X2, X3, X₁ on 1062 one of the lines will have four corresponding points 1 , 2, 3 , on the other determined by the last equation , and the two sets ofpoints will have equal anharmonic ratios . PROOF. This may be shown by actual substitution of the value of each z in terms of ', by (1061) , in the harmonic ratio { x }.

1063

If the distances of four points on a right line from a

point O upon it , in order, are a, a , ß, ẞ', where a , ß ; a', ẞ are the respective roots of the two quadratic equations ax² + 2hx + b = 0 ,

' ′ = 0; a'x² + 2hx + b

the condition that the two pairs of points may be harmonically conjugate is 1064

ab' + a'b = 2hh'.

INVOLUTION.

227

PROOF.- The harmonic relation, by (1057) , is (a − a') (b − f') = (a − ẞ') (a' — ß). Multiply out, and substitute for the sums and products of the roots of the quadratics above in terms of their coefficients by ( 51 , 52 ) .

1065 If u . u, be the quadratic expressions in ( 1063 ) for two pairs of points, and if u represent a third pair harmonically conjugate with u, and u , then the pair of points u will also be harmonically conjugate with every pair given by the equation u₁ + Au₂ = 0, where A is any constant. For the con dition (1064) applied to the last equation will be identically satisfied . Involution . ', QQ ' , &c. , on 1066 DEFINITIONS . -Pairs of inverse points PP the same right line , form a system in involution , and the rela tion between them, by (1000) , is

OP . OP = OQ.OQ ' = & c. = k². A

0

P Q

B

Q

P '

The radius of the circle of inversion is k , and the centre O is called the centre of the system . termed conjugate points.

Inverse points are also

When two inverse points coincide , the point is called a focus. 1067 The equation OP² = 12 shows that there are two foci A, B at the distance k from the centre , and on opposite sides of it, real or imaginary according as any two inverse points lie on the same side or on opposite sides of the centre.

1068

If the two homographic systems of points in (1058)

be on the same line, they will constitute a system in involu tion when B = C. PROOF.-Equation ( 1060) may now be written Axx +H (x +x') + B = 0 , H² B Η H or = k², (x + 1 ) ( x² + = 42 A H a constant. Therefore is the distance of the origin O from the centre A of inversion. Measuring from this centre, the equation becomes representing a system in involution.

= k³,

1069 Any four points whatever of a system in involution on a right line have their anharmonic ratio equal to that of their four conjugates .

228

ELEMENTARY GEOMETRY.

PROOF.- Let p, p′ ; I, q' ; r, r' ; s, s' be the distances of the pairs of inverse points from the centre. In the anharmonic ratio of any four of the points, for instance { pqrs}, substitute p² ÷ p', q = k² ÷ q, &c., and the result is the anharmonic ratio { p'qr's' } .

1070 Any two inverse points P, P ′ are in harmonic relation with the foci A, B. B A P 0 P PROOF. - Let p, p' be the distances of P, P from the centre 0 ; then p k = " therefore p'+ k = k +P ; pp'k', therefore P k k-P p' -k ' AP AP = that is, (933) BP BP

1071 If a system of points in involution be given, as in (1068 ) , by the equation 0 .... (1) ; Axx + H (x + x) + B = 0 and a pair of conjugate points by the equation ax² +2hx + b = 0 ......

..... (2) ;

the necessary relation between a , h , and b is 1072

Ab +Ba = 2Hh.

PROOF. -The roots of equation (2) must be simultaneous values of x, x′ in (1 ) ; therefore substitute in ( 1 ) 2h b and xx' = x+ x = (51) a a 1073 COR. A system in involution may be determined from two given pairs of corresponding points. Let the equations for these points be ax² +2hx + b = 0 and a'a + 2hx + b′ = 0. Then there are two conditions ( 1072) , Ab +Ba = 2Hh and from which A, H, B can be found.

Ab' + Ba′ = 2Hh',

A geometrical solution is given in (985) . C, D ; C, D are, in that con struction, pairs of inverse points, and I is the centre of a system in involution defined by a series of coaxal circles ( 1022 ) . Each circle intersects the central axis in a pair of inverse points with respect to the circle whose centre is O and radius d.

1074 The relations which have been established for a system of collinear points may be transferred to a system of concurrent lines by the method of (1056 ) , in which the distance between two points corresponds to the sine of the angle between two lines passing through those points.

METHODS OF PROJECTION.

229

The Method of Projection. DEFINITIONS .-The projection of any point P in space (Fig. of 1079 ) is the point p in which a right line OP, drawn from a fixed point O called the vertex, intersects a fixed plane called the plane ofprojection. 1075

If all the points of any figure, plane or solid , be thus pro jected, the figure obtained is called the projection of the original figure.

1076 Projective Properties. The projection of a right line is a right line. The projections of parallel lines are parallel . The projections of a curve , and of the tangent at any point of it, are another curve and the tangent at the corresponding point. 1077

The anharmonic ratio of the segments of a right line

is not altered by projection ; for the line and its projection are but two transversals of the same anharmonic pencil . ( 1056 ) 1078 Also , any relation between the segments of a line similar to that in (1015) , in which each letter occurs in every term, is a projective property.

[ Proof as in ( 1056) .

1079 Theorem .-Any quadrilateral PQRS may be projected into a parallelogram . CONSTRUCTION . B Produce PQ, SR to meet in A, and PS, QR to meet in B. Then, with any point 0 for vertex, project the quadri 0 upon any lateral plane pab parallel to OAB. The projected figure pqrs will be a parallelogram . PROOF . -- The planes OPQ, ORS in tersect in OA, and they intersect the plane of projection which is parallel to OA in the lines pq, rs. Therefore pq and rs are parallel to OA, and therefore to each other. Similarly, ps, ada qr are parallel to OB.

P

R น

230

ELEMENTARY GEOMETRY.

1080 COR. 1.-The opposite sides of the parallelogram pqrs meet in two points at infinity, which are the projections of the points A, B; and AB itself, which is the third diagonal of the complete quadrilateral PQRS, is projected into a line at infinity. Hence, to project any figure so that a certain line in it may pass to 1081 infinity-Take the plane of projection parallel to the plane which contains the given line and the vertex.

1082 COR. 2.-To make the projection of the quadrilateral a rectangle, it is only necessary to make AOB a right angle. On Perspective Drawing. 1083 Taking the parallelogram pqrs, in ( 1079 ) , for the original figure, the quadrilateral PQRS is its projection on the plane ABab. Suppose this plane to be the plane of the paper. Let the planes OAB, pab, while remaining parallel to each other, be turned respectively about the fixed parallel lines AB, ab. In every position of the planes, the lines Op, Oq, Or, Os will intersect the dotted lines in the same points P, Q, R, S. When the planes coincide with that of the paper, pqrs becomes a ground plan of the parallelogram, and PQRS is the representation of it in perspective. AB is then called the horizontal line, ab the picture line, and the plane of both the picture plane. 1084 To find the projection of any point p in the ground plan . RULE.- Draw pb to any point b in the picture line, and draw OB parallel to pb, to meet the horizontal line in B Join Op, Bb, and they will intersect in P, the point required . In practice, pb is drawn perpendicular to ab, and OB therefore perpendi cular to AB. The point B is then called the point of sight, or centre of vision, and O the station point. To find the projection of a point in the ground plan , not in the original plane , but at a perpendicular distance c above it. 1085

RULE.- Take a new picture line parallel to the former, and at a distance above it = c cosec a, where a is the angle between the original plane and the plane of projection. For a plane through the given point, parallel to the original plane, will intersect the plane of projection in the new picture line so constructed. Thus, every point of a figure in the ground plan is transferred to the drawing. 1086 The whole theory of perspective drawing is virtually included in the foregoing propositions. The original plane is commonly horizontal, and the plane of projection vertical. In this case, cosec a = 1, and the height of the picture line for any point is equal to the height of the point itself above the original plane. The distance BO, when B is the point of sight, may be measured along AB, and bp along ab, in the opposite direction ; for the line Bb will continue to intersect Op in the point P.

METHODS OF PROJECTION.

231

Orthogonal Projection. 1087

DEFINITION . - In orthogonal projection the lines of

projection are parallel to each other, and perpendicular to the plane of projection . The vertex in this case may be consi dered to be at infinity . 1088 The projections of parallel lines are parallel, and the projected segments are in a constant ratio to the original seginents . 1089

Areas are in a constant ratio to their projections .

For, lines parallel to the intersection of the original plane and the plane of projection are unaltered in length, and lines at right angles to the former are altered in a constant ratio. This ratio is the ratio of the areas, and is the cosine of the angle between the two planes .

Projections ofthe Sphere. 1090 In Stereographic projection , the vertex is on the sur face of the sphere , and the diameter through the vertex is perpendicular to the plane of projection which passes through the other extremity of the diameter . The projection is there fore the inversion of the surface of the sphere ( 1012) , and the diameter is the constant k.

1091 In Globular projection, the vertex is taken at a dis 2, and the tance from the sphere equal to the radius plane of the to perpendicu is vertex the through lar diameter projection.

1092 In Gnomonic projection , which is used in the construc tion of sun-dials , the vertex is at the centre of the sphere.

1093 Mercator's projection , which is employed in navigation , and sometimes in maps of the world , is not a projection at all as defined in ( 1075) . Meridian circles of the sphere are represented on a plane by parallel right lines at intervals. equal to the intervals on the equator. The parallels of lati tude are represented by right lines perpendicular to the meridians , and at increasing intervals , so as to preserve the actual ratio between the increments of longitude and latitude at every point. With r for the radius of the sphere, the distance, on the chart, from the equator of a point whose latitude is A, is = r log tan (45° + ^).

ELEMENTARY GEOMETRY.

232

Additional Theorems. 1094 The sum of the squares of the distances of any point P from n equidistant points on a circle whose centre is 0 and

radius r

= n (r² + OP ) . PROOF . - Sum the values of PB , PC², &c. , given in (819), and apply ( 803) . This theorem is the generalization of (923 ) .

1095 In the same figure, if P be on the circle, the sum of the squares of the perpendiculars from P on the radii OB, OC, & c. , is equal to nr . PROOF.- Describe a circle upon the radius through P as diameter, and apply the foregoing theorem to this circle. COR. 1.- The sum of the squares of the intercepts on the radii be 1096 (I. 47) tween the perpendiculars and the centre is also equal to ur³. COR . 2. - The sum of the squares of the perpendiculars from the 1097 equidistant points on the circle to any right line passing through the centre is also equal to ½nr³. Because the perpendiculars from two points on a circle to the diameters drawn through the points are equal. 1098 COR. 3. - The sum of the squares of the intercepts on the same right line between the centre of the circle and the perpendiculars is also (I. 47) equal to nr².

If the radii of the inscribed and circumscribed circles of a regular polygon of n sides be r, R, and the centre 0 ; then , I. The sum of the perpendiculars from any point P upon the sides 1099 is equal to nr. 1100 II . If p be the perpendicular from O upon any right line, the sum of the perpendiculars from the vertices upon the same line is equal to np. 1101 III. The sum of the squares of the perpendiculars from P on the sides isn ( ² + OP²) .

IV. The sum of the squares of the perpendiculars from the vertices 1102 n (p² + 1 R³). upon the right line is PROOF. In theorem I., the values of the perpendiculars are given by r- OP cos ((0 0+ +2

) , with successive integers for m .

Addtogether the n

values, and apply (803 ). Similarly, to prove II.; take for the perpendiculars the values

p - R cos ((0+ 0+ 2mx). To prove III. and IV. , take the same expressions for the perpendiculars ; square each value ; add the results, and apply ( 803 , 804) . For additional propositions in the subjects of this section , see the section entitled Plane Coordinate Geometry.

GEOMETRICAL

CONICS .

THE SECTIONS OF THE CONE .

1150

DEFINITIONS .-A Conic Section or Conic is the curve

AP in which any plane intersects the surface of a right cone. A right cone is the solid generated by the revolution of one straight line about another which it intersects in a fixed point at a constant angle . Let the axis of the cone, in Fig. ( 1 ) or Fig. ( 2 ) , be in the plane of the paper, and let the cutting plane PMXN be perpendicular to the paper. (Read either the accented or unaccented letters throughout. ) Let a sphere be inscribed in the cone, touching it in the circle EQF and touching the cutting plane in the point S, and let the cutting plane and the plane of the circle EQF inter sect in XM. The following theorem may be regarded as the defining property of the curve of section. 1151 Theorem.— The distance of any point P on the conic from the point S, called the focus , is in a constant ratio to PM, its distance from the line XM, called the directrix, or PS : PM = PS ' : PM' = AS : AX e, the eccentricity. [ See next page for the Proof.] 1152 COR. - The conic may be generated in a plane from either focus S, S' , and either directrix XM, X'M' , by the law just proved . 1153

The conic is an Ellipse, a Parabola , or an Hyperbola,

according as e is less than, equal to , or greater than unity. That is , according as the cutting plane emerges on both sides of the lower cone, or is parallel to a side of the cone, or in tersects both the upper and lower cones . 1154 All sections made by parallel planes are similar ; for the inclination of the cutting plane determines the ratio AE : AX. 1155 The limiting forms of the curve are respectively-a circle when e vanishes, and two coincident right lines when e becomes infinite.

2 H

GEOMETRICAL CONICS.

234

PROOF OF THEOREM 1151. -Join P, S and P, O, cutting the circular section in Q, and draw PM parallel to NX. Because all tangents from the same point, O or A, to either sphere are equal, therefore RE = PQ = PS and AE AS. Now, by (VI. 2) , RE : NX = AE: AX and NX := PM; therefore PS : PM = AS : AX, a constant ratio denoted by e and called the eccentricity of the conic.

(2)

(1)

S ' M F 12

S

F R

N

X

R M

S

R

R FOKLE YAS

P M

sats

Referring the letters either to the ellipse or the hyperbola in the subjoined figure, let C be the middle point of AA' and N any other point on it. Let DD, RR' be the two circular sections of the cone whose planes pass through C and N; BCB and PN the intersections with the plane of the conic. In the ellipse, BC is the common ordinate of the ellipse and circle ; but, in the hyperbola, LC is to be taken equal to the tangent from C to the circle DD'. 1156 is

The fundamental equation of the ellipse or hyperbola PN : AN . NA' = BC² : AC .

PROOF.- PN = NR . NR and BC = CD. CD (III. 35, 36) . Also, by ' : CD = A'N : A'C. similar triangles (VI . 2, 6), NR : CD = AN : AC and NR Multiply the last equations together.

235

THE ELLIPSE AND HYPERBOLA.

1157

N

COR. 1.- PN has

R

K

equal values at two points equi- distant from AA. Hence the curve is sym metrical with respect to AA and BB. These two lines are

B' D

called the major and mi nor axes, otherwise the transverse and conjugate axes of the conic . When the axes

B R

R

are D

equal, or BC = AC, the ellipse becomes a circle, and the hyperbola be or comes rectangular

B

equilateral.

1158 Any ellipse or hyperbola is the orthogonal projection of a circle or rectangular hyperbola respectively. PROOF.-Along the ordinate NP, measure NP' = AN . NA' ; therefore by the theorem PN : P′N = BC : AC. Therefore a circle or rectangular hyper bola, having AA' for one axis, and having its plane inclined to that of the conic at an angle whose cosine BC ÷ AC, projects orthogonally into the ellipse or hyperbola in question, by ( 1089) . See Note to ( 1201 ) . 1159 Hence any projective property (1076-78 ) , which is known to belong to the circle or rectangular hyperbola , will also be universally true for the ellipse and hyperbola respec tively.

THE ELLIPSE AND HYPERBOLA.

Joint properties of the Ellipse and Hyperbola. 1160

DEFINITIONS . -The tangent to a curve at a point P

(Fig. 1166 ) is the right line PQ, drawn through an adjacent point Q , in its ultimate position when Q is made to coincide with P. The normal is the perpendicular to the tangent through the point of contact .

GEOMETRICAL CONICS.

236

In ( Fig. 1171 ) , referred to rectangular axes through the centre C (see Coordinate Geometry) ; the length CN is called the abscissa ; PN the ordinate ; PT the tangent ; PG the nor mal ; NT the subtangent ; and NG the subnormal. S, S are the foci ; XM, X'M' the directrices ; PS, PS the focal dis tances, and a double ordinate through S the Latus Rectum . The auxiliary circle (Fig. 1173) is described upon AA ' as diameter . A diameter parallel to the tangent at the extremity of another diameter is termed a conjugate diameter with respect to the other. The conjugate hyperbola has BC for its major, and AC for its minor axis (1157).

The following theorems ( 1162 ) to ( 1181 ) are deduced from the 1161 property PS : PM = e obtained in ( 1151) . The propositions and demonstrations are nearly identical for the ellipse and the hyperbola, any difference in the application being specified.

1162

CS : CA : CX,

P

M

M'

XA

SAX

and the common ratio is e.

M

M

X

SAX

AS

B

B

PROOF.-By (1151) , e =

1163

(A'S ± AS) A'S AS CA CS = = = or CA } (A'X ± AX) A'X AX CX®

In the ellipse the sum, and in the hyperbola the dif

ference, of the focal distances of P is equal to the major axis , or PSPS AA' .

PROOF. With the same figures we have, in the ellipse, by (1151 ), AA PS +PS AS +A'S PS +PS = and also e = e= therefore & c. XX' XX" AX +A'X PM+PM' =

For the hyperbola take difference instead of sum.

237

THE ELLIPSE AND HYPERBOLA.

1164

CS2

AC² - BC

in the ellipse. [ For BS

CS

AC² + BC

in the hyperbola. [ By assuming BC.

AC, by ( 1163) . See (1176).

BC2 = SL.AC.

1165

(1151 , 1162) PROOF. (Figs. of 1162) SL : SX = CS : CA, .. SL . AC = CS . SX = CS (CX ~ CS) = CA³ ~ CS (1162) = BƠ (1164). 1166 If a right line through P, Q, two points on the conic , meets the directrix in Z, then SZ bisects the angle QSR. z'

M

M N Z

M'

N Z

S

S '

R

R

PROOF.-By similar triangles, ZP : ZQ = MP : NQ = SP : SQ ( 1151 ) , therefore by (VI. A.) 1167

If PZ be a tangent at P, then PSZ and PSZ ' are

right angles . PROOF.- Make Q coincide with P in the last theorem. 1168 The tangent makes equal angles with the focal dis tances . PROOF.-In ( 1166) , PS : PS ′ = PM : PM' (1151 ) = PZ : PZ '; therefore, when PQ becomes the tangent at P, 4 SPZ = SPZ, by ( 1167) and (VI. 7) . 1169 The tangents at the extremities of a focal chord inter sect in the directrix . PROOF. (Figs. of 1166) . Join ZR ; then, if ZP is a tangent, ZR is also, for ( 1167) proves RSZ to be a right angle. 1170

PROOF.

therefore

therefore

CN.CT

AC².

TS PS PM' NX' (Figs. 1171.) = (VI. 3, A.) = PM (1151 ) = NX' TS PS TS + TS 20T 20X NX + NX = = or 2CS TS - TS 2CN' NX - NX CN.CTCS . CX = AC².

(1162)

238

1171

GEOMETRICAL CONICS.

If PG be the normal , GS : PS

GS ' : PS' = e.

M' M

M

CGNS

TXA SN

AX T

~

A

)

PROOF. -By (1168 ) and (VI. 3, A.) , 2CS GS + GS GS = GS = = = e. PS PS PS + PS 2CA

(1162)

But, for the hyperbola, change plus to minus.

The subnormal and the abscissa are as the squares of axe the s , or NG: NC BC² : AC². 1172

PROOF.

(Figs. 1171. )

Exactly as in ( 1170 ) , taking the normal instead CG CN CN CX CA of the tangent, we obtain CS = CX' .. CG = CS = CS (1162), CN ~ CG CA CS NG BC = or .. = (1164) . CN CA NO AC2

1173 The tangents at P and Q, the corresponding points on the ellipse and auxiliary circle , meet the axis in the same point T. But in the hyperbola , the ordinate TQ of the circle being drawn, the tangent at Q cuts the axis in N. Q

B n

74 B

G

X

PROOF. For the ellipse : Join TQ. Then CN . CT = CQ' ( 1170) ; there fore CQT is a right angle (VI. 8) ; therefore QT is a tangent. For the hyperbola : Interchange N and T.

239

THE ELLIPSE AND HYPERBOLA.

PN: QNBC : AC.

1174

PROOF.--(Figs . 1173 ) . NG . NT = PN², and CN.NT = QN². Therefore NG : NC PN : QN ; therefore, by ( 1172) .

(VI . 8)

This proposition is equivalent to ( 1158 ) , and shows that an ellipse is the orthogonal projection of a circle equal to the auxiliary circle. 1175 COR.- The area of the ellipse is to that of the auxiliary circle as BC : AC ( 1089) .

1176

PN2 : AN . NA' = BC² : AC².

PROOF.- By ( 1174) , since QN² = AN . NA′ (III . 35 , 36 ) . An independ ent proof of this theorem is given in ( 1156) . The construction for BC in the hyperbola in ( 1164 ) is thus verified. Cn . Ct

1177

BC².

PROOF.-(Figs. 1173.) PN2 Ct PN Cn . Ct = PN2 PN2 = = .. (III. 35, 36). (VI. 8) = CT ΑΝ . ΝΑ ' NT' CN.CT CN.NT QN

Therefore, by ( 1170 ) and ( 1176) , Cn . Ct : AC² = BƠ² : AC³. 1178 If SY, S'Y ' are the perpendiculars on the tangent, then Y, Y' are points on the auxiliary circle , and SY . S'Y ' BC².

W

E

Y

E

A

S

S

Ꮓ XX PROOF -Let PS meet SY in W. Then PS = PW (1168 ) . Therefore SW= AA′ ( 1163 ) . Also, SY = YW, and SC:= CS'. Therefore CYS W = AC. Similarly CY' AC. Therefore Y, Y' are on the circle. Hence ZY' is a diameter ( III . 31 ) , and therefore SZ = S'Y', by similar triangles ; therefore SY . SZ := SA . SA' ( III. 35, 36) = CS ~ CA (II . 5 ) = BC (1164). 1179 then

COR .- If CE be drawn parallel to the tangent at P, AC. PE = CY

1180 PROBLEM.- To draw tangents from any point 0 to an ellipse or hyperbola. CONSTRUCTION.- (Figs. 1181. ) Describe two circles, one with centre 0 and radius OS, and another with centre S and radius == AA', intersecting in M,

240

GEOMETRICAL CONICS.

M. Join MS, M'S. These lines will intersect the curve in P, P ', the points of contact. For another method see (1204) . PROOF.- By ( 1163 ) , PS ± PS = AA' = SM by construction. There fore PS = PM, therefore OPS = OPM (1. 8) , thereforo OP is a tangent by ( 1168). Similarly P'S = P'M', and OP ' is a tangent.

1181 focus .

The tangents OP, OP subtend equal angles at either

P

A S'

M

off

PROOF. The angles OSP, OSP are respectively equal to OMP, OM'P, by (I. 8), as above ; and these last angles are equal, by the triangles OSM, OSM' , and ( I. 8). Similarly at the other focus.

Asymptotic Properties of the Hyperbola . — 1182 DEF. The asymptotes of the hyperbola are the diagonals of the rectangle formed by tangents at the vertices A, A , B, B'.

1183

If the ordinates RN, RM from any point R on an

asymptote cut the hyperbola and its conjugate in P, P', D, D ', M E

A A

B

R

N

241

THE HYPERBOLA.

then either of the following pairs of equations will define both the branches of each curve RN2 - PN2 = BC² = P'N² — RN²

(1) ,

RM² - DM²

(2) .

AC

D'M² - RM²

PROOF.

Firstly, to prove ( 1 ) : By proportion from the similar triangles RN2 BC2 PN2 = = RNC, OAC, we have CN AC CN -AC AN . NA = CN² —AC².

by (1176 ) , since

RN2-PN2 BC2 = AC AC2 RN2 -PN = BC .

Therefore therefore

By (II. 6) by the theorem (69) ;

Also, by (1176 ) , applied to the conjugate hyperbola, the axes being now AC² CN CN2 = = reversed, by similar triangles ; PN -BO RN BC2

therefore

or P'N'- RN² = BC².

P'N'- BC = RN²

Secondly, to prove (2) : By proportion from the triangles RMC, OBC, we DM² A03 RM2 = have = CM²-BC" BC CM² by (1176) , applied to the conjugate hyperbola, for in this case we should have BM.MB CM²— BC². Therefore

RM2- DM2 AC = ; BC2 BC2

therefore RM² — DM² = AƠ³.

Also, by (1176) , since CM, D'M are equal to the coordinates of D ', CM² BC2 CM² === = similar triangles ; D'M -AC AC2 RM2' by

therefore

1184

D'M²-AC² = RM²

or D'M² - RM² = AC².

COR. 1. - If the same ordinates RN, RM meet the

other asymptote in r and r' , then PR.Pr

BC²

and

DR . Dr' = AC².

(II. 5)

1185 COR. 2.-As R recedes from C, PR and DR con tinually diminish. Hence the curves continually approach the asymptote . 1186

If XE be the directrix , CE = AC. — CX : CA = CA : CS and CS = CO. CE : CO PROOF.-

1187

PD is parallel to the asymptote . RN2 BC RN2-PN2 PN = = PROOF. (1183) = DM2 (69) . RM²- DM AC RM2

Therefore RN : PN = RM : DM; therefore , by (VI . 2) . 21

( 1164)

242

1188

GEOMETRICAL CONICS.

The segments of any right line between the curve and

QR = qr .

the asymptote are equal, or

V

I D

E Ή

S P

K

O

7u

PROOF.

QR: QUqR : qU Qr : Qu = qr : qu' S '

and

Compound the ratios, and employ (1184).

1189

COR . 1.- PL

1190

COR . 2.— CH = HL .

1191

QR . Qr = PL² = RV²— QVª = — Q'V² —RV² .

QVqV.

PL and

Because PD is parallel to IC. (1187)

PROOF.- QR : QU = PL : PE ? Compound theratios. Therefore, by (1184) , and Qr : Qu Pl : Pe S QR. QrPL . PI = PL ( 1189) .

1192 PROOF. and

4PH.PK PH : PE = C0 : 00 ) PK : Pe Co : Oo

CS².

: . PH.PK : PE . Pe = C0³ : 0°² = CS : 4BC ; therefore, by ( 1184) .

Joint Properties of the Ellipse and Hyperbola resumed. If PCP' be a diameter , and QV an ordinate parallel to the conjugate diameter CD ( Figs . 1195 and 1188 ) . 1193

QV² : PV . VP ′ = CD² : CP2.

This is the fundamental equation of the conic, equation (1176) being the most important form of it.

THE ELLIPSE AND HYPERBOLA.

243

Otherwise :

In the ellipse ,

QV : CP - CV

CD : CP2.

In the hyperbola , QV² : CV²— CP² — CD² : CP² ; and

Q'V : CV²+ CP2

CD2 : CP2.

PROOF.-(Ellipse. Fig. 1195 . ) - By orthogonal projection from a circle. If C, P, P', D, Q, V are the projections of c, p, p , d, q, v on the circle ; qu² = pv.vp and cd = cp . The proportion is therefore true in the case of the circle. Therefore &c., by ( 1088 ) .

(Hyperbola. Fig. 1188 ) RV +PL PL2 RV2 CD2 QV = = = = CV_CP * CVCP CP2 CP CV2

or

Q'V₂ CV*+ CP*

(1191 )

1194 The parallelogram formed by tangents at the extremi ties of conjugate diameters is of constant area , and therefore , PF being perpendicular to CD ( Figs . 1195) , PF.CD

AC . BC.

PROOF.- (Ellipse. )-By orthogonal projection from the circle ( 1089 ) . (Hyperbola. Fig . 1188 . ) - CL . Cl = 4PH.PK = CO . Co ( 1192) ; there fore, by (VI. 15) , ALCI = OCo = AC . BC.

If PF intersects the axes in G and G' , PF.PG

1195

BC2 and PF.PG'

AC2. K

G'

L

D

F

n

n

C

N

G

K

PROOF. PF.PG. 1197

1

PF.PG = PK.PN

Cn . Ct

BC

( 1177) .

COR.- PG.PG' = CD² = PT.Pt.

Similarly for

By (1194)

244

GEOMETRICAL CONICS.

1198 The diameter bisects all chords parallel to the tangent at its extremity . PROOF.- (Ellipse. Fig. 1195 .) - By projection from the circle (1088 ) QV = VQ'. (Hyperbola .) By ( 1189. ) 1199 COR . 1. - The tangents at the extremities of any chord meet on the diameter which bisects it. PROOF.- The Secants drawn through the extremities of two parallel chords meet on the diameter which bisects them (VI. 4) , and the tangents are the limiting positions of the secants when the parallel chords coincide. COR . 2.- If the tangents from a point are equal , the (I. 8) diameter through the point must be a principal axis. 1200

1201 COR. 3.- The chords joining any point Q on the curve with the extremities of a diameter PP' , are parallel to con jugate diameters , and are called supplemental chords. For the diameter bisecting PQ is parallel to PQ (VI. 2) . Similarly the diameter bisecting PQ is parallel to PQ.

1202 Diameters are mutually conjugate ; If CD be parallel to the tangent at P, CP will be parallel to the tangent at D. PROOF. (Ellipse. Fig. 1205. ) - By projection from the circle (1088) . NOTE . Observe that, if the ellipse in the figure with its ordinates and tangents be turned about the axis Tt through the angle cos¹ (BC ÷ AC) , it becomes the projection of the auxiliary circle with its corresponding ordinates and tangents . (Hyperbola. Fig. 1188 . ) -By ( 1187, 1189) the tangents at P, D meet the asymptotes in the same point L. Therefore they are parallel to CD, CP (VI. 2.)

If QT be the tangent at Q, and QV the ordinate parallel to the tangent at any other point P, 1203

CV.CT =

CP .

(2)

(1)

W R JV

T 20

P

с

T

P

PROOF. - CR bisects PQ ( 1199) . Therefore PW is parallel to QR. Therefore, by (VI. 2) , CV : CP = CW : CR = CP : CT.

THE ELLIPSE AND HYPERBOLA.

245

1204 COR .- Hence, to draw two tangents from a point T, we may find CV from the above equation, and draw QVQ ' parallel to the tangent at P to determine the points of contact Q, Q '.

Let PN, DN be the ordinates at the extremities of con jugate diameters , and PT the tangent at P. Let the ordinates at N and R in the ellipse , but at T and C in the hyperbola , meet the auxiliary circle in p and d ; then

1205

CN = dR ,

CR = pN.

B

P

B

T

AN

R.

PROOF.-(Ellipse.) Cp, Cd are parallel to the tangents at d and p ( Note to 1202). Therefore pCd is a right angle. Therefore pNC, CRd are equal right-angled triangles with CN = dR and CR = pN.

AC² ( 1170) , CN.CT (Hyperbola.) and DR . CT = 2ACDT = 2CDP = AC . BC ( 1194) ; CN CN = AC PN ᎠᎡ ᏟᎡ .. = (1174) , .. (similar triangles) . = BC ᎠᎡ TN PN PN PN CN But = PN (VI. 8) ; .. CR = pN. Also Cp = Cd ; therefore the tri PN TN

angles CpN, dCR are equal and similar ; therefore CN = dR and dR is parallel to pN. 1206

COR.

DR : dR = BC : AC.

PROOF.- (Ellipse. ) By (1174 ) . (Hyperbola .) By the similar right-angled triangles, we have dRpN = CR : TN = DR : PN ; therefore AC : BC ( 1174) . dR: DR pN : PN

In the same figures , CN2 + CR2 = AC2 ; DR + PN² = BC².

1207

(Ellipse .)

1209

(Hyperbola .) CN2 — CR² = AC² ; DR² — PN² = BC².

PROOF.-Firstly, from the right-angled triangle CNp in which pN = CR (1205) . Secondly, In the ellipse, by (1174), DR +PN : dR + pN' BO : AỞ, and dR +pNAC , by (1205). For the hyperbola, take difference of squares.

246

GEOMETRICAL CONICS.

1211

(Ellipse .)

CP² + CD² = AC²+ BC².

1212

(Hyperbola .)

CP² - CD² = AC² — BC².

PROOF.- (Figs. 1205. ) triangles CNP, ČRD.

By ( 1205-1210) and ( I. 47) , applied to the

The product of the focal distances is equal to the square of the semi- conjugate diameter, or PS . PS'

1213

CD³.

(PS + PS') ' — PS² – PS" 2PS . PS PROOF. (Ellipse. Fig. 1171.) = 4A0-20S -2CP (922, i.) = 2 (AC + BC - CP ) ( 1164) = 2CD² ( 1211 ). ′ = PS² + PS" — (PS′ —PS)² = &c . (Hyperbola. ) — Similarly with 2PS . PS 1214

The products of the segments of intersecting chords

QOq, Q'Oq' are in the ratio of the squares of the diameters parallel to them, or OQ . Oq : OQ' . Oq' = CD2 : CD's. PROOF. (Ellipse. ) By projection from the circle ( 1088) ; for the propor tion is true for the circle, by (III. 35, 36) . (Hyperbola . Fig. 1188. ) Let O be any point on Qq. Draw IOi parallel to Ee, meeting the asymptotes in I and i ; then OR . Or - OQ . Oq = QR . Qr (II. 5) = PL (1191 ) ......... (1) . PL OR . Or Pl Or PL CD2 OR Now = = = = and ; .. ΟΙ PE . Pe ΟΙ . Οι Pe Oi PE' BC (1184). Therefore

CD2 CD OR . Or-PL OQ.Oq ; or, by (1), OI.Oi- BC2 = BO = B02 OI.Oi-BO²

Similarly for any other chord Q'Og drawn through 0. Therefore ′ = CD³ : CD². OQ . Oq : OQ . Oq 1215 COR. - The tangents from any point to the curve are in the ratio of the diameters parallel to them. For, when O is without the curve and the chords become tangents, each product of segments becomes the square of a tangent.

1216 If from any point Q on a tangent PT drawn to any conic (Fig. 1220) , two perpendiculars QR, QL be drawn to the focal distance PS and the directrix XM respectively ; then

SR : QL = e. PROOF. therefore

Since QR is parallel to ZS ( 1167) , therefore, by (VI. 2 ), SR : PS = QZ : PZ = QL : PM; SR : QL PS : PM = e.

COR.-By applying the theorem to each of the tangents from Q, a proof of ( 1181) is obtained.

Adidas

THE ELLIPSE AND HYPERBOLA.

1217

247

The Director Circle.- The locus of the point of inter

section, T, of two tangents always at right angles is a circle called the Director Circle.

T

P

2

T

S

XX

PROOF. Perpendiculars from S, S' to the tangents meet them in points Y, Z, Y', Z', which lie on the auxiliary circle. Therefore , by ( II. 5 , 6) and (III. 35, 36) , TO ~ ACTZ.TZ' SY . S'Y' = BƠ². (1178) Therefore TC² = AC² ± BC³, a constant value.

NOTE. -Theorems (1170) , ( 1177) , and ( 1203) may also be deduced at once for the ellipse by orthogonal projection from the circle ; and, in all such cases, the analogous property of the hyperbola may be obtained by a similar projection from the rectangular hyperbola if the property has already been demonstrated for the latter curve.

1218 If the points A, S ( Fig. 1162 ) be fixed , while C is moved to an infinite distance, the conic becomes a parabola . Hence, any relation which has been established for parts of the curve which remain finite , when AC thus becomes infinite , will be a property of the parabola.

1219

Theorems

relating to the ellipse may generally be

adapted to the parabola by eliminating the quantities which become infinite, employing the principle that finite differences may be neglected in considering the ratios of infinite quantities. EXAMPLE.-In (1193) , when P ′ is at infinity, VP ′ becomes = 2CP ; and in (1213 ) PS becomes = 2CP. Thus the equations become QV2 CD2 2CD2 = and PS = 2CP CP PV

Therefore QV = 4PS . PV in the parabola.

248

GEOMETRICAL CONICS.

THE

PARABOLA.

If S be the focus , XM the directrix , and P any point

M

on the curve, the defining pro perty is

Y

Z PSPM

1220

and 1221

(1153 ) T

e = 1.

X

S

A

N

G

Hence M' AXAS .

1222

The Latus Rectum = 4AS. SL = SX ( 1220) = 2AS.

PROOF.

1223 If PZ be a tangent at P, meeting the directrix in Z, then PSZ is a right angle. As in (1167) ; theorem ( 1166) applying equally to the parabola.

PROOF.

1224

The tangent at P bisects the angles SPM, SZM.

PROOF. - PZ is common to the triangles PSZ, PMZ ; PS = PM and ▲ PSZ = PMZ ( 1223) .

1225

COR.

ST

SP = SG.

(I. 29, 6)

1226 The tangents at the extremities of a focal chord PQ intersect at right angles in the directrix . PROOF.- ( i . ) They intersect in the directrix, as in ( 1169) . (ii.) They bisect the angles SZM, SZM' ( 1224) , and therefore include a right angle.

1227

The curve bisects the sub-tangent .

PROOF.-

1228

ST = SP (1225) = PM = XN,

and

AN = AT. AX = AS.

The sub-normal is half the latus rectum.

PROOF. - STSP = SG

and TX = SN ( 1227) .

NG = 2AS.

Subtract.

THE PARABOLA.

1229

PN2

249

4AS . AN.

PROOF.- PN TN . NG (VI. 8) = AN . 2NG (1227) = 4AS . AN ( 1228). Otherwise, by ( 1176 ) and ( 1165) ; making AC infinite. See ( 1219) .

1230 The tangents at A and P each bisect SM, the latter bisecting it at right angles . PROOF.- (i .) The tangent at A, by (WI. 2) , since AXAS. (ii ) PT bisects SM at right angles, by (I. 4) , since PS 4 SPY = MPY.

PM and

1231

COR.

1232

To draw tangents from a point O to the parabola.

[By similar triangles.

SA : SY : SP .

CONSTRUCTION.- Describe a circle, centre O and radius OS, cutting the directrix in M, M'. Draw MQ, M'Q ' parallel to the axis, meeting the parabola in Q, Q. Then OQ, OQ will be tangents.

M

PROOF - OS, SQ OM, MQ ( 1220) ; 0 therefore, by (I. 8 ) , ▲ OQS = OQM ; there fore 0Q is a tangent ( 1224) . Similarly OQ' is a tangent . Otherwise, by ( 1181 ) . When S moves to infinity, the circle MM' becomes the directrix. 1233

COR. 1.-The triangles SQO, SOQ' are similar, and SQ : SO : SQ' .

PROOF. Similarly

4 SQO = MQO = SMM' = SOQʻ. SQ'O = SOQ.

(III. 20)

1234 COR. 2. - The tangents at two points subtend equal angles at the focus ; and they contain an angle equal to half the exterior angle between the focal distances of the points . PROOF. Also

LOSQ = OSQ, by (Cor. 1 ). 4QOQ ' = 8OQ + SQO = −OSQ = QSQ.

1235 DEF. -Any line parallel to the axis of a parabola is called a diameter .

1236

The chord of contact QQ' of tangents from any point O is bisected by the diameter through 0.

PROOF. This proposition and the corollaries are included in ( 1198–1200), by the principle in ( 1218 ) . An independent proof is as follows. 2 K

GEOMETRICAL CONICS.

250

The construction being as in ( 1232) , we have ZM = ZM' ; therefore QV= VQ (VI. 2).

M

al Ꭱ

1237

COR . 1. - The

tangent Z P

RR' at P is parallel to QQ' ; and OPPV. PROOF. -- Draw the diameter RW. QW = WP; therefore QR = RO (VI. 2). Similarly Q'R' = R'O.

1.

V

w' R

M 1238

COR . 2.-Hence , the dia

meter through P bisects all chords parallel to the tangent at P.

If QV be a semi-chord parallel to the tangent at P, QV24PS.PV.

1239

This is the fundamental equation of the parabola, equation (1229 ) being the most important form of it. PROOF.-Let QO meet the axis in T. By similar triangles ( 1231 ), 4 SRP SQR = STQ = POR ; and 4 SPR = OPR ( 1224) . Therefore T PRPS.POPS.PV and QV = 2PR. Otherwise : See ( 1219) , where the equation is deduced from ( 1193) of the ellipse. 1240

S

COR. 1.- If v be any other point, either within or

without the curve , on the chord QQ' , and pv the corresponding diameter , 1241

vQ.vQ' = 4pS.pv.

COR. 2.- The focal

(II. 5)

chord parallel to the diameter

through P, and called the parameter of that diameter , is equal to 4SP. For PV in this case is equal to PS.

1242 The products of the segments of intersecting chords , QOq, Q'Oq' , are in the ratio of the parameters of the diameters which bisect the chords ; or

OQ.Oq : OQ' . Oq' = PS : P'S. PROOF.-By ( 1240) , the ratio is equal to 4PS.p0 : 4PS . p0. PS.PS CD2 = Otherwise : In the ellipse ( 1214) , the ratio is = PS.P'S (1213) CD" PS = when S' is at infinity and the curve becomes a parabola ( 1219) . P'S'

251

ON CONSTRUCTING THE CONIC.

1243 COR. - The squares of the tangents to a parabola from any point are as the focal distances of the points of contact . PROOF.

As in ( 1215 ) .

Otherwise, by ( 1233) and (VI. 19) .

1244 The area of the parabola cut off by any chord QQ ' is two-thirds of the circumscribed parallelogram, or of the tri angle formed by the chord and the tangents at Q, Q'. PROOF.- Through Q, q, q' , &c., adjacent points on the curve, draw right lines parallel to the diameter and tangent at P. Let the secant Qq cut the diameter in O. Then, when q coincides with Q , so that Qq becomes a tangent, we have OP = PV (1237). Therefore the parallelogram Vq = 2Uq, by (I. 43) , applied to the parallelogram of which OQ is the diagonal. Similarly vq = 2uq', &c. Therefore the sum of all the evanescent par allelograms on one side of PQ is equal to twice the corresponding sum on the other side ; and these sums are respectively equal to the areas PQV, PQU. (NEWTON, Sect. I. , Lem. II.)

U u

P

Practical methods of constructing the Conic. 1245

To draw the Ellipse.

Fix two pins at S, S (Fig. 1162) . Place over them a loop of thread having a perimeter SPSSS +AA'. A pencil point moved so as to keep the thread stretched will describe the ellipse, by (1163). 1246 Otherwise.- ( Fig. 1173. ) Draw PHK parallel to QC, cutting the axes in H, K. PK = AC and PH = BC (1174) . Hence, if a ruler PHK moves so that the points H, K slide along the axes, P will describe the ellipse.

1247

To draw the Hyperbola .

Make the pin S' ( Fig. 1162) serve as a pivot for one end of a bar of any convenient length . To the free end of the bar attach one end of a thread whose length is less than that of the bar by AA' ; and fasten the other end of the thread to the pin S. A pencil point moved so as to keep the thread stretched, and touching the bar, will describe the hyperbola, by ( 1163) . 1248 Otherwise :-Lay off any scale of equal parts along both asymptotes (Fig. 1188) , starting and numbering the divisions from C, in both positive and negative directions. Join every pair of points L, l, the product of whose distances from C is the same, and a series of tangents will be formed ( 1192) which will define the hyperbola. See also ( 1289) .

252

GEOMETRICAL CONICS.

To draw the Parabola.

1249

Proceed as in ( 1247) , with this difference : let the end of the bar, before attached to S', terminate in a " T-square," and be made to slide along the directrix (Fig. 1220) , taking the string and bar of the same length.

1250 Otherwise :-Make the same construction as in ( 1248) , and join every pair of points, the algebraic sum of whose distances from the zero point of division is the same. PROOF. - If the two equal tangents from any point T on the axis (Fig. 1239) be cut by a third tangent in the points R, r ; then RQ may be proved equal to rT, by ( 1233) , proving the triangles SRQ, SrT equal in all respects. 1251 SQT.

COR.-The triangle SRr is always similar to the isosceles triangle

1252 To find the axes and centre of a given central conic. (i.) Draw a right line through the centres of two parallel chords. This line is a diameter, by (1198) ; and two diameters so found will intersect in the centre of the conic. (ii.) Describe a circle having for its diameter any diameter PP' of the conic, and let the circle cut the curve in Q. Then PQ, PQ are parallel to the axes, by ( 1201 ) and ( III. 31 ) .

1253

Given two conjugate diameters , CP, CD, in position and magnitude : to construct the conic.

On CP take PZ = CD² ÷ CP ; measuring from C in the ellipse, and towards C in the hyperbola (Fig. 1188) . A circle described through the points C, Z, and having its centre O on the tangent at P, will cut the tangent in the points where it is intersected by the axes. PROOF. -Analysis : Let AC, BC cut the tangent at P in T, t. The circle whose diameter is Tt will pass through C (III. 31), and will make

CP.PZ = PT . Pt (III. 35 , 36) = CD² (1197). Hence the construction .

Circle and Radius of Curvature. 1254

DEFINITIONS . -The circle which has the same tangent

with a curve at P (Fig . 1259 ) , and which passes through another point Q on the curve, becomes the circle of curvature when Q ultimately coincides with P ; and its radius becomes the radius of curvature .

CIRCLE OF CURVATURE.

253

1255 Otherwise .- The circle of curvature is the circle which passes through three coincident points on the curve at P. 1256 Any chord PH of the circle of curvature is called a chord of curvature at P. 1257

Through Q draw RQ' parallel to PH, meeting the

tangent at P in R, and the circle in Q' , and draw QV parallel to PR. RQ is called a subtense of the arc PQ.

1258 THEOREM. -Any chord of curvature PH is equal to the ultimate value of the square of the arc PQ divided by the subtense RQ parallel to the chord : and this is also equal to QV² ÷ PV. PROOF. - RQ RP² ÷ RQ (III. 36) . And when Q moves up to P, RQ ' becomes PH; and RP, PQ, and QV become equal because coincident lines. 1259 In the ellipse or hyperbola, the semi- chords of cur vature at P, measured along the diameter PC, the normal PF, and the focal distance PS, are respectively equal to CD2 CD2 CD2 CP'

PF'

AC ;

the second being the radius of curvature at P.

R D R

Մ

E

TA H

2CD2 VP.CD QV¹ in the (1193) = PROOF.-(i . ) By (1258), PH = PV = CP CP3 limit when VP ' becomes PP = 2CP. (ii.) By the similar triangles PHU, PFC (III. 31 ) , we have PU.PF CP . PH = 2CD , by (i.) (iii.) By the similar triangles PIU, PFE (1168) , we have PI.PE PU . PF = 2CD , by (ii.) ; and PE = AC ( 1179) .

GEOMETRICAL CONICS.

254

1260 In the parabola , the chord of curvature at P (Fig. 1259) drawn parallel to the axis , and the one drawn through the focus , are each equal to 4SP, the parameter of the dia meter at P (1241 ) . PROOF.- By ( 1258 ) . The chord parallel to the axis = QV¹ ÷ PV = 4PS ( 1239) ; and the two chords are equal because they make equal angles with the diameter of the circle of curvature.

1261

COR.- The radius of curvature of the parabola at P

(Fig. 1220) is equal to PROOF. (Fig. 1259.)

1262

2SP2

SY.

PUPI sec IPU = 2SP sec PSY (Fig. 1221 ) .

The products of the segments of intersecting chords

are as the squares of the tangents parallel to them ( 1214-15 ) , ( 1242-43) . 1263 The common chords of a circle and conic (Fig. 1264) are equally inclined to the axis ; and conversely, if two chords of a conic are equally inclined to the axis , their extremities are concyclic . PROOF. The products of the segments of the chords being equal (III. 35, 36), the tangents parallel to them are equal (1262) . Therefore, by (1200).

1264 The common chord of any conic and of the circle of curvature at a point P, has the P same inclination to the axis as the tangent at P. T

PROOF. - Draw any chord Qq parallel to the tangent at P. The circle circum scribing PQq always passes through the same point p (1263 ) , and does so, there fore, when Qq moves up to P, and the circle becomes the circle of curvature.

R

1265 PROBLEM.-To find the centre of curvature at any given point of a conic. First Method . (Fig. 1264.) Draw a chord from the point making the same angle with the axis as the tangent. The perpendicular bisector of the chord will meet the normal in the centre of curvature, by ( 1264) and ( III. 3) . Second Method. -Draw the normal PG and a perpendicular to it 1266 from G, meeting either of the focal distances in Q. Then a perpendicular to the focal distance drawn from Q will meet the normal in O, the centre of curvature.

MISCELLANEOUS THEOREMS.

PROOF.- (Ellipse or Hyperbola .) By (1259), the radius of curvature at P = CD2 = CD2 PG (1195) = AC PG (1194) BC PF PF2 = PG sec S'PG = PO. For PE (by 1179) . AC (Parabola.)

For 2SP

255

P

S

G 0

By ( 1261 ) . The radius of curvature = 2SP ÷ SY = 2SP sec SPG PO.

PQ,

because SP = SG (1125 ) .

Miscellaneous Theorems. 1267 then

In the Parabola (Fig. 1239 ) let QD be drawn perpendicular to PV, QD² = 4AS.PV. (1231 , 1239)

1268 Let RPR be any third tangent meeting the tangents OQ, OQ′ in R, R; the triangles SQO, SRR, SOQ are similar and similarly divided by SR, SP, SR ( 1233–4) .

1269

COR.

OR . OR = RQ.RQ.

1270 Also, the triangle PQQ 20RR. With the same construction and for any conic, 1271

OQ : OQ' = RQ . RP : RQ. PR. RSR = 1QSQ.

(1244)

(1215, 1243)

1272

Also the angle

1273

Hence, in the Parabola, the points O, R, S, R are concyclic, by (1234).

(1181)

1274

In any conic (Figs . 1171 ) , SP : ST = AN : AT. AN PROOF. SP = SP = CA (69, 1163) = CN (1170) = AT (69). ST S'T CA CT

1275 COR.- If the tangent PT meets the tangent at A in R, then SR bisects the angle PST (VI. 3) . 1276

In Figs. (1178 ) , SY' , S'Y both bisect the normal PG.

1277

The perpendicular from S to PG meets it in CY.

1278 If CD be the radius conjugate to CP, the perpendicular from D upon CY is equal to BC. 1279

SY and CP intersect in the directrix.

1280 If every ordinate PN of the conic ( Figs. 1205 ) be turned round N, in the plane of the figure, through the same angle PNP, the locus of P' is also a conic, by ( 1193 ) . The auxiliary circle then becomes an ellipse, of which AC and BC produced are the equi-conjugate diameters. If the entire figures be thus deformed, the points on the axis AA' remain fixed while PN, DR describe the same angle. Hence CD remains parallel to PT. CP, CD are therefore still conjugate to each other.

256

GEOMETRICAL CONICS.

Hence, the relations in (1205-6 ) still subsist when CA, CB are any con jugate radii. Thus universally, PN : CR = DR : CN or PN . CN = DR . CR.

1281

1282 If the tangent at P meets any pair of conjugate diameters in T, T ', then PT . PT is constant and equal to CD³. PROOF.- Let CA, CB (Figs. 1205 ) be the conjugate radii, the figures being deformed through any angle. By similar triangles , PT : PN = DRY therefore PT . PT : PN . CN = CD³ : DR . CR. PT : CNCD : CR } , Therefore PT . PT ′ = CD, by ( 1281 ) . 1283 If the tangent at P meets any pair of parallel tangents in T, T, then PT. PT CD , where CD is conjugate to CP. PROOF. Let the parallel tangents touch in the points Q, Q. Join PQ, PQ, CT, CT. Then CT, CT are conjugate diameters ( 1199 , 1201 ) . There fore PT.PT CD (1282) . 1284

COR.-QT. QT = CD², where CD is the radius parallel to QT.

1285 To draw two conjugate diameters of a conic to include a given angle. Proceed as in ( 1252 ii . ) , making PP in this case the chord of the segment of a circle containing the given angle (III. 33 ) . 1286 The focal distance of a point P on any conic is equal to the length QN intercepted on the ordinate through P between the axis and the tangent at the extremity of the latus rectum. PROOF.-(Fig. 1220) . QN : NX = LS : SX = e and SP : NX = e. · (1162 , 1164 ). 1287 In the hyperbola ( Fig. 1183 ) . CO : CA = e. If a right line PKK' be drawn parallel to the asymptote CR, cutting the one directrix XE in K and the other in K' ; then ' = e . CN+ AC. 1288 SP = PK = e . CN - AC ; SP = PK PROOF.- From CR = e . CN ( 1287) and CE = AC ( 1186 ) . 1289 COR. -Hence the hyperbola may be drawn mechanically by the method of ( 1249) by merely fixing the cross- piece of the T- square at an angle with the bar equal to BCO. 1290

DEFINITION.-Confocal conics are conics which have the same foci.

1291 The tangents drawn to any conic from a point T on a confocal conic make equal angles with the tangent at T. PROOF.

(Fig. 1217. ) Let T be the point on the confocal conic. SY : SZ SZ : SY ′ ( 1178) . Therefore ST and ST make equal angles with the tangents TP, TQ ; and they also make equal angles with the tangent to the confocal at T ( 1168 ) , therefore &c. 1292 In the construction of (1253) , PZ is equal to half the chord of curvature at P drawn through the centre C ( 1259 ) .

DIFFERENTIAL

CALCULUS .

INTRODUCTION .

1400 Functions . -A quantity which depends for its value upon another quantity a is called a function of x. Thus , sin x, The notation log x, a*, a² +ax +x² are all functions of x. y=f(x) expresses generally that y is a function of a. is a particular function .

y = sina

1401 f(x) is called a continuous function between assigned limits , when an indefinitely small change in the value of a always produces an indefinitely small change in the value of f(x). A transcendental function is one which is not purely algebraical, such as the exponential, logarithmic , and circular functions a* , log æ, sinæ , cos æ, &c . If ƒ(x) = ƒ( −x) , the function is called an even function . If f(x) = -ƒ(-a) , it is called an odd function . Thus, and cos az are even functions, while a³ and sin x are odd functions of ; the latter, but not the former, being altered in value by changing the sign ofx.

1402 Differential Coefficient or Derivative.- Let y be any function of a denoted by f(x) , such that any change in the value of a causes a definite change in the value of y ; then a is called the independent variable , and y the dependent variable . Let an indefinitely small change in a, denoted by da, produce a corresponding small change dy in y ; then the ratio

, in

the limit when both dy and de are vanishing, is called the differential to x.

coefficient,

or

derivative,

2 L

of

y

with

respect

DIFFERENTIAL CALCULUS .

258

1403 THEOREM .-The ratio dy : de is definite for each value of a, and generally different for different values . PROOF. ―― Let an abscissa ON (Defs . 1160 ) be measured from 0 e equal to x, and a perpendicular or R dinate NP equal to y. Then, what S ever may be the form ofthe function y = f(x), as x varies, the locus of P will be some line PQL. Let OM = x', MQy' be values of x and y near to the former values . M N T Let the straight line QP meet the axis in T; and when Q coincides with P, let the final direction of QP cut the axis in T. PN Then QS or y - y = And, ultimately, when QS and SP vanish, NT x'--x SP dy PN they vanish in the ratio of PN : NT'. Therefore dr = NT = tan PTN, a definite ratio at each point of the curve, but different at different points. 1404

Let NM, the increment of a, be denoted by h ; then, dy_f(x +h) -f(x) =ƒ " (x), h dx

when h vanishes ,

a new function of.a, called also the first derived function. The process of finding its value is called differentiation .

1405

dy Successive differentiation . — If d or f ' (x) be differ de

entiated with respect to a, the result is the second differential coefficient of f(x) , or the second derived function ; and so on to any number of differentiations . These successive functions may be represented in any of the three following systems of notation : dry ; d³y d'u dy d'y ...... dx' dx da dx dx f ( c ),

f " (ư ) , f '' ( 2 ) ,

Yx

Y2x

Y&x

fi ( ), ...... ƒ" (x) ; …………… . Yne Yax

The operations of differentiating a function of a once , twice, or n times , are also indicated by prefixing the symbols d

d2

dx'

dx²

dn

or, more concisely,

d

d

d

; or

...

dam de, ders

dx' ( ) dx dnx .

* See note to ( 1487 ) .

( );

259

INTRODUCTION.

1406 If, after differentiating a function for æ, æ be made zero in the result , the value may be indicated in any of the d dy , f' (0) , Yxo , following ways : " dro dx . dxo If any other constant a be substituted for a in y , the result may be indicated by. ya .

— 1407

Infinitesimals and Differentials .

The

evanescent

quantities de, dy are called infinitesimals ; and, with respect toa and y, they are called differentials . da , d'y are the second differentials of x and y ; da³ , d³y the third , and so on. 1408 The successive differentials of y are expressed in terms of de by the equations dy = f (x) dx ;

d'y = f" (x) dx² ;

& c. ,

and dry = ƒ" (x) dx” .

Since f' (x) is the coefficient of de in the value of dy , it has therefore been named the differential coefficient of y or f(x) . * For similar reasons f ' (x) is called the second, and f" (a) the nth differential coefficient of f (x) , &c . 1409 Two infinitesimals are of the same order when their ratio is neither zero nor infinity. If dæ, dy are infinitesimals of the same order, dæ² , dy³ , and dedy will be infinitesimals of the second order with respect to dx, dy ; dx³ , dx²dy , &c . will be of the third order , and so on . da, da² , &c. are sometimes denoted by x, x, &c.

1410 LEMMA.-In estimating the ratio of two quantities , any increment of either which is infinitely small in comparison with the quantities may be neglected . Hence the ratio of two infinitesimals of the same order is not affected by adding to or subtracting from either of them an infinitesimal of a higher order .

EXAMPLE . the ratio

dy- dx² = dy - dx = dy " for de is zero in comparison with dx dx da . Thus, in Fig . (1403), putting PS = de, QSdy ; we have

Therefore ultimately, by (1258 ) , QR = kda², where k is a constant . PN = RS = dy -kdx² = dy the limit, by the principle just enunciated ; PS NT' dx dx in that is, QR vanishes in comparison with PS or QS even when those lines them selves are infinitely small. The name is slightly misleading, as it seems to imply that f' (x) is in some sense a coefficient of f (x) .

DIFFERENTIAL CALCULUS.

260

DIFFERENTIATION.

DIFFERENTIATION OF A SUM, PRODUCT, AND QUOTIENT.

Let u, v be functions of a, then 1411

d (u +v) dx

du

dv +

dx

dx

1412

dv du d (uv) = v • +u dx dx dx

1413

d - dv သ d x v = (v du Z (H) u do dx:) ) +0°. dx

PROOF.-(i. ) d (u + v ) = (u + du + v + dv) — ( u + v) = du + dv. (ii.)

d (uv) = (u + du) (v + dv) —uv = v du + udv ― du dv,

and, by (1410) , dudu disappears in the ultimate ratio to dr. u + du vdu- udv น = = d (iii. ) v (v + dv) v ' (뜸) v + dv

therefore &c., by ( 1410) ; vdu vanishing in comparison with v³. Hence , if u be a constant = c, 1414

dv d (cv) = C dx dx

c

dv

dx v² d ( ) ===

dx

and

DIFFERENTIATION OF A FUNCTION OF A FUNCTION.

If y be a function of z , and z a function of x,

dy = dy dx dz

1415

dz dx

PROOF . Since, in all cases, the change de causes the change dz, and the change dz causes the change dy ; therefore the change da causes the change dy in the limit. Differentiating the above as a product, by (1412) , the successive differ ential coefficients of y can be formed . The first four are here subjoined for the sake of reference. Observe that (y :) = Y2=%x•

1416

Yx = Yz≈x

1417

Y 2x = Y 2x 2 ~² + y z~2x •

1418

Y3.x = Y 3z ≈x3 + 3y zz ~ x ~ 2x + yz ~3x •

1419

Y₁v = Y₁₂ ~ + 6Y3₂ ~* ~2x + Y2z ( 3 ~2x + 4≈¸µ≈3x) + Yz≈µ •

261

DIFFERENTIATION.

DIFFERENTIATION OF A COMPOSITE FUNCTION . If u and v be explicit functions of x, so that u = p (x) and v = f (x),

dF (u, v)

dF du

dx

du dx

1420

dF dv + dv

dx

Here dF in the first term on the right is the change in F (u, v) produced by du, the change in u ; and dF in the second term is the change produced by dv , so that the total change dF(u, v) may be written as in ( 1408)

dF

dF

du +

dF₁ + dF₂ =

-dv.

dv

du

DIFFERENTIATION OF THE SIMPLE FUNCTIONS.

__f(x + h) —f (x) when h vanishes , we have the Since dy da h following rule for finding its value : 1421 RULE.-Expand f (x + h) by some known theorem in ascending powers of h ; subtract f (x) ; divide by h ; and in the result put h equal to zero. The differential coefficients which follow are obtained by the rule and the theorems indicated .

y = x".

1422

dy = n-1 dx

PROOF.—Here ƒ (≈ +h) —f(x) = (x + h)" - x" = na" -¹h + C (n, 2) x" -²² + h h h (125) = no" -1 + 0 (n, 2) 2" -2h + ... = na"-1, when h vanishes. 1423

COR. d'y = n (n − 1) ... (n − r + 1) x² -r.

dx

1424

y = loga x ;

dry dx"

n

1 dy = • dx x log, a 1

Proor .— By ( 145) , log , (z + ) — log. h

{ log. ( 1+

=

) } + h.

x log, a

Expand the logarithm by (155) .

1425

COR.

dry = ( -1 ) -¹ [ n - 1 dxn x" log a

Put n = -1 in (1423) and rn- 1.

DIFFERENTIAL CALCULUS.

262

dy = a* log, a. dx

y = a* ;

1426

-

= a* (a - 1) h

PROOF. h

1427

dry = a* (log, a)". dan

COR.

Function.

Derivative.

Method ofProof by Rule (1421) and Limits (753).

Expand by (627, 629) , and h put 1 - cosh = 2 sin² 2.

1428

sin x.

cos .

1429

cos x.

- sin x.

1430

tan x.

sec² x.

1431

cot x.

- cosec²x.

1432

secx.

1433

cosecx.

- cota coseca.

1434

sin-¹ x cos -1x

士

1436

1438

tan x secx.

1 √ (1 − x²)*

-1 tan-¹x cot-1 x

1 1+

sec¹x cosec-1 x } ·

If sin xy, x = sin y, therefore da = cos y = √1 − x² ; dys 1 therefore dy = dx √1-2

1 ± x √ (x² −1) *

Similarly for the rest.

1 2x

n == 2" +1° (- ) = ( -) =---- =

1441

1442

Expand by (631 ), observing (1410). 1 By cota = tan a and (1415) . 1 " and (1415). By seca = COS2 Similarly.

EXAMPLES . (√ /x); = (x³), = }æ¯ ‡ =

1440

Expand a by (149) .

(1422)

(1422)

{ (a + x²)³ ( b + x³) ² } = 3 ( a + x²) ² 2x ( b + x³)² + 2 ( b +x³) 3x² (a +x²) ³ = 6x (a +x²) (b + x³) (b + ax + 2x³) . (1412, '15, '22)

e. 1443

1444

4 = (e +e *) ( * + e ) - (e - e-*) ( e* —e¯ *) = 2 ete / (e*+ e¯*)³° (e*+ e-*)³ ( 1413 , 1426)

1 4 log tan a (1415, '24, '30) . d. (log tan a) = 2 log tan a tanx seca =" şin 2x

263

SUCCESSIVE DIFFERENTIATION.

Some differentiations are rendered easier by taking the logarithm of the function. For example,

1 - x2 1445

therefore log y =

y=

log ( 1 - x ) — 3 log ( 1 + x² ) ;

(1 +2²)3 1 dy

therefore therefore

1446

therefore therefore

1 -2x - 3 2x = y da 2 (1 - x ) 2 (1 +x²) - 2x (2 - x²) = -2x (2 - x²) dy y dx = 1 - x¹ (1 + ∞²) ³ (1 − ∞²) ** y = (sin x) ; therefore log y = log sin x ; • 1 cos x ; yx = log sin x + (1415 , '24, '28) sin x y = (sin x) (log sin x +x cot x).

Otherwise, by (1420) , y . = x ( sin x) -¹ cos x + ( sin x) * log sin x = (sin x) (x cota + log sin x) .

SSIVE SUCCESSIVE

1460

(1426)

DIFFERENTIATION.

Leibnitz's Theorem. - If n be any integer ,

(yz) nx = Y nxz + ny(n −1) x ≈x + C (n , 2) Y(n− 2) x ≈2x + ... ... + C (n, r) Y(n−r) x ≈rx + ... +ynx PROOF.-By Induction (233) .

Differentiate the two consecutive terms

C (n, r) Y(n - r) x Zrz + C ( n, r + 1 ) Y(n −r− 1) z ²(r + 1 ) z9 and four terms are obtained, the second and third of which are C (n, r) Yu - r) z Z (p + 1) z + C ( n , r + 1 ) Y (n - r) z z ( +1) = { C (n, r) + C ( n , r + 1 ) } Y (n - r ) z (r + 1 ) z = C ( n + 1 , r + 1 ) y ( 71-771 ) ( + 1 )2 by (102) . This is the general term of the series with n increased by unity. Similarly, by differentiating all the terms the whole series is reproduced with n in creased by unity.

DIFFERENTIAL COEFFICIENTS OF THE nth ORDER.

1461

(sin ax) nx = a" sin (ax + ¼nπ) .

1462

(cos ax) nx = a" cos (ax + ½µπ) .

1463

(ex)nx = an eªx.

1464

(exy)nx = eax (a + dx)" y,

By Induction and (1428 ) .

( 1426)

where , in the expansion by the Binomial Theorem , dry is to be replaced by Yra ( 1460, '63)

DIFFERENTIAL CALCULUS .

264

1465

(ex cos bx) nx = r²eªx cos (bx +no),

where

arcos

PROOF.- By Induction.

and b = r sin p.

Differentiating once more, we obtain

y” eªx { a cos (bx +np) −b sin ( bx + ng) } = p²+ ¹ª² { cos & cos (bx + ng) — sin ø sin (bx + ng) } = p² +1ear cos (bx + n + 14) .

Thus n is increased by one.

1466

(x² -¹log x) nx = | n − 1 ÷ x.

1467

= (-1)" 2 (1 +x)n+1 [n. 1 + xnx

1468

(1460) , ( 283)

(1423)

(tan-¹x)nx = ( -1 ) " - ¹ | n − 1 sin " ✪ sin no,

where

0

cot-¹x.

PROOF.-By Induction. Differentiating again, we obtain (omitting the coefficient) (n sin" -10 cos 0 sin n✪ + n cos no sin” 0) 0, = sin"-10 (sin no cos 0+ cos no sin 0) ( -sin' 0). 0, = - (1 + x²) -1 = -sin² 0. Since, by (1437), Therefore

(tan¹ ) (n + 1)

= ( −1 ) " [ n sin" + ¹0 sin (n + 1 ) 0,

n being increased by one.

1469

(1

)Nx

1470

(

)

= (-1)" n sin +1

sin ( n +1 ) 0 .

(1436, 1468)

= = ( −1) ” | n sin” +¹ 0 cos (n + 1) 0.

PROOP - By (1460), Then by ( 1469) .

1471

(

1 =x n (----) ร+ (----) ) NE = 1

2:

Jacobi's Formula.

: d(n −1) x ( 1 − x²)" −¹ = ( −1 ) " − ¹ 1 . 3 ... (2n - 1 ) sin (n cos¯¹x) ÷ n.

PROOF.- Let y = 1 - a2 ; therefore (y" + ¹) nz = − ( 2n + 1 ) (xy " ) (n - 1) z•

Also (y"

) ,z = (yy" -}) nz•

Expand each of these values by ( 1460) and eliminate (y" - )( - 2) , the deriva tive of lowest order. Call the result equation ( 1 ) . Now assume ( 1471 ) true for the value n. Differentiate and substitute the result, and also (1471 ) on the right side of equation (1 ) to obtain a proof by Induction.

THEORY OF OPERATIONS.

265

1472 Theorem .- If y, z are functions of x, and n a positive integer , Zynx = (YZ)nx —N (YZx) (n− 1) x + C (n , 2 ) (y≈2x) (n - 2) x ... + ( − 1 )" Y≈nx• PROOF.-By Induction. Differentiate for x, substituting for zyne on the right its value by the formula itself.

PARTIAL

DIFFERENTIATION .

1480 If u = f(x, y) be a function of two independent vari ables, any differentiation of u with respect to a requires that Y should be considered constant in that operation , and vice versa. d'u Thus , or a signifies that u is to be differentiated succes dx² sively twice with respect to x, y being considered constant . d'u * 1481 The notation or Uar sy signifies that u is to be da dys differentiated successively twice for x, y being considered constant, and the result three times successively for y, a being considered constant . 1482 The order of the differentiations does not affect the final result, or Uxy = Uyx • ƒ (x + h. u ) —f (x, y) in limit. (1484) PROOF.- Let u = f(xy) ; then u₂ = h Uxy = du, = _ f(x + h, y + k) −ƒ ( x, y + k) —ƒ (x + h, y) +ƒ (x, y) in limit. hk dy

Now, if u had been first formed, and then u,, the same result would have been obtained. The proof is easily extended . Let u₂ = v ; then Wzxy = V zy = v.yx = Uy ; and so on .

THEORY

OF

OPERATIONS .

1483 Let the symbols , Y, prefixed to a quantity, denote operations upon it of the same class , such as multiplication or differentiation . Then the law of the operation is said to be distributive, when

• (x +y) = Þ (x) + Þ (y) ; * See note to ( 1487) . 2 M

266

DIFFERENTIAL CALCULUS.

that is , the operation may be performed upon an undivided quantity, or it may be distributed by being performed upon parts of the quantity separately with the same result. 1484

The law is said to be commutative when

p¥x = Yox ; that is , the order of operation may be changed , upon

a producing the same result as

operating

operating upon Þæ.

1485 pm denotes the repetition of the operation & m times , and is equivalent to ❤❤ ... a to m operations. This definition involves the index law,

††" x = Þ” +” x = Þ” +” x, which merely asserts , that to perform the operation n times in succession upon a, and afterwards m times in succession upon the result, is equivalent to performing it m + n times in suc cession upon æ. 1486

The three laws of Distribution , Commutation, and the

law of Indices apply to the operation of multiplication , and also to that of differentiation ( 1411 , '12) . Therefore any algebraic transformation which proceeds at every step by one or more of these laws only, has a valid result when for the operation of multiplication that of differentiation is sub stituted . 1487

In making use of this principle, the symbol of dif d ferentiation employed is > or simply dx prefixed to the dx quantity upon which the operation

of differentiating with

respect to a is to be performed . The repetition of the opera de ds indicated tion is by da' d ' dady³ 15 ' & c . , prefixed to the function. An abbreviated notation is de, da , dзx, dzxsy , &c . Since dxdd in the symbolic operation of multiplication , it will be requisite , in transferring the operation to differentia tion, to change all such indices to suffixes when the abbreviated notation is being used .

NOTE. The notation Y. , Y , U23y, daзy, &c. is an innovation. It has, how ever, the recommendations of definiteness, simplicity, and economy of time in dou writing, and of space in printing. The expression requires at least dx dys fourteen distinct types, while its equivalent 23y requires but seven. For

THEORY OF OPERATIONS.

267

such reasons I have introduced the shorter notation experimentally in these pages. ...... All such abbreviated forms ofdifferential coefficients as y'y" y" ... .. or yyy..., though convenient in practice, are incomplete expressions, because the inde pendent variable is not specified . The operation day, and the derived function sy, would be more accurately represented by (d ) and (u ) , the index as usual indicating the repetition of the operation. But the former notation is simpler, and it has the advantage of separating more clearly the index of differentiation from the index of involution. In the symbols y' and y2 , the figure 2 is an index in each case : in the first, it shows the degree of involution ; in the second, the order of differentia tion. The index is omitted when the degree or the order is unity, since we write Y and yz The suffix takes precedence ofthe superfix. y means the square of Yx. dz (y ) would be written (y ) , in this notation . As a concise nomenclature for all fundamental operations is of great dy assistance in practice, the following is recommended : dx or y, may be read "yfor x," as an abbreviation of the phrase, "the differential coefficient of y d²u d'u day or the shortened for, or with respect to, a." Similarly, dx ' dxdy' dx dy³¹ 99 66 forms Yaz, Uzy, U23y, may be read "y for two x," u for xy,'," "u for two x, three y," and so forth. The distinction in meaning between the two forms yn and y , is obvious . The first (in which n is numerical and always an integer ) indicates n succes sive differentiations for ; the second indicates two successive differentiations for the variables x and z. The symbols dy or yxo and dzy or 220, may be read, for shortness , dxo dr² "y for x zero," "y for two x zero "; dzzz3y y (xy) can be read " d for two a three y of p (xy) .' Although the notation v, is already employed in a totally different sense in the Calculus of Finite Differences, my own experience is that the double signification of the symbol does not lead to any confusion : and this for the very reason that the two meanings are so entirely distinct . Whenever the operation of differentiation is introduced along with the subject of Finite Differences, the notation dy must of course alone be employed . da

Thus , in differentiation , we have

1488

THE DISTRIBUTIVE LAW

d (u +v) = d¸u + d¸v .

1489

THE COMMUTATIVE LAW

d. (d¸u)

= d, (d¸u)

dzy u

= dyxu.

d™ dru

= dm+nu,

dma dnx u

= d(m+n) x U.

or

1490 that is,

THE INDEX LAW

(1411)

(1482)

(1485)

268

1491

DIFFERENTIAL CALCULUS.

EXAMPLE.—

ry + d2y. (d₂ — d,) ² = (d¸ —− d¸ ) ( d¸— d₁) = d¸d₂ - d₂d - d¸d¸ + d¸d₁ = d₂ - 2dxy (1489 ) Here dd - dd, or d , = dy , by the commutative law. dd, = d by the index law. (1490) Also (d - 2dy + dy ) u = d₂u - 2du +du, by the distributive law. Therefore, finally, (d - d, ) u = du - 2du + dzu . Similarly for more complex transformations.

1492 Thus d, may be treated as quantitative, and operated upon as such by the laws of Algebra ; d being written de, and factors such as dd,, in which the independent variables are different, being written d¸y, &c.

EXPANSION OF EXPLICIT

FUNCTIONS .

TAYLOR'S THEOREM.-EXPANSION OF ƒ (x +h) .

1500

hn

h2

f(x+h) =f(x) +hf ' (x) + 1.2ƒ " (x) + ... + 1n ƒ" (x + 0h), where is some quantity between zero and unity, and n is any integer. PROOF. (i.) Assume f(x +h) = A + Bh + Ch³ +&c. Differentiate both sides of this equation,-first for a, and again for h,-and equate coefficients in the two results. 1501 (ii. ) Cox's Proof.—LEMMA.—If ƒ ( 2 ) vanishes when x = a, and also when x = b, and if f (x) and f' ( x) are continuous functions between the same limits ; then f (x ) vanishes for some value of a between a and b. For f ( ) must change sign somewhere between the assigned limits (see proof of 1403 ) , aud , being continuous, it must vanish in passing from plus to minus.

1502

Now, the expression ƒ(a + x) →ƒ(a) − xƒ′ (a) — ...

—

n -f" (a)

{( )

:

+

2"+1 [ n + 1 {ƒ (a +h) −ƒ (a) —hf ( a) — vanishes when x = 0 and when x = h. Therefore the differential coefficient with respect to a vanishes for some value of a between 0 and h by the lemma. Let Oh be this value. Differentiate, and apply the lemma to the resulting expression, which vanishes when 0 and when a = 00h. Perform the same process n + 1 times successively, writing Oh for 00h, &c. , since ✪ merely stands for some quantity less than unity. The result shews that In + 1 h" f" +¹ (a + x) — 1 +17 { ƒ (a + 1) − ƒ (a) — 15 ″ (a) —... - Hƒ «) } (a) h +1 n

vanishes when x = 0h. is proved .

Substituting Oh and equating to zero , the theorem

EXPANSION OF EXPLICIT FUNCTIONS.

1503

269

The last term in ( 1500) is called the remainder after

n terms. It may be obtained in either of the subjoined forms , the first being due to Lagrange ,

h" n -Jin (+ 0h )

h” or n

-1 (1—0)n-¹ƒ” (x + Oh) .

hn 1504

diminishes at last without

Since the coefficient n

limit as n increases (239 , ii. ) , it follows that Taylor's series is convergent if f (x) remains finite for all values of n.

1505

If in any expansion of f (x + h) in powers of h some

index of h be negative, then f(x), ƒ' (x) , ƒ'' (x) , &c . all become infinite. 1506

If the least fractional index of h lies between n and

n + 1 ; then f +1 (a ) and all the following differential coeffi cients become infinite. PROOF . -To obtain the value of f" (x) , differentiate the expansion n times successively for h, and put h = 0 in the result.

MACLAURIN'S THEOREM . Put 0 in ( 1500) , and write a for h ; then , with the notation of ( 1406 ) ,

222 xn ƒ" 1507_ƒ(x) = ƒ (0) + xƒ “ (0) + 1.2“ ' (0) + ... + n -ƒn (0x) , where 0, as before , lies between 0 and 1 . Putting y = f(x) , this may also be written

1508

y = yo + x

X3 x² d²y dry + &c. dy + + 3 1.2.3 da dxo 1.2 da

1509 NOTE. If any function f (x) becomes infinite with a finite value of æ, then ƒ ′ (x) , ƒ” ( x) , &c. all become infinite. Thus, if ƒ ( x) = sec¹ (1+ ) , f ' (x) is infinite when a == 0 ( 1438 ) . Therefore ƒ" (0 ), ƒ''' (0) , &c. are all infinite, and ƒ (x) cannot be expanded by this theorem .

Bernoulli's Series.-Put ha in ( 1500) ; thus ,

— 1510_ƒ(0) =ƒ(x) −xƒ (x) + 12ƒ˜ ′ (x) − 1.2.3ƒ¨ (x) + &c.

270

1511

DIFFERENTIAL CALCULUS.

If

PROOF. therefore

(y + k) = 0 and

(y) = x ; then Xx3 x² k= Y& +&c. ―xyx + 1.2 Y2x 1.2.3 Let

y = ¢ ¹ (x) = ƒ (x) , and let y + k = f (x + h) ; x + h = (y + k) = 0. 2:3 -Therefore y + k = ƒ (0 ) = ƒ (x) −xƒ′ (x ) + 122 ƒ″ (x ) − &c., by ( 1510) ; which proves the theorem .

EXPANSION OF ƒ (x + h , y + k) .

Let f(xy) = u.

Then, with the notation of ( 1405 ) ,

1512 f(x + h, y + k) = u + (hu₂ + ku,) +

1 1.2 (h³U2x +2hku.xy + k²U2y)

1.2.3 (h³U3x +3h³kUqxy + 3hk³U12y + k³U3y ) + &c. 1513

The general term is given by

(hd¸ +kd ) " u , n

where, in the expansion by the Binomial Theorem, each index of dx and d, is changed into a suffix ; and the coefficients 2x9 &c. are joined to u as symbols of operation (1487) ; da, dae, 3 thus H is to be changed into us .

PROOF. - First expand f (x + h, y + k) as a function of (x +h) by ( 1500) ; 1 thus, f(x + h, y + k) = f (x, y + k) + hfx ( x, y + k) + 1.2 h²ƒ2x (x, y +k) + &c. Next, expand each term of this series as a function of (y + k) . Thus, writing u for f(xy), 1 1 732 34 12 Uw u + f(x, y + k) = -k⭑Usy + ... ku, + +3 +141 Wus 1 hf, (x, y + k) =

hu₂ +

hku

+ 2

•hk³uzz x29y

hk³uzзy + ...

+

...

.:.

...

:.

. :

...

... :.

...

:.

1 ƒœ (x, y + k) = 4 h⭑Usx +

...

...

:

1 1 h²uzz + h² kuzzy + h²k² Uzzzy + f2x (x, y + k) 22 | _2 224 =2 / 12 h³ h³uzz + h³kuszy + ... ... ... 13 ·fx (x, y + k ) = 3 1 13 h¹

...

13

...

The law by which the terms of the same dimension in h and k are formed, is seen on inspection . They lie in successive diagonals ; and when cleared of fractions the numerical coefficients are those of the Binomial Theorem.

EXPANSION OF EXPLICIT FUNCTIONS.

271

The theorem may be extended inductively to a function of three or more variables . Thus , if u = f (x, y , z ) , we have 1514

f(x+ h, y +k, z + 1) = u + (hu + ku, + lu₂)

+1 (h²u2x + k²U2y + l²u2z + 2kluyz + 2lhuz.x + 2hku, xy ) + ….. , the general term being obtained as before from the expression

1 (hd, +kd¸ +ld )" u . n

1515

COR .- If u = f (xyz) be a function of several inde

pendent variables , the term (hu + ku, + lu ) proves , in con junction with ( 1410 ) , that the total change in the value of u, caused by simultaneous small changes in x, y , z , is equal to the sum of the increments of u due to the increments of x , y , z taken separately and superposed in any order. This is known as the principle of the superposition of small quantities.

1516

To expand f (x, y) or f (x, y, z, ... ) in powers of x, y, &c., put x, y , z each equal to zero after differentiating in (1512) or ( 1514) , and write x, y , ... instead of h, k , &c .

1517 Observe that any term in these series may be made the last by writing x + 0h for x, y + 0k for y , &c . , as in ( 1500) . SYMBOLIC FORM OF TAYLOR'S THEOREM. The expansion in ( 1500 ) is equivalent to the following 1520

f(x +h) = ehdf(x).

PROOF.- By the Exponential Theorem ( 150 ) , writing the indices of d, as suffixes (1487), ehdzf(x) = (1 + hd₂ + } h²d₂ + :.. )ƒ(x) = f(x) + hƒz ( x) + ¥ h²ƒ½ (x) + ..., by (1488) .

COR.

▲f(x) = f(x +h) —ƒ( x) = ( ehdɛ — 1 )ƒ(x) ,

therefore

▲³ƒ(x) = ( ehd, −1)²ƒ(x) , hd and generally "f (x) = (e¹d, −1 ) "ƒ(x), the index signifying that the operation is performed n times upon f(x).

Similarly 1521 PROOF.

hd kd ƒ f(x + h, y + k) = e' + y (x, y) .

ehd₂ +kdyf(x, y) = { 1 + hd₂ + kd, + } ( hd, + kd,) ² +

(hd, + kd, ) ³ + ... } ƒ (x, y)

=f(x +h, y + k) , by ( 150) and (1512).

DIFFERENTIAL CALCULUS.

272

And, generally , with any number of variables , 1522 f(x +h, y + k, + l ... ) = hd, + kd + ld₂+ ... f( ,y , ... ) . COR.

As in ( 1520) ,

Af(x, y , z ... ) = (ehd¸ +kd¸ +……..−1 ) f ( 2,3, ...) .

1523 If u = f(x, y) = p (r , 0) , where a = rcos0, y = rsin0 ; and if arcos (0 + w) , y'rsin (0 +w) ; then f(x'y' ) is expanded in powers of ய by the formula

= p² e (xd, −ydz)ƒ(x, y) . f(x' , y') = PROOF. -By ( 1520) , r being constant, ø (r, 0 + w) = ewdo q (r, 0) = ewd,ƒ ( x, y) . Now a and y are functions of the single variable 0 ; therefore Ug = UxXq + UyYə = u₂ ( ―r sin 0) + u, (r cos 0) = 2lyy The operation d, will be transformed by the same law ( 1492 ) ; therefore dad, -yd ; therefore ƒ(x', y′) = ew(xd, −yd¸) f(x, y) = 1 + w (xu , −yu₂) + } w²(x²u», — 2xyu»y + y²U2 ) + &c.

1524 EXAMPLES . -The Binomial, Exponential, and Loga rithmic series for ( 1 + x)" , a *, and log ( 1 + x) , ( 125 , 149 , 155 ) , are obtained immediately also the series for sin a The mode of proceeding , shewn in the following

by Maclaurin's Theorem ( 1507) ; as and cos a (764) , and tan-¹a (791 ) . which is the same in all cases , is example ; the test of convergency

(1504) being applied when practicable. 62 2 17 x05 + x² + 1525 tan x x + &c . x= = x x + + 1/3 203 + 283 15 5 315

Obtained by Maclaurin's theorem , as follows :-Let ƒ (x) = tan x = y Therefore y = z and z , = 2yz ; y and z being used for shortness. ′ (x) = sec² x = % ƒ ƒ" (x) = 2 sec² x tan x = 2yz, ƒ"" ' (x) = 2 (zy, + yz,) = 2 (z² + 2y³z) , fiv (x) = 2 (4y² + 4yz² + 4y³z ) = 8 (2yz² + y³z) ,

f (x) = 8 (2% + 8y²z² + 3y²x² + 2yz ) = 8 (2z³ + 11y²x² + 2y^z) , fvi (x) = 8 (12yz³ + 22yz³ + ... ) = 272yz³ + ..., f (x) = 272z + ... & c., the terms omitted involving positive powers of y, which vanish when a is zero, and which therefore need not be computed if no term of the expansion higher than that containing a is required."

273

EXPANSION OF EXPLICIT FUNCTIONS.

Hence, by making ∞ = 0, and therefore y = 0 and z = 1 , we obtain ƒ(0) = 0 ; ƒ ' (0 ) = 1 ; ƒ " ( 0) = 0 ; ƒ" (0 ) = 2 ; ƒiv ( 0) = 0 ; ƒ' ( 0 ) = 16 ; fi(0) = 0; fvil (0) = 272. Thus the terms up to 27 may be written by substituting these values in (1507) . In a similar manner, may be obtained 1526

5 61 24 + X6 + &c . sec x = 1 + 1/2 a² + 24 720

Methods of expansion by Indeterminate Coefficients. 1527

RULE I.- Assume f ( x ) = A + Bx + Cx² + &c .

entiate both sides of the equation .

Differ

Then expand f' (x) by some

known theorem , and equate coefficients in the two results to determine A, B, C, &c.

1.3.5 x 1.3 a 1 x3 -1 sin¹x = x + + &c . + 2.45 + 2.4.6 7 23 Obtained by Rule I. Assume

1528

Ex.

sin ¹a

A + Вx + Сx² + Dæ³ + Ex¹ + Fæ³ + --$ Therefore, by ( 1434), ( 1 − x²) ¯ = B + 2Cx + 3Dx² + 4Ex³ + 5Fc* + . 1.3 1.3.5 2 But, by Bin. Th . (128) , ( 1 − x² ) ¯ * = 1 + ' æ² + 2.4 2*+ 2.4.6 +........ 1.3 1 Equate coefficients ; therefore B = 1 ; C = 0 ; D = = , ;;; E = 0 ; F = 2.4.5 2.3 &c. By putting = 0, we see that A = ƒ ( 0 ) always. In this case A sin-10 = 0. In a similar manner, by Rule I. ,

x2 1529

esin

2

1530

3x

8.x5

3.8

5

6

-

= 1 + x +

...... & c... + &c 4

RULE II. - Assume the series , as before, with unknown

coefficients. Differentiate successively until the function re appears. Then equate coefficients in the two equivalent series . 1531

Ex.- To expand sin x in powers of x.

Assume sinx = A + Bư + Cử +D +Ec + F + 3 Differentiate twice, COS = B + 20x + 3D² + 4Ex³ + 5Fx + .... 20 + 3.2Dx + 4.3Ex² + 5.4Fa³ + -sina 0 in the first two equations ; therefore A = 0, B = 1. Put Equate coefficients in the first and third series. 2 N

274

Thus

Therefore

1532

DIFFERENTIAL CALCULUS.

-20A,

. 0 = 0;

-3.2DB,

-4.3E = C,

.. E = 0;

-5.4FD,

sin x = x

1 ; 2.3 1 &c. . F= 1.2.3.4'

.. D 1

25 23 + - &c., as in ( 764). 1.2.3 1.2.3.4.5

RULE III . -Differentiate the equation y = f (x) twice

with respect to x, and combine the results so as to form an equation in y, yx, and Y2x. Next assume y = A + Bx + Cx² + &c. Differentiate twice, and substitute the three values of y, yx , yx Lastly, equate coefficients so obtained in the former equation. in the result to determine in succession A, B , C , &c. 1533 of sin

Ex.-To expand sin me and cos me in ascending powers or cos 0.

These series are given in ( 775-779) . They may be obtained by Rule III. as follows : Put x = sin 0 and y = sin mo sin (m sin -¹æ) . m -1 ..... (i.) Therefore Y₂ = cos (m sin¨¹ ) ( 1434) √1 - x³ m mx -1 -1 Y2 = -sin (m sin¯¹x) + cos (m sin-¹x) 1 - x2 (1 -x²) *

Therefore, eliminating cos (m sin- ¹a) , (1 - a²) Y 2x − xY₂ + m²y = 0 .........(ii ) Let (iii.) y = A + A₁x + Agµ³ + ...... + A„æ” + . Differentiate twice, and put the values of y, Y., and Y2 in equation (ii.) ; = ( A + A₁x + A₂x² + А¸æ³ + ... + A„ x” + ………. thus 0m² ..) -x − (4 + 24, +34 +... +n4 + .) + (1-2 ) { 24, + 2.3A

+

+ (n - 1 ) nA² + n (n + 1 ) An + 1æ" ~ ¹ + ( n + 1 ) ( n + 2) An + 2x" + ... } . Equating the collected coefficients of a" to zero, we get the relation n² - m³ An+2 = ........ (iv.) An ..... (n + 1 ) (n + 2) Now, when = 0, y = 0 ; therefore A = 0 , by (iii. ) . And when x = 0, y = m, by (i. ) ; and therefore A , = m, by differentiating (iii. ) . The relation (iv. ) furnishes the remaining coefficients by making n equal to 0, 1, 2, 3, &c. in succession. Cos me is obtained in a similar way.

RULE IV .- Form the equation in y, yx, and yx, as in 1534 Take the nth derivative of this equation by applying Rule III. Leibnitz's formula (1460) to the terms, and an equation in

EXPANSION OF EXPLICIT FUNCTIONS.

Put x = 0 in this ; and

x and ynx is obtained . Y(n+ 1) x9

Y(n+2) x

275

employ the resulting formula to calculate in succession y3x0 , Yo, &c . in Maclaurin's expansion (1507) . a² x² + a (a²+ 1) X3 Ex. easin¨¹x = 1+ ax + 1535 1.2 1.2.3

+

a²(a² + 22) 14 2 + a (a² + 1) (a² + 32) x + &c . 1.2.3.4 1.2.3.4.5

Obtained by Rule IV.

Writing y for the function, the relation found is ( 1 − x²) Y 2x − XYx — a³y = 0.

Differentiating n times, by ( 1460 ) , we get ( 1 − x²) Y( n + 2) x '− (2n + 1 ) xу(n + 1) x − ( a² + n²) Ynz = 0. Therefore ( +2 ) 0 = (a² + n²) Ynro , a formula which produces the coeffi cients in Maclaurin's expansion in succession when yo and Y20 have been calculated.

1536

ARBOGAST'S METHOD OF EXPANDING $ (≈) ,

az z ≈ = a + a₁x + 1 .2 x2 +

where

Let y = $ (z ) . When x = 0, x= Maclaurin's theorem (1508 ) ,

a3 x³ + & c . 1.2.3 y =

( a) ;

......... (i .)

therefore, by

Xv3 x2 y = $ (a) + xyxo + 1.292.x0 + 1.2.3 Ysxo + &c.

...... (ii .)

Hence , in the values of Ya, Y. , & c . , at (1416 ) , æ has to be put = 0. Now, when a = 0 , z = a ; therefore yz, Y2z, &c . become ' (a) , " (a) , &c.; and Zr0, 220, 230, &c . become a , a , as, &c . Hence

Y20 = · ' (a) a₁ , Y2x0 = $ '' (a) a² + f ' (a) a₂ ,

Y3.x0 = 4″ (a) a³ + 34" (a) a¸a , + $ ' (a) az, &c.

1537

EXAMPLE.- To expand log ( a + bx + cx² + dx³ + &c.) . 1 '" (a) = 2. 4″ (a) = a² 4 6d 6bc 263 + Y3x0 = a³ a a

Here a = b, a‚ = 2c, a¸ = 6d, p′ (a) = — , Therefore

b Yx0 =11/17 a

2c 22 Y2x0 = - a² + " a

Therefore, substituting in (ii.) , we obtain as far as four terms, bc b с log (a + bx +cx² + ... ) = log a + a x + (~ -12/24 )x²+(30 3as a² a

212 )20³+

DIFFERENTIAL CALCULUS.

276

1538

To expand (a + a¸æ + ª‚x² + ... + ª‚ ") in powers of x.

Ex. 2.

Arbogast's method may be employed ; otherwise, we may proceed as follows. Assume (a +a +aµx² + ... α„x” ) ′ = ¸0 + ‚¤ + µò + Differentiate for a ; divide the equation by the result ; clear of fractions, and equate coefficients of like powers of x.

BERNOULLI'S NUMBERS .

x2 15391-

+ B

-- B

Xce +B

2

- &c. 6

where B2, B4, & c. are known as Bernoulli's numbers .

Their

values, as far as B₁ , are

1

1

= B=

B₂ = 6'

1

691

42'

2730'

66 43867

B18 =

B18 = 6'

5 B10=

30' 3617

7

B14 =

B12 =

1

By:=

B& = 30'

510 '

798

They are found in succession from the formula 1540

nB -1 + C ( n , 2) B„ -2 + C ( n , 3) B₂-3 + ….. ... .. + C ( n , 2) B¸ — ¹n + 1 = 0 ,

the odd numbers B3, B5, &c . being all zero . J PROOF. -Let y = Then, by ( 1508) , Τ 23 +11.00 + &c. y = yo + Yox + Y220 + Y320 3 2 14 Here yey + . Therefore

= ( -1 ) + B . Now yo = 1 and yo =-1, by ( 1587) . Therefore, by ( 1460 ) , differentiating n times, e² { Y nx ++ NY ny nμ - 1) x + C ( n , 2 ) Yin - 2) x + ... + ny, + y } = Ynz e* NY(n - 1 ) 20+ C ( n , 2 ) Y (n - 2 ) 10 + ... + nyo + Yo = 0.

Also

Substitute B -1, B - 2 , &c., and we get the formula required. B3, B5, B7. &c. will all be found to vanish. It may be proved, à priori, that this will be the case : for X x = x e²+ 1 + 1541 ex - 1 2 2e - 1 Therefore the series ( 1539 ) wanting its second term is the expansion of the expression on the right. But that expression is an even function of a ( 1401 ) ; changing the sign of a does not alter its value. Therefore the series in ques tion contains no odd powers of x after the first.

1542 The connexion between Bernoulli's numbers and the sums of the powers of the natural numbers in (276) is seen by expanding ( 1 - e²) -¹ in powers of e', and each term afterwards by the Exponential Theorem (150) .

EXPANSION OF EXPLICIT FUNCTIONS.

277

1543 x x x2 ་ (2−1) 7 + &c. , (2-1) — —— B ~+ 1 = ོ 2 + B,(2' − 1) ——B 4 et+ 1

1544

~ = ] = 2 { B.(2-1) ——B .(2−1) 7+ B ‚ (2−1) 7−&c. }. 6 x

PROOF.

= e*+1

1545

PROOF .

1+ 2

x et- 1

2x e -1 =1 and e²x-1 e*+1

2 x a e* +1 , and by (1539). x e²

22n-12n B 1 1 2n + 32 + + ........ = 42n 1.2 ... 2n

In the expansion of

a e*+ 1 ( 1540) substitute 2i0 for 2, and it 2 e -1

becomes the expansion of 0 cot 0 ( 770) . Obtain a second expansion by differ entiating the logarithm of equation (815, sin 0 in factors) . Expand each term of the result by the Binomial Theorem, and equate coefficients of like powers of in the two expansions .

STIRLING'S THEOREM. 1546

(x + h) —Þ (x) = hp' (x) + A¸h { p′ (x + h) −4′ (x) } + A₂h² { 4″ (x + h) −p ” (x) } + &c.,

where

A2n = ( −1 ) " B2n + 2n

and

A2n +1 = 0.

PROOF.-A1, A2, A3, &c . are determined by expanding each function of +h by (1500) , and then equating coefficients of like powers of x. Thus

-A₁ = 0 ; 12-4,

1 3

A₁ -A₂ = : 0; 2

1 - A₁ 4 3

A2 -A, = 0; 2

&c.

(x) = e", To obtain the general relation between the coefficients : put since A1, A2, &c . are independent of the form of p. Equation ( 1546)then h produces · = 1 — А₂h — Аgh³ — Agh³- &c .; = e" -1

and, by (1539 ) , we see that, for valucs of n greater than zero, = A2n +10 and A2n = ( -1 )"B₂n ÷ | 2n.

BOOLE'S THEOREM .

1547

(x + h) —4 (x) =

' (x + h) +☀p (x)} A¸h { $ + A¸h² { 4” (x +h) + $" (x) } + &c.

PROOF.-A,, A , A,, &c . are found by the same method as that employed in Stirling's Theorem.

278

DIFFERENTIAL CALCULUS.

For the general relation between the coefficients, as before, make and equation ( 1547) then produces e -1 = A¸h + A₂h² + Azh³ + &c. ; e +1

(x) = e*,

and, by comparing this with ( 1544) , we see that 22-1 and A2n-1 = (-1 ) -1B2n A2n = 0 2n

EXPANSION OF IMPLICIT FUNCTIONS .

1550 DEFINITION . -An equation f(x, y) = 0 constitutes y an implicit function of a. If y be obtained in terms of a by solving the equation , y becomes an explicit function of a.

1551 LEMMA. ――― If y be a function variables x a and z,

of two independent

d¸ { F (y) yz} = d. { F (y) yx } • PROOF. By performing the differentiations, we obtain F' (y) YzY: + F (y ) Yzx and F' (y) Y: Y₂ + F (y) Y which are evidently equal, by ( 1482) .

LAGRANGE'S THEOREM. 1552 Given y = x + xp (y) , the expansion of u =f(y) in powers of a is xn dn-1 ° (≈) + ... + ƒ(y) = ƒ(≈) +x$ (≈) ƒ " (~) ] + 'n dz −1 [ { $ ( ~ ) } " ƒ PROOF.-Expand u as a function of x, by ( 1507) ; thus, with the notation x2 x" of ( 1406) , u = Up + x Uxo + 2 U2x0 + ... + n Unxo + &c. Here u is evidently ƒ( z) . Differentiating the equation y = 2 + xø (y) for x and z in turn, we have Yx = $ (y) + x¢´ (y) Yz and y₁ = 1 + xp' (y) Y Therefore , (y) y ; and, since u, f' (y) y, and u, = ƒ' (!) Y₂, therefore also

Uz = $ (y) Uz.

(i.)

The following equation may now be proved by induction, equation (i. ) being its form when n = 1. Assume that (ii.) Unx = d(n- 1) x [ { † (y) } " u.] Therefore - 1 ) , d, [ { p (y ) } " u, ] ( 1482 ) U(n + 1) = d(n d(n−1)zdz = d₁n - 1), d: [ { p (y ) } " u, ] ( 1551 ) = dn: [ { p (y) } " + ¹u; ] , by ( i.)

279

EXPANSION OF IMPLICIT FUNCTIONS.

Thus, n becomes n + 1 . But equation ( ii. ) is true when n = 1 ; for then it is equation (i . ) ; therefore it is universally true. Now, since in equations ( i . ) and (ii. ) the differentiations on the right are all effected with respect to z, a may be made zero before differentiating instead of after. But, when x = 0, u, = ƒ′ (z) and (y ) = (z) , therefore equa tions (i. ) and (ii. ) give

uzo = 9 (z) ƒ' (z) ;

1553

Unzo = d(n − 1) z [ { 4 ( z) } "ƒ˜ ′ (z) ] •

Ex . 1.— Given y³ - ay + b = 0 : to expand log y in powers of a

Here y = x=

1 ; a

b 2,3 + a a - ; therefore, in Lagrange's formula, b ; f(y) = logy ; 2= and + (y ) = y³ ; a

Therefore

u₁ = log z ;

y = z + xy³.

Ux0 = ~ 23 1 = 2 ; 2

3n - 1) 1) (3n (3n —2) — 2 ) ... ( 2n + 1 ) z²”. Unzo = d( n−1) 3 ( 2ª¹n 1 ) = = ((3n 3n − Therefore, substituting the values of x ≈ and z, (1552) becomes b2 1 b (3n - 1 ) (3n − 2) ... ( 2n + 1 ) b² ″ 1 +.…………… log y = log + a² a + ... + 1.2 ... n a a²" an

1554

Ex. 2.

Given the same equation : to expand y" in powers of na.

f(y) is now y", and, proceeding as in the last example, we find b2 1 b" + n (n + 5) b 1 + n (n + 7) (n + 8 ) 7° 1 1 +n. y" = a² a {1. 1.2.3 a as 1.2 a¹ a²

+ n (n + 9) (n + 10 ) ( n + 11 ) b³ 1 + &c. } as 1.2.3.4 78 1 64 1 b b2 66 1 +12 +3 + &c. ...... +55 y= == 2/2 as a a² 2213 a (1+ 2/3 a

If n = 1,

1 CAYLEY'S SERIES FOR

中 (2) 1555 1 = A(≈)

n A" (- )" nz ~ :[ [( ~ $ ( ~) ] 2= + ... + 4 1 " ( — ~ ) .2 ... ~= n' [ ≈ { $ (~) } * ~ ¹]„₂ + … .. , 1.2

A2

1 where

A =

$ (0)* PROOF . Differentiate Lagrange's expansion ( 1552) for z, noting that dy 1 y ; and therefore = ' (y) = . Put f Replace a by dz 1 − xo'´ (y) ' (y) (y) 22 f ( ) := since ƒ is an arbitrary function. Then make y = 0. $ (z)'

280

DIFFERENTIAL CALCULUS.

LAPLACE'S THEOREM. 1556

To expand f (y) in powers of a when y = F { z + x$ (y) } .

RULE.Proceed as in Lagrange's Theorem, merely sub stituting F (z) for z in the formula .

1557 Here

Ex. 3. -To expand e' in powers of a when y = log (z +x sin y) . ƒ(y) = e" ;

F (z ) = log z ;

In the value of unro (1552 ) ,

( z ) becomes

f(z) becomes ƒ{ F (z) } = elog² = 2 ; Thus the expansion becomes

therefore ƒ′ (z) = 1 .

ez + sin log z + .. +

1558 Here

$ (y ) = sin y ; { F (z) } = sin log z ;

x" d -1) (sin log z) ". n

Ex. 4. — Given sin y = x sin ( y + a) : to expand y in powers of a. y = sin¹ (x sin y +a) , f(y) = y ;

F (z) = sin¹z ;

with z = 0 . 4 (y ) = sin (y + a) .

(z) in (1552) becomes $ { F (z ) } = sin (sin¯¹z + a) . f(z) becomes f { F (z ) } = F ( z ) = sin - ¹z ; therefore F " (z ) = ( 1-2 ) - §. Thus y = x sin (sin-¹z + a) ( 1 − z²) − ś -1 + ¼æ³d¸ { sin² (sin‍¹z + a) ( 1 − z²) −³ } + fx³d½23 ¦ sin³ ( sin¯¹z + a) (1 − ≈³) − $ } +

with z put = 0 after differentiating . The result is, as in ( 796) , sin 2a + æ³ sin 3a + &c. y = x sin a +

BURMANN'S THEOREM. 1559 To expand one function f (y) in powers of another function (y) . RULE .- Put x = 4 (y) in Lagrange's expansion , and there = (y) (y - z) ÷ 4 (y) ; therefore

fore

y- z 1560_ƒ ( y) = f(x) +4 ( v) { $ * ~ƒ ) } y= ,_.2 + ….. (3 ปT )° ® ... ..+

{4 (y) } " d"-1 dy1 { (47 )ƒ˜ ( v) } ,y=2 _, + &c. n Y

signifies that after differentiating z is to be sub Here y stituted for y.

EXPANSION OF IMPLICIT FUNCTIONS.

281

1561 COR. 1.- Since x = 4 ( y ) , y = 4-1 (x) ; therefore ( 1560 ) becomes, by writing a for (y) , n xn dn -1 y + ... ƒ{µ˜¹ (x) } = ƒ(≈) + ... + = , 3=2 ~~) ƒ * w ) }.... n dy • { ( But since the variable y is changed into z after differentiating, it is immaterial what letter is written for y in the second factor of the general term.

1562

COR. 2. - If ƒ (y) be simply y, the equation becomes n y xn dn-1 y- z + ... + 4¯¹ (x) = x +x — { (~~~~ y )" } ,_ n dyn-1 \+ (y) ' ‚ y =z + - + ~ U=S +

1563 COR . 3. - If z = 0 , so that y = xp (y) , we obtain the expansion of an inverse function , n xn dn-1 .-•+ ... + y) x (+( 4¹ (x) = x ( 税 ) y =0 y=.0 + ndy- {(xy )" } ) .. an_dy"-1

may be obtained by this

Ex. 5.- The series ( 1528) for sin-¹ 1564 formula ; thus,

Let sin-¹xy, therefore x = sin y = ↓ (y) , in ( 1563) ; therefore x² 23 yz y sin-1 = x + &c. = = = (1sin 2 )... 1 2 ( sin 1 ) yo + 1.2.3 ( sin ay) 240 , y=0 + 1.2

20

1565

=

Ex . 6. - If y = 1+ √ (1 - x )

-x theorem (1552) , since y = 84 2122x0 . +

we find

n+2 y" = ( 2 ) " + n ( 2 ) " ' ' + ... + Put

1 − √ ( 1—2²) , then, by Lagrange's x

nn + 2r - 1 x n+2r + ..... r n + r (232)

2t, thus

n \ n + 2r - 1 1566

(1-4 ) ) " = t" + nt" "' + ... + (¹ − √2

tn+r+ ir

n+ r

Change the sign of n, thus n _n \ n — r— 1 p ± '1+ √ ( 1—4t) 1567 := 1 − nt + ... ( −1 ) ″ 2 │r │n - 2r

n +1 or +1 terms, according as n is even 2 2 or odd, is equal to the sum of the two series, as appears by the Binomial theorem . 20

This last series, continued to

282

DIFFERENTIAL CALCULUS.

Also, by Lagrange's Theorem,

2r

2r - 1 + +... ( 2) )²+ log y = log 1568 ...+ log 2/2 + ( or, by putting = 2√t,

log

1569

1570

r 0+

1- √ /(1-4 ) = t+ ... + 2r - 1 ť + 2t rr

Ex. 7.- Given xylogy ; to expand y in powers of x.

The equation can be adapted as follows : therefore xy = xe² . y = exv, Put xy = y , therefore y = xe", from which, by putting z = 0 in ( 1552 ) , y may be expanded, and therefore y

Ex. 8. - To expand e" in powers of yet . Here x = yeb , ¢ (y) = y - z e-by, if we take z = 0. Therefore ye'y Ysby 1571 ev = 1 + aye™ +a (a − 2b) yey +a (a − 3b)² 2.3 + 1.2 1. ABEL'S THEOREM. 1572

If Φ (a) be a function developable in powers of e ; then

$ (x + a) = $ (x) +af ' (x + b) +

a (a − 2b) q″ (x + 2b) + ... 1.2 +

PROOF .-Let

a (a — rb) "-¹ 1.2 ... r

$ (y) = A + A₁e + ‚e² + Á¸3 ‹³ +

" (x +rb) + . (i. )

Put y = 0, 1 , 2, 3, &c. in ( 1571 ) , and multiply the results respectively by Ao, A₁e , Age , &c. Then the theorem is proved by equation ( i. ) .

1573

COR .- If

( x) = x" , Abel's formula gives

(x+a)” = x” + na (x + b) " -' + C (n , 2) a ( a − 2b ) ( x + 2b) " − 2 + …..

.... + C ( n , r) a ( a — rb) ”- ¹ (x +rb) "- " + &c.

INDETERMINATE FORMS .

0

1580 Forms

(x) >

RULE .— If

be a fraction which

0

↓ (x) takes either of these forms when x = a ; then º (a) = — o´ (a) or (a) (a) p" (a) the first determinate fraction obtained by differentiating " (a) '

INDETERMINATE FORMS.

283

the numerator and denominator simultaneously and substituting a for x in the result. 1581 But at any stage of the process the fraction may be reduced to its simplest form before the next differentiation . See example ( 1589) . PROOF.- (i .) By Taylor's theorem ( 1500 ) , since ¢ (a ) = 0 = ↓ (a), $ (a + h) = (a) + ho' (a + 0h) = (a + 0h) = ' (a) ' (a + th) (a + h) ' (a)' (a) + hp' (a + Oh) when h vanishes. 1 $ (a) = 1 (ii.) If (a) = (a) = ∞ , (a) (a)' (a) which is of the first form , and therefore ' (a) (a) ' (a) { (a)} ' (a) = Therefore $ (a) = ÷ ( 1414) = 4 (u)' { (a)} ' (a) ' (a) {4 (a) }³ ' { ø (a)

1582

Vanishing fractions in Algebra are of the indeter

minate form just considered , and may be evaluated by the rule, or by rejecting the vanishing factor common to the numerator and denominator.

1583

x³-as = 0 (x− a) (x² + ax +a²) 3a² 3 = = = a. 2a (n + x) (n − x)

Ex.-When x = a ;

Form 0 × ∞ .

RULE.- Iƒ p (x) × ↓ ( x) takes this form 1 which is of the when x = a, put p (a) × 4 (a) = ¢ (a) ÷ (a)'

. form 0

1584

Forms 0°, ∞ ° , 1 °.

RULE.— If {

(x) } (x) takes any of

these forms when x = a, find the limit of the logarithm of the expression. For the logarithm = (a) log (a) , which, in each

case, is ofthe form 0 × ∞ . . Form ∞∞

RULE. - Iƒ p (x) —4 (x) takes this form e- (a) 0 = when xa , we have e (a) - (a) = of 0 ; and if the value e-4(a) 1585

this expression be found to be c, by (1580) , the required value will be log c. 1586

(a) , which is of Otherwise : 4 (a) —4 (a) = ø (a) {1 – £(a)}} ,

the form ∞ X0 ( 1583) .

284

1587

DIFFERENTIAL CALCULUS.

Ex. 1. - With x = 0,

y=

x =0 = e -1 0

(1580) = 1.

Also, with a = 0, e - 1- re * -x -0 =- 1 = = 0 = e* — e* — xe* = Yx0 = 2e* 2 ( -1 ) e* 2 ( -1 ) (e*-1)* 0 1588

1589

Ex. 2. - With x = 1 ; 0 log = cot ( x) log x = tan ( x) Ex. 3.— With x = 0 ; (log )" = 00 2" (log æ)" = ∞ x -m

1 2

cos² ( x) (1580 ) = = T πX

In -m = 0, (—m )" x by (1581 ), differentiating n times and reducing the fraction to its simplest form after each differentiation.

n (log x)"−1 = -MX-m

1590

Ex. 4. - With x = 0 ; y = (1 + ax) = 1°. a ax) log 1 ( + = 0 = By ( 1584) , log y = (1580) = a ; x 1 + ax 0

..y = e .

z y = (π — x) sinx = 0º. log (π- x) -∞ sin x = = By (1584) , log y = sin x log (π- x) = ∞ cosec x (π - x) cos x 2 sin x cos x = 0; . ( 1580 ) = 0 1. = y .. − (π —x) sin æ —cos x -

1591

Ex. 5.-— With ∞ == ;

1592

If ƒ (x) and a become infinite together , then

ƒ(x) = ƒ ′ (x) =ƒ(x + 1) −ƒ(x) . x f(x) = ∞0 =f' (x) (1580) =f PROOF. = f(x (x + 1 )) -f(x) ( 1404) , æ 1 ∞0 since, when x = ∞ " h may be taken = 1 . Indeterminate forms involving two variables . 1592 RULE .- First : If the values x = a , y = b make the frac Φ (x, y) = 0 ; the true value is = ቀ. if , and 4, both 0 te 4 (x, y) vanish .

tion

1593 Secondly : If the fraction is k. PROOF.—( i .) By ( 1703)

x = 4x =

y : 4, = k,

the true value of

+ , and y being an arbitrary (x, y) = 4x+4yYz (x, y) function of x, —that is, independent of a, -the value of the fraction is inde terminate unless , and 4, both vanish. (ii.) If we substitute = ky, and ₁ = ky,, the fraction becomes = k.

285

JACOBIANS.

JACOBIANS .

1600 Let u, v , w be n functions of n variables x, y , z (n = 3 ) . The following determinant notation is adopted : Uix

Uy

Vx Vy

Xu Xv Xw

U Vz

=

d (xyz)

d (uvw ) "

Yu Yv Yw

=

d (uvw)

d (xyz) Zu Zv Zw

Wx Wy Wz

The first determinant is called the Jacobian of u, v, w with respect to x, y, z , and is also denoted by J (uvw) , or simply by J.

d (uvw ) 1601

THEOREM.

d (xyz)

= 1.

X

d (xyz)

d (uvw)

PROOF.- If the product of the two determinants be formed by the rule in (570) , first changing the columns into rows in the second determinant ( 559 ) , the first column of the resulting determinant will be and the whole Wu Vu Wu 100 Uz Xu + Uy Yu + UzZu = Uu determinant Uv Vv Wo = 010 = 1 . Uz Xo + Uy Y v + Uz Zv = Uv " 001 will be И го иго Иго Uz X w + Uy Y w + Uz Z ₂ = Uw

1602 If u, v , w are n functions of n variables a, B, ß, y (n = 3) , and a, ẞ, y, functions of x, y , z ;

d (uvw) X

d (aBy)

d (aBy) = d (uvw) • d(xyz)

d (xyz)

PROOF - Form the product of the two determinants, changing columns into rows in the second as in ( 1601 ) . The first column of the resulting determinant will be and the whole de B + u₁Y₂ = u₂ u_a₂ + u₂ẞ₂ terminant will u_a₁ + Up P²y + u,Y₁ = Uy { , be u_a₂ + uß₂ B + u₁Y₁ = U₂ since rows and columns may be transposed (559) . 1603

Ux Vx W x Uy Vy Wy Uz Vz Wz

= d (uvw) d (xyz) '

COR .- If a , ß , y are only given as implicit functions

of x , y , z, by the equations six variables ; then

d (4x4) X

d (aßy)

= 0, x = 0 , 40 , involving the

d (4x4) . d (aBy) = = (-1)8 d (xyz) d (xyz)

PROOF.-By (1737) , .ª₂ + ¢ ‚ ß₂ + 9,7% , where . is the partial derivative of . Thus u , v , we, in the determinant, are now replaced by -923 -X21-42 ; So with y and z ; and by changing the sign of each element, the factor (-1) is introduced ( 562) .

286

DIFFERENTIAL CALCULUS.

1604 If u, v, w , n functions of n variables x, y, z (n = 3) , be transformed into functions of E, n , by the linear substitutions = a₁§ + an + α¸} ]

then

y = b₁§ + b₂n + b;} } ; z = c₁§ +c₂n + c3}

d (uvw) = M d (uvw) d (sns) d (xyz)'

J ' = MJ ,

or

where M is the determinant (a, b , c ) called the modulus of transformation ( 573) . PROOF.

J=

u, ly Uz

M=

a₁ b₁ c₁

J' =

uz un us Из

Vz Vy Vz

ag by Ca

"

Wx Wy Wz

az bs cs

We wn η w5

"

"ટ્

Form the product MJ by the rule in (570) . The first element of the resulting determinant is u,a, + u, b, + u₂c₂ = U₂x¿ + UyY& + uzu . Similarly for each element. Then transpose rows and columns, and the determinant J' is obtained. 1605 When the modulus is unity , the transformation is said to be unimodular.

1606 If, in ( 1600) , 4 ( uvw) = 0 , where $ is some function ; then J (uvw) = 0 ; and conversely, PROOF. -Differentiate o for x, y, and z separately, thus Pubz + QvVz + PwWx = 0 ; similarly y and z ; and the eliminant of the three equations is J (uvw) = 0 .

1607 If u = 0 , v = 0 , w = 0 be a number of homogeneous equations of dimensions m, n, p in the same number of vari ables x , y , z ; then J (uvw) vanishes, and if the dimensions are equal J , J , J₂ also vanish .

PROOF.- By ( 1624), xux + yu, + zu, = mu xvx + yv , + zv₂ = nv xwx + ywy + zw₂ = pw

; .. Jx = A₁mu + B₁nv + C₁pw.

By (582), A₁, B₁ , C₁ being the minors of the first column of J. if u, v, w vanish, J also vanishes.

Therefore,

Again, differentiating the last equation, J+ xJ¸ = ømu₂ + B₁nv₂ + C₁pwz. Therefore, if m = n = p, J+ xJ₂ = m ( A₁u₂ + B₁v₂ + C₁w₂ ) = mJ. Therefore J, vanishes when J does.

If u = 0, v = 0, w = 0 are three homogeneous equations of the second degree in x, y , z , their eliminant will be the determinant of the sixth order formed by taking the eliminant of the six equations u, v , w , J , Jy , Je. 1608

287

QUANTICS.

PROOF. -J is of the third degree, and therefore J , Jy, J, are of the second degree, and they vanish because u, v, w vanish, by ( 1607) . Hence u, v , w , J., Jy, J, form six equations of the form (x , y , z ) ² = 0 .

1609 If n variables x , y , z (n = 3 ) are connected with n other variables , n, L, by as many equations u = 0 , v = 0 , w = 0 ;

then

dx dy dz = d (uvw ) = d (uvw) de dŋ dě d (sns) d(xyz)

PROOF.-By (562 ) we have Ux Uy Uz de dy dz Vx Vy Vs = de dn d Wx Wy Wz

Ux xz Uy Yn U= 25 Vx XE Vy Yn V₂ 25 W₂x

=

wyyn w =75

Из ипn из '' t V.η 25 w พ DE Uη AU;

QUANTICS .

1620 DEFINITION. -A Quantic is a homogeneous function of any number of variables : if of two , three variables , &c . , it is called a binary, ternary quantic , & c . The following will illustrate the notation in use. The binary quantic

ax³ + 3bx²y + 3cxy² + dy³ is denoted by (a, b , c, dxx, y ) when the numerical coefficients are those in the expansion of (x +y) ³ . When the numerical coefficients are all unity , the same quantic is written (a, b, c, dxx, y)³ . When the coefficients are not mentioned , the notation (x, y)³ is employed .

EULER'S THEOREM OF QUANTICS. If u = f (x, y) be a binary quantic of the nth degree, then

1621 1622

1623

xux +yu₂ = nu. x²u2x + 2xyuxy + y³U2 = n (n − 1 ) u. ... ... ... ... (xd¸ +yd¸) ” u = n (n − 1 ) ... (n − r + 1 ) u .

(1492)

PROOF.-In ( 1512) put hax, kay ; then, because the function is homogeneous, the equation becomes (1 + a)" u = u + a (xu₂ + yu,) + } a² (x²u₂ + 2xуuzy + y³Uzy) + ............ Expand (1 + a)", and equate coefficients of powers of a.

288

DIFFERENTIAL CALCULUS.

The theorem may be extended to any quantic, the quan tities on the right remaining unaltered . Thus , in a ternary quantic u of the nth degree, 1624 1625

xu¸ + yu₂ + zu₂ = nu ;

and generally

(xd + yd, +zd¸) ” u = n (n − 1 ) ... ( n − r + 1 ) u .

DEFINITIONS . 1626 The Eliminant of n quantics in n variables is the function of the coefficients obtained by putting all the quantics equal to zero and eliminating the variables (583 , 586 ) . 1627 The Discriminant of a quantic is the eliminant of its first derivatives with respect to each of the variables (Ex . 1631 ) .

1628

An Invariant is a function of the coefficients of an

equation whose value is not altered by linear transformation of the equation , excepting that the function is multiplied by the modulus of transformation ( Ex . 1632) . 1629 A Covariant is a quantic derived from another quantic , and such that, when both are subjected to the same linear transformation , the resulting quantics are connected by the same process of derivation (Ex . 1634) . 1630 A Hessian is the Jacobian of the first derivatives of a function . Thus , the Hessian of a ternary quantic u , whose first derivatives are u , uy , uz, is u2x Uxy Wxz d (u¸uu₂) = Uyx Uzy Wyz d (xyz) Uzx U zy U2z 1631 Ex. Take the binary cubic u = ax³ + 3bx³y + 3cxy + dy³ . derivatives are 3a 6b 3c 0 u₂ = 3αx² + 6bxy + 3cy² , 0 3a 6b 3c u₁ = 3bx² + 6cxy + 3dy³. 3b 6c 3d 0 Therefore (1627 ) the discriminant of u 0 3b 6c 3d is the annexed determinant, by (587) .

Its first

1632 The determinant is also an invariant of u, by ( 1638) ; that is, if u be transformed into v by putting a = a + n and y = a + ẞ'n ; and, if a corresponding determinant be formed with the coefficients of v, the new determinant will be equal to the original one multiplied by (aẞ' - a'ß) .

QUANTICS.

289

1633 Again, u₂ = 6ax + 6by, U₂ = 6cx + 6dy, Uzy = 6bx + 6cy. Therefore, by (1630) , the Hessian of u is Uzz Uzy - uzy = (ax + by) (cx + dy) - (bx + cy) = (ac- b³) x² + (ad - be) xy + (bd - c³) y³.

1634 And this is also a covariant ; for, if u be transformed into v , as before, then the result of transforming the Hessian by the same equations will be found to be equal to 22y -vy. See ( 1652) .

= 1635 If a quantic , u = f(x, y , z ...) , involving n variables can be expressed as a function of the second degree in X1 , X2 ... Xn- 1 , where the latter are linear functions of the variables , the discriminant vanishes . PROOF.-Let where

u = ¢ X} + & X, X₂ + XX₁Xz + &c., X₁ = α₁x + b₁y + c₁z + & c. The derivatives u , u,, &c. must contain one of the factors XX, ... X„ -1 in every term, and therefore must have, for common roots, the roots of the -1 = 0 ; n - 1 equations being simultaneous equations X₁ = 0 , X, = 0, ... X, -₁ required to determine the ratios of the n variables . Therefore the dis criminant of u, which (1627) is the eliminant of the equations u = 0, u, = 0 , &c., vanishes, by (588) .

1636 COR . 1.- If a binary quantic contains a square factor , the discriminant vanishes ; and conversely. Thus, in Example ( 1631 ) , if u has a factor of the form (Ax +By) , the determinant there written vanishes . 1637 COR . 2.- If any quadric is resolvable into two factors , the discriminant vanishes .

An independent proof is as follows : Let u = XY be the quadric, where Y = ( ax + b'y + cz + ….. ) . X = (ax + by + cz + ... ) , The derivatives u , u , u, are each of the form pX + qY, and therefore have for common roots the roots of the simultaneous equations X = 0, Y = 0 . Therefore the eliminant of u₂ = 0 , u, = 0, &c. vanishes ( 1627 ) .

1638

The discriminant of a binary quantic is an invariant .

PROOF.-A square factor remains a square factor after linear transforma tion. Hence, by ( 1636 ) , if the discriminant vanishes, the discriminant of the transformed equation vanishes, and must therefore contain the former discriminant as a factor (see 1628 ) . Thus the determinant in ( 1631 ) is an invariant of the quantic u.

The discriminant of the ternary quadric 1639

u = ax² + by² + cz²+ 2fyz + 2gzx + 2hxy 2 P

290

DIFFERENTIAL CALCULUS.

is the eliminant of the equations 1640

a

u = ax +hy+gz = 0 u,

that is , the determinant

hx + by +fz = 0

hb f

gf c

}u₂ = gx +fy + cz = 0 1641

h g

abc +2fgh- af — by² —ch² = A.

1642

The following notation will frequently be employed. The determinant will be denoted by A, and its minors by A, B, C, F, G, H. Their values are readily found by differ

entiating ▲ with respect to a , b, c , f, g , h ; thus , bc -f² = A , A = be

B = ca - g² = A ,

C = ab — h² = Ac,

F = gh— af=

G = hf- bg =

H =fg — ch = } → n •

1643

4,,

The reciprocal determinant is equal to ▲² or ah g 2 AHG

HBF

=

GFC

1644

4,,

By (575) .

h b f g ƒ c

The discriminant of the quaternary quadric u = ax² +by² +cz² + dw² + 2fyz + 2gzx +2hxy +2pxw + 2qyw + 2rzw

is the eliminant of the equations 1645

u = ax + hy +gz +pw = 0

that is ,

a h g p

u, = hx + by +fz + qw = 0

the deter minant

h bf q

Zuz = gx +fy + cz + rw = 0 zuw = px + qy + rz + dw = 0

gfcr

p q r d

The determinant will be denoted by A' , and , by decom posing it by (568) , we have

1646 1647

1648

A' = d . ▲ — Ap² — Bq² — Cr² — 2Fqr — 2Grp — 2Hpq. A' = } ',) + d.A . { (p Ap + q¹ + rA

THEOREM . —If ₫ (ry) be a quantic of an even degree,

$ (d,, — d¸) $ (x, y) is an invariant of the quantic.

291

QUANTICS.

PROOF. - Let the linear substitutions ( 1628) be x = a + bn, y = a'¿ + b′n

(i . )

Solve for

and n. Find Ex, y , z, 7 , and substitute in the two equations d₁ = d¿ ¿y + dn ny i d₂ = de Ex + dn nx. The result is d = { ad₂ + b (-de ) } ÷ M ;

-d₂ = { a'd, + b' ( —de ) } ÷ M...... (ii .) ,

where Mab - a'b, the modulus of transformation . Equations (i . ) and (ii. ) are parallel, and show that the operations d, and -d, can be transformed in the same way as the quantities and y ; that is, if p (x, y ) becomes ↓ (§, n) , then (dy, —dz) becomes (dŋ , −d§ ) ÷ M", where n is the degree of the quantic p. But ( d,, —d, ) p (x, y) is a function of the coefficients only of the quantic p, since the order of differentiation of each term is the same as the degree ofthe term ; therefore the function is an invariant, by definition ( 1628).

1649 EXAMPLE . Let (x, y) = ax + bx³y + exy +fy . The quantic must first be completed ; thus, p (xy) = ax + bx³y + cx²y² + exy³ +fy*, (c = 0) ; then • (d¸, −dz) p (x, y ) = (ad — bdзx + cdzy2x − edy3x +fd4x) 4 (x, y) = a.24f- b.6e + c.4c - e.6b +f. 24a4 ( 12af- 3be + c²) . Therefore 12af- 3be +c² is an invariant of p, and = ( 12AF − 3BE + C²) ÷ Mʻ, where A, B, C, E, F are the coefficients of any equation obtained from by a linear transformation. But if the degree of the quantic be odd, these results vanish identically.

1650 Similarly, if (x, y) , 4 (x, y) are two quantics of the same degree, the functions $ (dy, — dx) 4 (x , y) are both invariants .

and

(d,, — d¸) $ (x , y)

1651 Ex . If ¢ = ax² + 2bxy + cy² and 4 := a'x² + 2b'xy + c'y² ; then (ada - 2bdy + cd₂ ) (a'x² + 2b'xy + c´y³ ) = ac' + ca' -2bb' , an invariant.

1652

A Hessian is a covariant of the original quantic.

PROOF. Let a ternary quantic u be transformed by the linear substitu tions in ( 1604) ; so that u = (x, y , z ) = ↓ ( §, n , 5 ) . The Hessians of the

two functions are d (Uz Uy Uz) and d ( uz u uz) (1630) . Now d (Eng) d (xyz) d ( uz un u₁ uz) uz ( ท uz) = μd ƒƒ½d (µ‚µ‚µ₂ ) M = M d (1¸u„ u.) = = d (Ens) d ( xyz) d ($45) d (xyz ) The second transformation is seen at once from the form of the determi nant by merely transposing rows and columns ; the first and third are by Therefore, by definition ( 1629 ) , the Hessian of u is a theorem ( 1604) . covariant.

Cogredients . - Variables are cogredient when they are 1653 subjected to the same linear transformation ; thus, x, y are

DIFFERENTIAL CALCULUS.

292

cogredient with a ' , y' when x' = a§' + bn'

x = aέ+bn

and

y = c§ + dn

1654

y' = c§' + dn'

Emanents.-If in any quantic u =

(x, y) , we change

x into x + px' , and y into y + py' , where x' , y' are cogredient with x, y ; then, by ( 1512),

(x + px' , y + py') = u + p (x'd₂ + y'd, ) u + \p² (x'd¸ + y'd₁ ) ²u +&c.,

and the coefficients of p, p², p³,

are called the first , second ,

third , ... emanents of u.

1655 The emanents , of the typical form (a'd, + y'd¸)” u , are all covariants of the quantic u. PROOF. - If, in (x, y), we first make the substitutions which lead to the emanent, and afterwards make the cogredient substitutions, we change a into +px', and this into a + by + p (a + bŋ). And if the order of these operations be reversed , we change æ into a + bŋ, and this into a ( ¿ + pë′) + b ( n + pn') . The two results are identical, and it follows that, ifΦø (x, y) be transformed by the same operations in reversed order, the coefficients of the powers of p in the two expansions will be equal, since p is indeterminate. Therefore, by the definition ( 1629 ) , each emanent is a covariant.

1656 For definitions of contragredients and contravariants, see ( 1813-4) . For other theorems on invariants, see ( 1794 ) , and the Article on Invari ants in Section XII.

IMPLICIT

FUNCTIONS .

IMPLICIT FUNCTIONS OF ONE INDEPENDENT VARIABLE . If y and z be functions of a, the successive application of formula ( 1420) gives , for the first, second , and third deriva tives of the function $ (y, z) with the notation of ( 1405 ) ,

1700

Þx (YZ) = $vYx +

1701

2 (y ) = $zyYz + 2¢yzYx≈x +

z2x• 2x ≈x +

y Y 2x + 8=%2x·

293

IMPLICIT FUNCTIONS.

1702

3x (YZ) ( yz) = $syys + 3$2yzY& ≈x + 3&y2zYz ~+x + Þ3= ~3 Psx +3 (P2yYx + $yz≈x) Y2x + 3 (P2z≈x +ÞzyYx) ≈2x + QyY3x + $z≈3r•

By making z = quently z = 1 , z we obtain

in the last three formulæ, and conse

= 0 , or else by differentiating independently,

1703

pc( g) = $yyx+

x•

1704

$2x (xy) = $2yY% + 24.xvY± + $2x + QyY2x•

1705

By $3x (xy) = $3yys + 3+2yxYx + 3$y2xYx +$3x• +3 (

+

)

t

1706 In these formule the notation , is used where the differentiation is partial, while p (x , y) is used to denote the complete derivative of (x, y) with respect to x. Each successive partial derivative of the function (y, z ) (1700 ) is itself treated as a function of y and z, and differentiated as such by formula ( 1420 ) . Thus, the differentiation of the product 4, y, in (1700 ) produces (Py)zYz+ PyY2x = ( P2yYz + PyzZz ) Yz + PyYzx • The function , involves y and z by implication. If it should not in fact contain z, for instance, then the partial derivative 4 vanishes. On the other hand, Y., Y½ , &c. are independent of z ; and z,, Z2 , &c. are independent of y .

DERIVED EQUATIONS . 1707 If (xy) = 0 , its successive derivatives are also zero , and the expansions ( 1703-5 ) are then called the first, second, and third derived equations of the primitive equation (xy) = 0 . In this case , those equations give, by eliminating Y , 2 dy ―― 2x Þ3 — Þ¿y zy P32 d³y _ = 24.x, Þ. Þy — Þ22 1708 -$ ; dx dx2 3 $$

1710

Similarly, by eliminating y, and y2r, equation ( 1705 ) would give y in terms of the partial derivatives of Φ (xy) . See the note following ( 1732 ) .

1711

If If

1712

and

(xy) = 0

PROOF.-By (1708) , values of Y2x and yar

and

dy = 0 ; dx

dzy y =dx2

;

d³y = _ 32x¤xy — ¤3xÞy¸ dx $ 1 = 0.

Therefore (1704) and ( 1705) give these

291

DIFFERENTIAL CALCULUS.

1713

If x and ቀ. both vanish , yr in ( 1708 ) is indeterminate .

In this case it has two values given by the second derived equation ( 1704) , which becomes a quadratic in yr. 1714 If p2r, Pry, and pay also vanish, proceed to the third derived equation ( 1705 ) , which now becomes a cubic in yxı giving three values , and so on . 1715 Generally, when all the partial derivatives of p ( x, y ) of orders less than n vanish for certain values xa , y = b , we have , by ( 1512 ) ,

( a , b ) being zero , 1 = (hd + kdy)" (a + h , b + k) = n

(xy) , a, + terms

of

higher orders which may be neglected in the limit. (x, y are here put = a, b after differentiation . ) Now, with the notation of 1406 ) , therefore

họx,a + kpy.b = 0 ; hox,a k P.x,a = dy dx Py,b h

the values of which are therefore given by the equation 1716

(hd + kd,)"

(x, y), a,b = 0.

If Yx y, becomes indeterminate through a and y vanish dy = ing, observe that in this case, and that the value of dx x 1717

the latter fraction may often be more readily determined by algebraic methods .

If x and y in the function Φ (x, y) are connected by the equation (x, y) = 0, y is thereby made an implicit function of a, and we have Þz ( x, y) = Þ ¥y—¥ Þv 1718 4 1719

x + (P2x ¥y− ¥2x Oy) Vi 2x (x, y) = { (Pzv ¥y— ¥2v Py) ¥% yx yx −2 ( Put

)

x

:

PROOF. ( i. ) Differentiate both and for r, by (1703), and eliminate yx. (ii. ) Differentiate also, by ( 1704) , and eliminate y, and y

If u , y , z are functions of x, then , as in ( 1700) , 1720

$z (uyz) = $„ ux + QyYz + PzZx•

295

IMPLICIT FUNCTIONS.

1721

$2x (uyz) = $2uUx + 2yYz + $22≈2 +24yzYx =x + 2 zu ≈ x Ux + 20 uyuxYx + Þu Uzx + yY2x +ÞzZ2x •

1722 To obtain p above equations .

Let U =

(xyz) and

2 (xyz) , make u = x in the

(x, y , z , E) be a function of four variables con

nected by three equations u = 0, v = 0, w = 0, so that one of , may be considered independent .

the variables , 1723

We have, by differentiating for E,

$z X¿ + Þy Y € + $z ~ +d; − U¿ = :

dU

Uz xz + Uy y z + Uz ≈§ + U§

αξ

Vz Xz + Vy Yz + Vz ZE+ V§

where J = d (uvw)· d (xyz)

WxX¢ + WzY& + W₂≈z +w¢ dx

1724

== d (uvw ) 1 वह d (x ) Ji

== d (uvw) 1 वहु / d( (Eyz )1 J; ૫૪) dz

= d (puvw ) 1 d (xyz§) J³

d (uvw ) 1 ——

dé

d (xyέ) J

Observe that U stands for the complete and partial derivative of the function U.

for the

PROOF .- (i . ) U is found by taking the eliminant of the four equations, separating the determinant into two terms by means of the element -U§› and employing the notation in ( 1600 ) . (ii.) xg, Y and Z are found by solving the last three of the same equa

tions, by (582 ) .

IMPLICIT FUNCTIONS OF TWO INDEPENDENT VARIABLES .

1725 If the equation (x, y , z) = 0 alone be given , y may be considered an implicit function of a and z. Since a and z are independent, we may make z constant and differentiate for a ; thus , for a variation in a only, the equations ( 1703-5) Φ (x, y , z ) in the place of 4 (x, y) . are produced again with p 1726

x be made constant , z must replace x in those equa If a

tions as the independent variable . Again , by differentiating the equation ( xyz) = 0 first for , making z constant, and the result for z , making a constant ,

296

DIFFERENTIAL CALCULUS.

we obtain 1727

$xz + $yxYz + $yzYx + $zy Y x Yz + $y Yzz = 0 .

From this and the values of Yx y, and y., by (1708) , Þux ÞµÞz + ☀uz Þ« Þz¬Þx= 6; —Þ›r Þ- Þ2y¸ 1728 z= Φ 1729

If x, y, z in the function Φ (x , y , z ) be connected by the relation (x, y , z) = 0 , y may be taken as a function of two independent variables a and z. We may therefore make z constant , and the values of p (x, y , z ) and 2 (x, y , z ) are identical with those in (1718 , '19) if x, y , z be substituted for a, y in each function . 1730 By changing a into z the same formulæ give the values of (x, y , z) and p2z (X, Y , Z) . 1731 On the same hypothesis , if the value of ò̟x (x, y , z ) , in forming which z has been made constant, be now differen tiated for z while a is made constant, each partial derivative Px, y, &c. in (1718) must be differentiated as containing x, y, and z , of which three variables a is now constant and y is a function of z . The result is

1732

yx ¥y— ¥y= $y) ¥zVj pxz (x, y, z) = { (Pxz ¥y— ¥xz6v) ¥³ — ($y= — (Þyz¥y— ¥yzÞy) ¥x4% + (Þ¿y ¥y− ¥zy $y) ¥x ¥z ¥v } ÷ ¥}

In a particular instance it is generally easier to apply such rules for differentiating directly to the example proposed, than to deduce the result in a functional form for the purpose of substituting in it the values of the partial derivatives . (x, y , z ) = lx + my + nz and ↓ (x , y, z) = x² + y² + z² 1734 EXAMPLE . - Let = 1, 2 and z being the independent variables ; 9, (x, y, z) aud p .I ( e, y, z) are required. Differentiating p, considering z constant, 2 dy x == +2 == Pz (x, y, z) = 1 + m dy = l- m ; since dx dx y 4, 2 = ―m y² + P2x (x, y, z) = -m y❜ y³ a result which is otherwise obtained from formula ( 1719) by substituting the values 4₂ = 1, Py = m, P₂ = n ; P2x = P2y = $2 = 0 ; 4₂ = 2x, 4, = 2y, 4 = 2z ; 42x = 424:= 422 = 2. Again, to find P. (x, y, z), differentiate for z, considering a constant in the function du L. == 2 d'o xy do = 1 m23 = -m , since == — m² ; thus = +m dz dx dx dz y' y 4x

297

IMPLICIT FUNCTIONS.

1735

Let U = p (x, y , z , E , n) be a function of five variables connected by three equations u = 0 , v = 0 , w = 0 ; so that two of the variables , may be considered independent. Making n constant , the equations in (1723 ) , and the values obtained for Us, x , y , z , hold good in the present case for the varia tions due to a variation in E, observing that p, u, v, w now stand for functions of n as well as of E. 1736 The corresponding values of Uŋ , æŋ, Yn , Z, are obtained by changing into n.

IMPLICIT FUNCTIONS OF n INDEPENDENT VARIABLES . 1737 The same method is applicable to the general case of a function of n variables connected by r equations u = 0 , v = 0, w = 0 ... & c. The equations constitute any n - r of the variables we please, independent : let these be , n , .... The remaining r variables will be dependent : let these be x, y, z ... ; and let the function be Up (x, y, z ... E, n , L ... ) . For a variation in & only, there will be the derivative of the function U, and r derived equations as under .

&E = Uε 9 •x x¿ + dy Y€ + Þz~ + ... + &ε

1738

Uz X& + Uy Y ε + Uz ~ε + ... + uε = 0,

Vx XE + Vy Y # + V½2 ≈ε + ...· + vε = 0, ... ... ... ... involving the r implicit functions E, Y , Z , &c. of the requations , as in ( 1724) , gives

dx 1739 d¿

== d (uvw ... ) 1 ; d (Eyz ...) J

dy

where

dU Also

J=

d (xyz ...)

The solution

== d (uvw ...) 1 d (x§z ...) † ; &c.,

d (uvw ...) 1740

&c. ,

d (puvw ) 1

= d

d (xyzέ) J

The last value being found exactly as in (1723) . 1741 With replaced by n we have in like manner the values of x , y , Un ; and similarly with each of the inde pendent variables in turn .

1742 If there be n variables and but one equation • (x, y , z ... ) = 0 , there will be n - 1 independent and one depen 2 Q

298

DIFFERENTIAL CALCULUS.

dent variable.

Let y be dependent .

Then for a variation in

a only of the remaining variables , the equations (1703-5) apply to the present case , p standing for (x, y , z ... ) . If æ be replaced by each of the remaining independents in turn , there will be, in all, n - 1 sets of derived equations .

CHANGE OF THE INDEPENDENT VARIABLE .

If y be any function of a, and if the independent variable be changed to t , and if t be afterwards put equal to y , the following formulæ of substitution are obtained , in which p = y : C 1 dy Yt 1760 X't dx Xy Differentiating these fractions , we get J d'y X2y Yu Xt Xu Yt = PPy 1762 da xv ม d³y Xi {Y& Xi ― X3Y1} — 3X2 { Y2X1 — XuYi} ¸ 1765 das x.t 3x324 — XyX3¥ 1766 = = PP² + P²P²y • x5 y

ї Ex.— If x = r cos 0 and y = r sin 0 ; then ro¡ по dy

re sin 0+r cos 0

d'y

r。 cos 0 -r sin 0

dx²

dx

r² +2r² ― rrǝe =

=

1768

· (r, cos 0 —r sin 0)³

3.98 PROOF. -Writing 0 for t in (1760) and ( 1762) , we have to find 20 Y20 ; thus, —r sin 0 ; x = r. cos () X29 = 2, COS 0—2r, sin 0--r cos 0 ; Y20 = , sin 0 + 2r, cos 0 — r sin 0. y = r, sin 0 + r cos 0 ;

, yan

Substituting these values , the above results are obtained .

90 To change the variable from a to t in (a +ba)" ynx, where (a + ba) = e , employ the formula 1770

(a + bx) " Ynx = b" ( de — n — 1 ) ( d — n — 2) ... ( d — 1) yı.

in which, multiplication by d by the index law signifies the repetition of the operation d, (1492) .

CHANGE OF THE INDEPENDENT VARIABLE.

299

PROOF.Now be, therefore Therefore and finally But

Therefore

d { (a + bx) ” Y «x } = { n ( a + bx) " - ¹ bynx + (a + bx) ” Y(n+ 1) x} Xt• ea +ba. Substitute this, and denote (a + b ) " by Uni 1 de ( Un) = nUn + — Un - 19 or Ub (d - n) Un. b U = b (de — ñ — 1) U„ -1 and U„ -1 = b (d; —ñ — 2) Un- 2, &C., U₁ =: b (d, -1 ) U. U₁ = (a + bx) x = bx, ! '₂ = byt U„ = b” (d; — n − 1) (d — ñ — 2) ... ( de − 1) Yt.

1771 COR .- (a + x) " ynx and a" yn nx are transformed by the same formula by putting b = 1.

1772 Let V = F (x, y) , where a, y are connected with E , n by the equations u = 0, v = 0 . It is required to change the independent variables x, y to E, n in the functions V, and V„ . 1773

RULE .

To find the value of V -Differentiate V, u, v,

each with respect to x, considering E, n functions of the inde pendent variables x, y ; and form the eliminant of the resulting equations ; thus , ― √x 1774 Vε V V¢§x + Vnnx¬Vx η = 0 Ux .. = 0. Uε un Už Šx + Un Nx + U¸ = 0 Vz r Ex + vn Vn Nx + V₂ = 0

VE re

Vn

Vx

Similarly, to find Vy' 1775

Ex.- Let xr cos 0 and yr sin ; then sin 0 cos V₁r = V, cos 0 - V. V₁ = V, sin 0 + Vo 1° r

PROOF. ur cos 0 -x, v =r sin 0 - y, V, V. -V₂l and the determinant in ( 1774 ) takes the form r sin 0 -1 = 0. cos annexed by writing r and 6 instead of ¿ and ŋ . sin 0 0 r cos 0 A similar determinant gives Vy. sin 0 in the pl . ce of V in the value of To find V2 , substitute V, cos - V, r V ; and in differentiating for r and 0, consider V, and V, as functions of both and 9. Similarly, to find V, and V. Thus, sin2 0 siu2 0 2 sin cos 0 + √20 + Vr , cos 0+ 1777 V₂ = V22r 2.2 για (— V.—V.) r cos Ꮎ cos² 0 @ - ( V.-V.re) 2 sin cos 0 + V, + √20 , sin³ 0 — 1778 V2y = V22r 7.3 r r r

By addition these equations give 1779

V₁2x + V₁

= V 2r +

1/2 V₁ +

1 V 20. ‫زاده‬

‫راندر‬

300

DIFFERENTIAL CALCULUS.

known functions of Given V = f(x, y , z ) and E , n, 1780 x, y, z ; V , V , V, are expressed in terms of V , V , V₂ by the formulæ dV

PROOF. -Differentiate V as a function of with respect to independent variables , n, x, y, z. The annexed equations are the result. Solve these by (582) with the notation of ( 1600).

1781

d ( nV)

dV

dV _ d (¿ VC) = ÷ J, dn d (xy )

= d ( Vns) ÷ J, αξ d (xyz)

=

÷ J. d ( xyz )

dr

V&E + Vynz η + Vghz = V29 VεEy + V» »y + Vz5y = Vy, V¿§₂ + V_n₂ + V¿5₂ = V..

Given V = f (x, y, z ) , where x , y , z are involved with

, n, in three equations u = 0, v = 0, w = 0, it is required to change the variables to E , n , ¿ in V¸ , V„ , and V½. Applying Rule ( 1773 ) to the case of three variables , we have +V +V - V = Vεૐ V₁η V; − Vz ปะ g &x + un Nx + U¿ 5x + Ux = 0 ž un uz Uş uε Vx = 0. re Vn og §x + Vn Nx + V§ 5x + Vx = 0 V5 VE η Wx TUE η wę wn Ws wę Śx + w»ŋx + w¿5x + w¿ = 0 · V

The determinant gives V, in terms of V , V , V, and the deri vatives of u , v , w. V, and V, are found in an analogous manner . 1782

Similarly with n equations between 2n variables .

1783

Ex. - Given

x = r sin

cos & ;

y = r sin

sin

;

The equations u, v, w become r sin 0 cos - x = 0 ; r sin @ sin -y = 0 ; Writing r, 0,

≈ = r cos 0.

r cos 0 -z = 0.

instead of E, n, 4, the determinant becomes. V. sin 0 cos

sin 0 sin cos 0

V. r cos 0 cos o r cos o sin

V₁ ―r sin 0 sin p r sin 0 cos

-r sin 6

0

-V₂ −1

= 0. 0

From which V, is obtained . Similarly, V, and V.; and, by an exactly similar process, the converse forms for V , V., and V. The results are * In writing out a determinant like the above, it will be found expeditious in practice to have the columns written on separate slips of paper in order to be able to transpose them readily. Thus, to find the coefficient of V., bring the second column to the left side, and, since this changes the sign of the determinant, transpose any two other columns, so that the coefficient of V, may be read off in the standard form as the minor of the first element of the determinant.

301

CHANGE OF THE INDEPENDENT VARIABLE.

- V sing r sin u

1784

V = V, sin

cos

+ V.

cos e cos y

1785

V, = V, sin 0 sin

+ V.

cos @ sin ቀ cos + V₂ r sin e

1786

V,

1787

V₁ = V,

V, sino cos

1788

V₁ = V.

V₂r cos 0 cos

1789

V₁ = - Vr sin 0 sin

sin 0 V, cos 0 – V. ጥ + V, sin 0 sin

+ V, cos 0.

+ V, r cos 0 sin

- V₂r sin 0.

+ V, r sin ✪ cos p.

1790 To find V, directly ; solve the equations u, v, w, in ( 1783) , for r, 0, and ; the solution in this case being practicable ; thus, r = √ (x² + y² + z³) ,

0 = tan- ¹ √ (x² + y²), 22

$ = tan -¹ Y x

Find r₂, 0 , 0, from these, and substitute in V₂ = V‚ï₂ + V, 0₂ + V. Simi larly, V, and V. Also V, = Vxr + VyYr + V₂ #r. Similarly, V, and V.. 1791 To obtain V,, substitute the value of V, in the place of V, in the value of V,, in ( 1785) , and, in differentiating V, V , V , consider each of these quantities a function of r, 0, and ø .

2x + V2y + V289 To change the variables to r , 0, and 4 , in V2 the equations ( 1783 ) still subsisting . Result

1792

V2x + V2y + V23 1 2r + 2 V, ++ ( V₂ cot 0 + V₂ + V½ V₁r 24 cosec² 6).

PROOF.-Put r sin 0 = p, so that therefore, by (1779) ,

= cos and y = p sin , 1 Ve t p² V2 ......... + V₂ + V2y = √₁₂ + 17OB Р

(i.).

Also, since z = r cos 0 and p = r sin 0, we have, by the same formula, V₂ + V₂ = V₂ + g⁰= V₁ Vr + 1

29......

... (ii .) .

Add together (i. ) and (ii. ) , and eliminate V,, by (1776) , which gives cos Ꮎ · V, = V, sin 0+ V. r

If V be a function of n variables x, y , z ... connected by the single relation ,

1793

a² +y² + ~ + ... = p²

2r 1 , 71V V2x + V2y + V22+ & c. = V + n -

(i.) .

DIFFERENTIAL CALCULUS.

302

PROOF.

X since r₂ = VV, r₂ = = V,, V, " by differentiating (i .) , r ጥ

x r xvx V2 = Vr« — + V.T (1 - 23). r + V, " —2.2"'z = Var = 2.2

therefore

Similarly

— — 1/2 ) , &c. + V, ( r

V₁₂ = Var 2

Thus, by addition, n n- 1 · _ ² Var ²² + y² + - + V 2x + V₂ + &c. = √₂ V₁₂ V., ( ~T ~ ~² + y² + - - ) = √₂2r + " g ~ 1 y på

LINEAR TRANSFORMATION. 1794 If V = ƒ (x, y , z) , (1781) take the forms

and if the

x = a₁§ + b₁n + cs , y = a‚§+ b¸ŋ + c₂Š ≈ = a

+b

A >

then

+ cs

equations

u, v , w in

= A₁x + A₂y + Az~,

Δη = B₁x + B₂y+ B¸≈ Aŋ ~, sc = AL C₁ x + C₂ y + Czz ,

by (582 ) , ▲ being the determinant (a,b,c ) , and A, the minor of a₁, & c . 1795

The operations de, d,, d, will now be transformed by

the first set of equations below ; and de, dn, de by the second set.

d¸ = ( A‚d + B₁d,η + C₁d ) ÷ A )

d = a₁d₂ + ad¸ +ad¸

d₁ d, = ( A₂d¸ + B₂d„ + C₂d¿) ÷ ▲ d. = (Agde + B3d + C₂d;) ÷ A ·

d„ = b₁d¸ + b¿d„ +będz d¿ = c₁d₂ + c₂ d₂ +cz dz

= PROOF. By dde , + d,η n₂ + d₂ , and de = d values of ,,, & c. , from the preceding equations.

1797 From ( 1795) , again upon V , we have

+ dy + d.z ; and the

V = a₁V₂ + a₂Vy + az Ve •

Operating

2Æ = · (a₁d₂ + ady + azd₂) V¢ = α₁ ( V¢) x + α2 ( Vg) y + α3 ( V£) 29 V₁ , with V2 , and by substituting the value of V , and similarly V2259, we obtain the formulæ, 1798

25 = a; V₂ + a² V2y + a3 V₂ + 2a,az Vyz • V₂ ยะ + 2α, α₁1 Vx +2α₁α₂Vxy V2n = b; V2x + b² V2y +b² V₂ + 2b2b3 Vyz + 2b3b₁ Vzx + 2b₁ b₂ Vxy 2x + c₁₂ V₂ -24 +C3 V2z + 2C₂ C3 Vyz + 2C3 C1 Vx + 20₁₂ Va V26 * = c₁ V₂

CHANGE OF THE INDEPENDENT VARIABLE.

303

ORTHOGONAL TRANSFORMATION.

1799

If the transformation is orthogonal (584) , we have x² +y² + ≈² = §² +n²+ 5² ;

and since, by (582 , 584) ,

A = 1,

4₁ = a₁ ,

& c .;

equations

(1794) now become

1800

x = a₁§ + b₁n + c₂ b

§ = а₁x + a₂y + azz ·

y = a‚§ + b¸ŋ + c₂Š

n = b₁x + b₂y +b3≈

≈ = as$ + b₂n + cs ( )

[ = c₁ x + c₂ y + C3 ≈

And equations ( 1795 ) become 1802

d₂ = a₁d₂ + b₁d₁ + c₁d} ]

d = ad + ad, + ad

d₁ = a₂d¿ + b₂dη +c₂d¿

d₁η = b₁d¸ + b₂d₁ + bçd₂

d₂ = açd₂ + b3dη + csd

dz = c₁ dx + c₂ d₂ + cзdz

The double relations between x, y , z and E, n , Z , in the six equations of ( 1800-1 ) , and the similar relations in ( 1802-3) between dedd, and dedde, are indicated by a single diagram in each case ; thus , 1804

1806

≤

n

5

de d

dz

x

a₁ b₁ c₁

dx

a₁ b₁ c₁

y

az b₂ cg

dy

a, b,

2

az b3 C3

d.

az bz cz

c,

Hence, when the transformation is orthogonal , the

quantities x, y , z are cogredient with de , dy , d₂, by the defini tion (1653) . 1807 Extending the definition in ( 1629) , it follows that any function u = (x, y , z ) , when orthogonally transformed , has , That is , if by for a covariant, the function (dx, dy , d₂) u . the transformation , u = $ (x, y , z) = 4 (§, n , () ,

then also $ (dx , d¸ , d₂) u = ¥ (d¸, dŋ , d¿) u. 1808

But if u be a quantic, then , as shown in ( 1648 ) ,

(d¸, dy , d₂) u is always a function of the coefficients only of u , and the covariant is , in this case, an invariant.

1809

Ex - Let u or

(x, y, z) = ax² + by + cz² + 2fyz + 2gxx + 2hxy,

.. $ (d„ d„ d¸) u = au₁₂ + bu₂y + cu₂₂ + 2fuy + 2gu₂ + 2hury = 2 { a² + b² + c² + 2f + 2g + 2h² } , and this is an invariant of u.

304

1810

DIFFERENTIAL CALCULUS.

When V = ƒ (x, y , z ) is orthogonally transformed , V2x + V2y + V22 = V2¢ + V2n + V25 •

PROOF. By adding together equations ( 1798) , and by the relations a + b + c = 1 , &c., and a, a, + b₁ b₂ + c₂ C₂ = 0 , &c . , established in (584).

1811 If two functions u, v be subjected to the same ortho gonal transformation , so that u = $ (x, y, z) = • (§, n , then

1812

)

and

v = 4 ( x, y , z) = Y ( §, n , L) ;

$ (dx, dy , dz) v = ♣ (d¸, d„ , d¿) v .

Ex. -Let

u = ax² + by² + cz² + 2fyz + 2gzx + 2hxy = = a& ² + b'n² + c′5² + 2ƒ˜ » 5 + 2g*CE + 2h'En = &,

and let Then

v = x² + y² + z² = 5² + n² + 6² = ¥ and Y. 4 (d₂, d¸, d₂) v = av₂ + bv₂y + CV22 + 2ƒvya + 2gVzz + 2hvzy ,

and

℗ (d§, dn, dz) v = av2¿ + b'v2n + c' v25 + 2ƒ* vn5 + 2g´vs¢ + 2h´v§n* But V2 = 2, and = 0, &c. Hence the theorem gives = a + b + c' ; in other words, a + b + c is an invariant.

a+b +c

―――

1813

Contragredient.

1814

Contravariant .— If, in ( 1629) , the quantics are sub

When the transformation is not orthogonal, ( 1795) shows that d, is not transformed by the same, but by a reciprocal substitution , in which a₁ , b₁ , c₁ are replaced by the corresponding minors A1 , B1 , C₁. In this case de, dy, d, are said to be contragredient to x, y, z.

jected to a reciprocal transformation instead of the same, we obtain the definition of a contravariant.

1815 When z is a function of two independent variables a and y, the following notation is often used :

dx

dy

dp dy

=9 ,

= p,

=

dz dq = 8, = dx dx dy

SIS FIS

313 S13

dz

dz

dp = d2% = r, dx dx² d²

dq

dy

=

= t. dy

Let p (x, y, z ) = 0. It is required to change the inde pendent variables from x, y to z, y. The formulæ of trans

MAXIMA AND MINIMA.

305

formation are dx

d2x =

1 =

dz

1819

dx

1 =

1816

dy

d'.x _ 28pq — tp' —rq² ; 3 dy p

---

L; p

dz2

d²x dydz

= qr—ps 3 p⁹

PROOF. -Formulæ ( 1761 , 1763 ) give 2, and 2z, because, since y remains constant, may be considered a function of only two variables, a and z . Formule (1708-9) give x, and in terms of partial derivatives of ø, since z is now constant, and may be taken as a function of the two variables x and y. (x, y) -z = 0 ; and the partial (x, y, z) = 0 is equivalent to But with respect to x and y are the same as those of ; and derivatives of therefore the same as those of z when x and y are the independent variables. Hence z may be written for 4 in the formula. ďc dx IPx-Pqx 1 = gr-ps Lastly, = (-2) = (-1 ), da = 2 pi P dydz ps . The independent variable is here changed from z to z, without reference to the equation = 0 ; and this is allowable, because y is constant for the time being in either case.

MAXIMA AND

MINIMA.

Maxima and Minima values of a function of one independent variable. 1830

DEFINITION.-A function

when some value xa makes

(x) has a maximum value (x) greater than it is made

by any value of x indefinitely near to a. minimum value, reading less for greater.

Similarly for a

1831 ILLUSTRATION. — If the ordinate y in the figure be al E ways drawn = f (x) , it has maximum values at A, C, E, D and minimum at B and D ( 1403). NOTE. For the algebraic determination of maxima and minima values by a quadratic B ка к equation, see (58) .

Φ (x) is a maximum or minimum 1832 RULE I.- A function p when ' (x) vanishes, and changes its sign as x increases from plus to minus or from minus to plus respectively. 2 R

DIFFERENTIAL CALCULUS.

306

1833

RULE II.--Otherwise

(x) is a maximum or minimum

when an odd number of consecutive derivatives of and the next is minus or plus respectively.

(x) vanish,

PROOF.- (i . ) The tangent to the curve in the last figure becomes parallel to the x axis at the points A, B, C, D, E as a increases ; therefore, by (1403), tan 9, which is equal to f ' (x ) , vanishes at those points, while its sign changes in the manner described. (ii.) Let f (x) be the first derivative of f(x) which does not vanish when h" f z = a, n being even ; therefore, by ( 1500 ) , ƒ ( a ± h) =ƒ(a) + ** n ƒf" (a + 6h ) . The last term retains the sign of ƒ" (a) , when h is small enough, whether h be positive or negative, since n is even. Therefore f (x ) diminishes for any small variation of a from the value a if ƒ" (a) be negative, but increases if f"(a) be positive. Hence the rule.

1834 NOTE . - Before applying the rule for discovering a maximum or minimum, we may evidently (i.) reject any constant factor of the function ; (ii ) raise it to any constant power, paying attention to sign ; (iii ) take its reciprocal ; maximum becoming minimum, and vice versá ;

(iv.) take the logarithm of a positive function . 1835 Ex. 1.—Let (x) = 7-7-35x + 1, therefore ' (x) = 7xº - 28x³ - 35 = 7 (x³ - 5 ) ( x³ + 1 ) . Also " (x ) = 7 ( 6æ³ - 12x²) . Therefore x = 3/5 makes p′ (x) vanish, and p" (a) positive ; and therefore makes (a) a minimum. Ex. 2.- Let 1836 (x) = (x − 3) (x - 2 ) ". Here -- 3 ) ¹³ ( ≈ −2 ) ¹¹ + 11 ( ≈ −3)¹ (x p'′ (x) = 14 ( x − − 2 ) ¹ = (x −3 ) ¹³ (x − 2)¹º ( 25x — 61 ) , and we know, by (444) or by ( 1460) , that, when x = 3, the first thirteen derivatives of (x) vanish ; and 13 is an odd number. Therefore ( x) is either a maximum or minimum when x = 3. To determine which, examine the change of sign in ' (x) . Now (x - 3) ¹s changes from negative to positive as a increases from a value a little less than 3 to a value a little greater, while the other factors of p′ (x) remain positive. Therefore, by the rule, (2) is a minimum when x = 3 . Again, as a passes through the value 2, (e) does not change sign , 10 being even. Therefore 2 gives no maximum or minimum value of p (x). 61 Lastly, as a passes through the value 6 the signs of the three factors 25 in ' (x) change from ( - ) ( + ) ( − ) to ( − ) ( + ) ( + ) ; that is, ' (x) changes from + to ; and, consequently, p (a) is a maximum. 1837 of y.

Ex . 3. - Let p ( x, y) = x* + 2xy -y³ = 0.

To find limiting values

MAXIMA AND MINIMA.

307

Here y is given only as an implicit function of x. Differentiating, in order to employ formula ( 1708, 1711) , P₂ = 4x³ + 4xy, P₂ = 12x² + 4y, 4,y = 2x² - 3y' ; Solving this equation with 10 makes ₂ = 0. (x, y) = 0, we get z = ± 1 , y = −1 when y, vanishes. 12-4 And then y 8, positive ; therefore, when ±1, 2-3 Py -1 for a minimum value. y has -=

Similarly, by making y the independent variable, it may be shewn that, 8 4 when y = z has both the maximum and minimum values + √6. 9' 9 1838

A limiting value of

subject to the condition

is obtained from the equation

(x, y) , (i . ) , ¥ (x, y) = 0 ......... .. φ.Ψ. ...... (ii .) t Þx4y = $y¥x 中

Simultaneous values of a and y, found by solving equa tions (i .) and (ii . ) , correspond to a maximum or minimum value of p. PROOF.- By ( 1718) , ø being virtually a function of a only ; and, by (1832) , 9, (xy) = 0. 1839 Ex - Let (x, y) = xy and (x, y ) = 2x³ - xy + y³ = 0 ………….. (i .) Equation (ii. ) becomes y (3y² —x) = x (Cx² —y) . Solving this with (i . ) , we find y³ = 2x³ and æ² (4x — 3/2 ) . = 3/2, y = 1/4 are values corresponding to a maximum Therefore value of p . That it is a maximum, and not a minimum, is seen by inspecting equation (i . ) Most geometrical problems can be treated in this way, and the 1840 alternative of maximum or minimum decided by the nature of the case. (xy ) may be examined by formula ( 1719 ) for the Otherwise the sign of criterion, according to the rule.

Maxima and Minima values of a function of two independent variables . Φ (x, y) is a maximum or minimum 1841 RULE I. —A function 4 when Px and Ps both vanish and change their signs from plus to minus or from minus to plus respectively, as x and y increase. 1842

RULE II. - Otherwise , p , and 4, must vanish ; P2xP2y — Pxy

must be positive, and p2x or p2y must be negative for a maximum and positive for a minimum value of p. PROOF.- By ( 1512) , writing A, B, C for P2x, xy, P2 , we have, for small . changes h, k in the values of x and y, (x +h, y + k) -p (x, y) = hp₂ + kpy + ½ (Ah² + 2Blk + Cl2) + terms which may be neglected, by (1410) .

308

DIFFERENTIAL CALCULUS.

Hence, as in the proof of ( 1833 ) , in order that changing the sign of h or k shall not have the effect of changing the sign of the right side of the equa tion, the first powers must disappear, therefore 4, and o, must vanish . The next term may 2be written, by completing the square, in the form h だ 22 { ( 4 )k + B3)) ² + AC — B² } ; and, to ensure this quantity retaining its 2A sign for all values of the ratio hk, AC - B² must be positive . will then be a maximum or minimum according as A in the denominator is negative or positive . It is clear that A and B might have been transposed in the proof. Hence B must have the same sign as A.

1843

(x, y, z) ,

A limiting value of

subject to the condition .

¥ (x, y, z) = 0.……………….. (i .) ,

is obtained from the two equations

$24₁ = $ 42 ...... (iii .) ;

1844

$x¥y = = $v¥x· l ...... (ii . ) ,

1846

or, as they may be written ,

ป.

– Φε ...... (iv.) =Φ» = 41 42

Simultaneous values of x , y , z , found by solving equations (i . , ii . , iii . ) , correspond to a maximum or minimum value of . PROOF.- By ( 1841 ) , being considered a function of two independent variables a and z, and, by (1729, 1730) , (x, y, z) = 0 gives (i . ) , and p (x, y , z) = 0 gives ( iii. ) The criterion of maximum or minimum in ( 1842 ) may also be applied without eliminating y by employing the values of 2 and 2 in ( 1719 , '30 ) .

1847

Ex.- Let

and

(x, y , z ) = ax² + by² + cz² + 2fyz + 2gzx + 2hxy - 1 = 0 .. (i . )

$ (x, y , z ) = x² + y² + z²

Equations (ii . ) and (iii . ) here become 1 x 2 y = = = , say, (iv.) R gx +fy + cz ax + hy + gz hx + by +fz

Therefore, by proportion (70) and by (i . ) , x² + y² +z² = R¬¹ = p . From equations (iv. ) we have

hx + by +fz = Ry gx +fy + cz = Rz 1849

a -R

ax + hy + gz = Rx

;

..

h

g

h

b- R f

g

f

=

1848

= 0,

c- R

or (R - a) (R - b) (R − c) + 2fgh - (R - a) ƒ² - (R — b) g² — (R — c) h² = 0 , or (see 1641 )

R³ — R² (a + b + c) + R (be + ca + ab -f2 -g² - h²) —A = 0 ,

MAXIMA AND MINIMA.

309

This cubic in R is the eliminant of the three equations in x, y, z . It is called the discriminating cubic of the quadric (i .) , and its roots are the reciprocals of the maxima and minima values of a² +y + z² . 1850 To show that the roots of the discriminating cubic are all real.

Let R₁, R, be the roots of the quadratic equation R-b f 1851 R £ ¸ | = (R — b ) (R — c) — f² = 0.….. | f

... .. (v.)

R₁b and c, and b and c > R2. Make R = R₁ in the cubic, and the result is negative, being minus a square quantity, by (v.) . Make R = R , and the result is positive . There fore the cubic has real roots between each pair of the consecutive values + ∞ , R, R.; that is, three real roots . But since the roots are in order of magnitude, the first must be a maximum value of R, the third a minimum , and the intermediate root neither a maximum nor a minimum.

Maxima and Minima values of a function of three or more variables. Let 1852 Let p (xyz) be a function of three variables . P22, P2y , P2z, Pyz, Pr, Pry xy be denoted by a, b, c , f, g , h ; and let A, B, C, F, G, H be the corresponding minors of the deter minant A, as in (1642) . 1853

RULE I .—$ (x , y, z) is a maximum or minimum when z all vanish and change their signs from plus to minus or from minus to plus respectively, as x, y, and z increase. Otherwise Px,

y,

1854 RULE II .— The first derivatives of 4 must vanish ; A and its coefficient in the reciprocal determinant of ▲ must be positive ; and will be a maximum or minimum according as a is negative or positive . and b or C and c.

Or , in the place of A and a , read B

PROOF.-Pursuing the method of (1842) , let , y , be small changes in the values of x, y, z. By (1514) , $ (x + E, y + n, z +5) − p (x, y , z ) = {0₂ + ng + 59 ; + § ( a¿² + bn² + c$ ² + 2ƒn % + 2g + 2h&n ) + &c. For constancy of sign on the right, 4 , 4 , 4. must vanish. The quadric may then be re- arranged as under by first completing the square of the terms. in , and then collecting the terms in 5, 4, and completing the square . It 1 (Cn - F )2 BC-F2 thus becomes 24 { (a5 +hn +95)² + 2a 5 }. + 0 C

310

DIFFERENTIAL CALCULUS.

Hence, for constancy of sign for all values of E, n , 5, it is necessary that C and BC- F should be positive. This makes B also positive. By symmetry, it is evident that A, B, C, BC - F , CA - G, AB - H' will all be positive. is a The sign of a in the first factor then determines, as in ( 1842) , whether maximum or a minimum.

1855

The The condition may be put otherwise . Since BC- Fa▲ by (577) , the condition that BC2 - F2 must be positive is equivalent to the condition that a and ▲ must have the same sign. Hence we have also the following rule : 1856 RULE III . —px, y , p₂ must vanish ; the second of the four determinants below must be positive, and the first and third must have the same sign : that sign being negative for a max imum and positive for a minimum value of p ( x , y, z) .

1857

2x

фах ф.гу Pyx Pay

"

Pax Pry Pxz

9

P2x Pxy Prz Prw

Pyx Pay Puz

Pyx Pzy Pyzyw

Pzxzy Paz

Pzx Pay $22 $210

wx Pwy Pwr

wz Pew

1858 The theorem can be extended in a similar manner to 。 (x, y , z , W, ... ) , a function of any number of variables . Form the successive Hessians of ( 1630 ) for one , two , three, & c . variables in order as shown above ; then—

1859

RULE.-In order that p (x , y, z , w, ... ) may be a max

imum or minimum , Px , Py, Pz , Pw , &c . must vanish; the Hessians of an even order must be positive ; and those of an odd order must have the same sign , that sign being negative for a maximum and positive for a minimum value of the function p. For a demonstration in full, see Williamson's Diff. Calc. , 4th Edit., p. 433 .

1860

Ex -Required a limiting value of the function u = ax² + by² + cz² + 2fyz + 2gzx + 2hxy + 2px + 27y + 2rz + d. The condition in the rule produces equations ( 1 ) , (2 ) , (3 ) . Equation (4) results from Euler's theorem ( 1624) , thus ; introducing a fourth variable w, as in ( 1645) , we have xu₂ + yu, + zu₂ + wu, = 2u, which reduces to (4) by means of ( 1 ) , ( 2 ) , ( 3 ) , and the value of u, putting w = 1. 1861

Ju₂ = ax + hy + gz +p = 0 ...... ( 1) u, hx + by +fz + q = 0 ...... (2) ugx +fy + cz + r = 0 ...... (3) px + qy + rz + du...... (4)

...

ah g P q h bf = 0. r gf c p q r d- u

MAXIMA AND MINIMA.

311

The determinant is the eliminant of the four equations, by ( 583 ) , and is equivalent, by the method of ( 1724, Proof. i . ) , to A' — Au = 0, or u = A' ÷ A (Notation of 1646) . To determine whether this value of u is a maximum or minimum , either of the conditions in ( 1854, ' 6) may be applied ; and since, in this example, U₂ = 2a, U₂y = 2b, &c. , the letters a , b, c, f, g, h may be considered identical with those in the rule.

1862 To determine a limiting value of p (x, y , z, ... ) , a func tion of m variables connected by n equations u₁ = 0, u , = 0 , ... u₂ = 0.

RULE . ----- Assume n undetermined multipliers A1, A2, ... λα with the following m equations : — 4x + λ1 (U₁)x + λ₂ (U2)x + ... + λn (Un)x = 0 ), = ... 0, + λn (Un )y Py + λ1 (U1)y + λ2 (U2)3 + ... ... ... ... ...

making in all m + n equations in m +n quantities , x, y , z , ... and A1 , A2, ... An. The values of x, y , z , ... ,. found from these equations, correspond to a maximum or minimum value of p. PROOF. -Differentiate and u₁, U2, u, on the hypothesis that x, y, z, ... are arbitrary functions of an independent variable t. Multiply the resulting equations, excepting the first, by A₁ , λ , … .. A, in order, and add them to the value of p . The coefficients of x , y , z , ... may now be equated to zero, since the functions of t are arbitrary, producing the equations in the rule.

1863 Ex. 1.- To find the limiting values of r²x² + y² + z² with the conditions Ax² + By + Cz² = 1 and lx + my + nz = 0. Here m3, n = 2 ; and, choosing A and μ for the multipliers, the equa tions in the rule become 2x +2 Aλα + μι = 0 ...... ( 1 ) 22 2y+2Bλy + µm μm = 0 ...... (2)

Multiply ( 1 ) , (2 ) , ( 3 ) respectively by æ, y, z, and add ; thus μ disappears, and we obtain

2z + 2Cλz + μη µn = 0 ...... (3) ) x² + y² + z² + (Ax² + By² + Cz²) λ = 0,

- p³. therefore λ = —

Substitute this in ( 1 ) , ( 2) , (3 ) ; solve for x, y, z, and substitute their values in lx + my + nz = 0.

1864

The result is

n³ 72 m² = 0, a quadratic in r². + + Cr²-1 Ar² -1 Br2-1

The roots are the maximum and minimum values of the square of the radius vector of a central section of the quadric Aa² + By + Cz² = 1 made by the plane la + my + nz = 0.

1865 Ex. 2. To find the maximum value of u = (x + 1 ) ( y + 1 ) ( z + 1 ) , subject to the condition N = a* b* c*.

312

DIFFERENTIAL CALCULUS.

This is equivalent to finding a maximum value of log (x + 1 ) + log (y + 1 ) + log ( z + 1 ) , subject to the condition log N = x log a + y log b + z log c. The equations in the rule become 1 1 1 + λ log a = 0 ; +λ log b = 0 ; 2 + 1 + λ log c = 0. x+ 1 y+1 By eliminating A, these are seen to be equivalent to equations (1846) . Multiplying up and adding the equations, we find A, and thence +1 , y + 1 , z + 1 ; the values of which, substituted in u, give, for its maximum value, u = { log (Nabe) } ÷ 3 log a log b³ log c³ . Compare (374) , where a, b, c and x, y, z are integers . Continuous Maxima and Minima. 1866

If pr and p ,, in ( 1842 ) , have a common factor , so that $x = P

......... Φυ = = Q4 (x, y) …… ........ ( i . ) ,

(x, y) ,

where P and Q may also be functions of x and y ; then the equation (x, y) = 0 determines a continuous series of values of x and y. For all these values $ is constant, but , at the same time, it may be a maximum or a minimum with respect to any other contiguous values of p , obtained by taking æ and y so that

(xy) shall not vanish.

1867 In this case, p2rPaypy vanishes with P2xP2y -Pry criterion in Rule II. is not applicable .

, so that the

PROOF. -Differentiating equation ( i. ) , we have ry = P₁ ++ P↓,? y $2 = P₂ + P↓ ) 42y = Qy4 + Q4, Š Pyz = Qx4 + Q&s If from these values we form the expression.

2x02y - Pzy × Pyx ,

will appear as a factor of

1868

Ex.--Take z as (xy ) in the equation z² = a² — b² + 2b √ ( x² + y³) − x³ — y³ (i.) , b b 22x = x and − zz, = y' ( √(x² +y²) − − 1) 1) . · ( √(x² + y² ) − The common factor equated to zero gives x² + y² = b², and therefore z = ±a ….. (ii.) Here a is a continuous maximum value of 2, and -a a continuous minimum. Equation (i. ) represents, in Coordinate Geometry, the surface of an anchor ring, the generating circle of radius a having its centre at a distance b from the axis of revolution Z. Equations (ii. ) give the loci of the highest and lowest points of the surface. For the application of the Differential Calculus to the Theory of Curves , see the Sections on Coordinate Geometry.

INTEGRAL

CALCULUS.

INTRODUCTION.

1900 The operations of differentiation and integration are the converse of each other. Let f(x) be the derivative of (x) ; then (x) is called the integral of f(x ) with respect to x. These converse relations are expressed in the notations of the Differential and Integral Calculus , by

do (x) = =f(x) dx

and by Sƒ(x) dx = $ (x).

1901 THEOREM. -Let the curve y = f (x) be drawn as in (1403) , and any ordinates Ll, Mm , and let OL = a , OM = b ; then the area LMml = $ (b) —p (a). PROOF. - Let ON be any value of x, and PN the corresponding value of y, and let the area ONPQ = A ; then A is some function of x. Also, if NN' = dx, the elemental area NN'P′P = dA dA = yde in the limit ; therefore = y. Thus A da is that function of x whose derivative for each value of x is y or f(x) ; therefore = (x) + C, where C is any constant. Consequently the area LMml = (b) — ( a) , whatever C may be.

m

Q

I

NN'

M

The demonstration assumes that there is only one function (x) corres ponding to a given derivative ƒ (x) . This may be formally proved. If possible, let (x) have the same derivative as ( x ) ; then, with the same coordinate axes, two curves may be drawn so that the areas defined as above, like LMml, shall be ( x) and (2) respectively, each area vanishing with 2. If these curves do not coincide, then, for a given value of x, they have different ordinates, that is, g' (x) and ' (x) are different, contrary to the hypothesis . The curves must therefore coincide, that is, p (x) and ↓ (x) are identical. 2 s

INTEGRAL CALCULUS .

314

1902 Since Φ ( b ) — ( a) is the sum of all the elemental areas like NN'P'P included between Ll and Mm , that is , the sum of the elements yda or f(x) de taken for all values of a between a and b, this result is written b S" f(x) dx = $ (b) —

( a) .

1903 The expression on the left is termed a definite integral because the limits a, b of the integration are assigned. * When the limits are not assigned , the integral is called inde finite.

1904

By taking the constant C = 0 in ( 1901 ) , we have the

area ONPQ = 4 (x) = √ƒ(x) dx. NOTE. In practice, the constant should always be added to the result of an integration when no limits are assigned .

MULTIPLE INTEGRALS . 1905

Let f(x, y , z ) be a function of three variables ; then

the notation S**S**S** ƒ(x, y, z) dx dy dz

is used to denote the following operations . Integrate the function for z between the limits z = 21 , Z = %29 Then , considering the remaining variables x and y constant . whether the limits Z1 , Z2 are constants or functions of x and y , Next, consider the result will be a function of x and y only. ing a constant, integrate this function for y between the limits y₁ and 2, which may either be constants or functions of x. Y1 The result will now be a function of a only. Lastly, integrate æg. a, and Xq. this function for a between the limits X1 Similarly for a function of any number of variables .

1906

The clearest view of the nature of a multiple integral

is afforded by the geometrical interpretation of a triple in tegral.

* The integral may be read " Sum a to b, f (x) dx " ; f signifying " sum.”

315

INTRODUCTION.

12

n

m X

N M

#

Taking rectangular coordinate axes , let the surface z = p (x, y) (XYom in the figure) be drawn , intersected by the = 4 (x) (RMNnm) , and by the plane cylindrical surface y = The volume of the solid OLMNolmn bounded x = x (LSml) . by these surfaces and the coordinate planes will be ( (x) x1 (¥ (x) C $ (x, y) dx dy dz. С+ →)So [*' 'S* (*) 4 (x, y) dx dy = ' S0+( 0, 0) S* S* (* PROOF. - Since the volume cut off by any plane parallel to OYZ, and at a distance from it, varies continuously with a, it must be some function of x. Let V be this volume, and let dV be the small change in its value due to a change da in x. Then, in the limit, dV = PQqp × da, an element of the solid shown by dotted lines in the figure. Therefore (4 (x) dv = PQqp = [* dæ 0 (*) (x, y) dy, by ( 1902) , being constant throughout the integration for y. The result will be a function of a only. Making a then vary from 0 to x, we have, for the whole ( (x, y) volume,

[ " { 0

(x, y) dy } dax = ' [* 0 ”) dz] dy } dz, 0' { [*0 (*) [ ˜*(*.

since (x, y) = z. With the notation explained in ( 1905 ) , the brackets are not required, and the integrals are written as above.

1907 If the solid is bounded by two surfaces z₁ = 41 (X, Y ) , %, = ¢, (x, y) , two cylindrical surfaces ₁₁ (x) , y₂ = 4, (x) , and two planes x = x1 , x = X2, the volume will then be arrived at by taking the difference of two similar integrals at each integration, and will be expressed by the integral in (1905 ) . If any limit is a constant, the corresponding boundary of the solid becomes a plane.

316

INTEGRAL CALCULUS.

METHODS

OF

INTEGRATION .

INTEGRATION BY SUBSTITUTION.

1908

The formula is

√

(x) dx = §4 (x) dx dz,

where z is equal to f (x), some function chosen so as to facili tate the integration . RULE I.- Put x in terms of z in the given function , and multiply the function also by x ; then integrate for z. If the limits of the proposed integral are given by x = a, x = b , these must be converted into limits of z by the equation x = f¹ (z) .

The following rule presents another view of the method of substitution, and is useful in practice . 1909

RULE II .— If

(x) can be expressed in the form F ( z) z、;

then S4 (x) dx = S F (2) z,dx = SF ( 2) dz.

dx Ex . 1. —To integrate² + v²) · therefore

Ex. 2. Here

Substitute

= x + √ (x² + a³) ;

r 2 dz = x + √(x² + a²) = = 1 + dx √(x² + a²)' √(x² + a² ) √ (x² + a²) dz dx 1 dz = log z = log { x + √ ( x² + a²) } . √ √(x² + a²) dz d² = [ ²2 = 5 + 2a³ 5x-6 +2x-3 dx = dalog (x +x˜³) . 5 s x +x S x + x¹ 1 -·(5x - 6 + 2x - 2) . x = x˜³ + x¯², F (z) = -—-— " ‫ܠܰܐ‬ 2

1 - cx2

dx

=

Ex. 3.

x-2 - c

dx

( 1 + ca² √ (1 + ax² + c²x*)

Sx² + cx √ ( c²x² + x - ² + a) d, (x1+cr) dr

or

= -Si( x¯ ¹ + cx) √ { (x¯¹ + cx ) ² + a − 2c } 1 ¸x √ (a − 2c) + √ ( 1 + ax² + c²x*) ¸ log 1 + cx² ✓(a- 2c) 1 cos -1 x √(2c —a) 1+ ca³ √ (2c - a)

By (1927) or (1926) .

Here x = x¯¹ + cx.

METHODS OF INTEGRATION.

317

In Examples (2 ) and (3) the process is analytical, and leads to the dis covery of the particular function z, with respect to which the integration is effected. If z be known, Rule I. supplies the direct, though not always the simplest, method of integrating the function. INTEGRATION BY PARTS . 1910

By differentiating uv with respect to a, we obtain the

general formula

Suvda = uv — -Suv_dx.

The value of the first integral is thus determined if that of the second is known .

RULE . Separate the quantity to be integrated into two factors . Integrate one factor, and differentiate the other with respect to x. If the integral of the resulting quantity is known, or more readily ascertained than that of the original one, the method by " Parts " is applicable. NOTE.- In subsequent examples, where integration by Parts is 1911 directed, the factor which is to be integrated will be indicated. Thus, in example ( 1951 ) , “ By Parts Jezda " signifies that ea is to be integrated and sin bæ differentiated afterwards in applying the foregoing rule. The factor 1 is more frequently integrated than any other, and this step will be denoted by dx. INTEGRATION BY DIVISION.

1912

A formula is

xn S(a+bx")" dx = aS(a+bx" ) " -' +bSx" ( a +bx")" =¹ dx : The expression to be integrated is thus divided into two terms , the index p in each being diminished by unity, a step which often facilitates integration .

Similarly,

S (a + bx” +cx™) º dx

= a (a + bx" + cx™) ” - ¹ + bx” (a + bx" + cx™ ) ” - ¹ + cx” ( a + bx" + cx” ) p-!.

1913

Ex -To integrate

[ √ (x² +a² ) dæ.

x dx √√(x² + a²) dx = x √ (x² + a³) − √ √ √ (x² + a³)* x2dx a❜dx By Division, ac²) + √ (² 4 a²) [ √ (x² + a²) dx = √ √(x²+ √ √ (2 + a³) Therefore, by addition, a²dx /11 √ √ (x² + a³) dz = $x √ (x² + a³) + +S [ √ √ (x² + a³) = {} x √ (x² + a³ ) + } a² log { x + √ ( x² + a²) } , by (1909, Ex. 1) .

By Parts ( dz,

INTEGRAL CALCULUS .

318

INTEGRATION BY RATIONALIZATION. 1914 In the following example, 6 is the least common de nominator of the fractional indices . Hence , by substituting z = x+, and therefore x = 6z³ , we have

78 dx x -1 da [ x³ xt= { dz - 1 d² dz dz = 6 [ = -1 dx = [ z²=1 1 — z³ = 6 [ (~ +2 − 2 +2 −2 + 1 — —— — 1) de. Each term of the result is directly integrable by ( 1922 ) and (1923 ) . For other examples see ( 2110) .

INTEGRATION BY PARTIAL FRACTIONS . 1915 Rational fractions can always be integrated by first resolving them into partial fractions . The theory of such resolutions will now be given . 1916 If (x) and F (x) are rational algebraic functions of æ, (a) being of lowest dimensions , and if F (x) contains the factor ( -a) once , so that (1) ;

F (x) = (x− a) ¥ (x) .

(x 1917

A =

then

F (x)

+ x-a

X (x) and A = $ (a) F' (a) (x)

(2) .

PROOF. -Multiply equation (2 ) by ( 1), thus • (x) = A ¥ (x) + ( x − α ) x ( x) . Therefore, putting xa, (a ) = A (a) . _ Also, by differentiating ( 1 ) , and putting aa afterwards, F (a) = (a) . Therefore A = (a) ÷ F' (a) .

1918

Again, if F (x) contains the factor ( -a) , n times , so

that

F(x) = (x − a)" ↓ (x) . =

F (x)

A2

A₁

(x) Assume

+ (x-a)"

A + ... +

(x− a) ”–1

x-a

X(x) (x)

To determine A₁, Ag ... A. Multiply by (x - a)" ; put x = a and differentiate, alternately .

1919

If F (x) = 0 has a single pair

of imaginary roots

319

STANDARD INTEGRALS.

aiß ; then , applying ( 1917 ) , let $ (a- iẞß)

(a + iß)

= A ― iB ;

= A + iB ;

.. F' (a— iß)

F' (a + iß)

and the partial fractions corresponding to these roots will be

A - iB

A+iB

2A (x − a) +2BB

x -a + iß

(x- a)² + B³

+

x— a— iß

For practical methods of resolving a fraction into partial fractions in the different cases which occur, see (235-238) .

INTEGRATION BY INFINITE SERIES .

When other methods are not applicable , an integral may sometimes be evaluated by expanding the function in a con verging series and integrating the separate terms . Ex.

a²x² a³x³ a*x* eas + &c. + [ 2 dx = log x + ax + 1.22 + 1.2.3 1.2.3.4

STANDARD

(150)

INTEGRALS .

1921 Some elementary integrals are obtained at once from the known derivatives of simple functions . Thus the deri vatives (1422-38) furnish corresponding integrals . The fol ――― lowing are in constant use :

1922

1924

x √ a dx = Sam

xm + 1 • +1

a* Sadx = Loga 1

dx Sv x √ (x²—a²)

dx

Se= e. a

cos -1

=

1926

= log x. Sda x

a

1

a

X

.

sin- 1

or -

x

1 [ Sube. x

Xx

a log a+ √ (a²± ²) 1927 S√(a²± ² ) ==

1

[ Substitute

320

INTEGRAL CALCULUS.

dx 1928

S√ (x³± ³)

1929

√√α -a³)

= = log { x + √ ( x² ± a²) } .

dx

1 x = sin -¹ a

or

(1909, Ex. 1)

-cos -1 x a

(1434)

x dx = ± √ (a² ±x²).

1930

√√( ² ± ²) @

1931

By Parts, Division , and adding results (1913 ) , we obtain

√ √(x² ± a²) dx = } x√ (x² ± a²) ± } aª log {x + √ (x² ± a²) } . 1932

By Parts , Division, and difference of results , x² dx = \ x√ (x² ± a²) ‡ ¦a² log { x + √ ( x² ± a²) } . }

S√ ( x + a²)

1933 1934

-1 X + ½ x √ (a² — x²) . √√(a²—x²) dx = { a² sin¬¹ a [As in (1931 ) x² dx x = a² sin -¹ ½ x √ (a² — x²). a S= √(a² —x³) [As in (1932) 1

dx =

1935

a

Six² + a²

1 x or - -cot-1 a α

x

tan -1 a

1

dx =

1936 S x² -a²

x- a log

2a =

a+ x

sin x dx = — cos x. 1938 Ss 1940 Stanada =

1942

·

log 2a

S a² -x²

[ By Partial fractions x+a

1

dx 1937

(1436)

log cos æ.

secxdr = log tan se (1 + 1)

METHOD .-( 1940, '2) , substitute cos x.

[

Do.

a- x

cos x dx = sin x. Sco

cot x dx = log sin x. Sco x cosecx dx = log tan 2 Sco ( 1941, ' 3), substitute sin x.

1944

sin‍¹ x dx = x sin¯¹ x + √ (1 − x²) . S=

1945

cos¹x dx = x cos¹x - √ (1 - x²) . Sco

321

CIRCULAR FUNCTIONS.

-1 tan - ¹x dx = x tan-¹a 1946 Stan

log (1 +x²) .

cot-1 x dx = a tan¹a + Scot 1947 S

log (1 + x²) .

1948

sec¯¹ x dx = x sec¯¹ x −log { x + √ ( x² − 1 ) } . S sec S cosec¯¹ x dx = x cosec¯¹x + log { x + √ ( x² − 1 ) } .

1949 S METHOD.- ( 1944) to ( 1949) , integrate by Parts, Sdx.

1950

log x dx = xlog x - x. Slog

dr

2

a + b cos x

√ (a² — b³)

tan -1

1951 1

1952

[ By Parts, Sda

{ tan√( + )}

√ (b + a) + √/ (b - a) tan

r

log

or ✓ (b² — a³)

√ (b +a) −√ (b — a) tan ½ x

according as a is > or < b. [ Subs . tan

VARIOUS

, and integrate by (1935 or '37)

INDEFINITE

INTEGRALS .

GENERALIZED CIRCULAR FUNCTIONS .

cosec" dr. So

cos" dx. Sc

1954 Ssin" da.

METHOD. -When n is integral, integrate the expansions in ( 772-4) . Otherwise by successive reduction, see ( 2060) . For cosec” da, see (2058) . n-3 x tan"

tan" -1r x 1957

tan” rdr = Sta

n-5 x tan"

-

- &c .

+ n-l

n-3

n- 5

PROOF.-By Division ; tan" x = tan " -2x sec² a -――― tan" -2x, the first term of which is integrable ; and so on. dx 1958

S (a+b cos a)"

-1 = a² —b²) -4 S (a − b cos x) ¹¹ dz. -

2 2 By substituting tan 2 = tan 2 √( sin 20 in the place of cos x, substitute 7 - x.

METHOD.

cos x dx. 9 Scos nx 195

cos x dx. sin nx 2 T

6) . Similarly with

sin" x

sin' x dx.

Scos t nx

dx. Ssin nx

322

INTEGRAL CALCULUS.

METHOD. -By ( 809 & 812 ) , when p and n are integers, the first two functions can be resolved into partial fractions as under, p being < n in the first and < n - 1 in the second. The third and fourth integrals reduce to one or other of the former by substituting -2. cospx 1 ሽ 1 ) @ cos? (2r - 1 ) 0 with @ = π (-1) +1 sin (2= " 1963 2n n Σ**】 cos nx cos x - cos (2r − 1) 0 1964

COSP x 1 = sin ne n sin x

r-1

- (- 1) +1 sin² rû cos² rů with 0 = π ገሊ cos c - cos ro'

The fractions in ( 1963) are integrated by ( 1952 ) ; those in ( 1964) by (1990) .

Formula of Reduction. cos nx

dx co 'cos (n − 1 ) x dx - ' s (n − 2) x 2 S cos -1x Sc cos x

dx =

1965 Sco

cos nx

S

'cos (n − 2) x dx sin (n - 1) x dx + sing -1x Sco sing x

S

'sin (n - 2) x cos (n − 1) x dx + S dx sing x sin -1x

dx = --2 1966

sin x Sco

sin nx 2

dx = 1967 S sin' x

sin n--1) x 'sin ( n − 2) x dx dx 2 fsi cosp-1x Jsi cos c

sin nx dx = 1968

S cose c

PROOF. - In (1965). Similarly in ( 1966-8 ) .

1969

2 cos x cos (n - 1 ) x = cos nx + cos (n − 2) x,

&c.

sin x sin nx dx Ssi

sin x cos nx

p

+

p+ n

sin² -¹x cos ( n - 1) x dx. p+n S

cos x sin nx dx Sco

1970

cos x cos nx

+ p+ n

cos -1a sin ( n - 1 ) x dx. p+n S So

sin x cos nx dx S si

1971

sin" a sin nx

-

=

sin -1

sin (n - 1) x dx.

p+n S

p+n 1972

p

cos² x cos nx dx Sec

cos x sin nx

n +

p+n

cos - ¹ x cos (n - 1) x dx. p+n S

323

CIRCULAR FUNCTIONS.

PROOF.- (1969) . By Parts, S sin ne dæ. In the new integral change COS NA COS into cos (n - 1 ) x- sin næ sin x . By successive reduction in this way the integral may be found . Similarly in ( 1970-2 ) . Otherwise, expand sine or cos" in multiple angles by ( 772-4) , and integrate the terms by the following formulæ. 1973-1975

nx dx = 1 ( sin (p — n) x sin px pa sin na 121 ( Ssi p- n

sin (p+n) x). p+ n

and so with similar forms , by (666-9) .

1976

S

cos px dx cos nx

s

and

sin px dx are found from Ssi cos nx

zp+n−1 dz i cos pr — sin pæ dx = 2 s 1 + z²n cos nx

when p and ʼn are integers , by equating real and imaginary parts after integrating the right side by ( 2023 ) . PROOF.-Put cos x + i sin x = z ; therefore iz dx = dz . Multiplying nume rator and denominator of the fraction below by cos na + i sin næ, we get cos pa + i sinpå = 2 cos (p + n) x + i sin ( p + n ) x = 2 zp+n ; 1 + 22n 1+ cos 2nx + i sin 2nx cos nx 2P + n -1 dz cos pr + i sin pa dx = — therefore 2i 2n nx cos s S 1 + z²n

cos px dx 1978 S sin nx

and

dx sin px pa da are found in the same (sisinnx

zp + n - 1 dz cos pa px +i sin px dx = 2 S 1-22n sin nx s

way from

PROOF. As in (1976), by multiplying numerator and denominator of cos pa + i sin pa by cos nx — i sin næ. sin næ sin æ dr

cos x dx and

1980

S= * (cos nx)

(cos nx) i sin æ " we find tan na = i ( 1 — y") , and therefore */(cos nx)

cos Putting y =

ydx = -idy 2 —y"`

Hence, multiplying by i, we have

- sin x i cos xdy dx = J: (cos nx) S 2 - yn The real part and the coefficient of i in the expansion of the integral on the right by (2021 , ' 2) , are the values required .

324

INTEGRAL CALCULUS.

dx 1 = tantan x + b sin' a √(ab) . √a) dx 1 = a+b tande a²+ b² { b log (a cosx + b sin x) + ax}.

1982

[ Subs. tan 2

a cos 1983

[ Subs . tan e

1984

1 dr tan‍¹ √(a + b) = a + b sin² x cot x va √(a² + ub)

1985

1 sin x cos² a dx cos x = 3 tan - ¹ ( a cos x) a² a S 1 + a² cos² æ

1986

1 a sin x cos x dr = tan -1 √ (1 — a²) a³ √ (1 — a²) S 1 - a² cos²x

1987

1 cos ≈ √ / ( 1 — a² siu² æ) dæ = 1 sine √ / ( 1 − a² sin² ) + 2a sin ¹ ( a sin x) . S [ Substitute a sin a

1988

1989

By

[ Substitute a cos x

sin [ Subs . sin a a²

sin x √ (1 — a² sin² æ) dx = — − } cosæ √ ( 1 — a³sin³æ) I sin 1 - a². log a cosx + ( 1 - asina) . 2a 3 dæ = − } cos x (1− sinx) sin a ( 1 − aa² sin *a ) *da

[Subs . a cos x

(1 - a ) [ sin ᏍᏆ.

[ Subs. a cos a

+

1990

[Substitute cot a

( 1 - a² sin³æ) dæ.

dx = log { (u cosecæ + b cotæ) ° tanª½æ}¸ a²-b² sina (a +b cos x) 1 a- b cos x 12 sin x = (a - b ) sin x sina (a + b cos x) (a³ — b²) (a + b cos x)

tan a da

1

=

1991

√(b− a)

√(a + b tan² a)

1992

COS¸-1 cos x ✅ (b − a) √6 [ Subs . cos x

(b− a)

√b cos x (a +b sin x) dxb = cos-1 sin æ √ (a + b) -a log {

a cota +

(a cosec² a + b) }.

METHOD.-By Division ( 1912 ) , making the numerator rational, and in tegrating the two fractions by substituting cote and cos a respectively.

1993 where m =

dx dr dx = c ( m {S 2c cosx + b - m S2c cosx +b + m;}, a + 2b cosx + ccos 2x {b - 2c (a - c) } .

dx

Then integrate by (1953) .

de 1 √√(a² +b²)cos 0 + c · b = a - a, where tan a = a

1994 Sa cos a + b sin a + c

METHOD. -Substitute

(1953)

EXPONENTIAL AND LOGARITHMIC FUNCTIONS.

325

F (sin x, cos x) dx Saa cos x + b sina +c

1995

F being an integral algebraic function of sin x a and cos x. Substitute 02 -a as in ( 1994) , and the resulting integral sin (cos 0) f( sine, cos 0 ) 20 (cos A) 10 takes the form 0 + =11 A cos 0 + B ( A cos + B S A cos +B ' METHOD.

since ƒ contains only integral powers of the sine and cosine, and may there fore be resolved into the two terms as indicated . To find the first integral on the right, divide by the denominator and integrate each term separately. To find the second integral, substitute the denominator. F (cos x ) dr

"

1996

S (a₁ +b¸ cos x) (a, +b₂ cos x) ... (a +b, cos x

where F ' is an integral function of cos x. METHOD . - Resolve into partial fractions . dr form (1951 ) . B cos x + A Į

Each integral will be of the

A cosa + B sina + C dx.

1997

S a cos x + b sin a + c METHOD. Let (x ) = a cos x + b sina + c ; .. q′(x) = -a sina +b cos x. Assume A cos + B sin æ + C = Xp ( x) + µ q' (x) + ". Substitute the values of ₫ (x) and p′ (x) , and equate the coefficients to zero to determine A, μ, v. The integral becomes V ν λαμ ' (x) + dx, de Xa +μ log (x) + +打 } [ {" (x) $ (+) $ (1) and the last integral is found by (1991) .

EXPONENTIAL AND LOGARITHMIC FUNCTIONS.

1998

Se F (x) da can be found at once when F (x) can be

expressed as the sum of two functions, one of which is the derivative of the other, for

Se * {$ (x) +$ ' (x) } dx = e*¢ ( x).

1999

ear cos" bx dx and = sin" ba da are respectively 1 = Sear Senz

a cos bx + nb sin br

e¹ cos" -¹ bx + a² + n²b²

n (n - 1 ) b² ex cos"-2 bx dx. a² + n°b³ S Se

326

INTEGRAL CALCULUS.

and a sin br - nb cos bx

n (n - 1 ) b²

ear sin" -1 bx + a² + n²b²

eax sin"-2 bx dx. a² + n²b² Se

PROOF. —In either case, integrate twice by Parts, Sex dx. Otherwise, these integrals may be found in terms of multiple angles by expanding sin" and cos" x by (772-4), and integrating each term by ( 1951-2).

2000

Se* sin" a cos " a da is found by expressing sin” æ and

cos " a in terms of multiple angles . Ex.: Se sin x cos x dx. Put ez in ( 768) , (2i sin x) (2 cos x)² = (z - z -1) ³ ((z≈ + z - ¹)² = (z — z ¯¹) ³ (z² — z˜²) ² = ( z7 — z¯7 ) —3 ( 25—2−5) + ( 2³ — z˜³) + 5 ( ≈z −z¯¹ ) ; 26 e sin x cos x = e* (sin 7x − 3 sin 3x + sin 3x + 5 sin x) . Then integrate by (1999) . 2001

THEOREM. -Let P, Q be functions of a ; and let =

= SPQda P3, &c. Then n- 2 ... ± n P, 3 P₁₂ +1• 1 2 SPQ" dx = P¸Q" —nP¸Q"-¹ + n (n − 1) P¸Q¹ -'— PROOF. -Integrate successively by Parts, JP dx, &c. SPdx = P₁,

2002

let

SPQada

P₂,

THEOREM . -Let P, Q, as before , be functions of x ; and

P₂ = P

1 P₁ = (( );

( ); =( P

Ps (4) r₁ = ( ( );,

Then

&c.

P₁

dx = & S

(n − 1) Q₂Q” P3

(n − 1) (n − 2) QnQ” 1 n-1 dx. + 3 n- SPA (n − 1) (n − 2) (n − 3) Q¿Q”

PROOF. —Integrate successively by Parts,

Qx dx. — (n − 1) Qr Qn

EXAMPLES . 2003

(log x)" dx Jam-1 n Xrm n n = == (bIn ― In-2. - 3 + " (n − 1) po − 2 1In-1 2 — ... + ( − 1) » | ?? Įn --s— m m m² m"

METHOD .

BY (2001 ) .

xm xm xm P₁ = 29 P3 = 739 &c. P₁ = m

ALGEBRAIC FUNCTIONS.

327

2004 dx = Sa™ {1 +nx log æ + } (næ log a)² +&c. } dx,

Sam +mz [ By ( 149) and each term of this result can be integrated by ( 2003 ) .

xm-1dx 2005

Am

Sa (log x)" l-n+2 m m² 1- n +3 + + +&c. ) n- 1 (n − 1 ) (n − 2) (n − 1) ( n − 2) (n − 3) n 1 m" x"m- 1 dx + n-·1 S log x

METHOD.-By (2002) .

Px = x , P₁ = mx™ -¹, P₁ = m²x™ -¹, &c.

In this

The last method is not applicable when n = 1. case, writing for log æ,

m-1 dx

m212 = log l + ml + 1.22 +

m $74 +

1.2.32

2006 S log x

1.2.3.42

+ & c.

em log a

xm -1 =

METHOD :

m³13

log x

Expand the numerator by ( 150) , and inte x log x

grate. See also (2161-6) for similar developments of the expo nential forms of the same functions .

PARTICULAR ALGEBRAIC FUNCTIONS .

dx =

2007

1 1 -} 1 = = 2=ni { (1 - x)"2-1

s x" (x - 1)" n + x=22-2 += N = ·2 {( (1-2) = = - = == } -( 1 − x )=" -−2

+

X n (n + 1 ) ... 2 (n − 1 ) lo g ® I-¿ 1.2 ... (n - 1 ) n being even.

(1918)

da =

2008

[ (1+ a²²)√ ( 1-2²)

cos -1

√ (1 + a³)

√ √ (17

). [ Subs.

2009

2010

dx (1-2)/(1 +2) = √12 log # √2 + √( 1 + z³) √ (1 - x³) dx x (1 + )/( - 1) = √1 / 2 log # √2 + √ (a²- 1)

√ (1-22) x

328

INTEGRAL CALCULUS.

dr

1 =

sin -1 br)), (2² (ae— ac √( ✓(ace - be²) (c + ex²) √ (u + bx*) 1 ¸c √ (a + bx² ) + x√ (be² — ace) = log √ (c + ex²) ✓ (be²- ace)

2011

2

dx

=

2012 x√ (a + bx")

log n√ a

(1 + x²) dx

=

2013 ( 1 − x²) √/ ( 1 + æ¹)

√(a + ba") -√a √2n

1 æ √2 + √ ( 1 + a³) V₂ log 1 −x² √2

ae > bc

ae < bc

[ Subs.

1 73 √2

[ Subs .

æ√2 1-x √2 1+x

2014

1 da 2 = sin -1 2√ √(1+2² ) √ (1+ ) 1 + x²° √2

[ Subs.

2015

√√(1+x) da = 17/18 { log +2") dx √( 1 -x

/2 }

2016

S(1-

= ( + )

(1 + 2 )+ 2/2 +sin- 1

1 4√2 { log [ Substitute

2017-2² dx = ) (1 − x²) (2x² − 1 )*

log

+ 2√2 - sin -12√2 } 1 -)x² ( +2 ( 1 + x¹ ) = x√2 in (2015-6)

(2x²-1) -2 (2x² − 1 )* + x

tan-1

x - 1)** (2x² —

[ Substitute a dx

≈ ( 2.c² — 1 ) *

X

= tan-1

2018

{√ / ( 1 + x*) -x − ∞² } $

lat (1 + e x*) { √ ( 1 + x*) − x² } *

2 √ (1 + æ³) --[ Substitute √√ / { − x* } 2019 dx

1

= log √ x +/- 2³ —

2 /1 - x³ -x

tan -1 √3

/ (1- ³)

[ Subs.

x √3

³/(1 − x³) x

dx

2020

x reduces to ( 2019 ) by substituting 1+ √ (1 + x) / ( 1 + 3x + 3 «°)

-1 INTEGRATION OF

" ±1 If I and n are positive integers , and 1-1 < n ;* then, n being even,

* If l = n , the value of the integral is simply

1 log (x* —1 ) . n

2-1 329

INTEGRATION OF

x² + 1°

x²-1 dx

1 =

2021

n

Sa

log (x − 1 ) + (−¹ ) ' log ( x+1) n

+ = Σcosrlßlog( x² —2x cosrß + 1) −2Σsinrlẞtan - 1

—cosr‍ß sin rß

π

where ẞ =

, and Σ denotes that the sum of all the terms n

obtained by making be taken .

n- 2 2 , 4, 6 ...n

r

successively , is to

If n be odd , x²-1dx 1 = n log ( x - 1) n Sadr

2022

+ = Σcosrlßlog (x² —2x cos rẞ + 1 ) −2Σ sin rlßtan‍¹ª — cos rß n sin rß with r = 2 , 4, 6 ... n - 1 successively .

If n be even, -1 2023

== Σ cos rlẞ log (x² -2x cos rẞ + 1 ) n n Sa +1

Σ sin rlẞ tan -1 + n

- cos rß sin rß

with r = 1 , 3 , 5 ... n - 1 successively. If n be odd ,

x²-1dx

(−1 ) ²-¹ =

2024

x +1

log (x + 1 ) n

- 1Σcos rlßlog ( a² — 2x cos rß + 1 ) + 2Σsinrlẞtan -1 x - cosrß n n sin rß

with r == 1 , 3 , 5 ... n - 2 successively . x²-1 into partial fractions by the method x" ± 1 a²-1 am (a) = of (1917). We have 1. The different " since a" nan -1 n F' (a) = values of a are the roots of " ± 10, and these are given by x = cos rẞ ± i sin rẞ, with odd or even integral values of r. ( See 480, 481 ; 2r and 2r + 1 of those articles being in each case here represented by r.) The first two terms on the right in (2021 ) arise from the factors 1 ; the remaining terms from quadratic factors of the type PROOF .- (2021-4) .

Resolve

(x -cos rß-i sin rẞ) (x - cos rß + i sin rß) = (x - cos r³) ² + sin² rß. These last terms are integrated by (1923) and (1935) . Similarly for the cases (2022-4 ) . 2 U

830

INTEGRAL CALCULUS.

2 2025

If , in formulæ ( 2021–4 ) ,

Σ ( 3π - ‚ß) sin rlß be

F n

added to the last term for the constant of integration , the integral vanishes with a, and the last term becomes

x sin rß F ² Σ sin rlẞ tan - ¹ n 1 - x cos rẞ'

reading

in (2021-2 ) , and

in ( 2023-4) .

m +1 n

dx 2026

Sandr ( 4) a+ bxn = 4 ab where az"

bx” .

dz, Sd 1 + x"

Then integrate by (2023-4) .

4

am - 1 + xn - m-1 dx =

2027

-1 x — cos rß Σ sin mrẞ tan

n

xn+1

sin rẞ

÷ n , and r = 1 , 3 , 5 , ... successively up to n - 1

where B

or n - 2, according as n is even or odd . 2028 xmm -1 ―――·xnn - m - 1

2

dx =

E cos mrß.log ( x² −2x cos rẞ+ 1 ) ,

n

x +1

with the same values of r ; but when n is odd , supply the additional term

( −1 ) m 2 log (x + 1 ) ÷ n .

PROOF. Follow the method of ( 2024, Proof) . Similar forms are obtainable when the denominator is a" -1 .

2029

1

x -1 dx

Σ cos (n − 1)

= x² -2x" cos no +1 -

n sin no

Σ sin ( n - 1) 4.log ( x² - 2x cos

. tan -¹ ª — cos sin +1 ) ,

I'π where

- 0+

and r = 0, 2 , 4, ... 2 (n - 1 ) successively. x sin o But if the integral is to vanish with a, write tan - ¹ 1 - x cos p as in (2025 ) . n

PROOF. By the method of ( 2024 ).- The factors of the denominator are given in (807), putting y equal to unity.

P 331

INTEGRATION OF Sx™ (a + bæ² ) ?.

2030

'x¹¹ (log x)" dx = F

ß)' (r (rß) ™ { cos ( rlß + 3mm ) n

Sa x + 1

-1 x - cos rẞ "

Xlog (x² -2x cos rẞ+ 1 ) −2 sin (rlẞ+ mπ) tan

sin rß with the values of ẞ and r in ( 2021-4) . PROOF. - Differentiate the equations (2021-4) m times with respect to 1, by (1427) and ( 1461-2) . If m be negative , integrate m times with respect to l, and the same formula is obtained by ( 2155–6) . In a similar manner , from (2027-8 ) and (2029) , the general terms may be found for the integrals 2032 x - m - 1} (log x)" x²-1 (log x)" dx { am - ¹ ± ( −1 )º¸n dx and Sa-2x" cos no + 1' Sham-2: x"±1

2035

INTEGRATION OF Sx™ ( ax+ bx”) dx. m+l RULE I.- When is a positive integer, integrate n 1

by substituting z = (a + bx" ) . Xpm" ( a +bx") " dx =

Thus

m + 1 -1 ༧ .n = a **** dz. 1 ( ~2 nb b SxP+9-1 ( 2S

Expand the binomial , and integrate the separate terms by (1922) . 2036 But if the positive integer be 1 , the integral is known at sight , since m then becomes n - 1.

m +1 + P n q substitute z = (ax - " +b) . Thus 2037

RULE II.- When

is a negative

integer ,

m +1 p -1 29-b n zp+q Sx" (a +bx")}" dx = − nu £ § 2º • -' (2º— ~ —b) * ~ * ~* ` dz. Expand and integrate as before. 2038

But, if the negative integer be -1 , the integral is

found immediately by writing it in the form m + np [ x " + " ( ax¯ * + b) " dx = [ x =" -1 (ax¯* +i) " dz

-

q na (p +q) (ax¯ * + b) * * !!

332

INTEGRAL CALCULUS.

EXAMPLES .

2039

To find Sa* ( 1 + x*) * da .

a positive integer. Therefore, substituting x = 6y³ (y³ - 1), and the integral becomes

[(

m +1 = =3, n

Here m = } , n = } , p = 2, q = 3 , y = (1 + *) *,

x = (y³ − 1)',

= dy61)° dy, -1) y³ da dy

the value of which can be found immediately by expanding and integrating the separate terms.

x³ (a + bx*)* dx =

2040

{

3 (a + bx*)*. 166

For m +1 = 1 (2036) ; that is, m + 1 = n, and the factor 2 is the derivative n of .

2041

[ (1 + √x)² dx, or ( 毹 (1 + x¹) ¹ dæ.

Here m = −}, n = }, p = 2,

m +1 Therefore, substitute + P = -2, a negative integer. n q y = ( x¯* + 1 ) ³, x = (y³ — 1 ) -², x¸ = — by³ (y³ — 1) -³ . Writing the integral in the form below, and then substituting the values, we have

q = 3,

Sx¯² (x¯ * + 1 ) * x, dy = — 6 S y* (y³— 1 ) dy, which can be integrated at once.

2042

dx = √x (a +bx" ) = √x¹ ( a + ba " ) -¹ de .

Here

m +1 + P ==-1; n q

therefore, by (2038) , the integral 1 -2-1 log (ax = √ x - " - ¹ (ax¯ " + b ) - ¹ dx = − na

+ b).

REDUCTION OF S x™ (a +bx”)º dx. When neither of the conditions in (2035 , 2037) are ful filled , the integral may be reduced by any of the six following rules , so as to alter the indices m and p, those indices having any algebraic values . 2043

I. To change m and p into m +n and p - 1 .

Integrate by Parts , 2044

II. To change m and р into m - n and p + 1 .

Integrate by Parts ,

2045

Sxm dx.

Sx -¹ (a + bx") " dx.

III. To change m into m + n.

Add 1 to p. Then integrate by Parts, Sx by Division , and equate the results.

dx ;

and also

REDUCTION OF S ” ( a + bæ″)² dx.

2046 IV. To change m into m - n. Add 1 to p, and subtract n from m. Parts, Sx

333

Then integrate by

dx ; and also by Division , and equate the results .

2047 V. To change p into p + 1 . Add 1 to p . Then integrate by Division , and the new integral by Parts , Sx" -1 (a + bx")".

VI. To change p into p - 1 .

2048

Integrate by Division, andthe new Sx"-¹ (a + bx")".

integral by Parts,

MNEMONIC TABLE FOR THE SAME RULES .

2049 I.

m +n,

p- 1

II.

m- n, m +n m- n

p + 1 | By Parts (p) . (p + 1 ) , Parts (m) and Division.

III. IV.

By Parts (m) .

(p + 1 , m - n) , Parts (m) and Division.

V.

p +1

(p + 1 ), Division, and the new integral by Parts (p) .

VI.

p- 1

Division, and the new integral by Parts (p) .

By applying the rules , Formula of reduction are obtained . Thus, any of the six values below may be substituted for the xm Sx™ (a+bx™)º dx.

integral

2050-2055 I.

___ bnp +¹ (a +bx ")" m+n xm (a + bx ") -¹ dx. m + 1J m+ 1 ‚m −n + ¹ (a +bx”)p + 1 xm

m- n+ 1

xm bn p +1 Sam- " (a + bx ") + ¹ dx.

II.

bn (p + 1 ) III.

Xm +1·¹ (a + bx”)p + 1 ___ b (m+ n + np + 1 ) +n +” (a +bx”)º dx. b (m + n + up + 1) S x” a (m + 1 ) a (m + 1 )

IV.

m - n +1 xm + ¹ ( a + bxn) p + 1 b (m + np + 1 )

a (m - n + 1 )

xm-n Sæ™ ¯" (a +bx”)” dx.

b (m + np + 1 )

V. - xm +1 (a +bxn)p+1 аст (a + bx")p +¹ dæ. + m + n + np + ¹ √x™ an (p + 1 ) an (p +1)

xm +1 (a + b.x")p

anp

VI. +

m + np + 1

Sam (a +bx")º -¹ dx.

m + up + 1

334

INTEGRAL CALCULUS .

EXAMPLES .

2056

To find

√ (a²->³) dr. x³ √

Apply Rule I. or Formula I .; thus

Sx˜³ (a² — x²) ¹ dx = — § x¯² (a² — x²) ¹ + } § x¯¹ ( a² — x²) ¯ * dx (1927 ) .

21 To find

2057

dr.

Apply Rule II . or Formula II .; thus

S(a² — x²) } Sæ¹ (a² — x²) −¹ dx = x³ ( a² — x²) − ¹ — 3 § x² ( a² —x²) -¹ dæ 2058

Scosecode.

(1934) .

Substituting sin , the integral becomes do -m dx = dx =S √ 2¯ " ( 1 − x² ) − + dx.

sin- 0 s

Apply Rule III.; thus, increasing p by 1 and integrating, first by Parts Sa - da, and again by Division ; 1 2¹- (1-2) 22 -m ( 1 − x² ) ¹ dx = dx, -m fr² = ( 1 − x³) −¹ 1 -m − √2-m + 1= (x-

da (1—2 ' )' de = √2 f * (1—22²) ') " + dz. S " (1—29) —+ dz— [2²

Equating the results, we obtain -m 21-1 -m -m x² - m ( 1 − x²) −¹ dx . x¯" (1 − x²) - * dx = 1 —m (1 −x²) ¹ + 12= 2059 -m = m ( x² - ( Š By repeating the process, the integral is made to depend finally upon or [x¯¹ (1 − x²) − ¹ dæ [(1 − 2² ) − + dz, according as m is an odd or even integer ( 1927, '29) .

2060 Ssinode is found in a similar manner by Rule IV. The integral to be evaluated is Sam ( 1 - x ) -de ; and the integral operated upon is Sam - 2 ( 1 - x²) ¹ dæ.

Otherwise apply Formula IV.

See also ( 1954) .

dx 2061

To find

Apply Rule V.

p = -r, and increasing p by

√(x²+1 )1, we have, first, by Division, -dx -* [ (x² + a²)! " de = [ 2² (x² + a²) * dz + a° f (x² + a²) · " dz. a²) ¹ -r Integrating the new form by Parts, Sx (x² + a²) - " da, we next obtain 1 [x² (x² + a²) - * dx = ² (x² + a³) ¹ - r - 2 (1— r) f (x² +a³)¹-"dz. [ 2 (1 - r) Substituting this value in the previous equation, we have, finally, 2062

dx

x

2r-3

dx

+ ( x²+a²) " = 2 (r − 1) a² (x² + a²) ” -¹

a²( 272 )! a² (2r − 2) ↓ (x² +a²) ” ¯¹°

"

335

REDUCTION OF Ssin" e cos" edo.

Thus r is This equation is given at once by Formula V. , reduction changed into r - 1 , and by repeating the process of the original integral is ultimately made to depend upon ( 1935 ) for its value if r be an integer .

Another formula for this integral is

2063 S

dr-1 (−1)”-¹ dx +8) = 1.2 1.2 ... (r −-11)) dB (

X tan -1 12/8 ) 8

PROOF.- Write ẞ for a² in ( 1935) , and differentiate the equation r— 1 times for by the principle in ( 2255 ) .

2064

To find √ (a² + æ²)³ dæ.

Apply Rule VI .

By Division, we have n- 1 dx = a* a² f (a² + z') ' -' dz + [ 2° ( @²+ 2°) b = ¹ dz. f( a² +z?) ' dz

The last integral, by Parts, becomes 1 n ¹dx - 1 x² — n ! { {((a² a? +2 !) !". != x ( a² + z ?) }" = [2° (a² + z ? ) !" —' da = n Substituting this value in the previous equation, we obtain

n 2065

+ 1 " + n+ S(a² + x²)¹* dx = ² (a² n +a²) na²1 § S (a² + x²)

¬¹dx ,

a result given at once by Formula VI . If n be an odd integer, we arrive , finally, by successive reduction in this manner, at

( a² +x² ) * dx

( 1931 ) .

2066 The integral Ssin" cos " do is reducible by the fore going Rules I. to VI. , if, in applying them, n be always put equal to 2 ; if p be changed into p ± 2 instead of p ± 1 ; and if Division be always effected by separating the factor cos 01 - sin20 . m PROOF. S sin cos 0 d0 = √ x™ (1 − x²) ³ » -¹) dx , where x = sin 0. Thus n = 2 always, and the index ( p - 1 ) is increased by 1 by adding 2 to p.

Thus, Rule I. gives the formula of reduction .

2067 sin +10 cos? -10

sin" cos" 0d0 = Ssir

+ m +1

sinm+20 cos? -20d0. m+1 S

INTEGRAL CALCULUS.

336

But the integral can be found by substitution in the fol lowing cases :

If r be a positive integer,

2068

cos2 +1 x sin x dx = cos Scos² + 1

2069

sin2r+1x cos? x dx = —- ( (1—2²) "z" dz , where z = cosx. S=

(1 - x )"' "dz, where z = sinæ .

If m +p2r,

2070

m S sin” si x cos” xdx =√ (1 + 2²)-- ¹z " dz, where z = tanz .

FUNCTIONS OF a + bx + ca². The seven following integrals are found either by writing 2071

a + bx + cx² = { ( 2cx + b) ² + 4ac - b² } ÷ 4c,

and substituting 2cx +b ; or by writing

2072

a + bx - cx² = {4ac +b²- (2cx — b) ² } ÷ 4c ,

and substituting 2cx - b.

1

2cx +b - √ (b2-4ac)

2073⋅ Sa+bx +cx² =

log 2cx + b + √ (b² -4ac)

√(b³ —4ac)

2

2cx +b

tan -1

or √(4ac- b³)

√(4ac - b³)'

according as b² > or < 4ac (2071 , 1935-6) . 1

dx

√ (b² +4ac) + (2cx —b)

= 2074

Sa+ba - ca

log b²+4ac

√ (b² +4ac) — (2cx — b) * (2072, 1937)

2075 dx S√ ( a + arter") === log { 2er +b +2√ /c√ / (a +bx +cx²) } . 1

=

2076

S√ (a +br - cr)

2cx - b

sin-1 √c

√ (4ac +b³)* (2071-2, 1928-9)

337

FUNCTIONS OF a + bx ± cx² .

2077

√ (a +bx +cx²) dx = }c ># ¿c¯ ‡ √ √ (y² + 4ac — b²) dy.

2078

S√(a +bx−cx²) dx = ¿ c¯¹ S√ (4ac+b² —y °) dy,

where y = 2cxb.

The integrals are given at ( 1931–3 ) .

dx

dy S (y² +4ac - b²)P'

= 22-1-1

2079

(a + bx + cx²) ²

[ By (2071 ) , the integral being reduced by (2062-3) .

(lx + m) dx 2080 (a + bx + ca²) ι

=

bl

(2cx + b) dx

-

+ 2cJ (a +bx+cx²)p

m (

2c S (a+bx + cu²) ³·

The value of the second integral is ( a +be + ca²) ¹ - P ÷ ( 1 − p) , unless p1 , when the value is log (a + be + ca² ) . third , see (2079) .

For the

METHOD. Decompose into two fractions, making the numerator of the first 2c + b ; that is, the derivative of a + bx + cx².

px² + q dx may be integrated as follows : da Saa + bx² + cx* -b ± √ (b² - 4ac) , and, I. If b² > 4ac , put a and ẞ for 2c

2081

by Partial Fractions , the integral is resolved into 1 a

{Spat

c (a- B) a

II. If b² < 4ac, put с

2 dx - SPB+q dx }.

(1936)

= n² and 2√ (ac) — b = m², and the C

integral may be decomposed into 2082

1 2mne {§ I—pn) x + qm dx dx} , · S pm) =n qm dx q= x mx x² + -S x² + mx + n

the value of which is found by (2080) .

III. If b³ = 4ac,

b tan (2062) . √3a) ~~(~~ Sa+ ba² + cx² = 2a + bx² + √( 2at) 2 x x

2083

1

338

INTEGRAL CALCULUS.

x dx 2084 Sa+be +car

2a -

b (2a + bx²)*

x²dx =

2085

Sa a+ ba² + ca

2a S

1

b√( 2ab)

X √za -tan~ (~ + bar} 20 +ba 4 ) - 2a ( ~ √22a ()

REDUCTION OF Sæ™ ( a + bæ" + cæ²" ) º dæ. NOTE. In the following Formula of Reduction, for the sake of clearness, xm x™ (a + bx” + cx²”) ” is denoted by (m , p) , and the integral merely by § (m, p) .

2086 (m + 1)f ' (m, p) = (m+ 1, p ) .. (1 ) .

—bnp S (m+n, p −1) —2cnp J ( m+ 2n , p − 1 )

2087

bn (p + 1) S (m, p) = (m − n + 1 , p + 1 )

− (m — n + 1) S(m — n, p + 1 ) −2cn (p + 1 ) √ (m + n, p) … ..(2) .

2088

2cn (p + 1 ) S (m, p) = ( m − 2n + 1 , p + 1 )

− (m −2n + 1) S(m−2n, p + 1) −bn (p + 1) {( m— n, p ) ...(83) . 2089

(m + np + 1) S (m, p) = (m+ 1 , p) . (4) .

+ anp S ( m, p − 1) —cnp S (m + 2n , p − 1) .. 2090

(m + 2np + 1) S (m, p ) = ( m + 1 , p) . (5) .

+2anpS(m, p− 1) +bnp S (m + n, p− 1)

2091

b (m +np + 1) S ' (m, p) = (m − n + 1 , p + 1 )

—a (m —n + 1) S(m — n, p) —c ( m + 2np +n + 1) S(m + n, p ) . (6) .

2092

bn (p + 1) § (m, p) = − (m − n + 1 , p + 1 )

+ (m +2np + n + 1 ) S(m− n, p + 1) −2an (p + 1 ) J (m −n , p ) ...... ... ( 7) .

339

REDUCTION OF Sæm (a + bæ" + ca² ) " dx.

2093 en (p + 1 ) √ ( m, p) = ( m —− 2n + 1 , p + 1 )

+an (p + 1 ) √(m− 2n , p) − ( m + np − n + 1) √(m − 2n , p + 1) ......(8) .

2094

an (p + 1 ) √ (m, p) = − (m + 1 , p + 1 )

+ (m + np + n + 1 ) S(m,p + 1) + en ( p + 1)√(m+ 2n , p ). … ..(9) .

2095

2an (p + 1) S (m, p) = − (m +1, p + 1)

+ (m + 2np +2n + 1) S (m, p + 1) −bn (p + 1) S (m +n, p) ...... .. (10) .

2096

a ( m + 1) ( (m, p)

= (m + 1 , p + 1 )

−b(m + np + n + 1 ){(m + n, p) —c(m + 2np + 2n + 1 )√(m + 2n ,p) ....... (11 ) .

2097

c (m+2np +1) S √ (m, p ) = ( m − 2n + 1 , p + 1)

−b (m + np —n + 1 )S(m —n,p ) —a (m −2n + 1) f(m− 2n , p) .. ( 12) . PROOF.- By differentiation, we have

2098

S ( m − 1, p ) + bnp S ( m + n − 1 , p −1 ) + 2cnp S ( m + 2n − 1 , p − 1 ) . '(m, p) = m√ Formula (1 ) , (2 ) , and (3 ) are obtained from this equation by altering the indices m and p, so that each integral on the right, in turn, becomes (m, p). Again, by division, 2099 S(m, p) = aS ( m, p − 1) + b S ( m + n, p − 1 ) + c√ (m + 2n, p − 1).… .. (A). And, by changing m into m -n, and p into p + 1 , 2100 S (m −n, p + 1 ) = a § S ( m − n , p ) + b § ( m, p) + c ¦ (m + n, p ) .....(B) . Formula (4) to ( 12) may now be found as follows :-: (4) , by eliminatiug S ( m + n,

p − 1) between ( 1 ) and (A) ;

(5) , by eliminating

(m + 2n, p − 1 ) between ( 1 ) and ( A) ;

(6), by eliminating

(m - n,

(7) , by eliminating

(m +n, p) between ( 2) and ( B) ;

p + 1 ) between ( 2) and (B) ;

340

INTEGRAL CALCULUS.

(8), from (4), by changing m into m - 2n, and p into p + 1 ; (9) , from (4) , by changing p into p +1 ;

( 10) , from ( 5 ) , by changing p into p + 1 ; (11 ) , from ( 6) , by changing m into m + n ; (12) , from (6) , by changing m into m - n. If a and B are real roots

of the quadratic

equation

a + bx² + cx²n = 0 , then, by Partial Fractions , m dx dx 1 dx ) dr - Sande } , √ 2 " S atba" Feat 2101 B n a x - ){ S c (a Ᏸ and the integrals are obtained by ( 2021-2) . But, if the roots are imaginary , m+1 m dx a 2n = 47 )= SI- 22 1( 2102 Sa+be + ca² a 2 " cos no +2 b

where

and

cos no =

2√ (ac)

2103

(2029)

2n 2= · = (-2 a) + x.

dx is reduced to ( 2079-80) by (2097) . Sa+bx+cx²)

dx

dy

, (2075)

2104 where

2105

2106

(x + 4 ) √ ((a a + bx + ce²) = -S −√ √(4+ By + Cy²) y

(x +h) −¹ ,

A = c,

Bb - 2ch,

C = a - bh + ch².

1 dx 1 -11 + 2 -11 +hx = = cosh COS x+h x+ h √1- h² √h' - 1 S (x + h) √x²- 1 1 + ha 1 dx = sin-1 [ By (2181 ) . x+h √h - 1 |( +11) VI- 403

METHOD. - Substitute (x + h) h = 1.

, as in ( 2104) . Observe the cases in which

dx

2107

yr-¹dy • √; √ (A + By + Cy³)' (x +h) " √ / (a + bx + cx²) = -S

with the same values for A, B, C, and y as in ( 2104) . integral is reduced by ( 2097) .

(lx + m ) dr 2108 S x² +ß²) √ (a + bx +cx²) *

The

341

INTEGRATION BY RATIONALIZATION.

METHOD .- Substitute 0 by putting a = ß tan (0 + 7) , and determine the constant y by equating to zero the coefficient of sin 20 in the denominator. L cos 6 + M sin 0 do. The resulting integral is of the form S √ (P + Q cos 20) Separate this into two terms, and integrate by substituting sin and cos in the second.

in the first

(x) dx SF'C (x) √ (a + bx + cx²) '

2109

where (a) and F (x) are rational algebraic functions of x, the former being of the lowest dimensions .

(x) into partial fractions . The resulting integrals F(x) are either of the form (2107) , or else they arise from a pair of imaginary roots (Ax + B) dx of F(2) = 0, and are of the type [ { (x − a ) ² +1) ²) √ (a + bx + ca³)' Substitute METHOD.- Resolve

z-a in this, and the integral (2108) is obtained.

INTEGRATION

BY

RATIONALIZATION .

In the following articles , F denotes a rational algebraic function . function

In each case, an integral involving an irrational of

is , by substitution , made to take the form

SF (z) dz. This latter integral can always be found by the method of Partial Fractions ( 1915 ) .

SF{ x , (a (F +b ) , (a +b );, &c. } dr. f+gx f+gx a + bx Substitute is the least common de z , where f+ gx nominator of the fractional indices ; thus , p lp 12-1 (bf- ag) fz¹ -a dx 1+ bx\ a = x = = 2 , & c. , (b -gz¹) dz b- gz¹¹ ( f+gæ) l 2110

=

the powers of z being now all integral.

p 2111

S

Fxmn - F { a " , (a+ ") , (a+b +hr g”

);, &c. } dr.

Reduce to the form of (2110) by substituting a".

342

INTEGRAL CALCULUS.

Subs . √ (a + √mx + n) . 2112

SF { √ (a + √mx + n) } dx.

b

Substitute x = 2113

SF{x, √ (bx±cx²)} dx. bz √ (bx ± cx² ) = z² = c'

And therefore

F Z² 2bz

dx dz

(z² + c) **

2114

SF { x, √( a + be +cx²) } dr. Writing Q for a + be + ca² , F may always be reduced to A+B√ Q the form in which A, B, C, D are constants or C + D √ Q' rational functions of x. Rationalizing this fraction , it takes the form

L + MQ.

Thus the integral

becomes

[ Lde + MQde, the first of which two integrals is MQ rational, while the second is equivalent to dx , which S √Q is of the form in (2075) .

2115 OTHERWISE .- (i . ) When c is positive, the integral may be made rational by substituting a - cz² a - cz2 dx = 2c (bz - cz² - a) x= 6 + ²). " √(a + bx + cx²) = √ //c 2cz -b ' dz (2cz- b)* 2cz - b (ii ) When c is negative, let a, ẞ be the roots of the equation a + bx - cx² = 0 , which are necessarily real (a, b, and c being now all positive ) , so that a + bx - cx² = c ( x− a) (ẞ — x) . The integral is now made rational by substituting caz² +B dx = 2 (a − ẞ) cz (B - a) cz x= √(a + bx — cz²) = cz² + 1' dz (cz³ +1)² cz² + 1 In each case the result is of the form SF (z) dz.

2116 Sx"F {x", √a+bx" +cx²" } dæ,

m +1 when

is an integer, is reduced to the form (2114) by

n substituting a".

2117

a + bx Substitute ²= {F {x, √(a +be), √(f+gr) } dæ.

ƒ+gx'

INTEGRALS REDUCIBLE TO ELLIPTIC INTEGRALS.

a -fz² and, therefore ,

343

) = z /(ag -bf

x =

√(a + bx)

gz² — b '

√ (gz² —b)

√(ag - bf)

dr

√ (gz - b)'

dz

= 2 (bf- ag) % (gz² — b)²

√( f+ gæ) =

The form SF , √ (gz² — b) } dz comprehended in (2114) .

is

obtained ,

which

is

2118

Sa#F {x" , √ (a + b²x²”) , bx" ± √ /( a + b²x²") } dx, m +1 when is an integer , is reduced to the form SF (2) dz n

by substituting z = bx" ± √ (a + b2a2") ,

and therefore

_________ 22-a xn =

z² + a √(a + b²x²n) =

; 2z

2bz

m + 1 -1 n

dr

xm dz = 216 1² 2nb ( 2 )

ta ) 22 ( + .. )

p 2119

xm ) Sa™ ( a +be" 1

2120

p -1 xm-¹ F { x™ , x" , (a +bx") " } dar , when Sa

is rationalized by substituting F ( x ) de 1 m +1 + P m+ 1 or either ( a + ba") or (ax + b) according as n n q is integral, whether positive or negative. m

is either a n

positive or negative integer, is rationalized by substituting 1 (a+bx ") ",

INTEGRALS REDUCIBLE TO ELLIPTIC INTEGRALS .

2121

F { x,

√ (a + bx + cx² + dx³ + ex¹) } dx .

Writing X for the quartic, the rational function F may always be brought to the form

P+QX

P +Q'X' and this again, by rationalizing the denominator, to the form

INTEGRAL CALCULUS.

344

M+ NX, where P, Q, P' , Q' , M, N are all rational functions Mdx has already been considered ( 1915 ) .

of x.

Rdx where R is rational.

√X'

SN√ / Xdx = SNX dx or

, and determining p and q so By substituting a = 2+ 1+y that the odd powers of y in the denominator may vanish, the last integral is brought to the form Rdy 2122

√√(a +by² +cy¹)* R being a rational function of y, may be expressed as the sum of an odd and an even function ; thus the integral is equivalent to the two

F₂ (y²) dy

yF (y ) dy 2123

+

√ (a +by³ + cy¹)

√ (a + by² + cy*)*

The first integral can be found by substituting √y.

a + bx² The second, by substituting

for y , can be made to

c+ da depend upon three integrals of the forms

2124

dx S √ a 1 1 - ka Si

1 - k²x² dx

S√I √1 - x² dr

(1 + nx²) √1 − x².1 — k²x²´

By substituting

2125

-1 = sin- ¹x, the above become

do S√1- k² sin² 4 '

1 - k² sino do, S√1-k

S(1 + n sin³ ) 1-1

sin³¿

These are the transcendental functions known as Elliptic Integrals. 2126

They are denoted respectively by F (k, 4),

E (k, 4),

II (n , k, 4) .

INTEGRALS REDUCIBLE TO ELLIPTIC INTEGRALS.

345

APPROXIMATIONS TO F (k , p) AND E (k, p) IN SERIES . When k is less than unity, the values of F (k, p ) and E (k,

), from the origin = 0 , in converging series , are 1.3.k k2 1.3.5.k A— A₂ + ·AB + ... 2127 _F (k, 4 ) = 4— 2.4.23 2.4.6.25 1.3.5 ... n - 1

kn

2.4.6 ... n

2-1

An, & c .

...· + ( − 1 ) ³n

1.k¹ 1.3.k k? A2 A₁+ A。 - ... E (k , 4) = & 4+ ½ 4₂ + 22 2.4.23 2.4.6.25

2128

+ (− 1)in +11.3.5 ... n — 3 kn An 2.4.6 ... n 2n -1

& c. ,

where [n being an even integer.

sin 20 A₂ A, =

- &,

2 4₁ =

sin 44 ― 4 sin 24 +36, 2 4

15 sin 24 _104 , A6== sin 64 -6 sin 46 + 4 2 6

...

...

...

...

...

...

An= sin non sin (n - 2) n n-2

, C ( n , 2) sin (n − 4) o + n-4 C (n, 3) sin (n· ·6) + ... + ( −1 ) ¹" ↓ C (n , ¹n) 4. n-6

PROOF. - In each case expand by the Binomial Theorem ; substitute from (773) for the powers of sing , and integrate the separate terms. The values of F (k, p ) and E (k , p) , between the limits $ = 0, $ = 1 , are therefore

2129 2 2 k2 4 ' 1.3 ) ' + (1.3.5)² ) = = { ¹ + ( )' ~' + ( 2.4 2.4.6 2

1

F ( k,

· + &c . }

2130 2

1.324

1.3.52k

- ( 1 )' ~ - )' ' - (2.4.6 3.5) ' - (2.4 E (k , 2) = 2 = { ( 1 −

}. - &c. }

But series which converge more rapidly are 2131

2 2 2 π n) • n² 2) = = (1 + " F (4, 5 ') { 1 + ( 2 1)' ~ ' + ( 2.4 1-3)'n' + (1.3.5)'n' + &c. } 2.4.6 2x CX)

346

INTEGRAL CALCULUS.

2132 2 2 no n² — 1+ ) { 1 + ( 3) " " + (2 2 . ) n ' + (2.4.6 113 ) "" + & c. };

π π E ( k, E (4 , 2 5) = = 2 (1 +

n = 1 1+√ √ (1 (1 k) )* −k

where

2133

F (x) dx when F ( ) can be ex S ST √ (a + bx + cx² + ba”+ax*) '

(x − 1 )

Ta

pressed in the form

(x+ 1 ) ,

is integrated

by

1 substituting

•

+· x

If b is is negative , and F (a) of the form (2+ 1 ) ƒ (x − 1) ; x 1 substitute xX

F (x) dx 2134

Substitute

√(a + bx+ cx² + dx³ +cx¹ +bx³ + ax“) * 1 2. x+ X

1

dx Hence

dz,

=

√₁·³

12/1 ( ≈12

2√3+ 2

and the integral takes the form ²−4 P+ Q √ ~

√≈ + 2 − √ ≈ −2 d≈,

√ { a ( ≈3—3≈) +b (≈²− 2) +cz + d}

2√2-4

where P, Q are rational functions of z . Writing Z for the cubic in z , we see that the integral depends upon

P dz and

√ Z (≈ ± 2)

Q (≈ ± 2 ) dz fe √Z (≈ ± 2)

the radicals in which contain no higher power of z than the fourth .

The integrals therefore fall under ( 2121 ) .

2135

√ (a + bx² + cx ' + bx" + ax`)* Expressing F(a ) as the sum of an odd and an even func tion, as in (2123 ) , the integral is divided into two ; and, by substituting a , the first of these is reduced to the form in ( 2121 ) , and the second to the form in ( 2134) with a = 0.

INTEGRALS REDUCIBLE TO ELLIPTIC INTEGRALS.

347

F (x) dx 2136 Sz (a +ber av +dx) Put a = y + a , a being a root of the equation a + be + ca² + da³ = 0 ; and , in the resulting integral, substitute zy for the denominator . The form finally obtained will be

S(P+ Q √a+ß~¹) dã, which falls under ( 2134) , P and Q being rational functions of z .

F (x) dx

2137

√ (a + bx² + ca¹) * Expressing F (e) as the sum of an odd and an even func tion, as in (2123 ) , two integrals are obtained . By putting the denominator equal to z in the first , and equal to az in the second , each is reducible to an integral of the form

S(P+ Q √a +B= ¹) dã,

which falls under (2121 ) .

dr F

2138

(

√12

) = √√√√x Sa /(1 x¹)

24 )

[Expand by (2127).

dx

2139 =

F 1 ( —2, 4).

√1 + ∞¹ PROOF.

Substitute cose in (2138) , and 2 tan - ¹ x in (2139 ) .

2140 2

dx

/b

2a

= √ (2ax — x²) (b − x)

ZF (√

2a ' F (√2u

6) ) or √

),

according as b is > or < 2a.

PROOF.

Substitute accordingly, a da

2a sin' p or a = b sin² p. ―b

2 F

( a+ c S√ √ (a + c) √(a −x) (x− b) (x + c) PROOF.-Substitute a = a— ( a − b ) sin² , a being b .

2141

$).

348

INTEGRAL CALCULUS.

SUCCESSIVE

INTEGRATION .

2148 In conformity with the notation of ( 1487) , let the operation of integrating a function v , once , twice , ... n times for a, be denoted either by

v, v, ... v, Seffe nx , 2x

or by

d_nx d29 d- 29 ... d- 12)

the notation d , indicating an operation which is the inverse of d Similarly, since yx , Y2x , Yзr , & c . denote successive derivatives of y, SO Y - x, Y - 25 Y - 3 , & c. may be taken to repre sent the successive integrals of Y with respect to a. 2149 Since a constant is added to the result of each in tegration, every integral of the nth order of a function of a single variable must be supplemented by the quantity n n- 2 a2xn а₁xr - 1 x . n- 1 + n - 2 + .. + a₂ -1 + a₁ = ( 0,

A3 ... a, are arbitrary constants. where a₁, ag, as

Examples. The six following integrals are obtained from ( 1922 ) and (1923) . When p is any positive quantity,

Xp + n

XP =

2150

SNX

(p + 1 ) (p + 2) ... (p + n)

0. + Co. nx

When p is any positive quantity not an integer , or any positive integer greater than n,

2151

(-1)" 1 = o +f nx Xp S Nx . (p − 1) (p − 2) ... (p— n) x²-n

When P is a positive integer not greater than n, the fol lowing cases occur

1

=

2152

C px XP

1

(-1) -1 lo g p− 1 \

Jpx0. +S

·1)p−1

2153

0,

(x log x − x) +

(p+1) x XP

p-1

(p + 1)

SUCCESSIVE INTEGRATION.

1

349

Xr ( + ) 0. ( -1) (aloga) == } + S x

1)p-1 (\ p-I

2154 S(p+r) x XP =

For the integral within the brackets , see (2166 ) .

The following formula is analogous to (1461-2)

sin

1 sin

ax =

2155 nx COS }

an cos

(ax— \ nπ) +(Nx0,

SUCCESSIVE INTEGRATION OF A PRODUCT. Leibnitz's Theorem ( 1460) and its analogue in the Integral Calculus are briefly expressed by the two equations 2157

Dnx (uv) = (dx + 8 )" uv ,

D_nx (uv) = (dx + dx)˜ ” uv ;

where D operates upon the product uv , d only upon u , and ♪ Expanding the binomials , we get only upon v.

2159

− 1) ,U(n- 2) xV2x + &c. Dnx(uv) = Unxv + nu (n- 1) xV₂ + n (n1.2

U_nxV 2160 D_nx(uv) D_µz(uv) = U_nsv— nu_ (n + 1) x Vx + n (n + 1) u_ (n + 2) x V2x 2x − &c . 1.2 PROOF. - The first equation is obtained in ( 1460 ) . The second follows from the first by the operative law ( 1483) ; or it may be proved by Induction, independently, as follows Writing it in the equivalent form

uvx +

NX ( v ) = {__uv nx — n ↓__

(n + 1) x

n (n + 1 ) Uv2x- &c. ...... (i.) , 1.2 Į(n +2) x

make n = 1 ; then " Jan "] = ( an )"

fc

UV 2x- & c. .... x 2xwe₂ + [3x

(ii.) ,

a result which may be obtained directly by integrating the left member suc cessively by Parts. Now integrate equation (i . ) once more for a, integrating each term on the right as a product by formula (ii. ) , and equation (i . ) will be reproduced with (n + 1 ) in the place of n.

2161

eax am = eax (a + d¸ ) ¯ "a" +S 0.

Or, by expansion ,

/

2162 ear earx an { com |__ªzym = ctx

nm m -2 x -1 + n (n + 1) m (m — 1) 0. 10m -² _ & c . } + [,nx 1.2 a² a

If m be an integer, the series terminates with ( −1) ™ nm) ÷ a” .

INTEGRAL CALCULUS.

350

Similarly, by changing the sign of m,

2163 ear = ear an Snx x m

1 กา n (n + 1 ) m ( m + 1 ) 0. + axmi + + & c. } + √,nx xm 1.2 a2x +2

PROOF.-Putting u = eªx, v = x² in ( 2158 ) , the formula becomes earxm = (dx + dx ) ¯neaxxm = (dxeªr + eªx d¸) -N −¹µm = eªx ( a + dx) ¯ˆx™ . Sበር Here ea is written before & within the brackets, because & does not operate upon ear. Observe, also, that the index -n affects only the opera tive symbols dx and d , but it therefore affects the results of those operations. Thus, since de produces aea , the operation d, is equivalent to aX, and is retained within the brackets, while the subject ea , being only now connected as a factor with each term in the expansion of (a + ) ", may be placed on the left.

2164

m ex · a x" dx = a {x" Se" Seam

eax

eax dx =

2165 S Xm

a

m 1 m + аxm +1

xm ) a - ³ — &c . } −1 (m -1+ m a²

+

m (m + 1 ) m + 2 +&c. }

PROOF.- Make n = 1 in ( 2162 ) and ( 2163 ) .

2166

x xm. S__a” (loga) ™ = √_e( Nx Nx P+") = a*.

[ Subs. log .

Hence the integral of the logarithmic function may be obtained from that of the equivalent exponential function (2161 ) . For another method , see (2003-5) .

HYPERBOLIC FUNCTIONS .

2180 DEFINITIONS . -The hyperbolic cosine, sine , and tan gent are written and defined as follows : 2181

cosh x == 1 (e² + e¯*) = cos (ix).

2183

sinh c =· 1 (e* —e¯ *) = −i sin (ix) .

2185

tanha =

(768)

e -e = -i tan (ix).

e² + e¯ *

(770)

351

HYPERBOLIC FUNCTIONS.

By these equations the following relations are readily obtained . 2187

sinh 0 = 0 ;

cosh 0 = 1 ;

cosh co – sinh

=

.

cosh r – sinh r = 1 .

2191

2192

ở coshy + cosha sinh y . sinh ( + y ) = sinh æ

2193

cosh (x +y) = cosha cosh y + sinh x sinh y.

2194

tanh +tanh y tanh (x +y) = 1+tanha tanhy

2195

sinh 2

2196

cosh 2x = cosh²

2197

= 2 sinh æ cosh . + sinh² x.

= 2 cosh r− 1 = 1+ 2sinh .

2199

sinh 3r = 3sinh + 4sinh .

2200

cosh 3r

2201

tanh2r =

4 cosh³r - 3 cosh x.

2 tanha 1 + tanh 3tanha +tanh3r 2202

tanh3x =

1+

x 2203

sinh

;

cosh

2

cosh v ―――――― 1

cosh x---1 =

sinh =

=

cosha + 1

2

+1

=

2

x tanh

cosh

cosh a - 1

= 2

2205

tanhör

cosh

sinh

+1

21 2208

cosh x = 1 + tanh ; 1 – tanh gử

2 tanh } sinh

·

=

1– tanh

INVERSE RELATIONS . 2210

2211 2212

Let u = coshx,

.. x = cosh¹ u = log (u + √ u² − 1) .

v = sinh æ , .. w = tanhx,

= sinh- v = log ( v + √ v² + 1) .

.' . ~ = tanh¯¹w =

log (

+ )

352

INTEGRAL CALCULUS.

GEOMETRICAL INTERPRETATION OF tanh S. 2213 The tangent of the angle which a radius from the centre of a rectangular hyperbola makes with the principal axis , is equal to the hyperbolic tangent of the included area. PROOF. - Let be the angle, r the radius, and S the area, in the hyperbola x² — y² = 1 or r² = sec 20 ; then

Therefore

S= sec 20 de = log tan ( +0). =5° es.-e -S 1 + tan 0 = tanh S. e28 = ; therefore tan 0 = es+e 1 - tan 0

(1942) (2185)

VALUE OF THE LOGARITHM OF AN IMAGINARY QUANTITY .

2214

log (a +ib ) = } log ( a²+b²) +itan¬¹b .

b a b 1-i a

1 +i a + ib PROOF.-

log

= log √(a² + b²)

DEFINITE

By (771 ).

= itan -1 a

INTEGRALS .

SUMMATION OF SERIES BY DEFINITE INTEGRALS.

2230

[f(x) dx = [f(a) +f(a+ dx) + ... +f(a +ndx) ] dx,

where n increases and dæ diminishes indefinitely , ndxb - a in the limit.

2231

2232

so that

Ex. 1. - To find the sum, when n is infinite, of the series 1 α 1 1 1 Put n = ; thus, + + dx n n + n n+2 n+ 1 2a dx dx dx dx dx = = log 2. + + + + 2a 2dx + a dx + a a a x

Ex. 2.- To find the sum, when n is infinite, of the series 1 n n n n Put n = then + da n²2 + 1² n²+ 22 + n²+32 + ...... + n² + n²² dx π dx dx dx = = (1935) + + ……………. + 4 +x² 1 1+ (ndx) 0 1+ (da)' ' 1+ (2dx)²

THEOREMS RESPECTING LIMITS OF INTEGRATION.

THEOREMS

RESPECTING

THE

LIMITS

353

OF

INTEGRATION .

2233

[ Substitute a - 2. Sp S & (x) dx =√˜p 0 (a−x) dx.

fa | S***p (x) dx, [ " $ (x) dx = 2(

2234

(a - x) for all values of a

or zero, according as p (x) = between 0 and a.

2

— Ex.

cos x dx = 0. [ sinædz = 2 fisin zdz.

S.

If (x) = p ( −x) , that is , if ( x ) be an even function (1401 ) for all values of a between 0 and a. a a 1 -α$ (x) dx. 2236_s_p *p(x) dx = ½§° -a (x) dx =S π 1 2 2 Ex. cos x dx. dx cos x dx = ·S² cosa da = }} " ,πT ‫؟‬. 2

x) , that is , if ( If Φ (x) = for all values of a between 0 and a.

2238

φ §°_p(x) dx = −S*$ (x) dx

Ex. Stinzda = -fisin zde

and

(a) be an odd function

and SÞ( p(x) S˚ x) dx = 0. ㅠ 2 sin x dx = 0. fini 2

Given a < c < b, and that x = c makes value of

( x) infinite, the

a (e) da may be investigated by putting μ = 0 ,

after integrating, in the formula Ητους 2240 __$$ (x) dx =[a "$ (x) dx +§* 4 (x) dx. If the function (a ) changes sign on becoming infinite , this expression, when u is an indefinitely small quantity, is called the principal value of the integral . 2 z

354

INTEGRAL CALCULUS.

dx dx 1 dx ပိ 2*2 + 1 2 − 1 + 2μ² x = - 3 C₁ 1 = 0, 1 , formula ( 2289) reduces F (x) to the value in which a is < 1 ; and when a lies between 1 and , formula (2313 ) reduces the function to the value in which a lies between 0 and 1. Values of (a) , when a lies between 0 and 1 , can also be and , made to depend upon values in which a lies between by the formulæ , I ( 음) T (2x) √π = 2318 г(x) = 21-2 /π T (x) : г (1 + x) ' 21- cos r ( 14) cos PROOF.---- To obtain (2318) , make n = 2 in ( 2316 ) . To obtain ( 2319 ) , put m = ( 1 + x) in ( 2313) , and change a into in ( 2318) , and then eliminate r (1+2).

Methods of employing the formulæ2320

(i . ) When a lies between

and 1 , reduce F ( x) to

T (1 - x) , by (2313 ) . 2321 (ii. ) When a lies between and , reduce by ( 2319 ) , the limits on the right of which will then be and 2322

(iii . ) When a lies between 0 and

( + ) will then involve the limits reducible by case (ii . ) If 2x is
or < p . PROOF.

Put year cos be and z = 2 in (1460), employing ( 1465) . 3 I

426

INTEGRAL CALCULUS.

MISCELLANEOUS EXPANSIONS . The following series are placed here for the sake of reference, many of them being of use in evaluating definite integrals by Rule V. (2249) . Other series and methods of expansion will be found in Articles ( 125–129 ) , ( 149–159 ) , (248-295) , (756-817) , ( 1460) , ( 1471-1472 ) , ( 1500-1573) . For tests of convergency , see ( 239–247 ) . Numerous expansions may be obtained by differentiating or integrating known series or their logarithms . These and other methods are exemplified below.

1

1

π+x

2π-x

1

1 2911

cot x =

+ π- X

X

1

3п - х

3π + x

By differentiating the logarithm of equation ( 815 ) .

1

1 2912

- &c .

+ 2π + x

PROOF.

1

1 -

+

T cot πα = 11 + x

1+

+ x+ 1

1

x-2 1 1 ·3 *x+3 + &c.

1

+ x+2 + PROOF. -By changing x into

1 ---

1 + π- x 1/2π + x 1

-

X

tan x =

17

2913

2 in ( 2911 ) .

PROOF.

By changing a into

cosec x =

in ( 2911 ) .

1

1 2914

-

+ x

10/271

1 1 =x + & c. == = = = = x -3π+x + ㅠ+ 닭 1

1

1

π+x

2π - x

π -X

1 +

1

1

1

3π - x

3π + x

4π - x

+& c .

+ 2π + x

PROOF. -By adding together equations (2911 , 2913 ) , and changing a into .

427

EXPANSIONS OF FUNCTIONS IN SERIES.

2915

1 πT = sin mr m +

―――― ・m

2916

1+m

By putting x = m

- .&c . 3+ m

in ( 2914) .

1

22Bx

x

2

2B

cot x =

For proof see (1545) . be to (1541 ) not ( 1540) .

1

1 1 + -m + 2+ m 3 m

—

PROOF.

1

1

2B5 -

4

- &c.

6

The reference in that article (first edition) should

2917

26 2 (21 ) B³ + 3 4

22 tan x = 2º (221 ) 2 1

2918

cosec x =

+ x

Bot" + &c. 6

2 (2-1) Box 2

+ 2 (29—1 ) B¸³ + 2 ( 25—1) B¸æ³ +&c . + 6 4 PROOF.- By (2916) and the relations tan x

cota - 2 cot 2x,

cosecx = cotx - cot x .

2919 1 - a³ = 1 + 2a cosx + 2a³ cos 2x + 2a³ cos 3x + &c .

1-2a cosx + a² cos x

2920 1-2a cosx +a² a

+ 1 +a² ( cosx + a cos 2x +a³cos 3x +a³ cos4x + &c...) ·a² 2921

sin x = sin x + a sin 2x +a² sin 3x +a³ sin 4x + &c .

1-2a cosx +a² PROOF. -BY (784-6) making a = 6 = x and ca.

When a is less than unity and either positive or negative ,

428

INTEGRAL CALCULUS.

2922 a³

a² cos 2x +

log (1 + 2a cosa +a³) = a cosæ

2 2 a²

a sin x

2923

a sin x -

tan-¹

1 + a cos x

cos 3x- &c.

3 a³

sin 3x- & c.

sin 2x + 3

2

PROOF. -Putting za (cos +i sin x) , we have log ( 1+z ) = log (1+ a cos x + ia sin x)

a sin x = =

log ( 1 + 2a cos x + a²) + i tan -¹

(2214),

1 +a cos x and also

log (1 + * ) = 2— 222 + 2 2- 22 + + &c. 3 4

Substitute the value of z and equate real and imaginary parts. 2924

Otherwise. —To obtain ( 2922) , log (1 +2a cosx + a³) = log ( 1 + aei*) + log ( 1 + ae - ix) .

Expanding by (154) , the series is at once obtained by (768) . 2925 Otherwise.-Integrate the equation in ( 786) with respect to a , after changing a and ẞ into x, and c into -- a.

2926

When a is greater than unity, put

log (1 +2a cosx + a²) = log a² + log ( 1 + 2a¹ cos x+a¯²) , and the last term can be expanded in a converging series by (2922) .

2927 cos 3x - 1 cos 4x + &c .

x=

cosx - cos 2x +

x =

cos x -

cos 2x - 3 cos 3x--

2929

1x =

sin x -

sin 2x + 3 sin 3x - 4 sin 4x + & c.

2930

(π- x) =

sin x +

sin 2x +

log 2 cos 2928 log 2 sin

PROOF.

2931

( 2927-30) Make a

sin x +

2932

sin 3x +

cos 4x - & c .

sin 4x + &c .

± 1 in (2922-3) .

sin 3x +

sin 5x + &c .

π = 2√✓√ /2 (1 + } −} ~ } + } + 1\ — & c .) .

PROOF.

Add together (2929-30) , and put a = π.

When n is less than unity , and a = 1 ± √ ( 1 + n³) , 2933

log ( 1 + n cos x) = log ( 1 +2a cosx + a²) —log ( 1 +a²) ,

EXPANSIONS OF FUNCTIONS IN SERIES.

429

and is therefore equal to twice the series in ( 2922) , minus log ( 1 + a²) . But if a be greater than unity, expand , as in (2926) , by

2934

log (1 + n cos x)

= = log ( 1 +2a¹ cos x + a- 2) + log a² — log (1 +a²) .

2935 (1 +2a cos x)" = 4 + 4, cos x + 4, cos 2x + 4,3 cos 3x + &c . , where A = 1 + C (n , 2) 2a² + ... + C ( n , 2p) C (2p, p) a² + ..., A₁ = 2a { n + C (n, 3) 3a² + ... + C (n , 2p + 1 ) C ( 2p + 1 , p) a² + ... } , and

A¸-1 (n − r + 1) a — rA,

Ar+1 = (n + r + 1 ) a

If n be a positive integer, the series terminates with the n + 1th term , and the values of A and A, are also finite .

PROOF. - Differentiate the logarithm of the first equation ; multiply up and equate coefficients of sin reæ after transforming by (666) ; thus A.+1 is obtained. To find A and A₁, expand ( 1 +2a cos x)" by the Binomial Theorem, and the powers of cos 2 afterwards by ( 772) .

LEGENDRE'S FUNCTION X 2936 with

(1-2ax + a²) ¯ * = 1 + X, a + X,a² + ... + X„a” +

1

dn

2nn

dxn

Xn = X₁

(x²- 1)".

PROOF.- Expand by the Binomial Theorem, and in the numerical part of each coefficient of a" express 1.3.5 ... 2n - 1 as 2n ÷ 2" | n •

Consecutive functions are connected by the relation

2937

d¸Xn+1 = (2n + 1 ) Xn + dxXn-1·

PROOF.- Differentiate the factor once under the sign of differentiation in Xn +1 and X -1 given by the formula for X, in (2936). the values of X1

A differential equation for X, is

2938

(1 - x²) d₂X -2x d¸X, + n ( n + 1 ) X₁" = 0.

430

INTEGRAL CALCULUS.

2939

When P is any positive integer, 1 +2 + 3 +. +(n − 1) NP +1

np

B₂n¹ -1

Bn -s

B₁n¹-5

- &c.

+ = 2 { p + 12p £

2 │p − 1

4

p − 3 + 6 \ p−5

concluding, according as n is even or odd , with (−1)¿p +1 B₁n Te

−¹n (p + 1) __Вpp1 ² (−1)$ (p + 1) p-1 2

or

x ent-1 ent- 1 = • Expand the left side by division, and e -1 x e -1 each term subsequently by ( 150). Again, expand the first factor of the right side by ( 150 ) , and the second by ( 1539) , and equate the coefficients of a in the two results . PROOF.

See (276) for the values of the series when p is 1 , 2, 3, or 4. general formula there is incorrectly printed.

But the

Let the series (2940-4) be denoted by San , Sen, Son , $2n + 1 , as under, ʼn being any positive integer ; then

2940

1 1 S₂n = 1 + 22n + 32

2941

S2 = 1-

1

1

S2n = 1+

22n -1-1

1

22n + 32 1

2942

22n- 1 1 B₂n⋅ 4² + &c. ...: = 2n ²

2n

42n

1

32n + 52n

21

++ &c .

2n Ben

+ &c.... =

2n Bin

N

2 2n

PROOF.- (i. ) San is obtained in (1545) . 127 (ii.) San 20 - S2n = 226 = = =-+ &c. ) = 2. (;1 + 14 (iii.) 82n = ( S2n + S₂n) .

2943

This give Să

1 1 πde(2n n - 1) x cotπx 82n = 1 + 2 + 52 ++ &c. = 422n - 1

x = 1. 1 2944

―――― S² +1 = 1

1

32n + 1 + 52 + 1

1 2n

+ & c. =

., cot πα πd2nx 42 +1 2n

x = 1. PROOF. - By differentiating equation (2912 ) successively, and putting x = in the result. To compute de cot , see ( 1525) .

EXPANSIONS OF FUNCTIONS IN SERIES.

431

2945 The following values have been calculated by formulæ (2940-4) . πT¹ πTS π-6 S S₁ = T , S& = S, = 9450 ' 90' 945'

S2 -

S₁ =

12'

3176

7π*

S' =

S =

H

-6 π

S2

"

17π³

||

161280

960' 575

$3 =

1536'

cos t — Cos a = ( 1-2 a ) [1 1 - cos a

2946

617 =

$5 = 32'

;

S8 =

S6 =

96

si = ===;

127TS 1209600

30240'

720'

184320

x2 (2x —a)³] [1– (2π + a)²

X ) ]... &c., ( × [1 - (kr—a) (4π + a)³ (4π- a ) ²] [1- (4-7 x2 cos x + cos a = 1 1. α ·] [ ¹ ——-—.)· α ] [ ¹- ( a ] 1+ cos a - [ ¹- ~ ~π -~ ~.) ( 3z3π «) (x+ a)²

2947

X c. [ 1- ( x² ] ... & × [ 1- ( 3x + a) ] [1–

COS 2 = cos a = sin ↓ (a + z) sin (a − 2) . Expand the sines by sin' a 1-cos a (815) . The two n + 1th factors of the numerator divided by the corresponding ones of the denominator reduce to 2ax -x² 1737 - a 17) (1 + 4n' (1-2x + -a PROOF.

=

x (1. 2ηπ -a) (1+ 2uπ+a) (1+ 2 πx². a) ( 1-2m² +a )

= (1- (2a) ) (1- (2n

+ a )').

Similarly with ( 2947) employing ( 816).

2948

2949

2x a 2x cos + tan - sin 2 = ( 1 + π-а 3π 382 28 12 22 ) ( + ) (1 + 37-a 2x & x ( 1–2 ) ( 1 + 2 ) ( 1–2 ... , 5πτ )

COST-- cot

2x 2x a 2x sin x = 2 27 + a ( 1 – 220) a ( 1+ 2π -a) (1 2x 2x x (1+ 2 ) (1-2 ) (1+ 2 )... &c.

a cos ( a - x) Expand the cosines by ( 816) , cos + tan 2 sin x = cos a and reduce. Similarly with (2949) , employing 815 .

PROOF .

432

INTEGRAL CALCULUS.

2950 2951

ex -e-* ' = x [ 1 + ( - ) ] [( 1 + ( - ) ] [( 1 + ( 3π -- ) ']).... 2 2 +

=

1+·( 2) ] ...... [1 + ( 2/2 ) *] [ ¹ + ( 2/2) *] [:

PROOF.-Change 0 into iz in ( 815 ) and ( 816 ) . e - 2 cos a + e¯* 2952

2 (1 - cos a) 2 2 &c . = [¹ + (~ - ) '] [ ¹ + ( —_— ² _. _ — ) * ] [ ¹ + ( -—-— ~ —-— ~_a ~) '][ ¹ + ( 4π-a _— _ — _ — _ — __) ']…… ..& +

e* +2 cos a + e * 2953

2 (1+ cos a) x ... &c. = [( 1 + ( ~ -~ ——-—~ ) ′] [ ¹ + ( —-—-—~~ ) ′] [ ¹ + ( 3——-_-a) *] [ ¹ + ( 3—— + a ) *] ··· PROOF.-Change ≈ into ix in (2946-7) .

FORMULE

FOR THE EXPANSION

OF FUNCTIONS

IN TRIGONOMETRICAL SERIES .

x has any value between 1 and -1, When a 2 =0 nã (v —x) dv …… . (i.) , = ... (x) = (v) cos n=1 S // S_ p (v ) do + = = == _-1$ ( ι

2955

where n must have all positive integral values in succession from 1 upwards But, if x = l or -7, the left side becomes lp (1) + lp ( −1) . PROOF. -By ( 2919 ) we have, when h is < 1 , 1- h = 1 + 2h cos 0 + 2h³ cos 20+ 2h³ cos 30+ &c.... 1-2h cos 0 + h³ T Put 0 =·* (v −r) ; multiply each side by ø ( v) , and integrate for v from ι -1 to 1; then make h = 1. The left side becomes, by substituting zv − x, cl - x (1 - h³) (v ) dv = (² - x ( 1 - h²) ( x + z) dz x π v --x) + h³ -1 1-2h cos -1-2 (1 - h) + 4h sin 21 l When h = 1 each element of the integral vanishes, excepting for values of v which lie near to z. Therefore the only appreciable value of the integral arises from such elements, and in these z will have values near to zero, both positive and negative, since z has a fixed value between 1 and -1. Let these values of z range from -ẞ to a. Then between these small limits we shall sin' TZ #222 have and ( + ) = (x), 21 4/2

=

EXPANSIONS OF FUNCTIONS IN SERIES.

433

and the integral takes the form a dz (1 − k²) 4 (x ) [˜_g (1− h)² + 1173-3

h γη = ( 1 +h) l ( ) ( tan - ¹ 7απ ( 1 − h) +tan -117(14)) = 214 2lp (x), πνι ( i h) when h is made equal to unity, which establishes the formula. π% vanishes at both limits, that In the case, however, in which æ = l, sin? 21 is, when z = 0 and when z = = -21. We have therefore to integrate for z from 6 to 0, and also from -21 to -21 +a, a and 3 being any small quantities. The first integration gives lo ( 1) as above, putting a = 0. The second integration, by substituting y = 2 + 21, produces a similar form with limits 0 to a, and with p (x - 27) in the place of (a ) giving lp ( -1) when 1. Thus the total value of the integral is lo (1) + lo ( -1) . The result is the same when x = — 1. That the right side of equation (i . ) forms a converging series appears by integrating the general terms by Parts ; thus ι S Nπ (v -x) nπ (v− x) dv == $ (v) sin $ (") cos l -2 ηπι し ( ၈) ၂ V ι ・Z de, - Nπ ' ' (C ) sin ™ ( v2) 1 -2 which vanishes when n is infinite, provided (v ) is not infinite. Hence the multiplication of such terms by h" when n is infinite produces no finite result when h is made = 1 , although 1 " is a factor of indeterminate value . 2955a

A function of the form

( x) cosne, with n infinitely

great, has been called " a fluctuating function," for the reason that between any two finite limits of the variable a, the func tion changes sign infinitely often , oscillating between the The preceding demonstration shows (e) . (a) and values that the sum of all these values , as a varies continuously between the assigned limits , is zero. By similar reasoning , the two following equations are obtained . 2956

If a has any value between 0 and 1 , 1 nã (v — x) dv Σ ... (2) . • (x) = 21. 1 √ 4 (v) dv + + ≥ 0 (v ) cos し

But if x = 0 , write 16 (1).

(0 ) on the left ; and if x = l , write

If a has any value between 0 and 1 , 1' 2957 0 = dv 21.Sp (r ) dr ++ ; Sp 0 (v ) cos "

3 K

(rι +r) dv ...(3) .

434

INTEGRAL CALCULUS.

But if x = 0 , write 1 (1).

( 0) on the left ; and if a = l , write

2 2958

+ ( x ) == √4 (v) dv +

cos cos "

ηπι COS "T ™þ (r ) dv cos dr 0 . (4) .

This formula is true for any value of a between 0 and 1 , both inclusive. But if x be > 1 , write ( x ~ 2ml) instead of ( x) on the left , where 2ml is that even multiple of 7 which is nearest to x in value . If the sign of a be changed on the right, the left side of the equation remains unaltered in every case . 2959

......….. ( 5) . ... • ( v) = = Σ7" sin "T " ("0 sin "T ι" $ (r) dv

This formula holds for any value of a between 0 and 7 exclusive of those values . If a be > l , write ± 4 (x ~ 2ml ) instead of Φ (x) on the left, + or - according as a is > or < 2ml , the even multiple of l which is nearest to a in value . But if a be 0 or 1 , or any multiple of 1 , the left side of this equation vanishes . If the sign of a be changed on the right, the left side is numerically the same in every case , but of opposite sign . PROOF. For (2958-9 ) . To obtain (4 ) take the sum, and to obtain (5) take the difference, of equations ( 2 ) and (3 ) . To determine the values of the series when a is > l, put a = 2ml ± x', so that a' is < l.

EXAMPLES . For all values of a, from 0 to inclusive, π 4 cos 3x cos 5x ――― cos a + 2960 x = + 32 2 52 '+&c.}. ( п { PROOF.-In formula ( 4) put p (x) = x and l = , then 2 [v sin uv cos nv v cos nv dv = = or 0, + n n² F ୮

2.

according as n is odd or even. Similarly, by formula ( 5) , equation (2929) is reproduced . For all values of a , from — 1 π to 1 π inclusive . 2961

x=

sin 5æ sin 3r 4 S + -&c.}. 52 π { sin a 32

PROOF. -Change a into

-

in ( 2961 ) .

435

EXPANSIONS OF FUNCTIONS IN SERIES.

sin x

π eax - e-ax 2 eª -e- ап

2 sin 2r ――

=

2962

a² + 1

3 sin 3x &c.

+ a²+ 22

a²+3²

PROOF. - In formula ( 5) put o («x) = ear -e-ax and l = ; then Cπ ( − 1)n + 1 ¹ (eª¤ —- ¸¯аñ ) — e - a ) sin nv dv = (ev a² + n* S 2963 If (x ) be not a continuous function between a = 0 and a = l , let the function be (a) from x = 0 to x = a , and (e) from x = a to x = 1 ; then , in formulæ ( 4) and ( 5 ) , we shall have (a ) or (c) respectively on the left side, according to the situation of a between 0 and a, or between a and l. But, if a = a , we must write member.

( a ) +4 ( a ) }

for the left

PROOF. In ascertaining the value of the integral in the demonstration of ( 2955 ) , we are only concerned with the form of the function close to the value of a in question . Hence the result is not affected by the discontinuity unless aɑ . In this case the integration for z is from -ẞ to 0 with (x) for the function, and from 0 to a with (a) for the function , producing (a) +4 (a).

2964

Hence an expression

involving

a in an infinite πX series of sines of consecutive multiples of may be found , ι such that , when a lies between any of the assigned limits (0 and a , a and b , b and c , ... k and 7) , the series shall be equal respectively to the corresponding assigned functions fi (x) ,

ƒ2 (x) ,

ƒ3 (x) ... f (x), fs

provided that the integrals

[0 sin(" T") ; (2 ) de,

[ "asin "," , ... [ sin (" T ") . (2) de ι f (x) de

can all be determined . 2965 The same is true , reading cosine for sine throughout, with the additional proviso [ as appears from formula ( 4) ] that the integrals

[fi (e) dit,

ffe(2) dx, ... S [fu (x) dix

can also be determined .

2966 Ex. 1.- To find in the form of a series of cosines of multiples of z a function of x a which shall be equal to the constants a, b, or y, according as a lies between 0 and a, a and b, or b and π.

436

INTEGRAL CALCULUS.

Formula (4) produces, putting l = ″, -b 1 dx dx = {[ " ade + [" B dx + [[vide }]

n= 1 nx " a cos nx na de + [" Becos ne de + [" youane de } π Σ΄・not N con ne {[ 1 = = π { a (a −ß) + b (b− y) 2 + n =* π Σ **

+ *y }

1 n²2 cos në ¦ (a − ẞ ) sin na + (ẞ− y ) sin nb + y sin na }.

2967 Ex. 2. To find a function of having the value c, when a lies between 0 and a, and the value zero when a lies between a and 1. By formula (4), we shall have Nπα ήπι ηπι dv = cl sin (v) dx = c COS COS ι Nπ ι 0 0 since

(v) = c from 0 to a, and zero from a to l. Ωπα Ωπη πX 1 2c Ta ca sin COS + sin COS + (x) = 2 π { ι ι ι ι ι υπο 3πα 1 sin + + COS -&c. }. ι ι 3

Therefore

When xa, the value is [ (a ) +0 ] = c, by the rule in (2963 ) . may be verified by putting a = -1 in ( 2923).

Ex. 3. - To find a 2968 a lies between 0 and 1, and By formula (4) , NπV (v) cos 7 dv = '0$ [ This reduces to

a when

function of a which becomes equal to equal to k ( l - a) when a lies between

17 kv cos

0 7.12 π2112 (2 cos 2575

This

1 and 1.

ηπύ ηπυ dv + k (lv) cos ι dv. い Si

4/12 cos nл -·11 ) == 9 or 0, = #²n²

according as n is, or is not, of the form 4m + 2. Also -1 17 11 k , (lv) dv = [₁₂ 4 [ + ( c) dv = 2 ( v dv + k [ 2 7:l 87:1 1 COS 2πx + 1 COS σπα + 1 COS 10x $ (x) == し +&c. }. 22 62 73 102 ι ι 4

APPROXIMATE

2991

Let

cb f (x) de

be the integral ,

and

let the

curve

By summing the areas of the trapezoids , +1 equidistant ordinates sides are the

y = f(a) be drawn . whose parallel

INTEGRATION .

437

APPROXIMATE INTEGRATION.

Yo, Y1

... Yn , we find , for a first approximation ,

b- a Sf(x) dx = b+ (yo + 2y₁ + 2y₂ + ... + 2y - 1 + yn ) ...... (i . )

SIMPSON'S METHOD . If y, be the ordinate intermediate between y = f(a) and y½ = ƒ (b ) , then , approximately ,

2992

b- a Sf(x) dx =

(ii .)

(y +4y + y2) 6

PROOF. Take n = 3 in formula (i . ) ; write y, for y , and suppose two intermediate ordinates each equal to ₁ The area thus obtained is equal to what it would be if the bounding curve were a parabola having for ordinates Yo, y₁, y, parallel to its axis. Otherwise by Cotes's formula ( 2995) . 2993 A closer approximation , in terms of 2n + 1 distant ordinates , is given by Simpson's formula,

equi

1

= (f(x) dx

6n [Yo+ Y2n + 4 (Y₁ + Ys + ... +Y2n−1) +2 (y₂ +Ys + ... +Y₂n -2) ] ……………. (iii .)

PROOF.-We have 1

2

dx. dx dx + [ 'ƒ (4) de+... + [ _ƒ(1) da [0 ƒ (e) de = ['^0 ƒ (6) de n 12 Apply formula ( ii. ) to each integral and add the results, denoting by y, the r value of y corresponding to a = 2n

2994 When the limits are a and b , the integral can be changed into another having the limits 0 and 1 , by sub stituting aa + (b − a) y . COTES'S METHOD . Let n equidistant abscissæ , be

ordinates ,

and

n

2995

> 1.

...

"

0,

corresponding

n-1

2

1

Yo , y1 , Y2 ... Yn - 1, Yn

the

and

n

n

A formula for approximation will then be

r [ƒ(x) dx = Aoo + A‚y₁+ ... + A‚Y » + … .. + AnYm (n) 1 -· 1 ) n + r where A -1dx. r A₁ = 1 ° n - r o nx - r

(iv.) ,

(2452)

INTEGRAL CALCULUS.

438

PROOF. -The method consists in substituting for f ( ) the integral function (nx) "} n -1 y . ... + ( − 1 )" ¥ (x) = ( −1 )" [ ( x) (2 ) Yo Yr ... ... Lnx n (nx − r) [ r [ n— r

(na)( )1 ... + (- 1)"

(ux- n )\ n

Yn"},

r taking all integral values from 0 to n inclusive . When er, we have (r) == y ; so that (a) has n + 1 values in common with f (x) . The approximate value of the integral is therefore (e) da, and may be written as in (iv.) By substituting 1-2, it appears that -1 (nx) ) (n )! ) dx = ( dr; 1)"(". nx- (n − - r) nx - r

Consequently it is only necessary to calculate and therefore 4, = Ahalf the number of coefficients in (iv.) 2996 The coefficients corresponding to the values of n from 1 to 10 are as follows. Every number has been carefully verified , and two misprints in Bertrand corrected ; namely, 2089 for 2989 in line 8, and 89500 for 89600 in line 11 . 1 n = 1 : 4 = 4₁ = / 2

n = 2 : A。 = A, ==1/1/0

n= 3:

Aq = 43 =

"

7 n = 4 : A¸ = A¸¯= 90 "

2 4₁ A1 = //

== 4₁ 4₁ = A2 = 3

A = 4Ag =

16 45

4A2=

2 15

n = 5 : 4 = 45 =

19 288'

25 A, = " 4, == 4 96

25 A₂ = Ag = 144

n = 6 : A = A。 =

41 840'

9 A₁ = 4₂ = 35

9 34 A₂ = 4₁ = 280' A3 = 105'

n = 7: AA, =

751 17280'

3577 A₁ = A& = 17280'

49 A₂ = A5 = 640' 989 n = 8 : 4 = Ag = 28350'

A =A =

5248 14175'

43 = 4 =

2989 17280'

A₁ = A =

2944 14175'

A₁ =

454 2835'

= A₂ = Ag

464 14175'

439

APPROXIMATE INTEGRATION.

n = 9 : A。 = A, = =

2857 89600'

1209 43 = Ag = 5600'

15741 A₁ = Ag = 89600'

A₁ = 4, =

27 A₂ = Ar = 2240'

2889 44800

16067 26575 A₁ = 4, = A₂ = Ag = = 598752' 149688' 5675 4825 A = A7 = 12474' A₁ = Ag = A5 == 11088'

n= 10 : Ao = A10 =

16175 199584' 17807 24948

GAUSS'S METHOD. 2997 When f (x) is an integral algebraic function of degree 2n , or lower, Gauss's formula of approximation is

Sf(x) = A « f(x ) + ... + A ¢ƒ(x,) + ... + Anƒ(xn)

(v. ),

where a。....... a, are the n + 1 roots of the equation (vi. ) , ¥ (x) = d(n+1) x { x²+¹ (x − 1 )n +1 } = 0 ...... 1 −x (x− x。) 。) ……. (x —x,-1 ) ( x — x, + 1) ….. (x — xn) dx and 4, = S0 (X‚— · . (X‚ —Xr−1) (X¸ — X¸+1) ... ... (vii. ) ……. (Xp — Xn) .... Xo) ……

The formula is evidently applicable to a function of any form which can be expanded in a converging algebraic series not having a fractional index in the first 2n terms. The result will be the approximate value of those terms . PROOF.-Let and let

if (x) = (xx ) ( x − x,) ... (x - x ), (viii. ), ƒ(x) = Q¥ (x) + R where f(x ) is of the 2nth degree, Q of the n - 1th, and R of the nth, since (a) is of the n + 1th degree . Then the method consists in choosing a function (x) of the n + 1th degree, so that '

(x) de shall vanish ; and a function R of the nth degree , which

shall coincide with f (x) when x is any one of the n +1 roots of

( x) = 0 .

(i. ) To ensure that ( Q4 (2 ) dx = 0. We have, by Parts, successively, 0 (x) , and with the notation of (2148 ) ,

writing N for

{ 2ºN = x² [_N— p ƒ (x² -¹ [_N)

-= 2º [| N — po² 1 [ _2xN + p (p − 1 ) ( ( 2~~~~ 2x ) &c. &c. 1 p - 2 3x = 2º {_N− p 2 " -¹ 2x N + 2 (p − 1 ) 2 - [ _N - ...± 2 ( (p„ .+1), x N

(ix.)

Now Q (x) is made up of terms like a (x) with integral values of p from 0 to n - 1 inclusive . Hence, if the value (vi . ) be assumed for (r) , wo

440

INTEGRAL CALCULUS.

see, by (ix .), that ( 24 (e) de will vanish at both limits, because the factors x and x - 1 will appear in every term . (ii. ) Let R be the function on the right of equation (v. ) Then, when x = x , we see, by (vii . ) , that 4, = 1 , and that the other coefficients all vanish. Hence R becomes f(x) whenever a is a root of ¥ (x) = 0 . The values of the constants corresponding to the first six values of n, according to Bertrand, are as follows. The abscissæ values, only, have been recalculated by the author.

n = 0:

Xo = *5,

A, - 1.

n= 1:

x = 21132487, 278867513.

A。 = A₁ = · 5 ,

log

n = 2:

211270167,

4, = 4, =

log = 94436975 ;

x₁ = 5 ; x = 88729833,

A₁ = 4 ,

206943184,

A, = A, = 1739274,

log = 9.2403681 ;

₁ = 33000948, A₁ = A, = 3260726, xg = •66999052 ; 393056916.

log = 95133143 ;

n= 3:

89,

log

9-6989700 ;

9.6478175.

n =4:

x = 04691008, A, = A, = 1184634, log = 9.0735834 ; x = 23076534, 4, = 4, = 2393143, log = 9.3789687 ; X'z = 5, A, = 2844444, log = 94539975 ; 2,76923466 ; x = 95308992 .

n = 5:

0856622, = 03376524, A, = A, 1803808 , æ₁ = 16939531 , 4, = 4 , x = 38069041 , A, = A, = 2339570, 61930959 ;

log = 8.9327895 ; log = 9-2561903 ; log = 9.3691360 ;

x = 83060469 ; a = 96623476 . As a criterion of the relative degrees of approximation obtained by the foregoing methods, Bertrand gives the following values of π dx = log 22721982613 . ' log (1+r) ( 8 0

Method of Trapezoids, Simpson's method, Cotes's 99 Gauss's ""

n = 10, n = 10 , n = 5, n = 4,

2712837. 2722012. 2722091 . 2721980.

For other formulæ of approximation , see also p . 357 .

CALCULUS

OF

VARIATIONS.

FUNCTIONS OF ONE INDEPENDENT VARIABLE .

3028 Let y = ƒ (x) , and let V be a known function of x, y, and a certain number of the derivatives Y., Yar, Yзr, &c. The chief object of the Calculus of Variations is to find the form of the function f (x) which will make U= ST Vdx xo a maximum or minimum.

.. (i.)

See (3084) .

Denote yr, Yaz, Yзx, &c. by p , q, r, &c . For a maximum or minimum value of U, SU must vanish. To find SU, let dy be the change in y caused by a change in the form of the function y = f (x) , and let dp , dq, &c . be the consequent changes in p, q, &c . Now, p = Yx.

Therefore the new value of p , when a change takes place in the form of the function y, is p + &p = (y + dy) x = Yx + (dy) .x,

therefore

Sp = (dy) ;

Similarly,

&q = (dp) x,

that is , 8 ( du ) = d (Sy) dx

dr = (8q)x, &c.

(ii.)

Now SU = (" SVda (1483 ) . Expand by Taylor's theorem , xo rejecting the squares of dy, Sp, dq, &c . , and we find 8U = [” (V„ ôy + V„ &p + V« ôq … + ..) dæ, Xo

or, denoting V , Vp, Vq, ... by N, P, Q, ..., 8U = [” (N&y + P&p + Q&q .…….) + dæ …….………………….(iii.) Xo 3 L

CALCULUS OF VARIATIONS.

442

Integrate each term after the first by Parts , observing that by (ii.) ( dp dæ = dy , &c. , and repeat the process until the final integrals involve Sy da.

Thus

[ Ndy dr is unaltered ,

[ Pôp dx = Pêy -

P8ydx,

[ Q&p dx = Q&p — Q₂dy + √ Q₂dy dx,

| Rôr da = Rôq― R₂dp + R₂dy —f R₂dy dx, √ ... ... ... ... 3029

Hence , collecting the coefficients of dy, dp, dq , &c . , Cxi 2x − R3 + SU = √ (N− P₂ + Q22 ...) Sydx xo

+dy₁ (PQ₂ + R₂ -... ) — Sy。 (P - Qx + R2x -... ) 。 + Sp₁ (Q - R₂ + S2x -... ) 1 - Spo ( Q − Rx + S2x − ….. ) 。 + dq₁ (R - S₂ + T2x -... ) 1—8qo (R - S₂ + T₂ -... ) 。 + &c. (iv.) The terms affected by the suffixes 1 and 0 must have a made equal to a, and a, respectively after differentiation . Observe that Px, Qx , &c. are here complete derivatives ; Y, P, q , r, &c. , which they involve , being functions of x. Equation (iv.) is written in the abbreviated form ,

3030

SU = √* K dy dx + H₁— H。 xo

(v.)

The condition for the vanishing of SU, that is , for mini mum value of U, is 3031

3032

K = N- P₂+ Q2x − R3x + &c.

.... 0 ......... (vi .) ,

H₁ - H₁ = 0 ........................ (vii .)

and

PROOF.- For, if not, we must have

Kdy de = H -H₁ ;

xo that is, the integral of an arbitrary function (since y is arbitrary in form) can be expressed in terms of the limits of y and its derivatives ; which is impos sible. Therefore H₁ -H = 0. Also K = 0 ; for, if the integral could vanish without K vanishing, theform of the function dy would be restricted, which is inadmissible .

FUNCTIONS OF ONE INDEPENDENT VARIABLE.

443

The order of K is twice that of the highest derivative contained in V. Let ʼn be the order of K, then there will be 2n constants in the solution of equa tion (vi. ) and the same number of equations for determining them. For there are 2n terms in equation ( vii . ) involving dy₁, dyo, dp₁ , &c. If any of these quantities are arbitrary, their coefficients must vanish in order that equation (vii. ) may hold ; and if any are not arbitrary, they will be fixed in their values by given equations which, together with the equations furnished by the coefficients which have to be equated to zero, will make up, in all, 2n equations.

PARTICULAR CASES . 3033 I.- When V does not involve a explicitly, a first integral of the equation K = 0 can always be found . Thus , if, for example, V = 4 (y, p, q, r, s) , a first integral will be

V=

Pp + QPx - QxP + RP22 - RxPx + R2xP + SP3x - Sx P2x + S2x Px - S3xP + C.

The order of this equation is less by one than that of (vi .)

PROOF.-We have

V₁ = Np + Pq + Qr + Rs.

Substitute the value of N from ( vi. ) , and it will be found that each pair of terms involving P, Q, R, &c . is an exact differential .

II.- When V does not involve y , a first integral can be found at once , for then N = 0 , and therefore K = 0 , and 3034

we have

P₂ - Q2x + R32- & c. = 0 ;

and therefore

P - Qx + R2x- &c . = A.

3035

III.- When V involves only y and p, by Case I. V = Pp + A ,

3036

IV.

When V involves only p and q, V = Qq + Ap + B.

PROOF. Also

See also (3046) .

K

- P₂ + Q₂ = 0, giving, by integration, P = Q₂ + A. V₁ = Pq + Qr = Aq + Qzq + Qr. Integrating again, we find V = Qq + Ap + B, a reduction from the fourth to the second order of differential equations.

444

CALCULUS OF VARIATIONS.

3037 Ex. To find the brachistochrone, or curve of quickest descent, from a point O taken as origin to a point y₁, measuring the axis of y down wards.* Velocity at a depth y = √2gy. Therefore time of descent

V=

Here

√ = Jos 1 t+ypp² e dia. Sxo √2gy 1 +p² = p³ + A, by Case III. √ { y ( 1 +p²) } y

By reduction, y (1 +p³) = 14

2a, an arbitrary constant.

That is, since p = tan 0, y = 2a cos² 0, the defining property of a cycloid having its vertex downwards and a cusp at the origin 1 H - H reduces to { (pdy) - (pdy) } = 0. √2a If the extreme points are fixed , dy, and dy, both vanish. The values ₁, Y₁, at the lower point, determine a. Suppose ₁, but not y , is fixed . Then dy, is arbitrary ; therefore its coefficient in ( 3) ( P - Q + &c .) , must vanish ; that is, (V ) , = 0, or p = 0, therefore p₁ = 0, which means that the tangent at the { √y ( +p²) 15 },1 lower point is horizontal, and the curve is therefore a complete half cycloid. 3038 In the example of the brachistochrone, it is useful to notice that— (i. ) If the extreme points are fixed, dyo, dy, both vanish. (ii ) If the tangents at the extreme points have fixed directions, Po, dp₁ both vanish. (iii.) If the curvature at each extremity is fixed in value, dpo do, dp₁, dqi all vanish . (iv.) If the abscissæ , only have fixed values, dyo, dy, are then arbitrary, and their coefficients in H₁ - H, must vanish. 3039

When the limits a , a, are variable, add to the value of

SU in (3029 ) PROOF.

3040

V₁da₁ - Vodx。.

The partial increment of U, due to changes in x and x , is dU dU da = Vida - Vodro ·dæ₁ + By (2253). da dxo

When x

equations ,

and y₁ ,

。 and y, are connected by given

Y₁ = 4 (x ) ,

Yo = x (xo).

RULE .-Put = Sy₁ = {

(x₁ ) —p₁ } da,

and

dy %

{x' (x ) —p } dæɔ ,

* The Calculus of Variations originated with this problem, proposed by John Bern ulli in 1696.

FUNCTIONS OF ONE INDEPENDENT VARIABLE.

445

and afterwards equate to zero the coefficients of da, and dæo, because the values of the latter are arbitrary. PROOF. -y, + dy, being a function of ₁, (y₁ + dy₁ ) + d¸¸1 (y₁ + dy₁ ) dx₁ = ¥ 4 (x, + dx₁ ) = 4 ( x₁) + 4′ (x₁) dæ₁ ; therefore dy, +p₁dx₁ = (x₁ ) dx,, neglecting dpdx . Ex.- In the brachistochrone problem (3037) , the result thus arrived at signifies that the cycloid is at right angles to each of the given curves at its extremities.

3041

If V involves the limits xo, 21 , yo,

1, Po, P₁ , &c. , the

terms to be added to SU in (3029) , on account of the varia tion of any of these quantities, are

1 dœ‚[*"' To { V₂, + V₁P₁ + Vp &1 + … ..} dæ +dx。 [* ...} dæ To { Vzo + V„«Po + V» 20 + 1 , &Yo + Vv₁ & Y ₁ + Vpo &Po + V„‚ §P₁ + &c . } dx. + [ { Vvo To In the last integral, Syo, dy , Spo , &c. may be placed outside the symbol of integration since, they are not functions of a. involves the limits x。 , x₁ , Yo, Y₁ , Po, P1 , &c. , Hence , when and those limits are variable, the complete expression for SU is

3042

SU = S* { N— P₂ + Q2¬R₁ + &c . } Sy dx To

+ { V₁ +S® ( V₂ + VnP₁ + Vm & + …… .) dx}' `dæ

— { Vo −

Po 20 + ( Vz + Vv Po + V₂ ...) dx} dx,

To + { (P− Q₂+ R20 −...) 1 +(* V₁dx} dy₁ Xo — {(P— Q. + R₂— ...).—S“, V, dx } dy. + { (Q − R₂ + S2 − ... )1+S* V, „, dx } Sp₁ Po dx } Sp₂ + &c . — { (Q - R₂ + S₂ -...) .— SV, To

CALCULUS OF VARIATIONS.

446

3043 Also, if y₁ = (x ) and y = x (x ) be equations re stricting the limits , put dy₁ = {4′ (x,) — p₁ } da,

and

Syo = { x' (xo) —Po } dx。 • dy,

(3040)

The relation K = 0 is unaltered , and , by means of it, the additional integrals which appear in the value of H₁- H。 become definite functions of a.

3044 Ex. To find the curve of quickest descent of a particle from some point on the curve yo = x (x ) to the curve y₁ = ¥ (x,) . 1 1 +p 1 dx, √ (2g)[* y -yo Equation ( 3042) now reduces to

As in (3037), t = P, and y..

SU =

V=

1 +p , and contains y, y -yo

[" (N−P¸) èy dz + V, dz, — { Vo —[ " Vv.Pode} dx。 . Vy, da } dy. ….….…...(1). + P¸dy, — { P。 - ["xo

Now K = 0 gives N- P, = 0 ; therefore V = Pp + A (3035) ; 2 2 1 +p² = p² therefore +4. y -yo √ (y - yo) ( 1 +p²) 1 this becomes Clearing of fractions, and putting A = ✓(2a)'

(y - yo) (1 +p²) = 2a... Also Hence

therefore

P = V, =

(2) .

p = P - yo) ( 1 + p*)} ✓(2a) √(y (y ·— √

(3).

V:= 1+ p² ; Vy。 V₁₁ = -V₁y = −N = -P, (by K = 0), √(2a)' [" ,Yode = P₁ - P₁ = Po (2a) -P To V

......... (4 ) .

Substituting the values ( 2) , ( 3 ) , (4), in (1 ) , the condition H₁ - H, produces (1 +p²) dx- (1 +Pop₁) dxo + p₁ dy₁ — P₁dyo = 0. Next, put for dy, and dy, the values in (3040) ; thus the equation becomes { 1 +p₁ ' (x₁) } dx, - { 1 + p₁ x' ( xo) } dx。 = 0 ........ dx₁, dr, being arbitrary, their coefficients must vanish ; therefore P₁ ' (x ) == = -1

and

..... (5) ;

P₁x (x ) = −1 .

That is, the tangents of the given curves and x at the points y and x,y, are both perpendicular to the tangent of the brachistochrone at the point x,y,. Equation (2) shews that the brachistochrone is a cycloid with a cusp at = . the starting-point, since there y = yo, and therefore p∞

FUNCTIONS OF ONE INDEPENDENT VARIABLE.

447

OTHER EXCEPTIONAL CASES . (Continued from 3036. )

3045 and

V. - Denoting Y, Y , Y2x ... ynx by y , P1 , P2 ... Pn ; y Vp, V₁₂

Vp ... V

by N, P1, P2 ... Pri

let the first m of the quantities y, P1, P2, &c . be wanting in the function V ; so that V = f (x, Pm, Pm+1 ... Pn) . = P₂ = 0 , Pm - d( m + 1) x· Pm + 1 + ... ( −1 )” -” dnx Pn Then Kdma which equation , being integrated m times, becomes Pm- da Pm +1 + dx Pm +2— ... ( − 1 ) ” - md(n - m) x Pn = co + c₁x + ... + cm- 1xm - 1 ...... (i.) , a differential equation of the order 2n - m .

3046

VI.- Let x also be wanting in V, so that V = f (Pm , Pm + 1 ... Pn) ;

then K = 0 is the same as before , and produces the same From that equation take the value differential equation (i .) of P , and substitute it in V = PmPm+ 1 + Pm + 1Pm+ 2 + ... + Pn P₂+1•

Each pair of terms , such as Pm +2 Pm + 3 - d2gPm + 2Pm+1 , is an exact differential ; and we thus find V = c + Pm+ 1Pm+ 1 + ( Pm+2 Pm+2¬d₂ Pm +2 Pm+ 1) + ... + (PnPn¬dzPnPn-1 + d²x PnPn-2) — ... ( −1 ) n - m− ¹d(n − m- 1) xPnPm+1 +f (co + cæ + ... + cm -12m −¹) Pm+1 dæ. The resulting equation will be of the order 2n - m - 1 , or m +1 degrees lower than the original equation .

3047

VII.- If V. be a linear function of p , that being the

highest derivative it contains , P, will not then contain P. Therefore dr P, will be, at most , of the order 2n - 1 . In deed, in this case, the equation K = 0 cannot be of an order higher than 2n - 2. (Jellett, p. 44.)

CALCULUS OF VARIATIONS.

448

3048 VIII.- Let Pm be the lowest derivative which V in volves ; then, if P = f(x) , and if only the limiting values of a and of derivatives higher than the mth be given, the problem cannot generally be solved . (Jellett, p . 49. )

3049

IX.- Let N = 0 , and let the limiting values of x alone

be given ; then the equation K = 0 becomes Px - Q2x + Rзx- &c . = 0 ,

or, by integration,

P - Q +R₂- &c . = c,

and the two conditions furnished by equating to zero the co efficients of Sy₁ , dyo, viz . , (P − Qx + &c.)₁ = 0 ,

I

(P − Qx + &c.)。 = 0 ,

are equivalent to the single equation c = 0, and therefore H₁ - H0 supplies but 2n - 1 equations instead of 2n , and the problem is indeterminate.

3050

Let U = [ " Vdx + V′ , where To

V = F (x, y , P , q ... )

and

V' = f ( xo , x1 , Yo , Y1 , Po , P₁ , &c . )

The condition for a maximum or minimum value of U arising from a variation in y, is , as before , K = 0 ; and the terms to be added to H₁ — Н。0 are

+ V₁₁dy₁ + & c . V₁ dx + Vข Sy。 + Vp Spo + ... + V₁₁dæ₁ x1 If the order of V be n , and the number of increments do , yo, &c . be greater than n + 1 , the number of independent incre ments will exceed the number of arbitrary constants in K, and no maximum or minimum can be found. Generally, U does not in this case admit of a maximum or minimum if either V or V' contains either of the limiting values of a derivative of an order or > than that of the highest derivative found in V. (Jellett, p. 72.)

FUNCTIONS

OF TWO DEPENDENT VARIABLES .

3051 Let V be a function of two dependent variables y , z, and their derivatives with respect to a ; that is , let

...... ( 1 ) , V = f (x , y , p, q ... z , p' ‚ q ' ... …… . ) .... …………………………………..

I

FUNCTIONS OF TWO DEPENDENT VARIABLES.

449

where P, p, q, q, ..., as before , are the successive derivative of y , s and p p'',, q q '' , ... those of z . Then , if the forms of the functions y , z vary , the condition for a maximum or minimum value of U or

" Vda is Xo

SU = ·S" xo ( Kdy + K'dz) dx+ H‚— H₂ + H¸'— H' = 0 ... (2 ) . Here K' , H' involve z , p ' , q' , ..., precisely as K, H involve y, p, q, ... ; the values of the latter being given in (3029) .

First, if y and z are independent, equation (2) ne cessitates the following conditions :

3052

K = 0,

K' = 0 ,

………….. (3) . H₂ - H + H₁' — H ' = 0 ......

The equations K = 0 , K ' = 0 give y and z in terms of x, and the constants which appear in the solution must be deter mined by equating to zero the coefficients of the arbitrary quantities

Syo, Sy₁ , Spo, sp₁ ... Szo , Sz1, spó' , Spí' , ... ,

which are found in the equation

H₁ - H₂ + H₂ - H,' = 0 ....

(4).

3053 The number of equations so obtained is equal to the number of constants to be determined .

PROOF.-Let

V = f(x, y , yz, Y2 ... Ynx, 2, 2 , 22x ... Zmx) ,

K is of order 2n in y, and .. of form 4 (x, y, Yx I ... Yinx, Z, Z ... %(m + n) x) ... (i.), K' is of order 2m in z, and .. of form 4 (x, y, Yx ... Y (m+ 2+ 2) 29 2, Zx ... Z2mx) ... (ii.) Differentiating (i. ) 2m times, and ( ii. ) m + n times, 3m +n + 2 equations are obtained, between which, if we eliminate z, x ... Z(3m +n)x ? we get a resulting equation in y, of order 2 (m + n), whose solution will therefore contain 2 (m +n) arbitrary constants . The equations for finding these are also 2 (m +n) in number, viz., 2n in H₁ - H and 2m in H¡ -H .

3054 NOTE. -The number of equations for determining the constants is not generally affected by any auxiliary equations introduced by restricting the limits. For every such equation either removes a term from (4) by an nulling some variation (dy, dp , &c . ) , or it makes two terms into one ; in each case diminishing by one the number of equations, and adding one equation, namely itself.

3055

Secondly, let y and z be connected by some equation 3 M

450

CALCULUS OF VARIATIONS.

+ (xyz) = 0 . y and z are then found by solving simultane ously the equations (x, y , z) = 0

and

K : 4, = K' :

..

PROOF (x, y, z ) = 0 , and therefore (x , y + dy , z + dz) = 0 , when the forms of y and z vary. Therefore p₁dy + p₂dz = 0 ( 1514) . Also Koy + K´dz = 0, by (2 ) . Hence the proportion.

3056 Thirdly, let the equation connecting y and z be of the more general form

$ (x, y , p, q ... %, ≈, p' , q′ ...) = 0 ............... (5) . By differentiation, we obtain Pï dy + p p dp +

q &q + ... $z ¢ z8% + ¢‚ „· &p' + ¢q· dq ' + ... = 0 ... ( 6) .

If (which rarely happens) this equation can be integrated so as to furnish a value of 8z in terms of dy, then Sp' , dq' , & c . may be obtained, by simple differentiation , in terms of dy, Sp. Generally, we proceed as follows : &V = N&y + P&p + Q&q + ... + N'dz + P'dp' + Q'dq' + ...

... (7) .

Multiply (6) by A , and add it to ( 7) , thus

SV & V = (N + λp,) dy + ( P + λpp ) dp + ... ... + (N' + λp₂) 8z + ( P' + λpp.) dp′ + ......... (8) . The expression for SU will therefore be the same as in (2 ) , if we replace N by N+ λp , P by P + App, &c. , thus

3057

SU = (" [ { (N + λ4, ) −(P +λp,) x+…..} dy + { (N' + λþ₂) — (P' + λpp ) x + … .. } dz] dx + { P +λøp− (Q + λ$q) x + ….. } 1₁dy₁ − {P + dø, − ( Q + d$q) x + ….. } 0ody。 + { Q +λ6, − (R +뤂)x +….. } 1 dp₁ — { Q+ λ4, − (R + λ¤r) x + ….. } odpo &c. &c . + similar terms in P, Q ... p' , q' ... &c. . , . (9) .

3058

To render &U independent of the variation dz, we must

FUNCTIONS OF TWO DEPENDENT VARIABLES.

451

then equate to zero the coefficient of dz under the sign of integration ; thus ·&c. &c. = 0 ...... ( 10),

N' +λo, — (P' + dpp') x + ( Q´ + λpg )22 the equation for determining λ.

3059

Ex. (i.) - Given V = F(x, y, p, q ... 2), where 2= = Svde and v = F (x, y, P, Q ...) . 0 or v — z, = 0,

The equation Φ is now z -Svdz

$y = vy Pe== 0,

?p = vp Pp= -1,

Pq = vq, &c., = 0, the rest vanishing.

Substituting these values in (9) , we obtain

¿ U = [ "' [ { N + \ v, − ( P + \ v₂ ) z + ( Q + λvg) 2− ... } ¿y + { N' +λ ; } 8: ] dz To + { P+ λv₂- (Q + λv q) z + ... } ₂ dy₁ = { P + λv, − ( Q + λ v q ) x + ... }o dYo + { Q + λvq− (R + λvr) x + ... } 1 dp₁ — { Q + λv , - ( R + λ v,) z + ... } 0。 &P₂ + &c. For the complete variation DU add V, da, - Voda . To reduce the above so as to remove dz, we must put N' + λ = 0, and therefore λ = - N'da. Let be the solution, u being a function of x, y, p, q ... z. Substituting this A expression for λ, the value of ¿ U becomes independent of dz . Ex. (ii .) —Similarly, if z in the last example be = Sv (2148) , o becomes -v- ,, = 0 ; and, to make N' + ,, vanish, we must put λ = -SN'.

Ex. (iii.)-Let U= √ √ 1+ y² + 2 da ...... To p' P P= Here N = 0 ; N' = 0 ; P = √1 +p² +p² √1 + p² + p²²

3061

(1) .

Q =0;

Q = 0; and the equations K = 0, K′ = 0 become P = 0,

P = 0,

or

P 12 = α, √1 + p² + p³

Solving these equations, we get yx = m ; 2x = n ; or

y = mx + A ;

p' = b. √1 + p² + p³

z = nx + B.

3062 First, if 21 , y1 , 21 , Xu, Yo, zo be given, there are four equations to determine m, n, A, and B. This solves the problem, to find a line of minimum length on a given curved surface between two fixed points on the surface. 3063

Secondly, if the limits a,, 2, only are given, then the equations

(P), = 0, (P)。 = 0, (P′) , = 0, (P ′)。 = 0, are only equivalent to the two equations m = 0, n = 0, and A and B remain undetermined .

452

CALCULUS OF VARIATIONS.

3064

Thirdly, let the limits be connected by the equations (X1, Y1, 1) = 0,

We shall have Substitute

4 (xo, Yo, ≈0 ) = 0 .

($x, + Pv. P₁ + Qz, Pí) dæ¸ + ¢„‚ Èy₁1 + ¥z₁dz₁ = 0. = M₁ $x,

: n ; thus $z₁ = n₁Px, P₁ = m, pi =

(1 + mm, + nn, ) dx, + m , ¿y, + n, dz₁ = 0.

Eliminate de, by this equation from V₁dx₁ + ( P) ₁ ¿y₁ + (P′), dz₁ = 0, and equate to zero the coefficients of dy, and dz₁ ; then m₁V₁ = (P) , (1 + mm₁ + nn₁) ;

n₁V₁ = (P ′) , ( 1 + mm, + nn₁) .

Replacing V₁, P,1 by their values, and solving these equations for m and n, we find m = m₂, n = n₂. Similarly from the equation ( o, Yo, Zo) = 0 we derive m = m。, n = n。. Eliminating 1, Y1, 1, xo , Yo, zo between these equations, and y₁ = mx₁ + A ;

z₁ = nx₁ + B ; (x1, y1, 1 ) = 0 ;

y = mx + A ; z₁ = nx¸ + B ; (to, Yo, Zo) = 0 ;

four equations remain for determining m, n, A, and B.

3065

On determining the constants in the solution of (3056) .

Denoting p, q, r ... by P1, P2, P3 ..., we have V = F (x, y, P1, P2 ... Pn, ≈, Pí , P2 ... P'm ) ;

and for the limiting equation,

(x, y, P1, P2, ….. Pw, Z, P1 , P2, ... pm ) = 0. Vis of the order n in y and m in z. is of the order n' in y and m ' in z. 3066

RULE I.- If m be > m' , and n either > or < n' , the

order of the final differential equation will be the greater of the two quantities 2 ( m + n ') , 2 (m' + n) ; and there will be a sufficient number of subordinate equations to determine the arbitrary constants . 3067 RULE II . —If m be < m' , and n < n' , the order ofthe final equation will generally be 2 (m' +n') ; and its solution may contain any number of constants not greater than the least of the two quantities 2 (m' -m) , 2 ( n ' - n) . For the investigation, see Jellett, pp. 118-127.

3068 If V does not involve explicitly, a single integral of order 2 (m + n) -1 may be found . The value of V is that given in (3033), with corresponding terms derived from z.

FUNCTIONS OF TWO DEPENDENT VARIABLES.

453

dV = Ndy + P₁dp, + ... + Pdp + N'dz + Pidpi + ... + Pmdpm Substitute for N and N' from the equations K = 0, K' = 0, as in (3033 ), and integrate for V. PROOF.

RELATIVE MAXIMA AND MINIMA. 3069

In this class of problems , a maximum or minimum

value of an integral, U₁ = [" V₁da, is required , subject to the To

condition that another integral , U, =

V, da, involving the

To same variables , has a constant value . ▬▬▬▬▬▬▬▬▬▬▬ Find the maximum or minimum value of the func RULE. tion U₁ + aU ; that is, take V = V₁ + aV₂, and afterwards determine the constant a by equating U,2 to its given value.

For examples , see (3074) , (3082) .

GEOMETRICAL APPLICATIONS. 3070

PROPOSITION I.- To find a curve s which will make

F ' (x, y) ds a maximum or minimum, F being a given function of the coordinates x, y.

The equation (5 ) , in ( 3056 ) , here becomes p³ +p² = 1 ; where px , py,, x and y being the dependent variables, and s the in dependent variable. In (3057) , we have now, writing u for F (x, y),

the rest zero.

N = U«, _N' = Uy, &p = 2p, Pp = 2p '; The equations of condition are therefore 5 " u,—d, (\x¸) = 0 and u, -d, ( ^y, ) = 0 ........

. (1) .

Multiplying by *,, y, respectively , adding and integrating , the result is λ = u, the constant being zero. * Substituting this value in equations ( 1 ) , differentiating ux, and uy,, and putting u, uz, + u, y,, we get ... (2) , y, (uzy, -u, x,) = UX28....... ...... . (3). x¸ (µ‚x¸ — UxYs) = UY28•••••• * See Todhunter's " History," p. 405.

454

CALCULUS OF VARIATIONS.

Multiplying (2) by y , and ( 3) by a,, and subtracting, we obtain finally u (Y, X28 — X8Y28 ) = UxYs — Uy X89 or

du dy

u =

3071 P

du

dx

(4), dx

ds

dy ds

p being the radius of curvature. To integrate this equation , the form of u must be known , and, by assigning different forms , various geometrical theorems are obtained .

3072

PROPOSITION II.-To find the curve which will make ......

...

(1)

[F(x, y) ds + [f(x, y) dx a maximum or minimum, the functions F and ƒ being of given form . Let F (x, y) and f (x, y) = v. Equation (1) is equivalent to f(u + vx ) ds. In (3057) we now have Vu +vp ; and for 4, p² +p" = 1 , as in ( 3070) . Therefore N = u₂ + pvz j P = V₂ = v ; Pp = 2p ; = ' ; the rest zero. N' = uy + pvy ; Pp2p Therefore, equating to zero the coefficients of dx and dy, the result is the two equations

u₂ + pv₂- (v + λp) , = 0, u₁ +pv₁ — (\p'), = 0 ; or

d, ( x ) + v₁ = U₂ + x , vz = U₂ + xzvy• d. (xy )

Multiplying by x , y, respectively, adding, and integrating, we obtain, as in (3070) , λ = u, and ultimately, du dy dv du dx 1 = ——— 3073 + u d dy dy ds 11 ( x ds P 3074 Ex . To find a curve s of given length, such that the volume of the solid of revolution which it generates about a given line may be a maximum. Here (y²x, -a²) ds must be a maximum, by (3069), a³ being the arbitrary constant. The problem is a case of (3072), u = a², u¸ = 0, u; = : 0, v = y³, v₁ = 2y. 1 = 2y Hence equation (3073) becomes Р a³

and integrating, the result Giving p its value, - (1 + p³)³ where P = dy), (wh PPy 1 y²+b² (y² + b² ) dy = ; from which x = √1+p³ a¹ S√a^ — (y² + b³)

FUNCTIONS OF TWO INDEPENDENT VARIABLES.

FUNCTIONS

3075

OF TWO

Let

455

INDEPENDENT VARIABLES .

V = f(x, y , z , p, q , r, s , t) ,

in which a, y are the independent variables , and p, q , r, s , t stand for Z , Z , Zax Zxy , Zy respectively ( 1815) , z being an in determinate function of a and y . x1 y1 U= Let ༢ (" Vdx dy, and let the equation connecting x and y at the limits be (x, y) = 0.

The complete variation of U, arising solely from

an infinitesimal change in the form of the function z , is as follows :

Let V , V , &c. be denoted by Z, P, Q, R, S, T. Let

Φ = (P— R¸ − 1 S₁ ) dz + 1S8q + Rdp, $ 4 = (Q — T, — S¸) 8z + S&p + T8q, x = (Z − P₂ - Qy + R2x + Sxy + T2y) Sz.

The variation in question is then

3076

dy dy -&y=v. dx — - Þv = du; SU = √ (4vy=y1 = -4vy== yo x + Þvy ==yo v. do Ax ) dx x =x1 +

(ທ

zo y [ S" yo + dy ] x = x0 J J X dx dy .

201 PROOF.- ♪ [ [" V dx dy = [ [ "&Vdz dy [[ " evdady ๔o Jo [xifyi = (Zdz + P&p + Qoq + Rèr + Sès + Tôt) dx dy Toyo dy + x } dady , dx + d = [ [ Yo { d² g as appears by differentiatin the values of $ and . But 1 yi d d d o ody + 4y- vo dy dx -Py-y₂ y dy = dx ["ody. dx yo dx

by (2257) , and for Yo dy dy = 40 - v₂ - 4v - vo Hence the result.

3077

The conditions for a maximum or minimum value of

U are, by similar reasoning to that employed in (3032) , Φ = 0,

4 = 0,

x = 0.

456

CALCULUS OF VARIATIONS.

GEOMETRICAL APPLICATIONS. 3078

PROPOSITION I.-To find the surface,

S, which will

make [[ F(x, y , z) dS a maximum or minimum, F'being a given

function of the coordinates x, y , z .

[Jellett, p. 276.

Here , putting F (x, y , z) = u, V = u√1 + p² + q² ; du uq up Q= z = √1 + p + q dz ; P = √1 + p2 +q² √1 +p³ = √1 +p +g + q² '

and V , V, V, are all zero.

du dP P = + √1 +p³ + q³ (de dx + d = √1+

du + u (1+ g²) r - pige ddz) + ( 1 +p² + q²) §

du dQ = t− pqs 9 (du u y +8 d dr dy = √1 dz ) + (1 + p ) -py + q³ ( dy + p² + √1 +x² (1 +p² + q³) s The equation x = 0 or Z - P - Q, = 0 gives du du - du 1 (1 + q ) r − 2pqs + (1 + p²) t q q² u√ p² + dx 1 dz = 0. dy dd) + (p₁= + + + (1 +p³ + q³) } If R, R' be the principal radii of curvature, and l , m, n the direction cosines of the normal, this equation may be written 3079

1 du 1/2 R + 1/2 R + = u (idu dx + m dy + n du) = 0 ,

and according to the nature of the function geometrical theorems may be deduced .

3080

u different

PROPOSITION II.-To find the surface S which will make

[ F ( , y , z) d8 +||f (@ , J ,≈ didy a maximum or minimum ; Fand ƒ being given functions of the coordinates x, y, z. Let F (x, y, z ) = u and ƒ (x, y, z) = v . (3078), we have

Proceeding throughout as in

V = u√l + p² + q³ + v,

Z = √1 +p² + q³ Uz + V29 and the remaining equations the same as in that article if we add to the resulting differential equation the term -

on the left.

น

APPENDIX.

457

This equation may then be put in the form ι 3081

R+ R ' =

du dv du du ―― de) dx +mdy + n dz dz .

where l, m , n are the direction cosines of the normal to the surface.

3082 Ex . To find a surface of given area such that the volume contained by it shall be a maximum. By (3069) , the integral [[ (x - a√1 +p³ +q³) dødy must take a maximum or minimum value. The problem is a case of (3080) . We have u = —a, v = z, Ux = 0 , Uy = 0, U₂ = 0, v₂ = 1 ; and the differential equation of the surface ( 3081 ) reduces to

q) ³ = 0 ; (1 +q³) r − 2pqs + (1 +p³) t + — ( 1 + p² + °

3083

or

1 1 1 = R + Ꭱ a

APPENDIX.

ON THE GENERAL OBJECT OF THE CALCULUS OF VARIATIONS. 3084 DEFINITIONS . -A function whose form is invariable is called determinate, and one whose form is variable , indeter minate. Let du be the increment of a function u due to a change in the magnitude of the independent variable , du that due to a change in the form of the function, Du the total increment from both causes ; then

Dudu + Su. Thus , in (3042) , the terms involving da, and da, constitute du, and the remaining terms du ; the whole variation being Du. Su is called the variation of the function u. 3085

A primitive indeterminate function, u, of any number

of variables is a function whose variation is of arbitrary but constant form ; in other words , & u = 0. 3 N

458

CALCULUS OF VARIATIONS.

3086

Letv

F.u be a derived function , -that is , a func

tion derived by some process from the function u ; F denoting a relation between the forms , but not between the magnitudes , of u and v.

The general object of the Calculus of Variations is to determine the change in a derived function v, caused by a change in the form of its primitive u. The particular derived functions considered are those whose symbols are d and [ , denoting operations of differ

entiation and integration respectively .

SUCCESSIVE VARIATION. 3087

Let the variation of the variation , or second variation

of V due to a change in the form of the involved function, y = f(x) , be denoted by 8 (SV) or 82V; the third variation by 83V, and so on.

By definition (3085 ) , y being a primitive indeterminate y = 0 ...... function, and dy its variation, (1). 3088 The second variation of any derivative of y is also zero, i.e., p, q, &c. all vanish .

• PROOF

( x) = d ( ¿Ynx) =· { d ( dy) } nx = ( d²y) nx = 0 by (1) .

3089 If V = f (x, y , p , q, r, &c . ... ) , where y is a primitive indeterminate function of x, then

SV = (Syd, + Sp d, + Sq d₁ + ... ) " V, where , in the formal expansion by the multinomial theorem, Sy, Sp, &c. follow the law of involution, but the indices of dy, dp, &c. indicate repetition of the operation d,, dp , & c . upon V.

PROOF.

First,

V = ( dy d + dpd₂ + dqdq + ... ) V.

In finding 2V, each product, such as dy d, V, is differentiated again as a function of y, p, q, &c.; but, since the variations of dy, dp, &c. vanish by ( 2) , it is the same in effect as though dy, dp, &c. were not operated upon at all. They accordingly rank as algebraic quantities merely, and therefore

V = (dyd, + op d₂ + dqdq + ... ) ' V. Similarly for a third differentiation ; and so on.

IMMEDIATE INTEGRABILITY OF THE FUNCTION V.

3090

DEF.- When the function

V

(3028)

is integrable

without assigning the value of y in terms of a, and therefore

APPENDIX.

459

integrable whatever the form of the function y may be, it is said to be immediately integrable, or integrable per se. 3091 The requisite condition for V to be immediately integrable is that K = 0 shall be identically true.

―――― PROOF.

V de must be expressible in the form Sva (x₁ Y₁ P₁₁ ... ) — (XoYoPo Lo ….. ),

where is independent of the form of y. Hence, a change in the form of y, which leaves the values at the limits unaltered, will leave င် Vda = 0 ; 8 [" vda =

that is, (" Kòy = 0.

But the last equation necessitates K = 0 , since dy is arbitrary. And K = 0 must be identically true, otherwise it would determine y as a function of x.

DIFFERENTIAL

EQUATIONS .

GENERATION OF DIFFERENTIAL EQUATIONS .

3050 By differentiating ordinary algebraic equations , and eliminating constants or functions , differential equations are produced. Some methods are illustrated in the following examples .

3051 From an equation between two variables arbitrary constants , to eliminate the constants .

and

n

RULE. -Differentiate r times (r < n) , and from the r + 1 equations any r constants may be eliminated , and thus C ( n , r) different equations of the rth order (3060) obtained, involving d'y dr-ly, &c . dx "' dxr-1 '

Only r + 1 , however, of these equations will be

independent. By differentiating n times and eliminating the constants, a single final differential equation of the nth order free from constants may be obtained.

3052

Ex.

To eliminate the constants a and b from the equation y = ax² + bx

dy = 2ax + b Differentiating, we find dx Eliminating a and b in turn, we get dy x + bx = 2y, x dy = ax² +y dx dx

(i.) (ii.)

………………………..

(iii., iv.)

Now, differentiating (iii .) and eliminating b produces the final equation of the second order, dy 23 dy -2x (v.) + 2y = 0 ...... dx dx² The same equation is obtained by differentiating (iv.) and eliminating a. 3053 To eliminate the function & from the equation z = 4 (v) , where v is a function of x and y . We have

%x = p' (v) vx,

Therefore

Zy = p' (v) vy.

Zx Vy = Zy Vx·

GENERATION OF DIFFERENTIAL EQUATIONS.

3054 To eliminate Φ from u = tions of x , y , z .

461

(v) , where u and v are func

Consider x and y the independent variables , and differ entiate for each separately , thus

Ux + Uz%x =

' (v) (Vx + V₂ ²x) ,

Uy + Uzzy = p′ (v ) ( Vy + Vzŵy ) , and , by division, p' ( v) is eliminated .

3055

To eliminate

1,

2, ... P from the equation

F (x, y , z , p1 (a₁) , P₂ ( α₂) , ... Pn (an) } = 0 , where a₁ , a , ... a, are known functions of x, y, z . RULE .-Differentiate for x and y as independent variables , forming the derivatives of F of each order, up to the (2n - 1)th in every possible way ; that is, F ; Fr , Fy ; F22, Fry , Fay ; &c . There will be 2n² unknown functions , consisting of 91 , 92, ... Pn and their derivatives, and 2n² + n equations for eliminating them .

3056

To eliminate

E , 41 (E) ,

E) 2 (E) , ... P„ ( (E)

between the

equations F { x, y , z, E, p1 ( §) , P2 ( E) ... P„ ( E ) } = 0 , ƒ {x, y , z , E , p1 (E) , P2 ( E) ... Pn ( E ) } = 0 . RULE . -Consider z

and

functions of the independent

variables x, y , and form the derivatives of F and f up to the 2n - 1th order in the manner described in (3055) . There will be 4n² +n functions, and 4n2 +2n equations for eliminating them .

3057

To eliminate

from the equation

F {x, y , z , w ,

(a , ẞ) } = 0 ,

where a, ẞ are known functions of x, y, z, w. RULE . - Consider x, y, z the independent variables .

Dif

ferentiate for each, and eliminate p, pa, Ps between the four equations.

462

DIFFERENTIAL EQUATIONS.

DEFINITIONS

AND

RULES .

Ordinary differential equations involve the derivatives of a single independent variable .

3058

3059

Partial differential equations involve partial deriva

tives , and therefore two or more independent variables are concerned . 3060 The order of a differential equation is the order of the highest derivative which it contains. 3061 The degree of a differential equation is the power to which the highest derivative is raised . 3062 A Linear differential equation is one in which the derivatives are all involved in the first degree.

3063 The complete primitive of a differential equation is that equation between the primitive variables from which the differential equation may be obtained by the process of differ entiation .

3064

The general solution is the name given to the complete

primitive when it has been obtained by solving the given differential equation . Thus, reverting to the example in (3051 ) , equation (i .) is the complete primitive of (v.) which is obtained from (i.) by differentiation and elimination. The differential equation (v. ) being given, the process is reversed. Equations ( iii . ) and (iv. ) are called the first integrals of (v.) , and equation (i.) the final integral or general solution . 3065 A particular solution , or particular integral, of a differential equation is obtained by giving particular values to the arbitrary constants in the general solution. For the definition of a singular solution , see ( 3068) . 3066 To find when two differential equations of the first order have a common primitive . RULE . - Differentiate each equation, and eliminate its arbitrary constant. The two results will agree if there is a common primitive, which, in that case, will be found by elimi nating y between the given equations. Ex.-Apply the rule to equations (iii. ) and ( iv.) in (3052) .

463

SINGULAR SOLUTIONS.

3067 To find when two solutions of a differential equation, each involving an arbitrary constant, are equivalent. RULE.-Eliminate one of the variables. The other will also disappear, and a relation between the arbitrary constants will remain.

Otherwise, if VC, v = c be the two solutions : V and v being functions of the variables , and C and c constants ; then dV dv

dV

dv

dy

dy

dx

dx is the required condition .

PROOF.- V must be a function of v. and V, = ,, ; then eliminate 。.

Let V = p (v) ; therefore V₂ = $vVæ

Ex. -tan- (x +y) + tan ' (x - y) = a and 3 + 2bx = y3 + 1 are both solu tions of 2xy y = x +y + 1 . Eliminating y, z disappears, and the resulting equation is b tan a = 1 .

SINGULAR

SOLUTIONS .

3068 DEFINITION. " A singular solution of a differential equation is a relation between x and y which satisfies the equation by means of the values which it gives to the differ ential coefficients yr, y2r, & c. , but is not included in the com plete primitive."

3069

See examples (3132-3 ) .

To find a singular solution from the complete primitive

(x, y , c) = 0 . RULE I - From the complete primitive determine c as a function of x, by solving the equation y = 0, or else by solving x = 0, and substitute this value of c in the primitive. The result is a singular solution, unless it can also be obtained by giving to c a constant value in the primitive. 3070

Ifthe singular solution involves y only, it results from

the equation y = 0 only , and if it involves x only, it results from x = 0 only. If it involves both x and y, the two equa tions x = 0, y = 0 give the same result .

UN ut

464

3071

DIFFERENTIAL EQUATIONS.

When the primitive equation 4 (xyc) = 0 is a rational

integral function, 4.0 may be used instead of x = 0 or Yc = 0. PROOF.- Let

(x, y, c) = 0 be expressed in the form ... (1) .

y = f (x, c) .....

Then, if c be constant, (2 ) ; Yx =fx and, if c varies , (3). Yz = fz +fccz When cс is constant, the differential equation of which ( 1 ) is the primitive is satisfied by the value of y, in ( 2 ) . But it will also be satisfied by the same value of y, when c is variable, provided that either f. = 0 or f, = ∞ , and in either case a solution is obtained which is not the result of giving to ca constant value in the complete primitive ; that is, it is a singular solution . But fe = 0 is equivalent to ye = 0, and f∞ makes y, = ∞ , and therefore 2 constant.

GEOMETRICAL MEANING OF A SINGULAR SOLUTION. 3072

Since the process in Rule I. is identical with that

employed in finding the envelope of the series of curves obtained by varying the parameter c c in the equation (x, y, c) = 0 ; the singular solution so obtained is the equa tion of the envelope itself. An exception occurs when the envelope coincides with one of the curves of the system .

Ex .- Let the complete primitive be C y = cx + √1 − c³, therefore y = x— ; vi 3073

20

y = 0 gives c = √1 + x²

Substituting this in the primitive gives y = √1 + 2², a singular solution . It is the equation of the envelope of all the lines that are obtained by varying the parameter c in the primitive ; for it is the equation of a circle, and the primitive, by varying c, represents all lines which touch the circle. See also (3132-3) . 3074 "The determination of c as a function of a by the solution of the equation y = 0, is equivalent to determining what particular primitive has contact with the envelop at that point of the latter which corresponds to a given value of x. " The elimination of c between a primitive y = f(x, c) and the derived equation y = 0, does not necessarily lead to a singular solution in the sense above explained. "For it is possible that the derived equation y. = 0 may neither, on the one hand, enable us to determine c as a function of x, so leading to a singular solution ; nor, on the other hand, as an absolute constant, so leading to a particular primitive.

SINGULAR SOLUTIONS.

465

" Thus the particular primitive ye being given, the condition y = 0 gives e0, whence c is + if a be negative, and -∞o if a be positive. It is a dependent constant. The resulting solution y = 0 does not then represent an envelope of the curves of particular primitives, nor strictly one of those curves . It represents a curve formed of branches from two of them. It is most fitly characterised as a particular primitive marked by a singularity in the mode of its derivation from the complete primitive. " [ Boole's " Differential Equations," Supplement, p . 13. DETERMINATION OF A SINGULAR SOLUTION FROM THE DIFFERENTIAL EQUATION. 3075 RULE II .—Any relation is a singular solution which , while it satisfies the differential equation , either involves y and makes p, infinite, or involves x and makes () , infinite. (금)

3076 " One negative feature marks all the cases in which a solution involving y satisfies the condition p₁ = ∞ . It is, that the solution , while expressed by a single equation , is not connected with the complete primitive by a single and absolutely constant value of c. " The relation which makes p, infinite satisfies the differential equation only because it satisfies the condition y. = 0, and this implies a connexion between c and x, which is the ground of a real, though it may be unimportant, singularity in the solution itself. "In the first, or, as it might be termed, the envelope species of singular solutions, c receives an infinite number of different values connected with the value of by a law. In the second, it receives a finite number of values also connected with the values of a by a law. In the third species, it receives a finite number of values, determinate, but not connected with the values of x .” Hence the general inclusive definition 3077 "A singular solution of a differential equation of the first order is a solution the connexion of which with the com plete primitive does not consist in giving to c a single constant 99 value absolutely independent of the value of x.' [ Boole's " Differential Equations,” p. 163, and Supplement, p. 19.

RULES FOR DISCRIMINATING A SINGULAR SOLUTION OF THE ENVELOPE SPECIES .

is made infinite by , 1) ( equating to zero a factor having a negative index, the solution 66 may be considered to belong to the envelope species." 3078

RULE III.- When Py or

3079

" In other cases , the solution is deducible from the 30

466

DIFFERENTIAL EQUATIONS.

complete primitive by regarding c as a constant of multiple value , its particular values being either, 1st, dependent in some way on the value of x, or , 2ndly, independent of x , but still such as to render the property a singular one." [Boole's " Differential Equations," p. 164. 3080 RULE IV. —A solution which, while it makes p, infinite and satisfies the differential equation of the first order, does not satisfy all the higher differential equations obtained from it, is a singular solution of the envelope species .

m-1 Ex.: yx = my m has the singular solution y = 0 when m is > 1. m -r ገቢ Now Yrx = m (m− 1) ... (m − r + 1) y and, when r is > m , the value y = 0 makes y, infinite. therefore, by the rule of the envelope species.

The solution is,

3081 RULE V.-—" The proposed solution being represented by u = 0, let the differential equation, transformed by making u and x the variables , be u、 +f (x , u) = 0. Determine the in du tegral f as a function of x and u, in which U is either equal to f (x, u) or to f (x , u) deprived of any factor which neither vanishes nor becomes infinite when u = 0. If that integral tends to zero with u , the solution is singular " and of the envelope species. [ Boole, Supplement, p . 30.

3082 Ex.- To determine whether y = 0 is a singular solution or par ticular integral of Yx = y (logy) ³. 1 dy =Here uy, and Soy (logy) log y

As this tends to zero with y , the solution is singular. Verification .—The complete primitive is y = e- , and no constant value assigned to c will produce the result y = 0. 3083 Professor De Morgan has shown that any relation involving both x a and y, which satisfies the conditions p,, Pr = , will satisfy the differential equation when it does not make yr, as derived from it , infinite ; that it may satisfy it even if it makes y , infinite ; and that, if it does not satisfy the differential equation, the curve it represents is a locus of points of infinite curvature, usually cusps , in the curves of complete primitives . [ Boole, Supplement, p. 35.

FIRST ORDER LINEAR EQUATIONS.

FIRST

3084

467

ORDER LINEAR EQUATIONS .

M+ N dy = 0, dx

or

Mdx +Ndy = 0 ,

M and N being either functions of x and y or constants .

SOLUTION BY SEPARATION OF THE VARIABLES. 3085

This method of solution, when practicable, is the

simplest, and is frequently involved in other methods .

Ex.

xy (1 + x³) dy = ( 1 + y²) dx, dx ydy = therefore 1+ y² (1 +x²)' and each member can be at once integrated .

HOMOGENEOUS EQUATIONS . Here M and N, in (3084) , are homogeneous functions of x and y, and the solution is affected as follows :

3086

RULE.-Put y = vx, and therefore dy = vdx + xdv, and then separate the variables. For an example, see ( 3108) .

EXACT DIFFERENTIAL EQUATIONS . 3087

Mdx +Ndy = 0 is an exact differential when

My = N , and the solution is then obtained by the formula SMdx + S { N − d, (§ Mdx) } dy = C. PROOF. If V = 0 be the primitive, we must have V, = M, V, = N; therefore Vry = M, = N. Also V = [ Mdx + (y ) , (y ) being a constant with respect to x. Therefore N = V, = d, Mdx +4′ (y) ,

therefore

(y) = { { N ― d, [Mdx} dy + C.

3088 Ex. (x³ -3x²y ) dx + (y³ — x³) dy = 0. Here M, ==-3x² = N. Therefore the solution is 4 −xy + [ { y'- 2' - d, ( —'— x'y ) } dy C= =

— = =* 4 y * —x³y . 4 — x*y + { y* dy = x * +

DIFFERENTIAL EQUATIONS.

468

3089 Observe that , if Mda + Ndy can be separated into two parts, so that one of them is an exact differential, the other part must also be an exact differential in order that the whole may be such. 3090 Also , if a function of x and y can be expressed as the product of two factors, one of which is a function of the integral of the other, the original function is an exact differ ential.

x

1

3091

Ex.-

dx

COS y

y

x COS x dy = cos 2ydx - xdy = 0. y* y У

y

x is the integral of the second factor.

Here

Hence the solution is

y x sin

= C. y

INTEGRATING FACTOR FOR Mdx + Ndy = 0 . When this equation is not an exact differential , a factor which will make it such can be found in the following cases .

-

When one only of the functions Mx + Ny or Mx - Ny vanishes identically, the reciprocal of the other is an integrating factor. 3092

I.

3093 II.- If, when Mx + Ny = 0 identically, the equation is at the same time homogeneous, then x - ( +1 ) is also an in tegrating factor.

3094

III.-If neither Mx +Ny nor Mx - Ny vanishes identi 1 cally , then , when the equation is homogeneous , Mx + Ny is an integrating factor ; and when the equation can be put in the 1 is an integrating form (xy) xdy + x (xy) y dx = 0 , Mx - Ny

factor. PROOF. - I. and III.-From the identity

x M dæ + Ndy = } { (Mæ + Ny ) d log xy + (Mz – Ny) d log # } ,

469

FIRST ORDER LINEAR EQUATIONS.

assuming the integrating factor in each case, and deducing the required forms for M and N, employing (3090) . II.—Put v = 2, M = x" p (v), _N = x" ↓ (v) , x Mdx +Ndy and Mx + Ny.

3095 The general Mdx +Ndy = 0 is

form

S

με

for

an

and dy = xdv + vdx

integrating

factor

in

of

My-Nz dv, No - Muy

where v is some chosen function of x and y ; and the condition for the existence of an integrating factor under that hypothesis is that

3096

must be a function of v. M - Nx Nv -Mvy

PROOF. -The condition for an exact differential of Muda + Nudy = 0 is (Mµ)y = (Nµ) x (3087) . Assume µ = ( v) , and differentiate out ; we thus φυ = M - N obtain φυ Nux - Mvy

The following are cases of importance. 3097

I.- If an integrating factor is required which is a

function of a only, we put up (x) , that is , v = x ; and the necessary condition becomes

M -N must be a function of x only. N

3098 II.- If the integrating factor is to be a function of xy, the condition becomes, by putting xy = v ,

M - N must be a function of xy only. Ny -Mx

3099

III.- If the integrating factor is to be a function of

y 1, the condition is x

a² (N₂ - My) must be a function of Y. x Mx +Ny If Ma +Ny vanishes , (3092) must be resorted to.

DIFFERENTIAL EQUATIONS.

470

In this and similar cases , the expression found will be a

function of v =

if it takes the form F (v) when y is re

x placed by vx.

3100 IV. - Theorem. - The condition that the equation Mdx + Ndy = 0 may have a homogeneous function of a and y of the nth degree for an integrating factor, is

x² (N¸ - M₁ ) + nNx_ = F(u) , Mx +Ny 3101

where u = Y. x

The integrating factor will then be obtained from µ = xn√P(u) du

PROOF.

Put μ = v = x" ↓ ( u ) in (3097) , thus

= M - Nx Nvx- Mvy Perform the differentiations , and, by reduction , we get ย

' (u) = x² (N¸ − M₁) + nNx Mx + Ny ( u) The right member must be a function of u in order that by integration.

(u) may be found

3102 Ex. - To ascertain whether an integrating factor, which is a homo geneous function of x and y, exists for the equation Here

(y³ + axy³) dy — ay³ dx + (x + y) (x dy —y dx) = 0. M = -− ( ay³ + xy + y³ ) , N = (y³ + axy³ + xy + x³) .

Substituting in the formula of (3100) , we find that, by choosing n = : −3, the fraction reduces to axy -3xy , and, by putting y = ux, it becomes α -3u y¹ a function of u. a a - 3u du = -3 logu, 26 u³ ax ar +3 log 82) μ = x --3³e = y -³₂=4, the integrating factor required. It is homogeneous, and of the degree -3 in x and y, as is seen by expanding the second factor by (150) .

3103 If by means of the integrating factor μ the equation = µMdx + µNdy = 0 is found to have VC for its complete primitive , then the form for all other integrating factors will be uf (V) , where ƒ is any arbitrary function .

FIRST ORDER LINEAR EQUATIONS.

471

PROOF. -The equation becomes μMf (V) dx + µNƒ ( V) dy = 0 . Applying the test of integrability (3087) , we have {uMf ( V ) }, = {pNf (V )} . Differentiate out, remembering that (-M ), = (uN )x and the equality is established . 3104

V₁ = µM,

V = pN ,

GENERAL RULE . -Ascertain by the determination of an

integrating factor that an equation is solvable, and then seek to effect the solution in some more direct way.

SOME PARTICULAR EQUATIONS. 3105

(ax +by +c) dx + (a'x + b'y + c') dy = 0 .

This equation may be solved in three ways . I.-Substitute

x=

-a,

y = n - ß,

and determine a and ẞ so that the constant terms in the new and ʼn may vanish. equation in

II.

3106

Or substitute ax + by + c = E ,

a'x + by + c' = n .

But if a : a'b : b' , the methods I. and II . fail.

The

equation may then be written as a function of ax + by. Put zaxby, and substitute bdy = dz - a dx, and after wards separate the variables x and z .

3107

III.-A third method consists in assuming (An + C) d§ + (A´E + C′ ) dn

= 0,

and equating coefficients with the original equation after sub

stituting

E = x + m₁Y ,

n = x + m₂y.

m₁, m₂ are the roots of the quadratic am² + (b + a') m + b′ = 0 . The solution then takes the form

1 amı-a' { (am, —a') (x + m, y) + cm, —c' }

1 {(am₂ — a') (x +m₂y) +cm₂ — c' }

= C.

DIFFERENTIAL EQUATIONS.

472

3108

Ex.

(3y - 7x + 7) dx + ( 7y - 3x + 3) dy = 0 .

Puta, yn - ß, thus (3n — 75) de + ( 7n — 34) dn = 0 ......

(i.) ,

with equations for a and ß, 7a - 36 + 7 = 0 ; 3a − 7ẞ + 3 = 0 ; therefore ẞ := 0........... a = -1 ,

.... (ii. ) = v , and therefore dŋ = vd + dv (3086) ; 7v-3 dz dv = 0. + (7v³ — 7) ¿ dë + (7v − 3 ) dv = 0 , or 7v² -7 E

(i.) being homogeneous, put ..

The second member is integrated, as in ( 2080) , with b = 0, and, after reduction, we find Putting

5 log (n + ¿) +2 log (ʼn —¿) = C.

-1 and ny, by (ii. ) the complete solution is (y +x - 1 ) (y - x + 1 ) ² = C.

3109 When P and Q are functions of a only, the solution of the equation

dy + Py = 0 dx

is

dx e y= C Ce-SP

(i.)

by merely separating the variables .

3110

Secondly, the solution of

dy + Py = Q dx

is ' + √ QcSP.d² d.x' } . y = e¯SPd² { C

This result is obtained by the method of variation of parameters. RULE.-Assume equation (i . ) to be the form of the solution , considering the parameter C a function of x. Differentiate (i.) on this hypothesis, and put the value ofy, so obtained in the proposed equation to determine C. Pdr -Py , Thus, differentiating (i.) , we get y₂ Yx = Сe¯fraz¸ Cz Pdr Q = O₂e¯[Pdx therefore C therefore Qe = √ Qc√ml³ dx +0.

Then substitute this expression for C in equation (i .). Otherwise, writing the equation in the form (Py- Q) dx + dy = 0, the Pdx Par may be found by (3097) .

integrating factor

5 3111

Yx + Py = Qy"

is reduced to the last case by dividing by y" and substituting z = y! ".

473

FIRST ORDER LINEAR EQUATIONS.

0. P₁dx + P₂dy + Q (x dy —y dx) = :

*3212

P1 , P.2 being homogeneous functions of a and y of the ph degree, and Q homogeneous and of the qth degree , is solved by assuming P₁ =

" + (2 ),

2 P₁ =

"4 (2 ),

Q = rx (2).

Put y = va, and change the variables to a and v.

The result

may be reduced to da

x4 (v)

x (v ) x − p + 2

p (v) + vf (v)

p (v) + v4 (v) '

+ dv

which is identical in form with ( 3211 ) , and may be solved accordingly.

3213

(A₁ + В₁x + С₁y) (x dy — y dx) — ( A + B

+ C₂y) dy - (A +B + Csy) dx = 0. To solve this equation, put = & + a, y = n + ß, and determine a and B so that the coefficients may become homogeneous, and the form of ( 3212 ) will be obtained .

RICCATI'S EQUATION .

3214

ux + bu² = cxm ...... (A) . Substitute yue, and this equation is reduced to the

form of the following one, with n = m + 2 and a = 1. It is solvable whenever m (2t + 1 ) = - 4t, t being 0 or a positive integer.

3215

xy -ay + by² = can ..

(B).

I. This equation is solvable, when n = 2a , by substituting y = va", dividing by aa , and separating the variables . We

dv thus obtain

-xa -1dx. c - bv2

Integrating by ( 1937 ) or ( 1935 ) , according as b and c in equation (B) have the same or different signs , and eliminating by yvaa, we obtain the solution

3216

2x (be) Ce a +1 Xra y = √ 2x (be) √ 1-3 -1 Ce a

... (1 ) ,

* The preceding articles of this section are wrongly numbered. Each number and reference to it, up to this point, should be increased by 100. The sheets were printed off before the error was discovered. 3 P

474

DIFFERENTIAL EQUATIONS.

3217

or

3218 n - 2a

II. - Equation

……….. ( 2) . y = / a ( —bec) } ..... = √ (− 1)la^ an tan { C_a^ ✓

(B)

may also be solved

whenever

= t a positive integer. 2n RULE .-Write z for y in equation (B) , and nt +a for a in the second term , and transpose b and c if t be odd.

Thus, we shall have ca" (when t is even) ...... (3) , xz- (nt + a) z + bz ― Xzxx (nt + a) z + cz² = ban (when t is odd) ...... (4) . Either of these equations can be solved as in case (I. ) , when n - 2a t . z having been de 2n

n = 2 ( nt +a) , that is , when

termined by such a solution , the complete primitive of ( B) will be the continued fraction

a

Xn

Xon

y =

n+a b +

+

Xn

Xon 2n+a + ... + (t − 1) n +a + ≈ k

(5) ,

where k stands for b or c according as t is odd or even.

3219 n + 2a

III. - Equation (B) t a positive integer.

can also be

solved whenever

The method and result will be

2n the same as in Case II. , ifthe sign of a be changed throughout Thus and the first fraction omitted from the value of y. Xon Xin xn y = Xon 2n -a n- a (t − 1 ) n - a + ... + ( 6) . + + k 1 b C xn " and equate PROOF. -Case II.-In equation (B) , substitute y = A + Y1 a the absolute term to zero. This gives A = or 0. b Taking the first value, the transformed equation becomes x dy, − (n + a) y₁ + cy = = bxn. da xn n Next, put y₁ = 2+ a + " and so on. In this way the tth transformed C Y2 equation (3) or (4) is obtained with z written for the tth substituted variable yt.

FIRST ORDER NON-LINEAR EQUATIONS.

475

Case III.-Taking the second value, A = 0 , the first transformed equa tion differs from the above only in the sign of a ; and consequently the same series of subsequent transformations arises, with -a in the place of a. The successive substitutions produce (5) and ( 6) in the respective cases for the values of y.

3220

u₂+ u² = cx-1

Ex.

du = dx

Putting u = 2 x

(3214)

x²

and the equation is reduced to xy − y + y² = cx n + 2a a = 1 , b = 1 , n = 3, and = 2, Case III. 2n

of the form (B) .

Here

By the rule in (3218) ,

changing the sign of a for Case III., equation (3) becomes xz, -z +z² = cx³.

+

= Solving as in Case I., we put zvx , &c.; or, employing formula ( 1 ) directly, Ceb√cx + 1 23 ; and then, by (6), y = % = ✓cx+ zi 1 Ceb√cx³ — 1 3c Z is the final solution.

FIRST ORDER NON-LINEAR EQUATIONS .

3221

Type n n- 1 dy dy dy = 0 ………… .. ( 1 ) , + ...+ P n -1 dx +Pn = (d dx)" + P, ( d dx)

where the coefficients P1, P2, ... P, may be functions of a and y .

SOLUTION BY FACTORS.

3222

If ( 1 ) can be resolved into n equations , yx -P₁ = 0,

yx -P₂ = 0, ... Yx -Pn = 0 ...... (2) ,

and if the complete primitives of these are V₁ = C₁,

V₂2 = C2 , ...

Vn = Cn ....

. (3) ,

then the complete primitive of the original equation will be

( V₁ — c) ( V₂ — c) ... ( V₂ — c) = 0 .

. (4) .

476

DIFFERENTIAL EQUATIONS.

Differentiate and PROOF. - Taking n = 3, assume the last equation . eliminate c. The result is (5) . (V₂- Vs)² (Vs — V₁ ) ' ( V₁- V₁ ) ' dV₁dV₂dv₂ = 0 ............... ......... By (2) , dV₁ = (y - p₁) dx, &c., where μ, is an integrating factor . Sub stitute these values in ( 5 ) , rejecting the factors which do not contain differ ential coefficients, and the result is (Yz —P₁) (Yx —Pr) (Yr —Ps) = 0, which is the differential equation (1 ) .

Ex. - Given

3223

y² + 3y, +2 = 0.

The component equations are y , + 1 = 0 and y, + 2 = 0 , giving for the complete primitive (y + x − c) (y + 2x − c) = 0 .

SOLUTION WITHOUT RESOLVING INTO FACTORS.

3224

CLASS I.- Type

(x, p):= 0.

When a only is involved with p, and it is easier to solve the equation for a than for p , proceed as follows .

RULE.-Obtain x = f (p) . by means of dy = pdx. of the original equation.

Integrate and eliminate p by means

Similarly, when y = f (p ) ,

3225

Differentiate and eliminate dx

eliminate dy, &c.

Ex - Given a = ay, + by ,

….. ( 1 ) , i.e., x = ap + bp³ ............. therefore dy = pdx = apdp + 2bp³dp,

deadp + 2bpdp, therefore

y = ap² + 2bp +0. 3 2

Eliminating p between this equation and (1) , the result is the complete primitive (ax +6by - be) = (6ay -4x² - ac) (a² +4bx) .

3226

CLASS II .- Type x¢(p) +y+ (p) = x (p) .

RULE.-Differentiate and eliminate y if necessary.

Inte

grate and eliminate p by means of the original equation . If the equation be first divided by simplified into 3227

(p), the form is

y = xp (p) +x (p) .

Differentiate, and an equation is obtained of the form x₁₂ + Px = Q, where P and Q are functions of p . This may be solved by (3210) , and p afterwards eliminated .

FIRST ORDER NON-LINEAR EQUATIONS.

477

3228 Otherwise, a differential equation may be formed between Y and P, instead of between a and p. 3229 Or , more generally, a differential equation may be formed between a or y and t, any proposed function ofP, after which t must be eliminated to obtain the complete primitive .

3230

Clairaut's equation, which belongs to this class , is of

the form

y = px +f(p).

RULE.-Differentiate, and two equations are obtained— (1) ...... P

= 0, and .. p = c ;

( 2) ...... x +ƒ' (p ) = 0 .

Eliminate p from the original equation by means of ( 1 ), and The first elimination gives y = cx + f (c) ,

again by means of ( 2) .

the complete primitive.

The second gives a singular solution .

PROOF.- For, if Rule I. (3169 ) be applied to the primitive y = cx +f (c) , we have x +f' (c) = 0 ; and to eliminate c between these equations is the elimination directed above, c being merely written for p in the two equations.

3231

Ex. 1 .

y = px + x√1 +p².

This is of the form y = xp (p), and therefore falls under (3227) . entiating, we obtain

adp + da √1 +p²+

Differ

xpdp = 0, √ (1 +p³)

since dy = pda ; thus

1 dx p = O₁ √ (1 +p²) + 1+µ³) dp + x in which the variables are separated . Integrating by ( 1928) , and eliminating p, we find for the complete primitive +y² = C..

3232

Ex. 2.

y = p² + √b² — a²p³.

This is Clairaut's form (3230) . Differentiating, we have dp = 0. x {x_ dda . √ (b²-a²p³) The complete primitive is y = cx + √ (b² − a²c²) ; and the elimination of p by the other equation gives for the singular solution a²y² - b²x² = a²b², an hyperbola and the envelope of the lines obtained by varying c in the complete primitive , which is the equation of a tangent. Ex. 3.-To find a curve having the tangent intercepted between 3233 the coordinate axes of constant length.

478

DIFFERENTIAL EQUATIONS.

The differential equation which expresses this property is

or

Differentiating gives

y √1 +p² _ x √1 + p² = a , P ap y =px + √1 + p² a ap {" dp =0 x+ dx (1 +p²) #.}

. (1) . (2) .

Eliminating p between ( 1) and (2) gives, ac y = cx + 1st, the primitive √1 + c²

2nd, the singular solution

(3);

+y = a}

(4) .

(3) is the equation of a straight line ; (4) is the envelope of the lines. obtained by varying the parameter e in equation (3).

3234

CLASS III .- Homogeneous in x and y.

Type

RULE .

Xn 0. x* p (21/2, x p) = ( Put yvx, and divide by x" .

eliminate p by differentiating y = vx ;

Solve for p, and

or solve for v, and

eliminate v by putting v = Ꭹ ; and in either case separate the X variables.

3235

Ex.

y = px + x√1 +p².

Substitute y = væ, and therefore p = v + xv . This gives v = p + √1 + p³. Eliminate p between the last two equations, and then separate the variables. The result is

from which

dx 2v dv + = 0, æ 1 + v² x (v² + 1 ) = C

or

a² +y = Cx.

The same equation is solved in ( 3131 ) in another way.

SOLUTION BY DIFFERENTIATION . 3236

To solve an equation of the form

F {þ (x, y, yx) , 4 (x, y, yx)} = 0. RULE. -Equate the functions and respectively to arbi trary constants a and b. Differentiate each equation, and eliminate the constants. If the results agree, there is a common

I

HIGHER ORDER LINEAR EQUATIONS.

479

primitive (3166) , which may be found by eliminating y, between = the equations = a, b, and subsequently eliminating one of the constants by means of the relation F (a , b) = 0 .

Ex.

x- YYx +f(y² - y²y ) = 0.

Here the two equations

-yya, ƒ (y² - y¹ya) = b,

on applying the test, are found to have a common primitive. eliminating y , we obtain ƒ{y³- (x− a)³ } = b. a + b = 0.

Also, by the given equation, Hence the solution is

ƒ{y³— (x + b)² } = b .

HIGHER ORDER

3237

Type

Therefore,

LINEAR EQUATIONS .

dry dy + p, d"-¹y 1 + ... +P (n - 1) x - + Pny = Q , dan -1 dan dx

. of a or cons P . P" and Q are eithe funct tant e ₁ .. r ions s LEMMA.- If Y₁, y₁ , Y2, Y2 , ... Y Yn be n different values of y in terms of x, which satisf (3237 ) , when Q = 0 , the soluti y on in that

wher

case will be

y = C₁y₁ + C₂Y₂ + ... + Cnyn •

PROOF.-Substitute y₁, y , ... y, in turn in the given equation. Multiply the resulting equations by arbitrary constants, C1, C2, ... C₂ respectively ; add, and equate coefficients of P1, P2, ... P with those in the original equation.

LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS. 3238 3239

Ynx + a₁y (n - 1) x + ... + a (n - 1) yx + any = Q ......... ... …….. (1) . Case I.- When Q = 0.

The roots of the auxiliary equation m" + a₁m" -¹ + ... + a₂ - 1m + an = 0

(2)

being my, my, ... m₂, the complete primitive of the differential equation will be 3240

y = C₁e

+ С₂e² + ... + Сen

(3) .

If the auxiliary equation ( 2) has a pair of imaginary roots

DIFFERENTIAL EQUATIONS.

480

(aib) , there will be in the value of y the corresponding terms

...... ( 4 ) . ……………..

Aer cos bx + Beax sin ba

3241

If any real root m ' of equation (2) is repeated corresponding part of the value of y will be

3242

times , the

(A + A₁x + 4x² + ... + 4, -1x - 1) em'x .

And if a pair of imaginary roots occurs times, substitute for A and B in ( 3241 ) similar polynomials of the r — 1th degree in x. PROOF.- (i .) Substituting y = Cema in ( 1 ) as a particular solution, and dividing by Cem , the auxiliary equation is produced, the roots of which furnish n particular solutions, y = С₁emix, y = C₁em , &c., and therefore, by the preceding lemma, the general solution will be equation (2). (ii.) The imaginary roots a ± ib give rise to the terms Ceax + ibx + Ceax - ibr which, by the Exp. values ( 766 ) , reduce to (C + C) ex cos bx + i (C - C') ex sin bæ. (iii.) If there are two equal roots m, = m₁ , put at first m, m₁ + h . The two terms Ce + Ce(mh) become ema ( C₁ + Ce ) . Expand ehr by (150 ) , and put C₁ + Cg = A, C,h = B in the limit when h = 0, C₁ = ∞ , C₁ = —∞ By repeating this process, in the case of r equal roots, we arrive at the form (A + A₁ + Aga² + ... + Ar - 12 - ¹ ) ex ; and similarly in the case of repeated pairs of imaginary roots.

3243

Case II.— When Q in (3238 ) is a function of x.

First method.- By variation of parameters . Putting Q = 0 , as in Case I. , let the complete primitive be y = Aa + BB + Cy + &c . to n terms ............ (6) , a, ẞ, y being functions of a of the form em . The values of the parameters A, B, C, ... , when Q has its proper value assigned, are determined by the n equations 3244

Aa

+ Bxß

+ton terms

Axɑx

+ B&B.x

+

""

Ayax

+ BxB2x ...

+

99

***

Axɑ (n− 1) x + Bxß (n -1) x +

0, = 0, - 0,

... ""

Q,₂ = Q₁

A , B , &c . being found from these equations , their integrals must be substituted in (6) to form the complete primitive .

HIGHER ORDER LINEAR EQUATIONS.

481

PROOF.-Differentiate (6) on the hypothesis that A, B, C, &c. are func tions of x ; thus Yx = ( Aαx + Bẞx + ... ) + ( 4¸α + Bxß + ... ) . Now, in addition to equation ( 1) , n - 1 relations may be assumed between the n arbitrary parameters. Equate then the last term in brackets to zero, and differentiate y, in all, n - 1 times, equating to zero the second part of each differentiation ; thus we obtain

Yx

= Aux

+ Bẞx

Y2x ...

= Aɑ2x ...

+ Bẞ2x ...

+ &c. + &c. ...

Y(n-1) x = Aα (n- 1) x + Bẞ(n- 1) x + &c.

and Axα and Ara ... and

A

+ &c. = 0, + B&B + &c . = 0, + BxBx ... ... ... -2) x + Bß(n-2) x + &c . = 0 .

The n quantities A , B , &c. are now determined by the n - 1 equations on the right and equation ( 1 ) . For, differentiating the value of y(n- 1) , we have ynx = { Aanx + Bẞnx + &c . } + { Azα (n -1) x + Bxß(n - 1 ) x + &c. } , and if these values of yx, Y2x1 ... Ynx be substituted in ( 1) , it reduces to Axα( n - 1) x + Bxẞ(n -1) x + &c. = Q, for the other part vanishes by the hypothetical equation Ynx + α₁Y(n - 1) x + ... + any = 0 , since the values of yx, ... Y(n-1) , and the first part of yx, are the true values in this equation .

3245

Case II.— Second Method . - Differentiate and eliminate

Q. The resulting equation can be solved as in Case I. Being of a higher order, there will be additional constants which may be eliminated by substituting the result in the given equation.

3246

Ex.- Given

Y2x - 7yx + 12y = x ......

......... (1 ) .

1st Method.- Putting ≈ = 0, the auxiliary equation is m³ - 7m +12 = 0 ; therefore m3 and 4. Hence the complete primitive ofy2-7y + 12y = 0 is …………………… . (2) . .... y = Ae² + Be¹* ........

The corrected values of A and B for the primitive of equation (1) are found from 3.x + 1-3x+ α. Ae³+ В₂e = 0 ) .. A = -xe-3 and 4 = 9 " = S 34, e +4B, ex 4x +1 B₁ = xeand B = b.. e -4x+ * +b 16 Substituting these values of A and B in ( 2) , we find for its complete primi 12x +7 tive y = ae³x + be¹² + 144

3247

2nd Method.

Y₂ - 7y. + 12y = x ...... 3 Q

…………………. (1).

482

DIFFERENTIAL EQUATIONS.

Differentiating to eliminate the term on the right, we get 3r + 1242 = 0. Y₁x -713x The aux. equation is Therefore

m -7m³ + 12m³ = 0 ; therefore m = 4, 3, 0, 0. y = Ae¹ + Be³ + Cx + D ..... (2) ; 64e Y2x1 +9B . y = 4Ae + 3B + 0 ; 1 7 Substitute these values in ( 1 ) ; thus C = ; D= 12 144

therefore, substituting in (2) , y = Ae + Bee +

3248

When a particular integral

x 7 as before. + 144 12

of the linear equation

(3238) is known in the form y = f(x) , the complete primitive may be obtained by adding to y that value which it would take if Q were zero .

Thus, in Ex. ( 3247) , y =

x 7 + is a particular integral of ( 1 ) ; and 12 144

the complementary part Ae¹ + Be³ is the value of y when the dexter is zero.

3249 The order of the linear equation (3238 ) may always be depressed by unity if a particular integral of the same equation, when Q = 0, be known.

Thus, if ...... (1) , Y3x + P1Y2x + Payx + P3y = Q .......... .. and if y = z be a particular solution when Q = 0 ; let y = vz be the solution of (1 ) . Therefore, substituting in (1 ) , 1 (≈3x + P₁₂x + Pqx + P₁z ) v + &c. = Q, the unwritten terms containing v , V2x, and vår: The coefficient of v vanishes, by hypothesis ; therefore, if we put v₂ = u, we have an equation of the second order for determining u . u being found, v = Ju dx + C.

3250

The linear equation

(a + bx)"ynx + 4 (a + bx) " ¯¹y (n -1) w + B (a +bx )" − ²y (n - 2) x+….. ... + Ly = Q, where A, B, ... L are constants, and Q is a function of a, is solved by substituting a + bae , changing the variable by formula (1770) , and in the complete primitive putting t = log (a + bx) .

Otherwise , reduce to the form in (3446) by putting bx a + beλ , and solve as in that article.

HIGHER ORDER NON-LINEAR EQUATIONS.

483

HIGHER ORDER NON-LINEAR EQUATIONS .

3251

Type

F (x, y, dy, dy dx'

...

dry = 0. dxn:) =

SPECIAL FORMS. 3252

F(X, Yrx, y(r +1) x ... ··· Ynx) = 0 .

When the dependent variable y is absent, and yra is the derivative of lowest order present , the equation may be de pressed to the order n - r by putting yaz. If the equation in z can be solved, the complete primitive will then be y = = [ræ%+ +0 ræ

3253

(2149) .

F (y , Yrx, Y(r + 1) æ .. Ynx) = 0.

If a be absent instead of y, change the independent vari able from a to y, and proceed as before. Otherwise , change the independent variable to y , make p ( y ) the dependent variable.

For example, let the equation be of the form 3254 F (Y, Yx, Y2x, Y3x) = 0 .......

and

...... (1) .

(i. ) This may be changed into the form

F (ỵ ,

u

au su) = 0

by (1761 , '63, and '66) ;

and the order may then be depressed to the 2nd by ( 3252) . will thus give in terms of y.

The solution

3255 (ii. ) Otherwise, equation (1 ) may be changed at once into one of the form F (y, P , Py, P2y) = 0 , by ( 1764 and '67) , the order being here depressed from the 3rd to the 2nd . If the solution of this equation be p = (y, c₁, c₁) , then, since dy = pdx, we get, for the com dy x= plete primitive of (1 ) , + Cg. So(v ,c, c)

3256

Ynx = F (x).

Integrate n times successively , thus

y = { nx_F (c ) + C₂x® −¹ + C₂am − ² + ... +¤n •

DIFFERENTIAL EQUATIONS.

484

3257

Y2x = F (y) .

Multiply by 2y, and integrate, thus 2 dy dy √ √( 2 ( F (y) dy +c,Ì + C2. = f = 2 [F ( y) dy + c₁ ( dv) x = dæ dy + c₁ } {2

3258

Ynx = F {Y(n-1) x} , an equation between two successive derivatives . (n-1) x = , then z, Put y 7x = F (z) , from which Y(n−1) dz x = +c == = √F (z)

.... (1) .

If, after integrating , this equation can be solved for z so that z = p (x, c) , we have y(n - 1) x =

(x, c) , which falls under

(3256) . 3259 But if z cannot be expressed in terms of x, proceed as follows :

dz Y(n-2) x = = √zda = √√( ) =; F a dz dz ... z; = Y(n-3)x = F(z) ĮF (z) F dz dz dz ... z; 2) 2) √ F ( y = √ F( F (2)

Y(n -1) x = 2 ;

Finally,

the number of integrations and arbitrary constants introduced being n - 1 . 3260

Ynx = F {Y(n−2) x} • =

Put Y(n - 2) ; then 2x = F(z) , which is ( 3257) , the solution giving a in terms of z and two constants . If z can be found from this in terms of a and the two constants , we get

ឌ

or

Y(n-2) x = p (x, C1 , C2) ,

which may be solved by (3256) . 3261 But if z cannot be expressed in terms of a, proceed as in ( 3259 ) . DEPRESSION OF ORDER BY UNITY.

3262

When

F (x, y , Yx, Y2x , ... ) = 0

is rendered homogeneous by considering

X, Y, Yxı Y2x1 Ysx , &c .

HIGHER ORDER NON-LINEAR EQUATIONS.

485

to be of the respective dimensions 1 , 1 , 0, -1 , -2 , &c .; put x = eº,

y = ze° , &c .

The transformed equation will contain the same power of e in every term , and will reduce to the form F (z, Ze, Z20 , ... ) = 0 ,

the order of which is depressed by unity by putting zu.

3263

When the original equation is of the 2nd order, the

transformed equation in u and z may be obtained at once by changing x, y, yx, y2x into 1, z, u + z , u + uu,, respectively. The solution is then completed , as in example (3264) . PROOF.-We have

Ya = %₂ + ;

we ;

Y2xe

(%26 + % ) ;

y = ze" ;

Y3x = e- 20 (23 -z.) ;

and so on.

The dimensions of x , y , yx, &c . with respect to e, are 1, 1, 0, −1 , −2, &c . Therefore the same power of e' will occur in every term of the homogeneous equation.

3264

Ex.:

* 2 = (y− y ) .

Making the above substitutions for x, y , yx, and Y2x, the equation becomes 220 + 20 = -73. Put % = u ; thus du therefore u2 +1 = -Uz u³ + u = -— Uo = -uuz = -dz, u² + 1

therefore tan- lua - z ( 1935) , therefore de

cot (a - z) dz,

therefore

therefore

0 = log x

But

ba

cosec

u = tan (a — z) ,

therefore z,

and or

== log b sin (a - z ) (1941) .

2= y2 x be

sec

(a— 11 x ),

( 0+ 1 ),

by altering the arbitrary constant.

3265

When

F (x, y, yx, Y2x, ... ) = 0

is made homogeneous by considering x, y, ya, Y2x, &c . to be of the respective dimensions 1 , n , n - 1 , n - 2 , &c.; put

x = eº,

y = zene,

and depress the order by putting z, = u, as in ( 3262) .

3266

When the original equation is of the 2nd order , the

486

DIFFERENTIAL EQUATIONS.

final equation between u and z may be obtained at once by changing x, y, yx, Yax into 1 , ≈ , u + nz, uu₂ + (2n − 1 ) u + n (n − 1) ≈, respectively.

3267

...... (1 ).

2x 20 = y ......

Ex.:

With the view of applying (3265 ) , the assumed dimensions of each member of this equation , being equated, give therefore n = 4. n − 2 + n − 1 = 1 + n, Thus

e;

y = ze¹ ;

Yx = e³ ( z, + 4% ) ;

Y2x = e²º (≈29 + 7 %. + 12%) .

Substituting in ( 1 ), e disappears ; and by putting z. = u, z₂ = uu,, the equa tion is reduced to (u² +4uz) du + (7u² + 40uz + 48%² - z) dz = 0, which is linear and of the 1st order. by the rule in (3266) .

3268

When

This equation is also obtained at once

F (x, y, yx, Y2x, ... ) = 0

put y = Sudx " is homogeneous with respect to y, ya, yax, &c. , put y = and remove e as before by division. The equation between u and a will have its order less by unity than the order of F.

3269

Ex.:

***** (1) ,

Y2x +Pyx + Qy = 0

P and Q being functions of x.

Here

y = eSuax ;;

Y = uy ;

2 =

(ux + u³) y.

Substituting, the equation becomes u +u + Pu + Q = 0, an equation of the 1st order. If the solution gives up (a, c), then f (x, c) dx = log y is the complete primitive of (1).

EXACT DIFFERENTIAL EQUATIONS .

3270

Let

dU = 4 (x, y, Yx, Y2x1 ... Ynx) dx = 0

be an exact differential equation of the nth order. derivative involved will be of the 1st degree .

The highest

RULE FOR THE SOLUTION ( Sarrus) .— Integrate the term 3271 involving yx with respect to y(n - 1) x only, and call the result U₁. dÜ – dÜ₁ Find dU , considering both x and y as variables. will be an exact differential of the n - 1th order.

MISCELLANEOUS METHODS.

487

Integrate this with respect to y(n -2) x only, calling the result U2, and so on. The first integral of the proposed equation will be U = U₁ + U₂2 + ... + U₂ = C.

3272 Ex.: Let dU = {y² + ( 2xy − 1 ) yx + xу2x + x²Yзx} dx = 0 . Here U₁ = x²y2x, dU₁ = (2xу2x + x²узx) dx ; dU- dU₁ = {y² + ( 2xy - 1 ) yx - xу2x } dx = 0 ; .. U = -xy , du ,= − (y + z )d , du - du - du, = (y +2yy ) đư ; therefore U₁ = xy³ , and U = x²у2x − xyz + xy² = C is the first integral. Denoting equation (3270) by dU = Vda, the series of steps in Rule (3171 ) involve and amount to the single condi tion that the equation

3273

N- P₂ + Q2x - R3x + &c . = 0 , with the notation in ( 3028 ) , shall be identically true . This then is the condition that V shall be an exact 1st differential . 3274 Similarly, the condition that V shall be an exact 2nd differential is

P- 2Q₂ +3R2-4S3 2x 3x + & c . = 0 . 3275

The condition that V shall be an exact 3rd differential 3.4.5 S T3x + & c. = 0 , Q - 3R , +3.48 zx1.2 1.2.3

is

and so on .

[ Euler, Comm. Petrop. , Vol. viii.

MISCELLANEOUS

3276

METHODS.

= 0 Y2x + Pyx + Qys = 0 ......

(1) ,

where P and Q are functions of a only.

-2 The solution is y = ( 2ef √ Qc − 2 [ Pd®dx) -¹dæ. = [e¯SPdx √e-S· PROOF. —·Put e√ Pdz = z, and multiply ( 1 ) by z ; then, since z = Pz, zy, + Qzy = 0. Put zy, = u ', .. uu, = Qz¯², Q₂ -², .. u = √ (2√ Qz¯²dx), &c .

3277

Y2x + Qy + R = 0 ......

where Q and R are functions of y only.

(1),

488

DIFFERENTIAL EQUATIONS.

Qdy The solution is x = ·√√ (2 ) Re² dy) * dy. Qdy PROOF.-Put = z, and multiply (1 ) by z. = .. (zy,), Rz, ..zy, (zy,), .. (zy,)² = 2 ( Rz²dy, &c . Rz²y,, 3278

Y2x + Pyx + Qy" = 0,

where P, Q involve a only. % , and the form (3211 ) is arrived at.

Put y 3279

Y2x + Pyz + Qy" = 0 .

This reduces to the last case by changing the variable from æ to y by (1763) .

3280

For a few cases in which the equation Ya + Py² + Qy + R = 0

can be integrated , see De Morgan's " Differential and Integral Calculus ," p . 690.

3281

Y2x = ax + by.

Put ax +byt ( 1762-3) . or (3257) .

3282

Result t₂ = bt.

Solve by (3239)

(1 − a ) 2 − y + y = 0 .

Put sin¹æ = t , and obtain y2t + q³Y = 0 . Solution ,

y = A cos (q sin¯¹a) + B sin (q sin¹æ) .

3283

(1 + ax²) Y2x +axyx ±q³y = 0.

da

Put

= t, and obtain y

± qy = 0 as above.

√ (1 + ax²)

3284

Liouville's equation,

Suppress the last term . vary the parameter .

The complete primitive is

F(y)

Se↓ P(1) dy dy = 3285

Y2x +f(x) yx + F (y) y2 = 0 .

Obtain a first integral by (3209) , and

-Ĵf(x) dx A dx + B. Se

Jacobi's theorem. -If one of the first integrals of the

equation

Y2x =f(x, y)

is

y = $ (x, y, c) …………………… . (i., ii.) ,

489

APPROXIMATE SOLUTION.

the complete primitive will be

Sp. (dy—4dx) = c. PROOF.-Differentiating ( ii. ) , we obtain + = f (x, y) , and differen tiating this for c, re+ y = 0. But, by (3187), this is the condition xc + PcPyy + Pyc for ensuring that p.dy - p.pdx = 0 shall be an exact differential ; therefore 4. is an integrating factor for equation (ii. ) , y, - (x, y , c) = 0 .

Equations involving the arc s, having given

3286

ds² = dx² + dy²

3287

or

Sx = √1 + yå.

s = ax +by.

Here √1 + y = a + by .

Find y, from the quadratic equation .

3288

X28 = a.

Change from s to a by (1763) ; .. — 87³8 'S2x 2x = a, .' . 87² = 2ax + c , 1 or

1 +y =

2ax +c

.. y 1 = √√ (2ax²+ + ec −1 ) da + c' .

APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS BY TAYLOR'S THEOREM. 3289

The following example will illustrate the method :

Given 2

=

x ty,

.. Yзx = (x² + 2 ) Yx +ixy.

Generally, let ynx = Anyx + Bny Anyx + Bny ;;

Y(n x= = An + 1Yx + B₂ +1Y. Y (n ++1) 1) x

But, by differentiation ,

20 = (An ~ + An + Bn) Yx + ( An + B'₁) y , ∞ Y(n +1) x ·· An+ 1 = An ~ + A + Bพ

But

Ax, B = 1 , A₁ = x³ + 5x,

and Bn +1 = An + B'n •

.. A = x² + 2,

B¸ = x ;

B₁ = x² + 3 , &c .

Now, when xa, let y = b and y = p ; then, by Taylor's theorem ( 1500 ) ,

+ (Ÿµ + B3b) (x−a)³ y = a +p (x − a) + (A‚p + B¸b) (x —a)² 21 31

+ &c . , which converges when a - a is small .

3 R

[De Morgan, p. 692.

DIFFERENTIAL EQUATIONS.

490

SINGULAR

SOLUTIONS

OF

HIGHER

ORDER

EQUATIONS .

DERIVATION FROM THE COMPLETE PRIMITIVE .

3301

Let

Ynx = $ (x, y , Yx Y2x ... Y(n - 1) x)

……………...... (1 )

be the differential equation , and let its complete primitive be ...... ( 2 ) ,

s) ......... y = f (x, a, b, c ... 8) containing n arbitrary constants .

3302 RULE . - To find the general singular solution of ( 1) , eliminate abc ... s between the equations y = f,

yx = f、, y₁

and

.. Y2x = fgx ...9

....... (3) Y(n − 1) x = f(n− 1) x ...

fa fax

fa²x ... fa (n- 1) x

fb

fbx ... fb (n - 1) x

fbx

= 0 ...

... fsS

... fsx

(4) .

...

fex fs2x ... fs (n- 1) x

The result is a differential equation of the n - 1th order, and the integral of it, containing n - 1 arbitrary constants , is the singular solution . PROOF. -Differentiate (2) , considering the parameters a, b ... s variable, thus y = fx +faªx + ... +ƒ‚§x. Therefore, as in (3171 ) , y = fx if fa az +fo b₂ + ... fs 8 = 0, Y2x = f2x if faxªz +ƒvxbx + ... fsx 8x = 0, as well ; = and so on up to ynxfnx Eliminating a , b , ... s, between the n equations on the right, the determinant equation (4) is produced with the rows and columns interchanged .

3303

Ex.:

Y− x Y x + 3 x ² Y 2x — Y₁₂ — ( Yx — XY2x ) ² = 0 ………………………………….. ( 1 ) . ax² + bx + a² + b² ..... y= The complete primitive is · (2). 2 From which

Yx = ax + b ......

. (3),

and the determinant equation is |

1x² +2a, x =0 x +2b, 1 |

or 94+ 2bx = 2a

.............. (4) .

SINGULAR SOLUTIONS ÓF HIGHER ORDER EQUATIONS. 491

Eliminating a and b from (2) , ( 3) , and (4) , we get the differential equation 4dy + (2x + x ) da

= √ (1 +x²) dæ

(5) ,

✓ (16y +4x + x*) the integral of which, and the singular solution of ( 1 ) , is √ (16y + 4x² + x*) = x √ ( 1 + x²) + log { ~ + √ ( 1 + x² ) } + C. [Boole, Sup., p. 49.

6 3304

Either of the two

first integrals ' (3064) of a second

order differential equation leads to the same singular solution of that equation . 3305 The complete primitive of a singular first integral of a differential equation of the second order is itself a singular solution of that equation ; but a singular solution of a singular first integral is not generally a solution of the original equa tion. Thus the singular first integral ( 5) of equation (1 ) in the last example has the singular solution 16y +4 + x = 0, which is not a solution of equation (1 ) .

DERIVATION OF THE SINGULAR SOLUTION FROM THE DIFFERENTIAL EQUATION. 3306

RULE . -Assuming the same form (3173) , a singular

solution of the first order of a differential equation of the nth d (ynx) order will make infinite ; a singular solution of the d (y(n-1) x) d (ynx) d (ynx) second order will make both infinite ; d (y(n - 1) x) ' d (y(n − 2) x) and so on. [Boole, Sup., p . 51.

3307

Ex. -Taking the differential equation ( 3303) again, Y− x yz + } x²у 2x - 12- (Y₂ - XY2x ) ² = 0 ................ ( 1 ) ,

and differentiating for y, and y2 only, {}x² + 2x (Yx − XY2x) — 2Y2x } d (Y2x) − { x + 2 ( Yx − xy ∞ ) } d (Yx) = 0 . The condition d (Y2x) = ∞ requires d (y ) }x² + 2x ( Yx − xY2x) — 2Y2x = 0. Substituting the value of y2 obtained from this in equation (1 ) , and rejecting the factor (x + 1) , the same singular integral as before is produced (3303, equation 5 ).

DIFFERENTIAL EQUATIONS.

492

EQUATIONS WITH MORE THAN TWO VARIABLES .

3320

Pdx +Qdy + Rdz = 0

....

(1 ) .

P, Q, R being here functions of x, y , z, the condition that this equation may be an exact differential of a single complete primitive is 3321

P (Q. - R„) + Q (R¸ − P₂) + R (P, —Q₂) = 0 .

PROOF. - Let μ be an integrating factor of Pd + Qdy + Rdz = 0. Then -μPdx = μQdy + μRdz, and, by (3187) , for an exact differential, we must have (μQ), = ( R) . Write this symmetrically for P, Q, and R, differentiate out, and add the three equations after multiplying them respectively by P, Q, and R.

To find the single complete primitive of equation ( 1 ) . 3322

RULE. - Consider one of the variables z constant , and

Integrate, and add 4 (z) for the constant of therefore dz = 0. Then differentiate for x, y, and z, and compare integration. with the given equation (1 ) . If a primitive exists , 4 (z) will be determined in terms of z only by means ofpreceding equations. The complete primitive so obtained is the equation of a system of surfaces , all of the same species, varying in position according to the value assigned to the arbitrary constant .

3323

Ex.:

(x − 3y — z) dx + (2y − 3x ) dy + (z − x ) dz = 0 ……………….

. (1 ) .

Condition (3321 ) is satisfied ; therefore, putting dz = 0, we have (x− 3y — z) dx + (2y — 3x) dy = 0 . Applying (3187) , M, = -3 = N , and integration gives x² - 3xy —zx + y² + 9 (z) = 0 . Differentiating now for x, y, and z, (x −3y - z) dx + ( 2y −3x) dy + { ø' (z) − x } dz = 0. Equating coefficients with ( 1 ), p´ (z) = z, therefore (z) = Hence the single complete primitive is

z² + C.

x²+2y² + x² - 6xу - 2x = C, the equation of a system of surfaces obtained by varying the constant C.

3324 When the equation Pdx + Qdy + R dz = 0 is homo geneous , put x = uz , y = vZ . The result , when the coefficient of dz vanishes , is of the form 3325

Mdu +Ndv = 0,

EQUATIONS WITH MORE THAN TWO VARIABLES.

solvable by ( 3184) .

493

Otherwise it is of the form

dz = M du + Ndv,

3326

2 and the right will be an exact differential if a complete primi tive exists .

3327

yz dx +zxdy + xy dz = 0 ........... Ex.: Condition (3321 ) is satisfied . Putting x = uz, y = vz, dx = udz + zdu, dy = vdz + z dv, dv = 0, dz + du + (1 ) becomes 3v 3u ። and the solution is

log (zu¹v³) = 0 C

or

.. (1) .

xyz = 0.

When the equation Pdx + Qdy + Rdz = 0 ……………

... (1)

has no single primitive : 3328

RULE . -Assume

4 (x, y , z) = 0 ........ ………….

(2)

and differentiate ; thus

dx + , dy + ₂dz = 0

..... (3) .

The form of being given , eliminate z and dz from (1 ) by (2) and (3) . The result, being of theform

Mdx + Ndy = , 0 , can be integrated, and the solution taken with (2) constitutes a solution of equation ( 1) , and represents a system of lines (by varying the constant of integration) drawn on the surface (x, y, z) = 0 .

3329

Ex.:

(1 + 2m) x dx + ( 1 − x ) y dy + z dz = 0.

The condition (3321 ) not being satisfied, assume x² +y² + z² = ² as the function , therefore ada + ydy + zdz = 0 ; and by eliminating z and dz, 2mdx - ydy = 0, the integration of which gives y² - 4mx = C, a cylindrical surface intersecting the spherical surface in a system of curves (by varying C), whose projections on the plane of xy are parabolas.

The condition that 3330

Xd + Ydy + Zd ~ + Tdt = 0 ,

where X, Y, Z, T are functions of x, y , z, t, may be an exact

DIFFERENTIAL EQUATIONS.

494

differential, may be shewn , in a manner similar to that of (3321 ) , to be expressed by any three of the equations 3331

Y (Z, −T₂ ) + Z ( T, − Y₁ ) + T ( Y₂ - Z₁ ) = 0 , Z (T₂ - X ) + T ( X₂ − Z₂ ) + X ( Z₂ — T₂ ) = 0, T ( X ,−Y ) + X (Y − T, )+ Y (T − X ) = 0 , X ( Y₂ — Z, ) + Y( Z , −X₂ ) + Z ( X, − Y₂ ) = 0 ,

the fourth being always deducible from the other equations. If this condition is fulfilled , the solution of equation (3330) is analogous to ( 3322) . 2 and t were constant, and therefore dz and Integrate as if z dt zero, adding for the constant of integration

(z , t) . Differentiate next for all the variables , and determine Φ by comparison with the original equation . 3332

If a single primitive does not exist , the solution must

be expressed by simultaneous equations in a manner similar to that of (3328 ) .

SIMULTANEOUS

EQUATIONS

INDEPENDENT

WITH

ONE

VARIABLE .

GENERAL THEORY. 3340

Let the first of n equations between n + 1 variables be Pdx +P₁dy + P₂dz + ... + P₂dw = 0

...... (1 ) ,

where P, P₁ ... P may be functions of all the variables . Let a be the independent variable . The solution depends upon a single differential equation of the nth order between two variables . Solving the n equations for the ratios de : dy : dz : & c . , let

dz

dx

dy = dx Q

=

= dy Q₁ Q

dw =

Q₂ dz

= Q₂ da Q

=

" Qn dw

Qu

= da

Q

Differentiate the first of these equations n - 1 times , substi tuting from the others the values of %, ... w , and the result

SIMULTANEOUS EQUATIONS.

495

is n equations in Y , Yar ... Yar, and the primitive variables x, y, z ... w . Eliminate all the variables but x and y , and let the differ ential equation obtained be

Ynx) = 0. F (x, y , Yx ... Ynx) Find the n first integrals of this , each of the form F (x, y , Yx ... Y(n -1) x) = C, and substitute in them the values of Yes Y2 ... Yne, in terms of x, y, z ... w , found by solving the n Thus a system of n primitives equations last mentioned. is obtained, each of the form F (x, y , z ... w ) = C.

3341

The same in the case of three variables .

Heren

Let the given equations be

2.

1 P₁dx + Q₁dy + R₁dz = 0, P₂dx + Q₂dy + R₂dz = 0 . da 3342

=

Therefore Q₁R₂- Q₂R₁

From these let Therefore

x = 4 (X, Y , Z) ,

R₁P₂ - P₂R₁

=

dz P₁Q₂ — P₂Q1

Zx = 4 (x, y , z) .

Y2x = Þx + PyYx + PzZx•

Substitute the value of 2,, and (x, y , z) . yx = is the result.

dy

eliminate

by means

of

x9 yax An equation of the 2nd order in x, y , yx, Let the complete primitive of this be

Then we also have p (x, y , z ) = d¸x (x , a, b) . These two equations form the complete solution .

y = x (x, a , b ) .

FIRST ORDER LINEAR SIMULTANEOUS EQUATIONS WITH CONSTANT COEFFICIENTS . 3343 In equations of this class , the coefficients of the dependent variables are constants, but any function of the independent variable may exist in a separate term. Such equations may be solved by the method of ( 3340) ,

but more practically by indeterminate multipliers .

3344

Ex. (1 ) :

dx + 7x - y = 0, dt

dy + 2x + 5y = 0. dt

Multiply the second equation by m and add. The result may be written 5m -1 d (x + my) = 0 .... ........ ..... ( 1 ) . + (2m + 7) { x + 2m + 7 y } dt

DIFFERENTIAL EQUATIONS.

496

5m - 1 = m. 2m + 7

To make the whole expression an exact differential , put

m =

gives

-1+i 2

m' =

1 -i 2

This

****.. (2) ;

d (x + my) + (2m + 7) (x + my) = 0, dt and the solution is a + my = ce'- (2m + 7) and a + m'y = c'e (2m '+7) t

(1 ) now becomes

Solving these equations, and substituting the values (2) , iy = ce - (6 + i) t — c´e- (6 - i) t = e − 6t { ( c - c′ ) cos t ― i (c + c ) sin t } , e- 6t iz = e . " • { ( 2

++2 ) cost + (

or yet (C cost- C'sin t) ,

3345

Ex. 2 :

+ 2 ) sint } ,

x a = let { ( C + C′) cost + ( C − C′) sin t} .

x₁ + 5x + y = e²,

Y; + 3y - x = e²t.

Multiply the second equation by m, and add to the first 1 + 3m d (x + my) + (5 − m) { } x + - m y } = e' + me²t. dt 5 1 + 3m Put = m, thus determining two values of m, and put a + my = z ; thus 5- m Zt+ (5 ― m ) z = e +met . This is of the form (3210) . The equations of this example, written in the symmetrical form dx dy = of (3342) , would be = dt. et -5x - y eat + x - 3y NOTE.

3346

General solution by indeterminate multipliers. dy

dx =

Let

P

= P₂

dz Ps

be given with P₁ = a₁x + b₁y + c₁ z + d₁₂ dą, + d29 P₂ = ax + b₂y + CqZ z+ P3 = а3x + b3y + C3 z + dz. Assume a third variable t and indeterminate multipliers l, m, n such that dt - Ida + mdy + ndz Idx + mdy + ndz = IP₁ +mP₂2 + nPs t λ (lx + my + nz + r)

exact differential, and , to determine λ, l, m, n, r, λl, λm, λη , λr,

a₁ -λ b₁ C1

Az b₂-λ Ca

az ba Cz- λ 5

The last fraction is an we have a₁l + am + a¸ n = b₁l + b₂m + b₂ n = c₁ l + c₂ m + c₂ n = d₁l + d₂m + d¸n =

(1) .

= 0.

497

SIMULTANEOUS EQUATIONS.

The determinant is the eliminant of the first three equations in l, m, n. The roots of this cubic in A furnish three sets of values of l, m, n, r, which, being substituted in the integral of ( 1 ) , give rise to three equations involving three arbitrary constants ; thus, 1 + nï nz + r ) ^ c₂t = ((lx 2, x + my + qt = (hx + my + mz + r,)}, 1 Czt = (lzx + mzy +Ng + rz) ^s. Eliminating t, we find for the solution two equations involving two arbitrary constants . A similar solution may be obtained when there are more than three variables.

dx 3347

=

P₂-yP

P₁ -xP where

dz

dy =

To solve

Pax + by + c,

&c . ... (1), P- P

P₁ = α₁x + by + c₁ ,

&c.

Assume

p = a + bn + c5, p₁ = α₁ + b₁n + cs, &c. , de = dn = d and take ..... (2) , P Pa Pi = a , n = ys, and the solution of which is known by (3346) . Substitute become equations these xd + ¿da_yd + ¿dy = ydb + ¿dy = — d¸ P1 P P2 Zda = Zdy = dz and therefore P P1 -xp Pa-yp Dividing numerators and denominators by 4, the first equation in (1 ) is pro duced, and therefore its solution is obtained by changing , in the solution of (2) into x and y .

Certain simultaneous equations in which the coefficients are not constants may be solved by the method of multipliers . Thus , 3348

Ex . (1) :

x + P (ax + by) = Q,

Yt + P ( ax + b'y) = R,

P, Q, R being functions of t. Multiply the second equation by m, add, and determine m as in (3344) . The solution is obtained from (a +ma') -(a+ma") Pdt e ma")[rdt ( Q + mR) dt } , x + my = e (3210) 1 pa² 10+ √ with two values of m.

2 3349

Ex. (2) : ∞, +

(x − y ) = 1 ,

1 Ye + } (x + 5y ) = t

are equations solvable in a similar manner, and the results are

x +y =· 1/2 ( a + 12 + 1) 4

x + 2y = 1/1 ( ( C₂ + 2 .+ 16 ) . S+응 [Boole, p. 307. 3 s

498

DIFFERENTIAL EQUATIONS.

REDUCTION OF ORDER IN SIMULTANEOUS EQUATIONS. THEOREM.-n simultaneous equations of any orders 3350 between n dependent variables and 1 independent variable are reducible to a system of equations of the first order by sub stituting a new variable for every derivative except the highest . 3351

The number of equations and dependent variables in

the transformed system will be equal to the sum of the indices of order of the highest derivatives . This will , therefore, in general be the number of constants introduced in integrating those equations.

If, after integrating, all the new variables

be eliminated , there will remain n equations in the original These equations variables and the above-named constants . form the complete solution . In practice, such reduction is unnecessary. are methods frequently adopted :

The following

3352 RULE I.-Differentiate until by elimination of a vari able and its derivatives an equation of a higher order in one dependent variable only is obtained. 3353

RULE II .-Employ indeterminate multipliers.

3354

Ex. (1) :

xax + by,

Y2 = ax + b'y.

By Rule I., differentiating twice for t and eliminating y and yet, we obtain Xst (a + b ) x2 + (ab' - a'b) x = 0,

which may be solved by (3239) . Otherwise by Rule II . , exactly as in (3344) , we find - b = 0, a'm² + (a − b') m − differential exact the and for (x + my) 21 = (a + ma') (x + my) , the solution of which, by (3239) , is +ma') t + O₂e - √(a + ma ') t x + my = C₁eva +n

in duplicate with the two values of m. 3355

Ex. (2) :

x₂ - 2ay; + bx = 0,

Y₂ + 2ax¿ + by = 0.

Differentiate, and eliminate y, yt, Yat ; thus Xst +2 (2a² + b) x2 + b²x = 0, and solve by (3239) . Otherwise assume y = n cos at- sin at, a = cos at + ŋ sin at, and the given equations reduce to E2t = (a (a³ + b) 4, which are solved in (3257) .

Nat =

(a² + b) n,

[Boole, p. 311.

499

PARTIAL DIFFERENTIAL EQUATIONS.

3356 Ex. (3 ) .— Let u = 0, v = 0, w = 0 be three equations in x, y, z, t, involving derivatives of t up to a Yet 27t To obtain an equation between 2 and t. Differentiate each equation 6 +7 = 13 times, producing 3 + 13 × 3 = 42 equations involving derivatives of t up to 16t, Y19t9 201 Between these 42 equations eliminate y, yt, ... Y19t 24, 2t9 ... %20t, in all 41 quantities , and an equation of the 16th order in and t is the result. [ De Morgan.

3357

If a number of equations involve the quantities x , xat,

æst, &C . , Yt , Yst, Yst, &c . , all in the first degree , these quantities may be eliminated by assuming

x = L sinpt,

y = M cos pt.

3358 If there be n linear homogeneous equations in n vari ables x, y , z , ... and their derivatives of the 2nd order only, the equations may be solved by putting x = L sin pt,

3359 Ex.: Putting

y = M sin pt,

x2t = ax + by,

z = N sin pt,

&c.

Yat = gx +fy. y = M sinpt,

x = L sin pt, (a +p²) L + bM = 02 " gL+ (f+p²) M = 0S

..

a +p², b ₁ ƒ+p² | = 0, 99,

p and the ratios L: M are thus found.

L = -kb

Suppose then

and

x = -kb sinpt, and k and t are arbitrary constants.

PARTIAL

3380

M = k (p² + a) , y = k (p² + a) sin pt,

DIFFERENTIAL

EQUATIONS .

An equation is termed a general primitive or a com

plete primitive of a partial differential equation , according as the latter is obtained from it by eliminating arbitrary functions or arbitrary constants, as illustrated in (3150-7) .

LINEAR FIRST ORDER P. D. EQUATIONS . 3381 u=

To

form

the

P.

D.

equation from the primitive

(v) , where u and v are functions of x, y , z.

500

DIFFER ENTIAL EQUATIONS .

RULE.- Differentiate for x and y in turn , and eliminate (v) .

See (3054) .

Otherwise.-Differentiate the equations u = a , v = b ; thus udx + udy + udz = 0 ,

v、 dx + v¸dy + v₂dz = 0. dx

dy

dz

P = d (uv) &c. d (yz) '

where

=

Therefore

R'

P = 37:

Then the P. D. equation will be

Pz₂ + Qzy = R. PROOF. Since z is a function of x and y, z,dx + zdy = dz. dy = kQ, dz kR, therefore kPz₂ + kQzy = kR.

3382

But dx = kP,

Ex .

The general equation of a conical surface drawn through the y- b point (a, b, c) is y=b =c), x- α = 4 (z xa the form of being arbitrary. Considering as a function of two independent variables x and y , differ entiate for x and y in turn, and eliminate as in (3154) . The result is the partial differential equation (x− a) z₂ + (y — b) z, + z — c = 0 . 3383

To obtain the complete primitive ; that is , to solve

the P. D. equation ,

P%₂ + Qzy = R,

P, Q, R being either functions of x, y, z or constants .

RULE.-Solve the equations dx

dz =

P3 = 31/03 Let the two integrals obtained be

Ꭱ ua,

v = b;

. then

u = $ (v)

will be the complete primitive.

Propositions (3381 ) and (3383 ) extended to any number of variables . 3384

To form the partial differential equation

primitive

$ (u , v , ... w) = 0

....

from

the (1) ,

where u, v, ... w are n given functions of n independent vari ables x, y, ... z and one dependent t.

PARTIAL DIFFERENTIAL EQUATIONS.

501

RULE.-Differentiate for all the variables thus, Pu du + p,dv + ... + wdw = 0 ....

...... ( 2) .

Therefore, since p is arbitrary, du, dv ... dw must separately vanish, giving rise to the n equations

du = u, dx + u, dy + ... +u, dt = 0 , dv = v, dx + v, dy + ... +v, dt = 0 , ... ... ... ... dw = w,dx + w,dy + ... + w.dt = 0 .

Solving these for the ratios, by (583 ) , we get dx

P

dy = = Q

dz

dt

=

...

R

(3) ,

S

P, Q ... R, S being functions of the variables or else constants. Now , t being a function of all the rest,

t, dx + t, dy + ... + t₂dz = dt therefore, by (3) required is 3385

3386

... (4) ,

and (4) , the partial differential equation

Pt + Qty + ... + Rt₂ = S.

If u, v ... w be n functions of n variables , x, y ... t, the

condition of interdependence of the functions or existence of some relation expressed by equation ( 1 ) is J (u, v ... w) = 0 (see 1606) ; that is, the eliminant of equations (2) must vanish.

3387

tion

Conversely, to integrate the partial differential equa ..... Pt + Qt, + ... + Rt₂ = S …… ..

(1).

RULE .- Solve the system of ordinary equations

dx P =

dt

dz

dy

=

= & c. = Ꭱ

.....

(2) ,

S

and let the integrals obtained be ua, vb, ... w = k ; then (u, v, ... w) = 0 will be the complete primitive.

If P, Q ... R, S are linear functions of the variables , the integrals of equations (2) can always be found by the method of (3346) .

DIFFERENTIAL EQUATIONS.

502

NOTE . -Suppose , in equation ( 1 ) , that any coefficients P, Q vanish ; then, by (2) , dx = 0, dy = 0, and therefore the cor responding integrals are a a, y = b. The complete primi tive thus becomes

$ (x, y, u, v ... w):= 0.

3389 When only one independent variable occurs in the derivatives of the partial differential equation, the equation may be integrated as though the others were constant , adding functions of the remaining variables for the constants of integration .

3390

(Ex . 1) : dz = У dx √y² — x²

Integrating for ≈ as though y were con

stant, the complete primitive is

-1 x z = y sin + (y) . y Some equations are reducible to the above class by a transformation. Thus :

Put zu, #zy = 20² + y². Ex. (2) : therefore Wy therefore u = , = x²y + }y³ ++ (x), u = x² + y³, 3391

therefore

z = {x³y + }xy³ + [ ¢ (x) dx + 4 (y ) ,

or

z = } ( x³y + xy³) + x ( x) + ¥ (y) .

3392

Ex. (3) :

Solving by ( 3283) ,

(x − a) zx + (y — b) z₁ = c — z . dz dx dy = = • z- c y x-a

The integrals are y- b log (y -b)-log (x - a) = log O — c = C' , " or = C, 2- a x- a log ( c) -log (x - a) = log C'S y- b therefore = 9 (2 ) is the complete primitive. x- a For the converse process in respect of the same equation, see (3382) .

3393 Ex. (4) .— To find the surface which cuts orthogonally all the spheres whose equations (varying a) are x² + y² +z² — 2ax = 0 …………….. (1). Let Φ (x, y, z ) = 0 be the surface . Then (x− α) z + y

+ zz = 0

by the condition of normals at right angles. (1 ) , and divide by ; thus,

Substitute the value of a from

(x² - y³ —z³) z₂ + 2xyz₁ = 2zx.

PARTIAL DIFFERENTIAL EQUATIONS.

da By (3383), z² x² — y² —-22

503

dz = dy = 2zx' 2xy

dy dz = gives y = c for one integral. 22 y x Substituting y = cz, we then have dx - dz 2zx' x³ — ( c² + 1 ) z² which, being a homogeneous equation in x and z, may be solved by putting x² +y² +2² = C. Hence the com z = vx (3186 ) . The resulting integral is 2 x² + y² + z² = Y plete primitive is 2 z ) and the equation of the surface sought. »·((212

3394

Ex. (5).- To find an integrating factor of the equation (x³y — 2y¹) dx + (xy³ — 2x¹) dy = 0 .......

...... (1 ).

Assuming z for that factor, the condition (Mz), = ( N≈) , ( 3087) pro duces the P. D. equation (xy³ — 2x¹) zx + (2y¹ —x³y) z , = 9 ( x³ —y³) z..

. (2).

The system of ordinary equations (3283) is dz dx = = dy xу³ - 2x+ 2y¹ - x³y 9 (x³ — y³) z The first of these equations is identical with ( 1) (and such an agreement x y +12/23 = C. always occurs). Its integral is x² y²

Also

dz y dx +xdy = xy¹ — 2x¹y + 2xy¹ — x*y 9 (x³ - y³) z

3dx 3dy dz + + = 0; x 2 y = c. and thus the second integral is a³y³z

which reduces to

Hence the complete primitive and integrating factor is 1 X = 2= 2³ys Φ ( 1/2 y2 + 1/17).

Any linear P. D. equation may be written as a homogeneous equation with one additional variable ; thus , equation (3387) may be written

3395

Pu₂ + Qu₂ + ... + Ru = Su .

SIMULTANEOUS LINEAR FIRST ORDER P. D. EQUATIONS . 3396 PROP. I. - The solution of such equations may be made to depend upon a system of ordinary 1st order differential

504

DIFFERENTIAL EQUATIONS.

equations having a number of variables exceeding by more than one the number of equations .

Let there be n equations reduced to the homogeneous form (3395) involving one dependent variable P and n + m inde pendent . Select n of the latter, x, y ... z , and let the remaining m be , n .... From the n equations find P , P, ... P, in terms of P , P,η ... Ps, and arrange the results as under : P¸x + a₁Pƒ + b₁P„η ... + k₁1 P¸ = 0 )

P₁ + a₂ P¿ + b₂ P, P₂ท .... + k₂P₂ = 0 ... ...

(1) .

n Pz + ɑnP¿ + b„P,η ...• + k₁₂P₂ = 0) Multiply these equations by A1, A2, ... A, respectively, and add ; thus ,

y .. λ¿Ð¸ + λ½Ð¸ …... + λ„P₂ + E (λa) P§ + Σ (№b ) P„ ... + Σ (λk) P¿ = 0 . ( 2) . From this , as in (3387) , we have the auxiliary system

dx

dz

dy ... =

= λι

λε

λ ∙n

d dn = ... = Σ ( λα) - Σ (26 ) Σ ( λι) dk

=

(3) ,

and, by eliminating A1, A2 ... Ans de - ada - ady ... -a, dz = 0 = 0 0 dn - b₁dx - body ... ― b₁dz = ... ... ... -dɩ — k¸dæ — kądy ... — k„ dz = 0

... (4) .

Then, if u = a, vb, &c . be the integrals of (4) , they will be values of P satisfying the equivalent system ( 1 ) , and the integral of that system will be

F ( u , v, ... ) = 0 .

3397 PROP. II .— To integrate a system of linear 1st order P. D. equations. Let

▲' = ad¸„ + bd, ... + kd.,

so that AP = 0 represents a homogeneous linear P. D. equa tion of the 1st order.

RULE .

" Reduce the equations to the homogeneous form (1) ;

express the result symbolically by ΔΡ = 0 ,

ΔΡ = 0 ,

... AP = 0 ,

PARTIAL DIFFERENTIAL EQUATIONS.

505

and examine whether the condition

(A;A; —A;A;) P = 0 is identically satisfied for every pair of equations of the system. If it be so, the equations of the auxiliary system (Prop. I.) will be reducible to the form of exact differential equations, and their integrals being ua, v = b, w = c , ... , the complete value of P will be arbitrary.

F (u , v, w, ... ) ,

the form

of F being

" Ifthe condition be not identically satisfied, its application will give rise to one or more new partial differential equations. Combine any one of these with the previous reduced system , and again reduce in the same way. "With the new reduced system proceed as before, and continue this method of reduction and derivation until either a system of P. D. equations arises, between every two of which the above condition is identically satisfied, or, which is the only possible = 0 , ... appears. alternative, the system P, = 0 , P, In the Py former case, the system of ordinary equations corresponding to the final system of P. D. equations, will admit of reduction to the exact form, and the general value of P will emerge from their integrals as above.

In the latter case , the given system

can only be satisfied by supposing P a constant."

3398

" Ex.:

Pa + ( + xy + xz) P₂ + (y + z - 3x) P₂ = 0, P₁ + (xzt + y − xy) P₂ + (zt −y ) P = 0 .

Representing these in the form A‚P = 0, A‚P = 0, it will be found that --▲,4₁) P = 0 becomes, after rejecting an algebraic factor, P₂ + Pt = 0, (4,4, — and the three equations prepared in the manner explained in the Rule will be found to be Pa+ (3x² + t) P = 0,

P, + yP = 0,

No other equations are derivable from these. one final integral .

P + xP₂ = 0.

We conclude that there is but

"To obtain it, eliminate Pa, P , P. from the above system combined with Pada +Pdy + Pdz + P₁dt = 0, and equate to zero the coefficient of P, in the result.

We find

dz- (t + 3x²) dx - ydy - x dt = 0, - } = c. the integral of which is a- t− − "An arbitrary function of the first member of this equation is the general value of P." [ Boole, Sup., Ch. xxv. For Jacobi's researches in the same subject, see Crelle's Journal, Vol . lx. 3 T

506

DIFFERENTIAL EQUATIONS.

NON-LINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS .

3399

Type

F (x, y, z, Z、, Z ) = 0 ...

CHARPITS'S

(1) .

SOLUTION . - Writing p , q instead of z, and zy,

assume the equations

dz

dx = dy =

-Xp

dp

= q-Pqp

(2) . qx + pqz

Find a value of p from these by integration, and the corres ponding value of a from the given equation , and substitute in the equation

dz = pdx + qdy.

. (3) ,

and integrate by (3322) to obtain the final integral. PROOF. Since dz = pdx + qdy, we have, by the condition of integrability, Puq . Express P and q on the hypothesis that z is a function of x, y ; p a function of x, y, z ; q a function of x, y, z , p ; considering a constant when finding p , and y as constant when finding 4. Equating the values of p, and 1. so obtained, the result is the equation Ap + Bpy + Cp: = D, Iz +PYz• +Plz in which A, B, C, D stand for -1 , 1, q - PL» Qx Hence, to solve this equation, we have, by (3387) , the system of ordinary equations (2) .

3400 NOTE.- More than one value of p obtained from equations ( 2) may give rise to more than one complete primitive. The first two of equations (2 ) taken together involve equation (3) .

DERIVATION OF THE GENERAL PRIMITIVE AND SINGULAR SOLUTION FROM THE COMPLETE PRIMITIVE . RULE . Let the complete primitive of a P. D. equation of the 1st order be z = f (x, y , a, b) ….. 3401

. (1) .

The general primitive is obtained by eliminating a

between the form of

= f { x , y, a , 4 (a) }

and

fa = 0 ....

(2 ) ,

being specified at pleasure.

3402 The singular solution is obtained by eliminating a and b between the complete primitive and the equations fa = 0, f₁

f =0

(3) .

PARTIAL DIFFERENTIAL EQUATIONS.

PROOF.

507

By varying a and b in ( 1 ) ,

p = fx +faªx +fvbx, q = ƒy +faªy +frb„ • Therefore, reasoning as in ( 3171 ) , we must have faaz +fbx = 0 and faa, +fb,, = = 0 0

........ (3),

therefore either fa = 0, f = 0, leading to the singular solution ; or, elimi nating fa, fb, ab - a, b, = 0, and therefore, by (3167) , b = 4 (a). Multiply equations ( 3) by dx, dy re = (a) in this and spectively, and add, thus fada +fdb = 0. Substitute bp in (1) , and the equations ( 2) are the result.

SINGULAR SOLUTION DERIVED FROM THE DIFFERENTIAL EQUATION. 3403 RULE.-Eliminate P and q from the differential equa tion by means of the equations

Zp = 0,

zq = 0. Zq

PROOF. Let the D. E. be z = f(x, y , p, q) , and the C. P. z = F (x, y, a, b) . Now P and q being implicit functions of a and b, we have, from the first = ZapPa + Zqar

equation,

Z = Zp Pr + % q qv•

Hence the conditions z = 0, z = 0 in ( 3 ) involve, and are equivalent to, z = 0, zq = 0.

3404

All possible solutions of a P. D. equation of the 1st

order are represented by the complete primitive , the general primitive , and the singular solution. [Boole, p. 343.

3405 To connect any given solution with the complete primitive . Let F(x, y , a , b) be the complete primitive , and z=

(x, y) some other solution . Determine the values of a and b which satisfy the three

equations

F = 4,

F₂ Fx =

x,

Fy =

y•

If these values are constant, the solution is a particular case of the complete primitive ; if they are variable so that one is a function of the other , the solution is a particular case of the general primitive ; if they are variable and unconnected , the solution is a singular solution . 3406 COR.-Any two solutions springing from complete primitives are equivalent .

different

508

DIFFERENTIAL EQUATIONS.

3407

z = px + qy +pq ....... dz dx p A = dy == D

Ex.:

By (3299) ,

(1). (2) ,

A == &p = xy + z q = 2 = p²² ; (p + y) = ; p +y 2pz + yz - p²x ; D = qx +Pqz = -p -P + p = 0. +P C = q - Pl₂ = P +y (p +y)3 (p + y)³ dx = dy = (p + y) dz = Hence (2) becomes = dp ; xy + z 2pz +yz - p³x 2 - ax .. dp = 0, p = a ; .. q = Substituting in dz = pdx + qdy, a +y z- ax dz = a dx + dy .... a+y adx dy + = 0, By (3322) , making constant, z - ax a+y and we have

therefore

. (3) .

.....· (4) .

-log (z - ax) + log ( a + y ) = ¢ (z)

Differentiate for x, y, z, and equate with (3) , thus ' (z) = 0, therefore (z) constant (say -log b) ; therefore, by (4), z = ax + by + ab, the C. P. of (1) . 3408 To find a singular solution by (3402 ) , we must eliminate a and b between a == 0, z = 0 ; that is, a +b = 0 and y + a = 0, therefore z = −xy ― xy + xy = xy is the singular solution . To find the general primitive by ( 3401 ) , eliminate a between the two equations z = ax + (y + a) o ( a) and x + (y + a) q′ (a) + ¢ (a) = 0 .

NON- LINEAR FIRST ORDER P. D. EQUATIONS WITH THAN TWO INDEPENDENT VARIABLES .

MORE

3409 PROP.- To find the complete primitive of the differ ential equation ..... (1) , F (X1, X2 ... Xn, Z, P1, P2 ... Pn) = 0 ...

dz

dz where

P₁ =

>

dx

3410

9

P2 =

&c .

dx

RULE . -Form the linear P. D. equation in dF\ dø dF / do do Στ r {(dF + Prdz / dpr Ca + Pr 2, dz dp, dx,

the summation

extending from r = 1 to r = n.

auxiliary system (3387 ) n - 1 integrals P₁ = α19

Pμ- 1 = An -1 P₂2 = A2, ... On-1

denoted by

= 0,

From

the

509

SECOND ORDER P. D. EQUATIONS.

may be obtained . From these equations, together with ( 1 ) , find P1, P2 ... Pn in terms of X1 , X2 ... Xn, substitute the values in

dz - P₁dx₁ + p₂dx, ... + Pdxn and the integral of this last equation will furnish the solution required in the form f (X1 , X2 ... X , Z, a₁, a ... a ) = 0. [ Boole, Diff. Eq., Ch. xiv., and Sup., Ch . xxvii.

SECOND ORDER P. D. EQUATIONS .

3420

Type

F ( x, y , z , zx , Zy , Z2x , ≈xy , ≈2y) = 0 .

The derivatives Zx ,

y , Zax , Zxy , Zey are briefly denoted by P, q, r, s, t respectively. z being a function of the two independent variables ≈ and y, the following values are of frequent use 3421

dz = pdx + qdy ;

dp = rdx + sdy ;

dq = sdx +tdy.

If u be any function of x , y , and z , the complete deriva tives of u are indicated by brackets , thus 3422

(ux) = ux + puz

(Uy) = U₂ + qU₂•

A linear 2nd order P. D. equation is of the type 3423

Rr +Ss + Tt = V

..... (1 ) ,

in which R, S, T, V are functions of x, y, z , P, q.

PROPOSITION.-Any P. D. equation of the 2nd order which has a first integral of the form u = f( v) , where u and v involve x, y , z , p , q , is of the form

3424

= V Rr + Ss + Tt + U (rt — s²) = V.......

(2) ,

where R, S, T, U, V are functions of x, y , z , p , q, and

3425

U = UpVq — UqVp ....

(3) .

PROOF . -Differentiate u = f (v ) for x and y separately, considering x, y, z, P, q all involved in u and v, and eliminate ƒ' (v) . The result is equation ( 2 ) , with the values

510

3426

DIFFERENTIAL EQUATIONS .

-µR = up (vy) — ( Uy ) vp, µT = vq (Ux) — (Vx) Uqɔ µS = v₂ (u¸) — (v¸) Up − Vq ( Uy) + (Vg ) Uqs µV = ( " ,) ( vx) — (Uz) (Vy) , µU = Up Vq — UqVµ,

with the notation ( 3422 ) , µ being an undetermined constant. 3427

COR. - The

condition to be fulfilled in order that

equation (1) may have a first integral of the form u = f ( r ) is UpVq— UqVp = 0. SOLUTION BY MONGE'S METHOD OF 3428

Rr + Ss + Tt = V.

RULE . -Write the two equations Rdy -Sdxdy + Tdx² = 0 …………..

( 1) ,

... Rdpdy - Vdx dy + Tdqdx = = 0 0 ....

( 2) .

Resolve (1) into its factors, producing the two equations dy - m, dx = 0

and

dy - m, dx = 0 .

From dy = mdx and equation (2)

combined,

if necessary,

with dz = pdx + qdy, find two 1st integrals u = a , v = b ; then u = f(v) will be one 1st integral of the given equation. Similarly from dy = m, dx find another 1st integral . 3429

The final 2nd integral may be found from one of the

1st integrals by Lagrange's method (3383 ) . 3430

Otherwise, determine p and q in terms of æ, y , z from

the two 1st integrals ; substitute in dz = pda + qdy, and then integrate by (3322 ) to obtain the final integral.

If equation (1) is a perfect square, there will be only integral, and Lagrange's method only is applicable . 1st one

3431

By (3427) we may put u = mup , PROOF. dz = pdx + q dy ( 3321 ) in the complete derivatives

vq = my,; and also

(du) = u , dx + u , dy + u, dz + u, dp + u, dq = 0 , (dv) = &c . = 0 ; ... by (3422) (ux) dx + (u„) dy + u, (dp + mdq) = 0 } (v.) dx + (v ) dy + v, (dp + m dq) = 0 )

(3) .

Solving these equations for the ratios de : dy : dp + mdq, we obtain at once dx dy +mdx mdy dp + m dq = = = ********* (4) , R T V S

with the values of R, S, T, V in (3426) .

SECOND ORDER P. D. EQUATIONS.

511

Equations (1 ) and ( 2) are the result of eliminating m from (4). These two equations with dz = pda + qdy suffice to determine a first integral of (3428) when it exists in the form u = ƒ (v) .

3432

Ex. (i .) : q ( 1 + q) r− (p + q +2pq) s +p ( 1 +p) t = 0 .

Solving the quadratic equation ( 1 ) , we find pdx + qdy = 0, or (1 + p) da + ( 1 + q) dy = 0 ...... …………………. ( 5 ) . dz = pdx + qdy = 0, .. z = A. First, Monge's equation ( 2 ) is q ( 1 + q) dp dy +p ( 1 +p) dq dx = 0, which, by pdx = -qdy, gives

Hence a first integral is

dp = dq ; and, integrating , 1 +2 = B. 1 +9 1 +q 1 +P

1+ P = 4 (2) ..... 1+q

.... (6) .

Next, taking the second equation of (5 ) with ..x + y + z = C. dz = pdx + qdy, dx + dy + dz = 0, Also, by (5 ) , equation (2) now reduces to gdp = pdq, and by integration , p = qD; therefore the other first integral is p q¥ (x + y + z ). For the final integral integrate p − q4 = 0 ; i.e., 2x - 42, = 0, by (3383 ) ; da + dy + dz dy = dz .. dx = ―――― .' . z = A , and da = 0' 1 −4 (x + y + z ) ` ¥ (x + y + z )

d (x + y + z) = F (x + y + 2) + B. S1−4 (x + y + z) Hence the second integral is a -ƒ ( x + y + z) = F (≈) . .. x =

3433

Ex . (ii.) :

≈2.x — a²22y = 0.

(i.) Here, in (3428) , R = 1, 80, T - a , V = 0 ; therefore ( 1 ) and (2) become dy³ — a³dx² = 0, dpdy - a'dqdx = 0 . From (1 ) dy +ade = 0, giving y + ax = c, and converting (2) into dp +ad0, which gives p +aq = c' ; therefore a first integral is (3) . p + aq = 4 (y + ax) ... Similarly, from ( 1 ) , dy - a da = 0 gives rise to another first integral ..... (4) . p -aq = (y - ax) Eliminating p and q by means of ( 3) and (4) from dz = pda + qdy, we find dz = (2a) -1 -¹ {p (y + ax) (dy + a dx) −4 (y — ax) ( dy — a dx) } , therefore, by integrating, z = ( y + ax) + ¥ (y − ax ) . For the symbolic solution of the same equation, see (3566) .

SOLUTION OF THE P. D. EQUATION.

3434

Rr + Ss + Tt + U (rt — s²) = V ................ (1) .

Let m₁, m, be the roots of the quadratic equation

3435

m²- Sm + RT+ UV = 0

(2) .

DIFFERENTIAL EQUATIONS.

512

= Let u₁ = a , v₁ = b, and u₂ = a' , v, b' be respectively the solutions of the two systems of ordinary differential equations . 3436

Udpm₂dy - Tdx Udq = m₁ dx - Rdy

(3) ,

dx Udpm, dy - T da Udq = madx - Rdy

dz = p dx + q dy

(4) .

dz = p dx + qdy.

Then the first integrals of ( 1 ) will be = f (u ),

g = f(" ).

To obtain a second integral : 3437 1st. ― When m₁ , m, are unequal, assign any particular forms to fi and f₂, then substitute the values of p and q, found from these equations in terms of x a and y , in dz = pdx + qdy, which integrate.

Otherwise , assign the form of one only of

the functions fi, f2, involving an arbitrary constant C, solve for p and q, and integrate dz = pdx + qdy, adding an arbitrary function of C for the constant of integration.

3438

-2ndly. When m₁, m, are equal, and therefore , by (2) , S¹2 = 4 (RT + UV) ...... (5) .

Equations reduce to

(3)

3439

and

(4)

coincide ,

and ,

Udp8dy - Tdæ ...... S dx - Rdy Udq = 48 dz = p dx + q dy ....

since

m = 1S,

. (6) , (7) ,

(8) . Here Py P₁ = 4 , and therefore the last equation is integrable if the values of p and q, obtained by integrating (6) and ( 7) , be substitut in it . Let u = a , v = b be the integrals of (6) ed

and (7) ; and let

z = p (x, y , a, b, c)

(9)

be the integral obtained from ( 8) . The general integral is found by making the parameters a, b, c vary subject to two conditions bf(a) , c = F (a) ; that is , by differentiating

z = 。 {x, y, a, f(a) , F(a) }

for a, and eliminating a. 3440 The general integral therefore represents the envelope of the surface whose equation is (9) .

513

SECOND ORDER P. D. EQUATIONS.

PROOF. (Boole, Sup., p. 147. ) Assuming a 1st integral of the form u = ƒ (v ) , eliminate u and v from equations (3126 ) by multiplying (i. ) by (u ) uq, (ii.) by ( u ) up, ( iv. ) by ( ux) (uµ) , ( v. ) by upuq, and adding. Again, eliminate μ and v by multiplying (i .) by (u ) ², (ii . ) by ( u„) ², ( iii . ) by (ux) (u,) , (v.) by (ux) u₂ + ( u„) uq , and adding.

The two resulting equations are

? q=0 R ( ux) uq + T ( u„ ) up− U ( u¸) ( u„ ) + Vup W

...... .. ( 10) . R (ux)² + S ( ux) ( U„ ) + T ( u„ ) ³ + V { ( ux) u₂ + ( uz ) uq} = o 0} Multiply the 2nd of these by m, divide by V, and add to the 1st equation ; the result is expressible in two factors either as ( 11 ) or ( 12) , {R (u,) + m, ( u, ) + Vu„ } { m, ( u₂) + T (uy) + Vuq } = 0 ......... ( 11 ) , {R (uz) + m₂ (u₁ ) + Vu, } { m, ( u₂) + T ( u , ) + Vu , } = 0 .....……….. ( 12) , m , m. being the roots of the quadratic (2) . By equating to zero one factor of ( 11 ) and one of ( 12) , we have four systems of two linear 1st order P. D. equations. Taking each system in turn with the equations

(u₂) + ru₂ + suq = 0, (u ) + su + tuq = 0,

V 0 2 8

10

R M1 and eliminating (u,) , ( u ) , up, u , we have the de terminant annexed for the case in which the 1st ma T factor of ( 11 ) and the 2nd of ( 12) are equated to 1 0 zero. In this case, and also when the 2nd factor 0 1 of ( 11 ) and the 1st of ( 12 ) are chosen, trans posing m₁, m, in the determinant, the eliminant is equivalent to

= 0, S t

V { Rr + Ss + Tt + U (rt — s²) — V } = 0, having regard to the values of mm, and m₁ + m, from (2). When the 1st factor of both ( 11 ) and ( 12) is taken , the 2nd order P. D. equation produced by the elimination is Vt - R (rt - 82) = 0, and when the 2nd factor of each is taken, the elimination produces Vr- T (rt - s³) = 0. Hence the hypothesis of a 1st integral of ( 1 ) , of the form u = ƒ (v) , involves the satisfying one or other of the systems of two simultaneous equa tions, ( 13 ) or ( 14) , below : R (u,) + m (u¸) + Vu₂ = 0 } m₂ (u₂) + T (u₂) + √u₂ = 0 } ... (13).

= 0} R (u ) + m₂ ( u ) + Vu„ = = 0 } ... (14). m₁ (u₂) + T ( u₂ ) + Vuq

Now multiply the 2nd equation of ( 13 ) by A and add it to the 1st . In the result, collect the coefficients of ur, uy, Up, Uq Uz• The Lagrangean system of auxiliary equations (3387 ) will then be found to be

dx =

R + ama

du dz dy -dp = dq = = my + XT V XV Rp + m ,q + λ (Tq + m,r) 0

Eliminating A, equations ( 3) are produced. the same way, equations (4) are produced. 3 U

Treating equations ( 14) ir

514

DIFFERENTIAL EQUATIONS.

POISSON'S EQUATION. 3441

P = (rt- s²)" Q,

where P is a function of p, q, r, s , t , and homogeneous in r , s, t ; Q is a function of x, y, z , and derivatives of z , which does not become infinite when rt - s2 vanishes , and n is positive. RULE .- Assume q = (p ) and express s and t in terms of p and r ; thus, rt - s vanishes, and the left side becomes a func tion of p, q, and qp. Solve for a 1st integral in terms ofp and q, and integrate again for the final solution.

8 = qx = YpPx = 9p² ;

PROOF.-

therefore rt -s² = 0.

t = qy = LpPy = q; r ; Also P is of the form (r, s, t)" = ™ (1 , Ip, 93) ™.

Hence the equation takes the form ( 1 , qp, q )" = 0. LAPLACE'S REDUCTION OF THE EQUATION.

3442

Rr + Ss + Tt + Pp + Qq + Zz = U ............ ………………………….. ( 1 ) ,

where R, S, T, P, Q, Z, U are functions of x and y only. Let two integrals of Monge's equation (3428) Rdy -Sdady + Tda² = 0 be

p (x, y) = a,

4 (x, y) = b.

Assume

E = p (x, y) ,

n = 4 (x, y) .

3443 To change the variables in equation ( 1) to ₹ and ŋ, we have

P =" = 7x =" P

t + " ;

¶ = %y = %¢ ¾y + %‚ ¹y ; ( 1701 )

1 * = 72x = %28 ++ 2 # §n Exnx + %2n nå +44 52x + %n Nex ; t = % 2y = %25 53 + 2 %¢n Ey¹y + Z2n ng + Z§ 52 +2 , 12y ;

8 = %xy = 72€ 5x5y + %±n NxNy + Zên ( Ex¹y + Eyŋx ) + Z§§xy + %ŋŋxy • The transformed equation is of the form

..... (2) ,

Ꮴ ...... Zεn + Lizε + M2₂ + Nz = V where L, M, N, V are functions of may be written in the form

and n.

This equation

(dg + M) (d„ + L) z + (N − LM — L¿ ) z = V …………….

If

(4) ,

N- LM - L¿ ' = 0 we shall have (d + M)

= V

with

(

+

. (3) .

)

= %,

LAW OF RECIPROCITY.

515

and the solution by a double application of ( 3210 ) is obtained from

2 = e -SLan

%' { Þ ( 5) + √ ½ ' e√Lan an } ,

z = e-SMAE { 4 (n) + [ VeS Ma£ JE} . By symmetry, equation (1) is also solvable , if

N- LM- M = 0

(5) .

But if neither of these conditions is found to hold, find z in terms of

from (3) .

It will be of the form

2 = A¿ + Bz' + C, Substitute this for z in and n . where A, B, C contain . (d, + L) z = z' , and the result is of the form

' %'s Zen + L # + M' %', + N'z' = V ' . The same conditions of integrability, if fulfilled for this equa tion, will lead to a solution of ( 1 ) , and , if not fulfilled , the transformation may be repeated until one of the equations , similar to (4) or ( 5) , is satisfied .

3444

COR . - The solution of the equation

Zen + azε + bz η + abz = V is

2:

' b e¯an ø ( 5) + e− ¹³ ↓ (n ) + e - an-of

ean +be Vdn de. Sse

3445 For the solution of equation (2 ) , when L, M, V contain also z, see Prof. Tanner, Proc. Lond. Math. Soc., Vol. viii. , p . 159 .

LAW OF RECIPROCITY .

3446

[ Boole, ch. xv.

Let a differential equation of the 1st order be (x, y, z, p , q) = 0

...... ( 1 ) .

Let the result of interchanging ≈ and p , y and q , and of changing ≈ into px + qy - z, be (p , q, px + qy — z, x, y) = 0 ............ then, if z

. (2) ;

(x, y) be the solution of either ( 1 ) or ( 2) , the

516

DIFFERENTIAL EQUATIONS.

solution of the other will be obtained by eliminating

and n

between the equations x = d;4 (§, n) , _y = d¸ ¥ (§, n) , 3447

Ex - Let

z = pq ******· ( 1) ,

x= = §x +ny— ¥ (§, n) . px + qy - z = zy ...... (2) ,

be the two reciprocal equations.

The integral of ( 2) is

z = xy + xj ( 2 ) , •• 4 (En) = En + xƒ ( 27)⋅

E, ŋ η have now to be eliminated between η x= n y = £ +ƒ' (7 ) , = " - } ƒ ( 2 ) + ƒ ( 2 ),

z = En ...... (3 ) .

Each form assigned to ƒ gives a particular integral of (1 ) . If ƒ ( 2 ) = a 2 + b, z = En ,

the equations (3) become an + b, y = & + a, and the elimination produces z = (x − b) (y — a) .

3448 In an equation of the 2nd order, the reciprocal equa tion is formed by making the changes in (3446) , and , in addition, changing - S t r r into s into t into ; rt rt-s22 rt - s2 then, if the 2nd integral of either equation be z = ↓ (x, y ) , that of the other will be found by the same rule.

3449

The above transformation makes any equation of the

form $ (p, q) r + 4 (p, q) s + x (p , q) t = 0 dependent for solution upon one of the form

x (x, y) r−¥ (x, y) s + ¢ (x, y) t = 0 .

3450

And, in the same way, an equation of the form Rr+ Ss + Tt = V (rt — s²)

is dependent for solution upon one of the form Rr + Ss + Tt = V. See De Morgan, Camb. Phil. Trans ., Vol. VIII.

SYMBOLIC METHODS .

FUNDAMENTAL FORMULÆ.

Q denoting a function of 0, 3470

(d. — m) -¹Q = emeSe -me Qdo .

SYMBOLIC METHODS.

517

PROOF. The right is the value of y in the solution of day - my = Q by (3210) . But this equation is expressed symbolically by (d. - m ) y = Q (see 1492 ) , therefore y = ( d. — m ) ¹Q. = Hence Let xe , therefore dead, and ad0dx. (3470) may be written

3471

(xd¸ — m) ¯¹ Q = x™ Sæ¬m-¹ Qdx.

3472

COR.

3473

or

(do - m) -¹0 = Ceme, (xd - m)-10 = Cxm.

Let F (m) denote a rational integral function of m ; then , 2 20 eme = m²eme, & c. , the operation d is since demememe , d2 Hence , in all cases , always replaced by the operation mx .

3474

F(do) eme = eme F (m).

3475

F (de) e¹º Q = emo F (d¸ + m) Q.

Formula ( 2161 ) is a particular case of this theorem . 3476

eme F(d.) Q = F ( d. — m) em² Q.

Also, by (3474-6) , 3477

F (m) = e -me F (da) eme.

3478

F (d₂ + m) Q = e -m° F (d.) emº Q.

3479

F (d.) Q = e−m² e-m° F (d. — m) e¹º Q.

To the last six formulæ correspond

3481

m F (xdx ) x¹ = x¹ F (m) . m F (xd¸) xQ = x™ F (xd¸ + m) Q.

3482

xm x "F (xd ) Q = F (xd, —m) x™ Q.

3483 3484

F (m) = x¯ ™ F (xdx) x™. -m F (xd, + m ) Q = x¯¹F (xd¸) x™ Q.

3485

F (xd¸) Q = x¯ ™F (xd¸ — m) x™ Q .

3480

If U = a + bx + cx² + &c . , then, by ( 3480 ) , 3486

F(xd ) U = F (0) a + F ( 1 ) bx + F ( 2) cx² + &c .

518

DIFFERENTIAL EQUATIONS.

3487 3488

F-1 (xd ) U = F-¹ (0 ) a + F -¹ ( 1 ) bx +F -¹ ( 2) cx² +&c . m z F(xd , yd¸, zd., ... ) x™y" ≈” ... = F (m, n , p ... ) x™y”≈º …

3489

x" unx = de ( d. — 1 ) ( d. − 2) ... ( d. − n + 1 ) u ,

or, more succinctly, writing D for de, D (D - 1 ) ... (D − n + 1) u 3490

Otherwise

or

Du

( 2452 ) .

" unx = xd (xd — 1 ) ... (xd¸ - n + 1 ) u .

PROOF. As in (1770) . Otherwise, by Induction, differentiating again, and remembering that x = x.

NOTE.

In the symbolic solution of differential equations ,

we may either employ the operator ad directly, or the Formulæ ( 3480–5 ) or operator d, after substituting e for a. (3474-9) will be required accordingly. 3491

{ + (D) ere} " Q

= enre $ (D + nr) $ (D + n − 1.r) ... $ (D + r) Q

= ¢(D) $ (D — r) $ (D − 2r ) ... Þ { D— (n − 1 ) r } enrº Q. PROOF.

By repeated application of ( 3475 ) or ( 3476) .

For ready reference , formulæ ( 1520 , '21 ) are reprinted here .

3492

f(x +h) = ehdzf(x).

3493

f(x+h , y +k) = ehd¸ + kdyƒ(x, y) .

Let

, x" = ƒ (x) , а + а₁ x + а₂x² ... + a n

then, denoting d. by D, 3494_ƒ ( D) uv = uf(D) v + uf (D) v + 1;½ƒ ' (D) v + &c., where f' (D) means that D is to be written for x a after differ entiating f(x) . PROOF.-Expand uv, D. uv, D³.uv ... D" .uv by Leibnitz's theorem (1460) ; multiply the equations respectively by a , a , a, ... a,, and add the results. 1 3495_uf(D) v =ƒ(D) .uv −ƒ ' (D) u‚¸v + 1.2 —¿ƒ “ D.u„ v—&c.

SYMBOLIC METHODS.

519

PROOF. - Expand uv,, uv₂ , UV39 uvs, ... uv, by theorem (1472) , and proceed as in the last.

sin F (der)

3496

sin mx = F (-m²)

COS

mx. COS

A more general theorem is 3497

F (T²) (Uim + U - im) = F ( —m²) ( U¡m + U_im) ,

where u and have the meanings assigned below (3499 ) , and i = √ - 1 .

Theorem. — If p and denote any algebraic functions of a and y, it may be shown, by ( 3474) and (3475 ) , that 3498

¥ (d +y) f(x) = $ (d, + x) ¥(y) .

3499 Let u , or, more definitely, " n, = (x, y, z , ... ) " , represent a homogeneous function of the nth degree in severable vari ables , and let

3500

π = xdx + yd, + zd₂ + &c .

Then , by (3480) ,

3501 3502

πи = nu, Hence

π²un²u ,

³u = n³u ,

&c.

F (T) u = F (n) u .

REDUCTION OF F ( π₁ ) TO ƒ( π) . 3503 Let u be any implicit function of the variables , and let = ₁ + 2 , where ₁ operates only upon a as contained in u, and , only upon a as contained in Tu, &c . after repetitions of the operation π. Then 3504

3506

пiu

— пи , ...

πu = ( π - 1 ) πε , ... ...

π¸u = (π — r + 1 ) ... ( π — 2 ) ( π — 1 ) πи .

PROOF.

πικ = (π - π. ) u = πω , since , has here no subject to operate upon. 2 T₁U = (T — T₂) TU = (π — 1) TU, for, zu being of the 1st degree, 7, and 1 are equivalent as operators . next step, and 2 are equivalent, and so on .

COR .

In the

When u is a homogeneous function , we have, by

520

DIFFERENTIAL EQUATIONS.

(3501 ) , π'u = n'u, therefore and n are equivalent operators upon u. Hence (3506) may be written 3507_u = (n − r + 1 ) ... ( n − 2 ) ( n − 1 ) nu = n } u ,

which is Euler's theorem of homogeneous functions ( 1625 ) , since in that theorem the operator is confined to u. 3508 As an illustration, let u = (xd, + yd, ) u = ₁u , then au = ( ² dax + 2xydzy + y² day) U, π²u ( πιμπ. ) που = πu + ποπι . Here ποπηu = (xd, + yd ) ( xd, + yd ) u , =

the operation being confined to a and y in the second factor ( 3503 ) , and there fore producing (xd, + yd, ) u merely . Hence 2u = (x²d2x + 2xy dxy + y²d2y + xdx + yd, ) u, which proves (3505) .

If U = u₂ + u₁ + u₂ + ..., a series of homogeneous functions of dimensions 0 , 1 , 2 , ..." then, by (3502 ) , 3509

F (T) U = F (0) u。 + F ( 1 ) u , + F (2 ) u₂ + ….. ,

3510

F-¹ (T) U = F−¹ (0 ) u。 + F− ¹ ( 1 ) u , + F−¹ ( 2) u₂ + ...

3511

Ex. 1 :

3512

a* U = u¸ + aµ₂ + a²u₂ + ……., a¯* U = u₂ + a¯¹µ₁ + a¯² u₂ + …..

Ex. 2 : If u have the meaning in (3499) , 3513 28

u2 +F ' (3n)

F (T) e" = F (0 ) 1 + F ( n) u + F ( 2n )

1.2

+ &c . , 1.2.3

-1 and similarly for the inverse operation F -¹ ( #) . PROOF. -By (3502 ) applied to the expansion of the subject by (150).

3514

where

rz ……. Un C (n, m) un = Σ a zd "d„ dpxy'd≈ x"d s m! p ! q! r ! .. p + q + r + ...

m,

and

p ! = 1.2 ... p .

PROOF. - Equate coefficients of a" in the expansion of ( 1 + a) * U = (1 + a) ,(1 + a) ,(1 + a) reducing by (3490) . 3515 The general F (do) u = Q is

symbolic

solution

u = F-¹ (do) Q + F- 1 ( de ) 0 ,

... U ,

of the

equation

by ( 1488–90 ) .

SYMBOLIC METHODS.

3516

521

The solution of the equation (3238 ) , viz . , Ynx + α₁y(n - 1) x + ... + a (n - 1) yx + any = Q ......... ( 1 ) ,

where Q is a function of a , is most readily obtained by the symbolic method . Thus m , m2, ... mn being the roots of the auxiliary equation in ( 3239) , and A , B, C ... Ñ the numerators of the partial fractions into which (m² + a₁m” -¹ + ... + an) ¯¹ can be resolved , the complete primitive will be

3517 -1 y = { A ( d¸ — m₁) -¹ + B (d¸¯m₂) ¯¹ ... + N (d¸ „ —m„) ¯¹ } ( Q + 0) , where

= em² emxSe - ma Qda, (d — m) -¹Q =

( 3470)

and the whole operation upon zero produces, by (3472) , for the complementary term ,

C₁emix +C₂ em² ... + Cnem .

3518

PROOF.-Equation ( 1 ) may be written (dx + a₁d(n - 1) x + azd(n - 2) x ... •• + an) y = Q, 2 or

(d₂ — m₁) (d₂ — m,) ….. (dx— mn) y = Q,

.. by (3515),

y = { (d₂— m₁ ) ( d₂ — m₂) ... (d¸ — m„ ) } ¯¹ ( Q + 0),

which, by partial fractions, is converted into the formula above.

If r of the roots m1, M2, ... are each

m , those roots give

rise in (3517 ) to a single term of the form 3519

-mx ... e Q. (A + Bd₂ + Cd2 ……. + Rd¸» ) em²[1x

PROOF. -By (1918) , the r roots equal to m will produce { A′ (d, —m) ¯ " + B′ ( d, —m) -r + ¹ ... + R' (d, —m) −¹ } Q, or 3520

(A + Bd₂ + Cd2 ... + Rd,x) ( dz - m) - " Q. -2 But, by (3470) , (d¸ - m) -² Q = ( dx — m) - ¹ emzJe emx e - m² Q dx da

e-mxemx = emx e'mx e mx Qda, ***[ { ' *' *'* Je for - ** Qile, m² Qde } de = " [_2x ' 3521

Yar -Y2x-5yz - 3 = Q.

Ex. (1) :

Here

and so on

m³ —m² — 5m - 3 = (m − 3 ) ( m + 1 ) , 1

and

1

1

1

=

16 (m + 1 ) 16 (m - 3) 4 (m + 1 )2' -1 -2 y = 1% (d, —3 ) ˜'Q— ' ( d₂ + 1 ) ´¹ Q − 4 ( d¸ + 1 ) −² Q 3x = 1fe² Se - ² Qda - 16 * Se * Qdx - 4e - SSe* Qd« x². 3 x

(m −3) (m + 1)❜ therefore

(3520)

522

DIFFERENTIAL EQUATIONS.

3522

u2x + a²u = Q, -1 u = (dx + u²) ¹ Q. -1 -1 (m² +a²) -¹ = (2ia ) -¹ { ( m — ia) -¹ — ( m + ia) ´¹ } , -1 u = (2ia) -¹ { (d, —ia ) -¹ Q− (d¸ + ia) − ' Q}

Ex. (2) :

therefore Here

therefore

(3470) = (2ia) -¹ {etaxSe-iaz Qdx — e- iax Seiax Qdx'} = a¹ sin ax cos ax Q dx - a¹cos axSsin ax Qdx, Sco by the exponential values ( 766-7) .

3523

COR. 1.- The solution of

u₂ + a²u = 0 is

u = A cos ax + В sin ax.

3524

COR . 2.— The solution of u₂ - a²u = 0 is u = Aeax + Be -ax.

Change a into ia in the fifth line of (3522) , and put Q = 0 .

3525

When Q is a function whose derivatives of the nth and

higher orders vanish, proceed as in the following example.

Ex. (3) : therefore

U₂ + a²u = (1 +x) ², u = ( d₂x + a²) -¹ ( 1 + x) ² + ( d₂ + a³ ) -¹ 0 - &c . ) ( 1 + 2x + x² ) + (d² + a³) -¹0 = (a¯² — a¯¹d² + a˜ºd₁x− = a² (1 + x) ² - 2a

+ A cos ax + B sin ax,

the last two terms by (3523) .

Exceptional Case of the Inverse Process.

3526

Ex . (4) :

U2x + a³u = cos nx,

.. u = (d₂ + a²) ~ ¹ ( cos nx + 0) = 1 ( d₂ + a²) -¹ (einx + e - inx +0) = } (einx + e - inx) ( - n² + a²) - ' + A cos ax +B sin az by (3474) and (3523) -1 = cos nx (a² — n³) -¹ + &c. Now, if n = a, the first term becomes infinite. : as follows :

In such cases proceed

cos nx- cos αx Put A = A′ - (a² — n²) -¹, and find the value of " when a²-n²2 x sin ax Thus the solution is na. By ( 1580) it is = 2a æ sin ax u= + A'cos ax + B sin ax. 2a

The same result is obtained by making Q = cos ax in the solution of (3522) . For another example, see (3559) .

523

SYMBOLIC METHODS.

3527

Y2x-9y +20y = x²²², 3x y = { (d, —4) (d, −5) } - ¹ x²e³ + { (d, -4) ( d, —5) } -¹0 =· e³ { (d, −1 ) ( d, −2 ) } - ¹ x² + Ae¹² + Bex. (3475, 3517, 3472) -1 (m²— 3m + 2 ) -- ¹¹ = † ( 1 — 3m—m³) ¨¹ 2

Ex. (5) :

therefore

Now

1 + 3 = }+ {{ 1+ =

3m -- m² 2 = m² + ·( 3m -- m² ) +&c. } 2

Hence the solution becomes

4x y = e² { } + 3d, + Jd2 + &c . } x² + Ae¹ +Bear 7 3x = e3x = o' { " + 3/2 2 + } } + de ' + Be“.

3528

Ex . (6) : (d₂ - a)" u = eаx, ear therefore u = (d, − a) ¯ " eax = eax ( d¸) ¯ " 1 (3476) nx . ( d,) - * 1 = e²² ( ~ 7 +С.0)

3529

Ex. (7) :

(2149)

(d, + a)³ y = sin mx,

therefore

y = (d, + a) ² sin mx + ( d, + a ) -20 =· (d¸ — a) ² (d²x— a² ) -² sin mx + e - ax ( d¸) -20 [ by ( 3478 ) with Q = 0] = ( -m³ — a³) -2 (d, -a) sin mx + e - ax (Ax + B) (by 3496) - 2am cos mx + a² sin mx ) + e¯ªx (Ax + B) . = (m² + a³) -² ( —m² sin mx —

REDUCTION OF AN INTEGRAL OF THE nth ORDER.

3530

S 9 =

{

a SQdx— (n − 1), x -¹SQxd.x

x² + C (n, 2), a*=³SQa²dæ... ±SQx™-1dx, where n - 1 ! = 1.2 ... n .

-110 PROOF. - By ( 3489 ) dxQ = e = "" (d. — n + 1 ) ( d, ―n + 2 ) ... d. Q ... ... ... ( 1 ) , therefore d_nxQ = { (d, ―n + 1 ) (d, −n + 2 ) ….. d. } ¯¹e¹º Q 1 = · { (d。 — n + 1 ) - ' -— ( n − 1 ) (d, —n + 2) -¹ n -1 ! -1 + C (n, 2) (d, ―n + 3) -¹- &c . } e" Q 1 = n = 1 } { e ™ -¹) • (d.) ¯ ¹ e* — (n − 1) e '* −2) • (d.) −¹ e² + &c . } Q.

(3517)

Then replace e' by x.

The equation 3531

axymx + bx" ynx + &c . = A + Bx + Cx² + &c. = Q

524

DIFFERENTIAL EQUATIONS.

may, by (3489) , be transformed into { a (xd¸) ) + b (xd¸)

+ & c . } y = Q_or

F (xd ) y = Q.

The solution is then obtained from 3532

y = F¹ (xd ) Q + F- 1 (xd ) 0 .

The value of the 1st part is given in ( 3487) . 3533

If a, ẞ, y, &c. are the roots of F (m) = 0, the second

part gives rise to the arbitrary terms

C₁x² + C₂x² + & c . 3534 If a root a is repeated terms are

times , the corresponding

a p №ª { C₁ (log x) ” -¹ + C₂ (log x) ”- ² + ... + C,} .

PROOF. The partial fractions into which F-1 (ad,) O can be resolved, as in ( 3517 ) , are of the type C (ad¸ - m) -¹0 , m being a root of F (x) = 0. But (xd, —m ) -¹0 = Cam (3473) , C being an arbitrary constant. For a root m repeated r times, the typical fraction is C (xd, —m) ˜º, p being less than r. Now (xd - m)" Cx (log x) = ( d. — m ) º Cem³ (jp-¹ = e™ ³ (d.) COP- 1 (3475) = 0, therefore (xd - m)-0 = Ca" (loga)P-1.

The equation 3535

ayme + byne + &c . = ƒ ' ( eº , sin 0 , cos 0)

is reducible to the form of (3531 ) by a

e ; or, substituting

from (768) , it may be written F (do) y = Σ (Ameme) , and the solution will take the form 3536

y = ΣAmeme F -1 (m) + F - ¹ (d.) 0 ,

for the last term of which the forms in ( 3533-4) are to be substituted with a changed to eº.

3537

x³узx = axm + bxn

Ex. (1 ) :

ad, ( dr− 1)( dr− 2) y = a.” + biển , y = { dax (2 ,−1)( d,−2)}- (a +b )+ { da ( d − 1)( d,−2)}-10 axm bxn = + A + Bx + Cx², + m (m - 1 ) (m - 2) n (n - 1 ) (n - 2 ) by (3180) and (3533 ) .

A result evident by direct integration.

do

で SYMBOLIC METHODS.Or

3538

Ex. (2) :

525

x²Y2x + 3xYx + y = ( 1 — x) −².

By (3490)

{xd (xd - 1 ) + 3xd, +1 } y = ( xd¸ + 1 ) ² y = 1 + 2x + 3x² + &c. , ..y = (xd + 1 ) ² ( 1 + 2x + 3x² + ) = ( 0 + 1 ) -² + 2 ( 1 + 1 ) ² ≈ +3 ( 2 + 1 ) −²x² + x B A log a = —— +3393 + & c. + (3480) = 1+ + 1 x x log (1 − x) + &c. x

Ex. (3) : Y2x + (4x − 1 ) Yx + ( 4x² −2x + 2) y = 0 .

3539

d, + 22.

Let

Then the equation may be written

..y = { ( -1 ) } - ' 0 = ( -1 ) -10 -- ¹0 . Let ( —1) -¹0 = u , .' . (π — 1 ) u = 0, or u₂ + (2x - 1 ) u = 0 , .. u = Ae** - *. π (π - 1 ) y = 0,

3540

The solution of a P. D. equation of the type a

+b

+ & c . = u₁ + u₂ + &c. ,

where u₁, U2, &c. are homogeneous functions of the 1st , 2nd degrees , &c . in x, y, and ₁ = æd, + yd, ( 3503 ) , is analogous to (3531 ) , and is obtained from that solution by substituting U1, U2, &c. for Ba, Ca² , &c .; and , for such terms as Ca", an arbitrary homogeneous function of x and y of the same degree.

3541

Solution of

F (T) u = Q,

n-2 1 F (π) = π” + A¸π” ¬¹ + Â½π” ¬² + An,

where

and

Q = u₂ + u₁ + u₂ + & c . ,

a series of homogeneous functions of x, y, z , ... of the respec tive dimensions 0, 1 , 2 , & c . Here 3542 general

u = F -¹ (π ) Q + F¬¹ ( π) 0 .

The value of the 1st term is given in (3510 ) . value of the last term (see

Proof

For the

of 3533 ) ,

let

F (m) = 0 have r roots = m ; then 3543

C (π - m) º0 = C { u (log x)p- ¹ +v (log x)” -2 ... + w } ,

where u, v, ... w are arbitrary functions of the variables all of the degree m.

3544

COR.

-1 (π— m) -¹0 = (x, y, …..)™,

that is , a single homogeneous function of the variables of the degree m (1620) .

526

DIFFERENTIAL EQUATIONS.

3545

Ex.:

x²≈2x + 2xYzxy + Y²Z₂, —α (xz¸ + yz „ ) + az = um + Un

um, un being homogeneous functions of the mth and nth degrees . The equation may be written

( ²а₁ + α) z = Um + Un ;

""n + " n = 2 ( 1

)(

)

or, by (3505 ) , 0 ₁−{ ( 1 − 1 ) ( 1 − 2 ) } + (" n + " n ) 、 - { ( I − 1) ( − 1) } = ≈

therefore

Ит

=

Un + + Ua + U1. (m — a) ( m − 1) ' (n — a) (n − 1 )

The first two terms by formula (3502) ; the last two terms are arbitrary functions of the degrees a and 1 respectively, and result from formula ( 3543) by taking p1 and ma and 1 .

3546 To reduce a P. D. equation , when possible , to the symbolic form n -1 ........ ( 1 ) , 1 ”~¹ + A₂II " -2 ... + A „ ) u = Q (II” + A₁II Q ... where

[ ] = Md + Nd, + &c . ,

and Q , M, N, &c. variables .

are any functions

of the independent

Consider the case of two independent variables , (Md₂ + Nd, ) ² u = M² U2x + 2MNUxy + N² Uzy

+ (MM₂ + NM₁) Ux + (MNx + NN₁ ) u ... (2) . Here the form of II is obtainable from the right by con sidering the terms involving the highest derivatives only, for these terms are algebraically equivalent to (Mdx + Nd₁)³. The reduction being effected, and the equation being

brought to the form of ( 1 ) ; then , if the auxiliary equation 3547

m" + A₁m² - ¹ + A₂m" -2 ... + A₂ = 0

...... (3)

have its roots a, b, ... all unequal , the solution of ( 1 ) will be of the form 3548

u= (

............ ……………….. (4) . -a) -¹Q + ( II — b ) -¹Q + &c . ....

The terms on the right involve the solution of a series of linear first order P. D. equations , the first of which is 3549

Mụx + Nu₂ + ... - au =

Q,

and the rest involve b, c , & c . If equal or imaginary roots occur in the auxiliary equation , we may proceed as in the following example.

SYMBOLIC METHODS.

3550

527

Ex.:

(1 +x²)² 2x - 4xy ( 1 + x²) %xy + 4x²y²zay + 2x ( 1 + x² ) %x + 2y ( x² − 1 ) % , + a²z = 0. Here II = (1 + x²) d - 2xydy, and the equation becomes ( II +a²) z = 0. Let the variables x , y be now changed to E, n, so that II = de. Therefore, since II (E) = 1, II (4) = (1 + x³) Ex - 2xy = 1. dx = dy -2xy = de, 1 + x²

Therefore, by (3383) ,

from which, by separating the variables and integrating, we obtain x²y + y = A ...... = Etana +B and, by (1436 ) ,

Also, since II ( n ) = n = 0 , Therefore

. (1) , (2) .

( 1 + æ³) n₂ - 2xy n₁ = 0. dx dy dn = == -2xy 1+x²

the solution of which is equation ( 1 ) . Thus and n = x²y + y. = tan The transformed equation is now

(d2 + a²) z = 0 ,

and the solution, by (3523) , is z = $ (n) cos a& +

(n ) sin ak,

arbitrary functions of the variable, which is not explicitly involved, being substituted for the constants (3389) . Therefore finally, z=

(x²y + y) cos ( a tan - ¹x) +

(x²y + y ) sin ( a tan - ¹x) .

MISCELLANEOUS EXAMPLES . 3551

U2x + U2y + U2z = 0 .

Put day + d₂ = a² . • is

Thus

2x + a²u = 0, the solution of which, by ( 3523) ,

u = 4 (y, z) cos ax +

(y, z) sin ax,

arbitrary functions of y and z being put for the constants A and B. Expand the sine and cosine by ( 764-5) ; replace a² by its operative equivalent, and , in the expansion of sin ax, put a¥ (y, z ) = x ( y , z ) ; thus 22 2 (day + d2z) $ u = 4 (y, z) —-2017 (day + drz) $ (y , z) + Φ (y, z) — &c. 4! 2! 23 28 (day + daz) x (y, z ) — &c . + xx (y, z) — 3! (day + d2z) x (y , z) + 5!

[See (3626) for another solution.

3552 ux + u₂ + u₂ = xyz .

Here

u = (dx + d₂ + d₂) ˜¹ (xyz + 0 ) = {d_z ― d_2 (d, + d₂) + d_з (dy + d₂ ) ³ — ... } (xyz + 0) .

Operating upon xyz, we get u = } x²yz —zx³ (z + y) + 17x*,

DIFFERENTIAL EQUATIONS.

528

the rest vanishing. For symmetry, take rd of the sum of three such expressions ; thus u == } { zz (x * + y * + z¹ ) − f ( x³y + xy³ + y³z + yz³ + z³x + zx³) + } xyz (x + y + z) } .

Operating upon zero, we have, in the first place, dx0 = ¢ (yz) instead of a constant, therefore d- 20 = xp (yz), &c. The result is { 1 − x (d„ + dz) + { œ² (d„ + d₂ ) ² — ... } ¢ (yz ) = e − x ( dy + dz} ¢ (yz) = ¢ (y − x, z — x) (3493) the complementary term.

3553

Otherwise, putting d, + d₂ = D , we have, by ( 3478 ) , (dx + D) −¹ xyz = e− xDd_xexDxyz = e − x®d_x { x (y + x ) ( z + x ) } , (3493) = e −xD { { x²yz + } x³ (y + z) + } x* } = x² ( y — x ) ( z — x) + } x³ (y + z − 2x) + 1x¹,

which agrees with the former solution .

3554

au¸ + bu + cu₂ = xyz.

Substitute x

ak, y = bn, z = c , and the equation becomes uε+ u + uz = abcns,

which is solved in (3552) .

The same methods furnish the solution of

3555

au¸ +bu, + cu₂ = x”y"zº.

3556

x≈x +yz₁ = 2xy√ /a² —z² .

Put

.. TZ = α cos v . TV,

3557

z = a sin v, .. πυ - 2xy,

..za sin (ay + c).

axu™ +byu, + czu, —nu = 0 .

Put

= ,

y= n,

.. Eu¿ + nu„ + Su¿— nu = 0,

3558

x= ; 1 1 1 z² )” ) ".. .. by (3544) u == (x , y , z

The solution of any P. D. equation of the type n ... F (cd », yd , dạ , ... )u = EA ” y”

is, by (3488) and (3557) , 1

Axmy"

u= Σ

0.

+ F (m, n , p, ...)

F (xd , yd , zd₂, ...

SYMBOLIC METHODS.

3559

Ex.:

where Q

529

xu +yu, ―au = Qm

(x, y) m ( 1620) .

Here u = (m — a ) ¹Qm + Ua. When am, this solution becomes inde terminate. In that case, as in ( 3526 ) , assume - Va Qa " Ua = :: u = Qm -Qa + Va. U₁ m -a m -a Differentiate for a, by ( 1580) , putting Qa first in the form

' { thus

F (1 ) +

(税 ) } ;

Qm (log x + log y) + Vm

u=

Similarly , the solution of 3560

xux + yu + zu₂ - mu = Qm

is

u=

3561

Qm (log

+ log y + log z ) + Vm²

xu¸ +yu₂ + zu₂ = c .

The solution, by ( 3560) , is u = c (log x + log y + log z ) + V。. 3562

≈2x - 2a≈xy + a²≈2y = =0 0

or (d - ad₁ ) ² z = 0. = (d - ady ) ¹ ¢ (y) eaxdy z = (d - ad,) -20

(3472 )

by putting ad, for m and (y) for C. The second operation produces, by z = eaxdy { xp (y ) + (y ) } = xp (y + ax) + (y + ax) . (3476) , (3492)

3563

X²≈2x − Y²Z2y + XZx — yz₁ = 0.

This reduces to Here therefore

(xd₂ + yd ) (xd, —ydŋ ) z = 0 .

xd, + yd,, and m = 0 in (3544) ,

0

* = (, ) +( ,y ) ,

the second term being obtained by substituting y¹y', and so converting the second factor into (xd, + y'd, ) . The above may also be written

2

F = P ( ) + ( y),

F and ƒ being integral algebraic functions.

3564

≈2x - a²≈2y + 2abz, + 2a²b≈, = 0.

Putting yan, this equation is equivalent to (d, ―dn + 2ab) (d, + dn) z = 0 ; 3 Y

530

DIFFERENTIAL EQUATIONS.

putting a

log a' and n = log n' , this gives, by (3544) , z = (e³, e-¹) - zab + ( e*, e" ) " = e -zabx & ( e² + " ) + ↓ (ext- ")

= e - abxF (y + ax) +ƒ (y - ax), the functions being algebraic and integral.

U2x — a²u₁₁ 2 = $ (x, y).

3565 ..

(3515) u = (d₂ — a²d2y) −¹ ¢ (x, y ) = (2ad¸ ) −¹ { (d¸ — ad¸) ´¹ — (d₂ + ad,) − ¹ } ¢ ( x, y)

(3470)

= · ( 2ad,) −¹ { eazdy fe ‑azdy 4 (x, y) dx —e-azdy feazdy o (x, y) dx }

(3470 )

= ·(2a) -¹ [ { •, (x, y + ax) − •¸2 (x, y — ax) } dy ,

since

caxdy 4 Φ (x, y) = 4 (x, y + ax)

Here

1 ℗, (x, y) = [ 4 (x, y — ax) dx + 4 (y) ,

(3492) .

2 ❤¸ (x, y ) = √ p (x, y + ax) dx +x (y ) .

3566

If Φ (x, y ) = 0 , the solution therefore becomes u=

(y + ax) +x₁ (y—ax)

[Boole, ch. 16.

For the solution in this case by Monge's method , see (3433) .

3567

≈x − az₁ = em

cos ny.

z = ( d¸ — ad¸) ¯¹êm² cos ny = card, fe - axdy em² cos ny da (3470) = eaxdy em.x cos n (y -ax) dx ( 3492) , and this by Parts, or by ( 1999) , is Se -1 = eaxdy emx {m m cos n (y - ax ) — an sinn (y — ax) } (m² + a³n²) -¹ +eardy ¢ (1 )

.*. % = emx

m cos ny —an sin ny } (m² + a²n²) ˜¹ + ¢ (y + ax), by (3492) .

Zt - Az2x = 0.

3568

%= z = (d - ad ) ¹0 eat d2p (p) , by (3472) , (x) taking the place of the constant C. Therefore z = 4 (x) + at 2x + }a²t²₁x + &c. ( 3492 ) Otherwise, to obtain z in powers of a , we have, putting b² = a¯¹ , 22 - b³zt = 0,

.. % = { (d₂ +bd}) (d¸ + bd}) } ~ ¹0 = ebxdip (t) + e - bædi ↓ (t) (3518 ) ; then expand by (150) .

3569

22x +

2y = cos nx cos my.

SYMBOLIC METHODS.

531

z = (d2x + day) ¹ cos nx cos my. Treating da, and cos my as constants, we have, by ( 3526) , putting day for a², 1 z = cos nx (d₂, —n²) -¹ cos my + A cos ax +B sin ax, or by (3496) , -1 = cos nx cos my ( — m² — n³ ) −¹ + ¢ (y ) cos (xdy ) + (y ) sin (xd,) , A and B becoming

3570

(y ) and

(y) .

≈24 + 2%64 +≈24 + a²z = cos (mp + n¥) .

Therefore where

( + ia) (π- iα) z = } { e! (m * + n ) + e −i ( ma + mv)} , = dd . Therefore, by ( 3510) , with xe , ye , e−i(mp+n¥) ei (mø +n¥) + (e³, ev) -ia + (e*, ev) ia, = } { a² — = (m + n)² + a² — = (m + n)" }}

z = cos (mo + n¥) + (e*, ev) -ia + (e*, e*)ia. a² - (m + n)²

or

PROP. I.- To transform a linear differential equation 3571 of the form (a +bx +cx² ... ) Unx + (a' + b'x + c'x² ... ) u (n - 1 ) x + &c . = Q ... ( 1 ) into the symbolical form

.... (2) , fo (D) u +fi (D) e'u +f2 (D) e²° u + &c. = T ...... where Q is a function of x, T a function of 0 , x = eº and D = d.. Multiply the equation by a" ; then the 1st term on the left becomes , by (3489) , (a + be + ce²º + ... ) D (D − 1 ) ... (D — n + 1 ) u . This reduces, by the repeated application of formula (3476) with the notation of (2451 ) , to 3572_aD ) u + b (D − 1 ) (" ) e° u + c (D − 2) (n ) e²º u + &c . The other terms admit of similar reductions .

3573 Conversely , to bring back an equation from the sym bolic form (2 ) to the ordinary form ( 1) , employ formula (3475) so as to transfer eme to the left of the operative symbol .

3574

Ex.:

20 ² (x²u₂ + 7xu₂ + 5u) =: e² { D (D − 1 ) + 7D + 5 } u = e20 (D3 + 6D + 5 ) u = e² (D + 1 ) (D + 5) u = (D − 1) (D +3) e² u

(3476) .

532

DIFFERENTIAL EQUATIONS.

For the converse reduction, the steps must be retraced, employing ( 3475) . See also example ( 3578 ) .

3575

PROP. II .-To solve the equation

u +a₂$ (D) e'u + a¸¢ ( D) $ (D − 1 ) e²º u ... + a„$ (D) $ (D − 1 ) ... $ (D − n + 1) e¹º u = U,

where U is a function of 0. By (3491 ) n $ (D) $ (D − 1 ) ... ¢ (D − n + 1 ) e¹º u = {+ (D) eº } " u . Putting pu for this , the equation becomes (1 + a

3576

+ a² ... + a, ç” ) u = U.

Therefore

u = · { A₂ ( 1 − 919) ˜¹ + A₂2 (1 − (29) ˜¹ ... + A „ ( 1—9„9 ) −¹ } U, where 41 , 42 ... qn are the roots of the equation an = 0 , q" + a₁q" ¹ + A27" -2 ... + α₂ 2-1 A, = (Ir − 41) (Ar − 42) ... ··· (Ir — In)`

and

The solution will then be expressed by 籍 u = ¸µ₂ + ¸µ½ ….. + Anun

where ur Изо is given by the solution of the equation 3577 (D) e' u, = U. u, —q,$

3578

Ex .:

(x² + 5x³ + 6x ) U₂ + (4x + 25x² + 36x³) u₂ + (2 + 20x + 36x³) u = 20x³. Putting ae , and transforming by (3489 ) , (1 + 5e + 6e²º) D (D − 1 ) u + ( 4 + 25e* + 36e²º) Du + ( 2 + 20e + 36e²) u = 20e³ . The first term = D (D − 1 ) u + 5 ( D − 1 ) ( D − 2 ) e ° u + 6 (D − 2) (D − 3) e² u by applying (3476 ) . The other terms similarly ; thus, after rearrangement, (D + 1) (D + 2) u +5 ( D + 1 ) ² e' u + 6D (D + 1 ) e² u = 20e³. Operating upon this with { (D + 1 ) ( D + 2 ) } - ¹ , we get D 20e39 D+ 1 e20u u= eᵒu +6 u +5 = e , by (3474) ; D+2 D+2 (3 + 1 ) (3 + 2 ) or if p = (D + 1 ) ( D + 2 ) -¹e' ; (1 + 5p + 6p²) u = e" , therefore u = { 3 ( 1 + 3p) -¹ −2 ( 1 + 2p ) ¯¹ } e³ = 3y — 2z,

if

y = (1 + 3p) ¹³

and

z = (1 + 2p) ¹ e³.

SYMBOLIC METHODS.

30 (1 + 3p) y = e³°

Hence therefore

or

533

y +3 (D + 1 ) ( D + 2 ) -¹ e ° y = e³ ;

(D + 2) y + 3 (D + 1 ) e ° y = e³ ( 3 + 2 ) ,

or

(D + 2) y + 3e° (D + 2) y = 5e³°,

that is,

(x +3x²) y₂ + 2 ( 1 + 3x) y = 5æ³. (x +2x²) % +2 (1 + 2x) z = 5æ³.

Similarly

by ( 3474) ,

by (3475 ) ;

Solve these by (3210) , and substitute in u = 3y - 2z.

3579

PROP. III .-To transform the equation u +4 (D) e¹ºu = U_into_v + 4 (D + n) e™º v = V, and

u = enov

put

PROOF.- By (3474) , because

3580

(D) e(n + r) • v = e¹º p (D + n) e** v.

PROP. IV.-To transform the equation into

u+ ¢ (D) erº u = U

put

3581

U = ene V.

u = P₁ ? (D) v (D)

where P Ф

and

v+

(D) e™º v = V,

Φ U = Pr + (D) V, (D)

Φ (D— ?) p Φ (D − 2r) ….. = $ (D) ø (D) 4 (D) 4 (D — r) 4 (D − 2r) *

PROOF.- Put u = f (D) v in the 1st equation, and erƒ (D) v = ƒ (D — r) e™ v (3476) . After operating with f-1 (D) it becomes v +¢ (D) ƒ (D — r) fƒ˜¹ (D) e²° v = ƒ˜¹ (D) U,

therefore therefore

ø (D) ƒ (D — r) f−¹ (D) = ↓ (D) ƒ(D) = • (D) ƒ (D − r) = ↓ (D)

and so in inf.

(D) (D)

by hypothesis ;

(D- r) f(D - 2r) , (D - r)

Also Uf (D) V.

3582 To make any elementary factor x (D) of ¢ (D) be come, in the transformed equation, x (Dnr), where r is an integer ; take

(D) = x (D ± nr) X1 (D) .

See example (3589) .

x (D) 3583

To make any factor of

(D) of the form

x (D ± nr) disappear in the transformed equation , take (D) = x₁ (D), where X1 (D) , in each case, denotes the remaining factors of See example (3591) . (D).

534

3584

DIFFERENTIAL EQUATIONS.

In the application

of

Proposition

IV. ,

differentia

tion or integration will be the last operation according as P $(D) (3581 ) has its factors , after reduction , in the " (D) numerator or denominator, and therefore according as (D) is formed by algebraically diminishing or increasing the several factors of Φ (D) . However, by first employing Proposition III. , the given equation may frequently be so prepared that the final operation with Prop . IV. shall be differentiation only. See example ( 1 ) . For further investigation, see Boole's Diff. Eq., Ch. 17, and Supplement, p. 187. 3585

To reduce an

equation

of the homogeneous

class

(3531 ) to a binomial equation of the same order of the form Ynx »a + q = X. The general theory of such solutions is as follows . the given equation be

Let

u + q { (D + a,) (D + a₂) ... ( D + a„ ) } ¯¹e¹º u = U ... (1) , a1, a2, ... a, being in descending order of magnitude. Putting ue-av, by Prop . III . , -1 v + q { D (D — ã₁ — α₂) ….. (D — ภ— α₂ ) } ¯¹ enº v = eª‚º U ….. (2 ) . To transform these factors , regarded as

(D) , by Prop. IV.

into (D) = D(D − 1 ) ... ( D - n + 1 ) , we convert D into D + rn (3582 ) , r being an integer . Hence for the pth factor we must have D + rna₁ + a₂ ap = D - p + 1 ,

...... (3) . and therefore a₁ - a, = rn +p − 1 ..... relation If this holds for each of the constants a ... an

3586

equation ( 1 ) is reducible to the form

3587

y + q { D (D − 1) ... ( D − n + 1 ) } ¯¹enºy = Y ...... (4) ,

which, by (3489) , is equivalent to Ynx + y = Y₂x nx = X. y being found in terms of a from the last equation, and , v being

P $ (D) n y (3580) , the solution will result from (D)

u = e−ap (D − 1 ) (D− 2) ... (D − n + 1) ,y; (D − a₁ +a²) ... (D − a₁ + an) while U and Y are connected by the same relation as u and y.

3588

SYMBOLIC METHODS.

3589

Ex. 1 : Given

535

a³uз + 18x²u₂ + 84xu¸ + 96u + 3x³u = 0.

Puttinge and employing ( 3489) , this becomes { D (D − 1 ) (D − 2) + 18D ( D − 1 ) + 84D + 96 } u + 3e³ u = 0, or

(D + 8) (D + 4) (D + 3 ) u + 3e³ u = 0,

therefore

u +3 { (D + 8) ( D + 4) ( D + 3) } - ¹e u = 0

...... ........

Employing Prop . III. , put u = e- 80 v, therefore (3476) v +3 { D (D - 4) ( D - 5) } - ¹ e³° v = 0

.. (1) .

........ (2) .

To transform this by Prop. IV. into y +3 { D (D − 1) (D − 2) } - ¹³ y = 0 ................ ( 3) ,

we have — (D − 1 ) (D − 2) , PP (D) = D (D- 1) (D - 2) (D - 3 ) (D − 4) ( D − 5 ) ··· = D (D- 4) (D - 5) ( D - 3 ) ( D − 7) (D - 8) ... (D) .. u = e-8 (D - 1 ) ( D - 2) y ......... (4), (D - 1) (D - 2) y, .. v and the solution is obtained by differentiation only, performed on the value of y as obtained by the solution of ( 3) , that equation being equivalent to D (D − 1) (D − 2 ) y + 3e³ y = 0,

or, by ( 3489) ,

Yзx + 3y = 0 .

If, however, Prop. IV. were used to pass directly from ( 1 ) to ( 3) , we should have PP(D) = D (D − 1) (D− 2) ( D − 3) (D − 4) (D − 5) ... (D + 8) (D + 4) (D + 3) ( D + 5 ) (D + 1 ) D ... (D) 1 = (D+ 8) (D + 5 ) (D + 4) ( D + 3 ) ( D + 2) ( D + 1 ) ' and equation (4) would involve integrations of y as high as D- y.

3590 NOTE. By the literal application of Rule IV. , the right side of equation (3) ought to be V = { (D − 1 ) ( D − 2 ) } -10 ; but no such term is required when the original and transformed equations are of the same order, for in such cases the arbitrary constants introduced by the operation upon zero disappear with the terms containing them in the final differentiation . The result is the same as if the operation upon zero had not been performed. In the following example, V has to be retained. 3591

Ex. 2 :

(x − x³) U₂ + ( 2−12x²) u, − 30xu = 0 …………………………………........... ( 1) .

Multiply by a, transform by (3489) , and remove e to the right of each function of D by (3476) , thus .... (2) . u (D+ 4) (D + 3) e²º u = 0 ........... D (D + 1) Transform this by Prop . IV. into

v

We have

D + 32v = V *****... D+ 1

*.***.... (8) .

D+4 u = P₂ v = (D + 4) (D + 2) v, D

V = { (D +4) (D + 2 ) } - ¹0 = Ae-2 + Be-"

(3518 ) .

The operation upon zero is required in this example (see 3590) , because (3)

536

DIFFERENTIAL EQUATIONS.

is of a lower order than (2) ; but only one term of the result need be retained , because only one additional constant is wanted . Hence (3) becomes (D + 1) v- (D + 3 ) e² v = (D + 1 ) Ae-2 Changing again to x, this equation becomes

- Ae-2 .

(x³ — x³) v¸ − 4x³v + A = 0 . The value of v obtained from this by (3210) will contain two arbitrary con stants . The solution of ( 1 ) will then be given by u = (D + 4) (D +2) v.

3592 Ex. 3 : n being an integer.

[Boole, p. 424.

u₂ - n (n + 1 ) x-² u — q³u = 0,

Multiplying by a and employing (3489) , this becomes u− q³ { (D + n ) (D — n − 1 ) } - ¹ e² u = 0. This is changed by Prop . III. into v - q' { D (D — 2n − 1 ) } - ¹ e²v = 0, and this, by Prop . IV. , into y - q { D ( D- 1 ) }

e2 y = 0

with u = e ="" ,

or

y2 - q'y = 0

(3489) .

y being found from this by ( 3524) , we then have D- 1 (n) P u = ene ₂ e- " (D - 1) y = x-" (xd, -1 ) y = e-ne D- 2n - 1 m F (xd, ―m) = x™ F (xd₂) x¯”, But, by (3484 ), ..

u = x¯” x (xd₂) x−¹ . x³ (xd¸) x−³ ... x²n−1 (xd₂) x−²n +¹y,

or

u = x¯(n+ ¹) (x³d¸) " x−2n + 1 .y = x-" -¹ (x³d,) " x -2-1 (Ae

+ Be *)

This may be evaluated by substituting z = x². Vol. XVII., p. 77.)

3593

y.

Ex. 4 :

(3525) .

(See Educ. Times Reprint,

— (n + 1 ) x²u = 0. Uzza¹³uzyn zx

The solution is derived from that of Example ( 2 ) , by putting q = ad,, and arbitrary functions of y after the exponentials instead of A and B ; thus x−²n u = x- " -¹ (x³d,) " x - 2n ++ 11 { eardy (y) + e - ard n ( y ) } = x= "-¹ (x³d,)" x− 2 + 1 { p (y + ax) +

(y + ax ) } , by (3492) .

[Boole, p. 425 .

3594

(1 + ax²) u2x +axu¸ ± n³u = 0 .

To solve this equation or its symbolical equivalent obtained by (3489) , viz . , a (D − 2)² ±n² e²º u = 0 . e20 3595 u+ D (D- 1) dx Substitute t = in the solution of u ±n²u = 0 , by (3523–4) . √√√(17 7 9237) +

SYMBOLIC METHODS.

3596

537

Similarly, to solve the equation

(x² +a) x²u₂x + (2x² +a) xu¸ ± n²u = 0, or, the same in its symbolical form,

3597

u+

(D − 1) (D − 2) e20u = 0. aD² ± n²

dx in the solution of un'u = 0. Sx√(x² + a) (3596) is obtainable from (3593) by changing 0 into -0. Substitute t =

3598

Pfaff's equation, (a + bx") x³u₂x + ( c + ex” ) xu¸ + (f+gx”) u = Q.

When Q = 0, the symbolical form becomes b (D— n) (D — n − 1 ) + e (D − n) + 9 eno u = 0 ………………………….. ( 1 ) . ......... น+ aD (D - 1) + cD + f If n be not = 2, substitute 20′ = n0, and therefore 2d. = nd.. 3599

Thus

u+

b (D - a₁ ) (D — -α ) 20' u = 0 ………………………………… .. a,) ……………... (2) . a (D - B₁) (D - B₂)

where a,, a, are the roots of the equation b ( nan) ( na - n - 1 ) + e ( nan) + g = 0 ........ (3) , and ẞ₁, ẞ, are the roots of a ền

( nß − 1) + cần +f = 0.

Four cases occur 3600

I.- If a₁ - a, and ß₁ —ß, are odd integers, (2 ) can be reduced by

Prop. IV. (3581) to the form v +

b (D - a, ) (D - a, -1) e20'v = = 0, a (D — ß₁) (D— ß₁ — 1)

and then resolved into two equations of the first order.

3601 II.-If any one of the four quantities a, -ß₁ , a₁ — ß₂, ɑ‚— ß1 , ɑ2 — ß₂ is an even integer, ( 2 ) can be reduced by Prop. IV. to an equation of the first order . 3602 III .-If ß, -6, and a, + u, --B, are both odd integers, then, by Props. III. and IV. , (2) is reducible to (3595) . 3603 IV.- If a₁ - a, and a₁ + a, -ß₁ - ẞ, are both odd integers, ( 2) is reducible in like manner to (3597) . [Boole, p. 428. NOTE. The integers may be either positive or negative, and when even may be zero.

3 z

DIFFERENTIAL EQUATIONS.

538

SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS BY SERIES .

3604

CASE I.- Solution of the linear differential equation

fo (D) u -fi (D) erº u = 0

or

f (xd ) u -fi (xd¸) x” u = 0 ,

in which fo (D) , fi (D) are polynomial expressions of the form а + a₁D + ª‚о ….. + a, D" and ƒ (D) = (D — a ) ( D — b ) ( D — c) …….. 3605

Let

3606

(D) = ƒ₁ (D) ÷ƒ 。 (D) , and let

(a) = 1 + ¢ ( a + r ) x² + ¢ ( a + 2r) ¢ ( a + r) æ²r +

(a + 3r )

(a + 2r ) $ (a + r) æ³r + &c .

Then the solution will be 3607

u = AxªÞ (a ) + Bx³Þ ( b) + Cxˆ • ( c) + &c .

PROOF.-Operating with f¹ (D) and writing p for

(D) erº,

u—pu = f (D) 0 = Ae + Bebe + &c. (3518 ) u-pu Therefore, by (3515 ) , u = ( 1 - p) -¹ ( Aeª® + Be¹º + ... ) = (1 + p + p²+ ... ) Aeª® + ( 1 + p + p + ... ) Beb + &c.

Now in each term substitute for p", the value in ( 3491 ) , and remove D by formula (3474).

CASE II.- Solution of 3608_ƒ (D) u +fi (D) e° u +ƒ½ (D) e²º u ... +ƒn (D) e¹º u = 0 ... (1), where fo (D) = (D — a) (D — b) (D — c) ...

Let

(a) = 1 + F₁ (a + 1 ) x + F₂2 ( a + 2 ) x² + & c . ,

where the coefficients F (a + 1) or v₁ , F (a + 2) or v₂, &c . are determined in succession by the formula 3609

f (m) vm +fi (m) vm-1 ... +fn (m) vm -n = 0 and v = 1 ......(2) . The solution will then be expressed by

3610

u = Axª ¥ (a) + Bæ³ ¥ (b ) + Сx° ¥ ( c) + &c .

PROOF.

From (1) ........ (3 ) , = { 1 + ¢ , (D) e ° ... + ‡n (D) enº } −¹ƒ˜¹ (D) .0 …………………………………..

where Here and

Pr (D) = fr (D) ÷ƒ。 (D) . f¹(D) 0 = { ( D — a) ( D — b ) ... } -¹0 = Aeª²º + Bebº + ... ( 3518) ; -1 { 1 +41 (D) e° ... + ¢n ( D) e¹º } −¹ = 1 + F, ( D ) e° + F, (D) e² + ...

SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS.

539

To determine F₁, F₁, &c. , operate upon each side with { 1 +¢ ( D) e ° + &c . } , and equate coefficients of powers of e ; thus formula ( 2) is obtained . (3) now becomes be u = { 1+ F₁ (D) e° + F₂ (D) e²º + ….. } ( 4eª® + B¸¹º + ... ) ......... (4) . Multiply out ; apply (3474) , and put a for e .

3611

Ex.:

x²u2x- (a + b - 1 ) xu + abu - qx³u = 0,

(D - a) (D - b) u - ge²º u = 0. or, by (3489) , Here f。 (D) = (D − a) (D — b) , fı (D) = 0, ƒ½ (D) =

Therefore (2) becomes

-9.

(m - a) (m — b) vm = qvm -2,

therefore F₁, F , &c. vanish, and F. (a) = 1, F, (a + 2) =

qF, (a) = (a + 2− a) (a + 2 — b ) = 2 (a +2

F. (a + 4) =

qF, (a + 2) q² = 4.2 (a + 4 - b) (a + 2 − b) ' (a +4 - a) (a + 4- b)

Therefore

(a) = 1 +

9224 qx² + ... + 4.2 (a - b + 4) ( a − b + 2 ) 2 (a +b - 2)

Similarly we find F, (b + 2) , F (b + 2) , &c. , and thence stituting in (3610), we have

น=

3612

(b) ; and, sub

Ax²+

Aq²x +4 Aqxa +2 + ... + 2 (a -- b- 2) 4.2 ( a − b + 4) ( a − b +2)

+ Bx² +

Bq2ab+4 Bqxb+2 + ... + 2 (b - a +2) 4.2 ( b - a + 4) ( b - a + 2)

The solution is arrived at more quickly by formula (3607) .

We

q $ (D) = (D— a) (D — b)' (D–

have

.. ¢ (a + 2) = 2 ( a + 2 — b ) '

$ (a + 4) = 4 (a +4—b) ' &c.,

producing the same series by the value of

3613

( a) .

When f (D) has r factors each

Similarly with ♣ (b) .

D - a, the corres

ponding part of the value of u in equation (4) will produce 3614

4 + 4, log x + A , (log x)² ... +A ,-1 (log x) ”−¹,

where the coefficients A , A1, ... are each of the form Coxª + C₁xa + ¹ + C₂α¶ + 3 + ... 3615 But if any one of the quantities F, ( a + r) = 0 ( 3608 ) , then C, = 0 also.

540

DIFFERENTIAL EQUATIONS .

PROOF.-f (D) now contains a term ofthe form e²º (c₂ + c₂0 ... + c,or - 1) = ev, say. The corresponding part of u in (4) is

{ 1 + F₁ (D) e ° + ... } eª°~ = { eª° + e( a + 1 ) • F₁ ( D + a + 1) + e(a+2)•F₂(D + a + 2) + ….. } v

by (3475 ) .

Expand each function F by Taylor's theorem in powers of D, operate upon v, and arrange the result according to powers of 0.

In practice, proceed as in the following example. 3616

Ex.:

xu2x + u₂ + q²xu = 0.

Multiplying by ≈ and changing by (3489) , this becomes D'u + q'e²º u = 0. D'u = 0 gives u = A + BO. Substitute this value and operate with D, considering A and B as variables, and equate to zero the coefficients of the powers of 0 ; thus D³A + q²e²ºA +2DB = 0 , D³B + q³e²ºB = 0. Then change D into m , and eºA into am-r, to obtain the relations m³am +q³am-2 + 2mbm = 0 ; m³bm + q³bm-2 = 0 , which determine the constants successively in terms of a and b, (which are arbitrary) in the equation v = ao tay ta *+ ... + logæ (b + bc + b *+ ...) , which thus becomes the solution sought. [Boole, Diff. Eq., p. 439.

SOLUTION BY DEFINITE INTEGRALS . *

3617

is

Laplace's method. - The solution of the equation x$ (d₂) u +¥ (dx) u = 0

(1)

dt t +Svet de ( pt) -¹ } dt u= C •cf{e +S

(2) ,

the limits being determined by Svet _zt+Su =0 0 .... ttdt = e

PROOF.-Assume u = ezt Tdt, and substitute in (1 ) , putting u = efc= = (t) et (3474) , thus

(3). (d,) et

√ e* 4 (t) Tdt = 0 . [ xe*t4 (t) Tat + f This method of solution is merely indicated here, and the reader is referred to Boole's Diff. Eq., Ch. xviii., for a complete investigation.

541

SOLUTION OF DEFINITE INTEGRALS.

Integrating the first term by parts, this becomes

T— eat (t) Tf ex [ d, { ø ( t) T } − ↓ (t) T] d -S

. .......... . .4.) , an equation which is satisfied by equating each term to zero. The second term thus produces a value of T by integration by (3209) , and this value substituted in the first term, and in the value of u, gives the results (3) and (2).

3618

Ex. ( 1 ) :

xu2x + au - q²xu = 0*

....

........ (4) .

Here (d ) = d , —q², 4 (d₂) = ad¸, ¢ (t) = tª — q², 4 (t) = at. (2) and (3) become -1 u = C | e≈t (tª³ — q') ; ´¹dt ; eªt ( t −q¹³)i = 0 ; a being positive, the limits are t

Hence

± q, and, putting tq cos 0, we find

u = 0 ew cos ° sin♥ -1 Ꮎ dᎾ ... ... of

(5) .

3619 The solution in series by (3608) is as follows. (2) of that article are in this case

Equations (1 ) and

D (D + a- 1) u - qeu = 0

and

Thus, a in (3608 ) = 0 , and b = 1 - a .

น

m (m + a − 1 ) vm − q²vm-2 = 0 . Therefore (3610) becomes

q²x² 9*2* = 4 { 1 + 2 (a + 1 ) + 2.4 (a + 1) (a + 3) + &c. } + Bx¹-

9*2* 1+ ..... (6) . + &c. } { 1 + 2 (3 — a) + 2.4 (3 — a) (5 —a)

Both series are convergent by ( 239 ii . ) .

The results deduced by Boole are these

3620 (5) is equivalent to the particular integral represented by the first series of (6).

3621 A second particular integral, by assuming u = el - a) v, is found to be, when 2 - a is positive, r COSO sin¹- 0do ……………….. (7) . u = O₂x¹-a ["ea 0

3622

When a lies between 0 and 2, the complete integral is r u = C₁ ea cose° sinª¬¹0d0 + С₂ï¹ - a e co sin¹S0 S0 afte

do ......... ………………….. (8) .

* The method by definite integrals is elucidated by Boole chiefly in the solution of this important equation.

542

3623

DIFFERENTIAL EQUATIONS.

But, if a = 1 , the solution becomes excose

น = S

(9).

4+B log (≈ sin³ 0) } do {A

3624 If a does not lie between 0 and 2, then, if a be negative, put aa - 2n, and replace the first term of (8) by earcose sing -¹do ... (10), C¸ (xd¸ + a ′ −1 ) (xd¸ + a ′ —3 ) ... (xd¸ + a ' — 2n + 1 ) (* the transformation being effected by (3580) .

3625 And if a be positive and > 2, put u = e(l- a) • v = l - av . This con verts a into 2 — a, a negative quantity, and the case is reduced to the last one.

3626

Ex. (2 ) .— To solve by this method the P. D. equation Uqx + U2y+ U2z = 0 ...............

..... ( 1 )

when r = √ (x² + y²) . Eliminating

and y, ( 1 ) becomes

ru₂ + ur + ruzz = 0

……………… (2) .

Now the solution of this equation is number (9) of Example (1) , if we change a into r, q into id₂, and A and B into arbitrary functions of z. We thus obtain

u=

…………… . (3) , Ser 0 ercoso id₂ { p (2) +4 (2) log ( r sin³ 0) } de …………....

or, by (3492) , u= { z + ir cos 0 } do ++ (z + ir cos 0) log (r sin³ 0) d☺………… .. (4) . S = &{ See ( 3551 ) for another solution.

3627 If u be the potential of an attracting mass at an external point, and if u F ( ) when r = 0 ; then, since logr∞ , (z) must vanish ; therefore

F (-) = ["4 0 (2) d0 = *4 (x) . Hence (4) reduces to

น =

== π

F { z + ir cos 0 } do. 0

Parseval's Theorem. 3628

and

If, for all values of u, A + Bu + Cu² + ... ==$ ( u) -2 A' + B'u-¹ + C'u² + ... = 4 ( u) ..... .....

(1) ,

DIFFERENTIAL RESOLVENTS.

543

then 1 AA'+ BB ' + ... = = 2π 2 =ƒ˜” [ ¤ (e¹ ) & (eº ) + ¢ (e¯ * ) * (e−*)] do PROOF. - Form the product of equations (1), and in it put u = e¹ and e - iº separately, and add the results. Multiply by de, integrate from 0 to π, and divide by 2π.

P.

D.

EQUATIONS

WITH

MORE

THAN

TWO

INDEPENDENT VARIABLES .

3629 By means of Fourier's theorem ( 2742) , the solution of the equation Uzt− h² (U2x + U2y + Uzz) = 0 may be deduced by a general method in the form

u = (1 + də)SSSSSS ei (4+Bht) ↳ ป (a , b, c) da db dc dλ dµ dv, the limits of each integration being ― ∞o to co , and the func tion being arbitrary and different in the two terms arising from the operator (1 + de) . Boole, Ch. xviii. , and more fully in Cauchy's Exercice d'Analyse Mathé matique, Tom. I., pp. 53 et 178.

3630 Poisson's solution of the same equation in the form of a double integral is π 2π u = (1+ (1 + dt) ("* S ** t sin (x +ht sink sinn, y +ht sin cos n , z + ht cos ) didn 0 0

with the same latitude in the interpretation of 4. [ Gregory's Examples, p. 504.

DIFFERENTIAL

RESOLVENTS OF ALGEBRAIC EQUATIONS .

3631 THEOREM I. (Boole) .-" If / y 1, Y2 ... Yn are the n roots Y₁, of the equation ...... (1), y" —ay" -¹ + 1 = 0 .....

and if the mth power of any one of these roots be represented

DIFFERENTIAL EQUATIONS.

544

by u, and if a = eº , then u as a function of differential equation

u− (n = ¹D + m −1 )" " (2 - m −1 ) [ D

satisfies the

] -¹e" u = 0,

and the complete integral of the same will be m m m72 u = Cig +Cu +… ... + Cu

3632

" COR. I.- If m = -1 and if n be >2 , the differential

equation

D(n-1) u

Ꭰ -1 )(" -" ) , eno u = 0 n 1 D - 1-1 n n (n =

has for its general integral

u = · С₁₁¹ + С₂у¹ ... + Cμ - 19-12 y , y ... Y - 1 being any n - 1 roots of (1) .

" If be changed into -0 , and therefore D into -D, the above results are modified as follows : 3633

" COR. II.- The differential equation

u− (D− 1) ' [ (" = ¹ D− m) "

' ( P + m )] ¯* e* u = 0

has for its complete integral m m u = C₁₂ + C₂y ... + Cnyn , Y1 Y2 ... Y

being the roots of the equation ay" —y" -¹ + a = 0

3634

" COR. III .-The differential equation \ eno u— n (D − 2)(* - ¹) [ (n=1 D+ 1 n ) * ¯¹ ]¯`*e* u = 0,

supposing n > 2 has for its complete integral u = Civil+ C, z

... + C -13-2 ,

Y₁, Y₂ ... Y.-1 being any n - 1 roots of ( 2) .

(2) .

DIFFERENTIAL RESOLVENTS.

545

" Theorem II. (Harley) . 3635

" The differential equation

m n a² ( xd,¸) '") u — ( " = " xd, + n − 1 ) " " (

a xd - m - 1) " xru

=0 is satisfied by the mth power of any root of the equation

y" —xy"- " + a = 0, u being considered as a function of a.

3636

“ COR. - The differential equation nn

(n - r) n-r ( " =" xd — m²) " ¯ " (xd ) () u r ጥ -(n - 1 ) (

(. ad.-- 1 ) au = 0

is satisfied by the mth power of any root of the equation

y" —ny"-" + ( n - 1 ) x = 0. ” [Boole, Diff. Eq. , Sup. 191--199. 3637 See also Boole, Phil. Trans . , 1864 ; Harley, Proc. of the Lit. and Phil. Soc. of Manchester, Vol . II .; Rawson, Proc. of the Lond. Math. Soc., Vol. 9.

4 A

CALCULUS

OF

FINITE

DIFFERENCES.

INTRODUCTION.

3701 In this branch of pure mathematics a function is denoted by u , and ( + ) consequently by x +h

( ) The

increment h is commonly unity. If Ar denotes the increment h , and Au, the consequent increase in the value of u , we have ▲Ux = Ux+ Ax— Ux•

3702

3703

When A

diminishes without limit, the value of

Aux

dux

Ux+ Ax ― Ux is

or Δ Ax

dx

Ax

3704 The repetition of the operation A is follows :

ΔΔu 2 = Δu ,

3705

3707

1 1 3 2

2 4 5 2

3 4 5 9 16 25 7 9 ... 2 ... ...

... :::

FORMULE

3706

and so on.

Ex.: Let u₂ = x²,

x = x² = Ax² = A²x² =

If

ΔΔ κα = A³ux,

indicated as

... ...

FOR FIRST AND nth DIFFERENCES .

*x =a

+b

+ ca ”- + & e .,

A'u = an (n − 1 ) ... ( n − r + 1 ) æ” ¯r + c¸æn -r-¹ + &c., Anu ∞

an (n - 1 ) ... 3.2.1 .

FORMULE FOR FIRST AND nth DIFFERENCES.

547

3708 Hence the nth difference of a rational integral func tion of the nth degree is constant .

An " = 1.2.3 ... n.

3709

So also

3710

NOTATION. - Factorial terms are denoted as follows :

UxUx- 1 ... Ux −m+ 1 = u(m) x " 1 = u ( m) · 20 Ux + m - 1 WxUx+1

3711

3712

Thus

x (x− 1 ) ... (x −m + 1 ) = x(m) "¸ 1

= x( m).

3713 x (x + 1 ) ... (x + m - 1 )

m , m !, and mm) are equivalent symbols .

Hence

3714 According to ( 2452) , (m) would here be denoted by am) . The suffix, however, being omitted , it may be understood that the common differ ence of the factors is always -1 .

3715

Ax(m) = mx(m - 1) ,

▲nx(m) = m (n) x(m− n) ,

m"(m), ▲ mx(m) = m

and, if m < n, Ana (m) = 0 , since Ac0 if c = constant .

3718

Anx( m) = (

= — mx(−m− 1) , Ax -m) —

m)(n) x -m- n) .

Au(m) = (ux + 1 - ux - m + 1 ) um− 1 "

3720

u¯m - m − 1) . Δι Au!-m) = (u₂ux+ m) u! Ex.: 3722 ▲(ax + b)( - ) = = —am - am (ax + b)( - m − 1), ▲ (ax + b)(m) = am (ax +b) (m - 1),

3724 3726

Alog u x = log { 1+ Aའu }, ླ

Δα* = (a − 1) a*,

Ux +1 A log um-1) = log 20 Ux - m +1

Anamx = (am — 1)" amx.

a 2 ein COS { ax +b + " (a2+*) } , COS (ax +b) = ( 2 sin =)"

sin

3728

ΔΗ

PROOF.

A sin (ax +b) = sin (ax + a + b)-sin (ax + b) a = 2 sin sin ( ax + b ++* ).

μ+D That is, ▲ is equivalent to adding a sine by 2 sin 2. 2'

to the angle and multiplying the

2

548

CALCULUS OF FINITE DIFFERENCES.

3729

Conversely, the same formula holds if the sign of n be changed throughout . EXPANSION BY FACTORIALS . 3730

If A"

(0) denote the value of A"

(x) when x = 0 ,

Δφ then $ (a) = $ (0 ) +Aḍ (0) x + A²º (0) ∞( v®2)+ + 4³ (0) x(3) + &c. 2.3 2 3731

If Ax = h instead of unity, the same expansion holds

good if for A" (0) we write (A" a0 after reduction .

(x) ÷ h") = 0 ; that is, making

PROOF .-Assume (x) = a + a₁x + ɑqx( ²) + ɑzx(³) + &c . Compute Ap (a) , ▲ªp (x) , &c., and put x = 0 to determine a , a , a,, &c. GENERATING FUNCTIONS . 3732 If ut be the general term in the expansion of (t) , then ( t) is called the generating function of u, or ( t) = Gu Ex.: ( 1 - t ) ² = G (x + 1 ) , expansion.

for

+1 is the coefficient of t

in the

Gux+n = $ (t) Gux+1 = $ (t), ... tn

3733

Gu = 4 (1) ,

3734

Gau , = (-—-— 1) 4 (1) , … .. GA" u, = ( — −1 )** (1).

PROOF.

GAU

Gux+1 - Guz, &c.

THE OPERATIONS E, A, AND dr. 3735

E denotes the operation of increasing a by unity, Eu¸x = Ux+1 = u¸ + Au¸ = ( 1 + ▲) u¸•

The symbols E and ▲ both follow the laws of distribution , commutation, and repetition (1488-90 ) .

3736 PROOF.-

E = 1 + A = ex

or

CD *

1 Eu = Ux + 1 = Ux + dxux + ½ d²x²x + 2.3 du₂ + &c. 1 x = eªzU uz• 2¹² 3 dax + &c.) u₂ = =· (1 + dx + } d 2x + 2.3

By ( 1520), Ae being unity. * The letter d is reserved as a symbol of differentiation only, and the suffix attached to it indicates the independent variable. See (1487) .

FORMULE FOR FIRST AND nth DIFFERENCES.

Ae" -1

3737

Hence

3739

Consistently with (3735) E-¹ denotes the diminishing E-¹ux 30

a by unity ; thus For Eux-1 =

x,

and

549

. Uz- 1

D = log E. D

Ux- 1 .

E- ¹u

ux+n in terms of u, and successive differences. 3740

Ux + n = Ux + n ^u» + C ( n , 2) ▲²u¸ + C ( n , 3) ▲³ux + &c .

PROOF. (i. ) By induction, or (ii. ) by generating functions, or (iii. ) by the symbolic law : n Gux+n (ii.) Gill, .. = ( -) " () = { 1+ ( −1 ) } ** ( 0. Expand by the Binomial theorem, and apply (3734) . Ux+ n = E" u₂ ux = ( 1 + A ) " u¸ ux•

(iii.)

Apply the laws in (3735) by expanding the binomial and distributing the operation upon u Conversely to express A" u, in terms of ur, U +19 Ux+2, &c. 3741

Au₂ = Ux+n − nux+ n -1 + C ( n , 2 ) Ux+n− 2 ... ( −1 ) ” Ux•

PROOF.

Aux = (E - 1 ) " (3736) . Expand, and apply ( 3735 ) as before. Putting x = 0 , we also have 3742

▲ Anuo "u。 = u₁₂ - nu₂ - 1 + Cn, 2un - 2 ... ( —1)" u。.

3743

▲" x" = (x + n) ” — n (xn − 1)" + C (n , 2) (x + n − 2)™ — & c .

3744

▲" 0" = n” —n (n − 1 ) " + C ( n , 2) (n − 2)™

-C (n , 3) (n - 3) 3745 Ex.: By (3717) ▲"0" = n ! obtained .

3746

▲" uv

+&c.

Hence a proof of theorem ( 285) is

= ( EE ' - 1 ) " u¿V.x

where E operates only upon u, and E' only upon vx. PROOF.

Aux — Ux + 1 Vx + 1 — Ux Vx = Eux . E'vx — u¸²¸ = ( EE′ — 1 ) UxVx·

Applications of (3746) . 3747

Ex. ( 1) :

xVx = ( -1 ) " ( 1 - EE') " UxVx• ▲" uv u₂ vz•

Expand the binomial, and operate upon the subjects u , v,; thus 3748

" u,v, = ( - 1 )” { Uz vz − NU₂+ 1 Vz + 1 + C (n , 2) Uz+ 2 Vz+2— &c. } .

CALCULUS OF FINITE DIFFERENCES.

550

3749

Ex. ( 2) : To expand a sine by successive differences of sin æ. n A" a sin x = { E ( 1 + ▲′) − 1 } " a ′ sin æ = { ▲ + E ▲' } " a² sin æ

= { A" + nA” -¹EA' + C ' (n , 2) ▲" -² E² ▲² + &c . } a sin e +1 -2 = ▲" a*. sin x + n▲ -¹az + ¹ ▲ sin x + C ( n , 2) ▲ n - ² a² + 2 A² sina + &c. = a* { (a − 1 ) " sinx + n ( a− 1 ) ” - ¹ a ▲ sin x + C ' (n , 2 ) ( a− 1 ) " -² a² ▲² sin x + &c. } , by (3727) , while ▲" sina is known from (3728) .

3750 Ex. (3) : To expand A" u, v , in differences of u, and v, alone : put E = 1 + A , E' = 1 + A′ in ( 3746) , thus ▲¹µ¿vx = (A + A′ + AA') ” Uz Vz which must be expanded.

Au, in differential coefficients of u . +1 n+ 2 3751 ▲" u₂ ux = d" u¸ + A₁d™ +¹ + ød™x ²u¸x + & c .

PROOF.

▲"u¸ = (eª² — 1 )” u, (3737) . Expand by ( 150) and ( 125) as if d, were a quantitative symbol. (3761) .

See also

d"u in successive differences of u . dxn dru

=

3752

{ log (1 + 4) } " u.

dan The expansion by (155) and (125) will present a series of ascending differences of u.

PROOF.

3753

edx = 1 + A,

Ex.: If n = 1 ,

..

du = Au dx

d₂ = log ( 1 + A) .

Δu A²u + 3 2

A¹u + &c. 4

If C be a constant , 3754

(D ) C = 4 ( ▲ ) C = p (0 ) C

and

( E) C = 4 ( 1 ) C,

Since every term of (D) , or of ( A ) C, operating upon C, produces 0 ; and every term of (E) operating upon C produces C.

HERSCHEL'S THEOREM. 3757 3758

$ (e²) = $ (E) eº.t =

t2 (1) +4 (E) 0.t + $ (E) 0² . 1.2 + &c .

INTERPOLATION.

551

(et) = A + A₂et ... +Anet

PROOF.- Let

= Aet + A₁Ee.t ... + A₂E"e" .t = ( A + A₂E ... + A„ E„) eº. ¿ 0212 = ¢ (E) eº . t = ¢ (E) { 1 + 0.t + 1.2+ &c. } , ( 1 ) by ( 3756) .

(E) 1

and

A THEOREM CONJUGATE TO MACLAURIN'S ( 1507) . 3759

$(t) = $ (D) e° .t

12 = 4 (0) +4 (d ) 0.t + 4 (d ) 0² 1.2 + & c .

3760

PROOF.

$ (t) =

(loge') =

= 4 (D) e.t (3738 ) =

(log E) et (3757)

(D)

1 + 0.t ++ &c. &c. } , { • ( D) 1 = ¢ (0 ) ( 3754 ) .

and

n being a positive integer , 3761

dru Anu =

An On + 1

dn + ¹u

AnOn +2

dn+2

+

+

dxn

1.2 ... (n + 1) dan+1 + 1.2 ... (n +2) dx +2

PROOF. -By ( 3758) , putting

( et) = (et — 1 ) ", t2 + &c. (e* —1 )" = (E — 1 ) " 0.t + ( E − 1 ) " 0º . 1.2

A" 0.t + A" 0² .

12 + &c . 1.2

Put td , and employ (3736 ) and (3737) .

INTERPOLATION.

Approximate value of u, in terms of n particular equi distant values. 3762

If u 2 is an integral algebraic function of the degree n − 1 , ▲" ux vanishes , and therefore by making a = 0 , and writing a for n in (3740) ,

3763

u₂ = u₂ + x^u₂ + Cx , 24³uo ... + Cx , n - 1 ▲ -¹u .

This is formula ( 265 ) .

The given values are u。, ▲u。 , ▲²u。,

& c . , corresponding to a , b , c , ... 3764 For an application of the formula to the problem of interpolation, see (267) , in which example = 154 and u = log 72:54.

552

CALCULUS OF FINITE DIFFERENCES.

3765

When the term to be interpolated is one of a set of = " u0, as in ( 3762) ;

equidistant terms , employ (3741) . therefore 3766

u - nu - 1 + Cn, 2 Un - 2 — Cn, 3

n -3 ... + ( − 1 ) ” u = 0 .

3767 Ex.: From sin 0, sin 30°, sin 45°, and sin 60° , to deduce the value of sine 15°. The formula gives sin 0-4 sin 15° +6 sin 30° -4 sin 45° + sin 60° = 0, or

-4 sin 15° 3−2√2 + } √3 = 0, sin 15° = } ( 6-4 √2 + √3 ) = " 2594.

from which

The true value is 2588 ; the error 0006.

LAGRANGE'S INTERPOLATION FORMULA. 3768 Let a, b, c, ... k be n values of a, not equidistant , for which the values of u x are known ; then generally 3769

ux = Ua

(x —b) (x −c) ... (x −k) (x — a)(x−c) ... (x − k) + ur (a—b) (a — c) ... a- k) b— a) (b − c) ... (b − k) (x− a) (x −b) (x − c) ... +Uk

(k— a) (k — b) (k— c) ….. '

PROOF.-Assume

... (x −k) u₂ = A (x − b) (x −c) …

+ B (x − a) (x − c) ... ( x − k) + C ( x − a) (x − b) (x − d) ... (x −k) + &c., and determine A, B, C, &c . by making a = a , b, c, &c . , in turn .

If the values of a , b, c, ... kare 0, 1 , 2 , ... n - 1 ,

( 3769)

reduces to x (x − 1 ) ... ( x − n +2) Ux = Un− 1 1.2.3 ... (n- 1)

3770

x (x− 1 ) ... ( x − n + 3) ( x − n + 1 )

--n- 2

1.1.2.3 ... (n — 2)

+ Un-s

x (x - 1 ) .. ... ( x − n +4) ( x − n + 2) ( x − n + 1) _ &c . , or 2.1.1.2.3 ... ( n −3)

3771 Ux =

x(n) Un-1 Cn - 1,2 Un –3 — - Cn - 1,1 Un - 2 &c . } . + 2 n x− 3 + x- n + 3 =1 (n = 1) : { x = n + 1

553

MECHANICAL QUADRATURE.

MECHANICAL

QUADRATURE .

= The area of a curve whose equation is yu, in terms of n + 1 equidistant ordinates u, u19 ... un, is approximately

3772

nu+ + 2 su x + nu+ +( (

− 1 ) + 4 +(

− x² + » ²) —

n5 3n¹ Δu + ( x²3y² +11m² '" 3 - 3n²) = 5 4!

no 35n¹ -2n³ + + (27/03 6 4

Ağu

50n³

+12m²) 3

5!

15n6 +17n³ — 225n * +(W -15 -

4

Au

274n³ ·60n²

+ 6

3

n PROOF.-The area is = ude. Take the value of u S0 U U₁ ... u₂ - 1 from (3763) and integrate.

6! in terms of

2

udx = u +4u₂ + u². 3 Su

3773

When

3774

n = 3,

Su,dr = 3 (n + 3u, +3u ;+us ),

3775

n = 4,

u₂dx = Sou

n = 2,

14 (u + u¸) +64 (u₁ + u3) + 24u2 45

3776 3 n = 6,

Su,dx = 10 { } { u + us + u , + us +5 ( u ,+us) +6us} •

In the last formula , which is due to Mr. Weddle, the co efficient of Au is taken as 10 3 instead of 146, its true value . These results are obtained from (3772) by substituting for each ▲ its value from ( 3742) .

COTES'S AND GAUSS'S FORMULE. These give the area of the curve directly in terms of 3777 fixed abscissæ. They are obtained by integrating Lagrange's value of " (3769-71 ) , and are fully discussed in articles (2995–7) . 4 B

CALCULUS OF FINITE DIFFERENCES.

554

LAPLACE'S FORMULA.

Su_dx = = "

3778

un " + " +", ... + 1/2

1 − 1 (Au¸— ▲u ,) + 24 ½ (A³µ‚— ▲²u,) —&c., the coefficients being those in the expansion of t { log ( 1 + t) } - ¹.

Δ Aw₂ = d₂ { log ( 1 + A) } wr,

PROOF.

by (3736) ,

Δ Δ' 1+ = d, { 1 +/1 / 2 + 21-19 24 720 A + & c. } w 2 > / Hence, putting u₂ = Awx,

file = " + 2 Suda

- A12 + A& c. , 24

U₂ uzdx 24 2 ., &c - A12 + AS "_dz = " +" and so on ; then add together the n equations.

3779 Formula (3778 ) contains Aun , A²un , &c . , which cannot be found from uo, U1 ... Un • The following formula does not involve differences higher than Aun -1. Աջ ... = + "/2 " ude = 1/2 + " + " [

3780

-

1 1—12 ( Au „ -1 — Au, ) — —4 (A²u „ -1 —A³u«) — &c. 24

Δ -AE-1 PROOF. — In the proof of ( 3778) , change log ( 1 + 4 ) into E log (1 -AE -¹)' and put E- ¹w, w₂-1 (3739 ) after expansion, and proceed as before.

SUMMATION

3781

Definition :

3782

Theorem :

Eu,

OF

u +

SERIES .

a+1 +

a + ... + u₂ - 1 ·

Σu₁₂ = A¯¹u₁₂ + C,

where C is constant for all the assigned values of x.

SUMMATION OF SERIES.

PROOF. -

Let

(2) be such that Ap (x) =

555

x, then

(x) = ▲ -1 ¯¹ux,

therefore u = ( a +1 ) − ¢ (a) . Write thus, and add together the values of Ua, ya + 1, ... Uz- 1 . Therefore, by (3781 ) , Eu, = p ( x ) — (a) = ▲ -¹u - q (a), and (a) is constant with respect to x. Taken between the limits xa , x = b - 1 , we have the notation , x= b- 1 Συ 3783 Zuz or Eu = Σκ - Συ = Δ--Δ - Μα x =α

Functions integrable in finite terms : x(m +1)

3784

Class I.

Σx (m) =

+ C. m+ 1

(ax+b) (m +1) 3785

+ C.

Σ (ax +b) (m) =

a (m + 1) (-m +1) 3786

Class II.

Σx(-m) =

+ C.

By (3718), i and notation (3711 ) .

-m + 1 (-m + 1) = C− (ax +b)

3787

Σ (ax + b) (- )

a (m− 1 )

*

Formulæ (3785) and ( 3786 ) are equivalent to the rules (269) and ( 271 ) . They are the direct results of theorem (3782).

ax Σa* Σα =

3788

[By ( 3726) . a-1

Class III -If u, be a rational integral function, 3789

Σa+x−¹U¸x ¹u₂ = { x + Cx, 2 ^ + Cx, 3 ^² + ... } ua. Σa+

PROOF. -By (3735) and (3736 ) ,

Ua + Wa + 1 ... + Wa + x - 1 = ( 1 + E + E² ... + Ex - ¹) va =

Ex- 1 (1 + A)* −1 Wa Wa = E- 1

= the expansion above.

3790 The formula has been given at (266) and an example of its appli cation. The series there summed is 1+ 5 + 15 +35 + 70 + 126 + to 100 terms . The function u, which gives rise to these terms is found by (3763 ) to be ux = (x² + 10x³ + 35x² + 50x +24 ) ÷ 24.

CALCULUS OF FINITE DIFFERENCES.

556

-1 3791 If this function be presented as u,, and u , be required, we first find u = 1 , u , = 5, u, = 15, &c.; then the differences Au , A³u , ... A⭑u = 1 , 4, 6, 4, 1 , and then , by ( 3789) , the required sum, as in the example re ferred to .

3792 For another example, let Σ" æ³ = 1 + 2³ ... + n³ be required . Here Ax³ = 3x² + 3x + 1 , A²x³ = 6x + 6, A³x³ = 6, therefore

401,

A306,

= ................. ……………. (1), 406

may now be expressed in factorials, and the summation may then be effected by (3784) . First, by (3730) , x³ = x + 3x (x − 1 ) + x ( x − 1 ) ( x − 2) ;

therefore, by (3784) ,

Σ" x" = Σ (n + 1)* (3783), — n² (n + 1)* Σ" x³ = n (n + 1 ) + 3 (n + 1 ) n (n − 1 ) + (n + 1 ) n ( n − 1 ) ( n − 2) = 2 4 3 4

Otherwise, by (3789) , taking a = 0, we have u₂ = x³, u = 0, Au = 1, A²u = 6, A³u, = 6, as above. Therefore -1 6n (n − 1 ) (n - 2) + En (n - 1 ) (n - 2) (n - 3) = n² (n - 1) n (n − 1) x" ~ ' u, = n ( n − 1 ) + 4 2 1.2.3 1.2.3.4

3793

therefore, changing n into n + 1 , Σu, =

(n + 1 )² n³ 4

3794 Class IV. -When the general term of a series is a rational fraction of the form A + Bx + Cx² +

"

where u = ax + b,

Ux Ux +1 ... Ux+ m and the degree of the numerator is not higher than a +m - 2 ; resolve the numerator into

A' + B'u₂ + C'u¿U¸x + 1 + ... + D'UzUx + 1Ux +2 ••• Ux + m− 29 by (3730) .

The fraction then separates into a series of frac

tions with constant numerators which can be summed by (3787).

3795

If the factors u ... U +m are not consecutive , introduce

the missing ones in the denominator and numerator, and then resolve the fraction as in the foregoing rule .

3796

Ex.: To sum the series

1 1 1 + to n terms. + + 2.5 3.6 1.4

SUMMATION OF SERIES.

557

1

(n + 1 ) (n + 2) n (n + 1 ) (n + 2) (n + 3 ) 2 1 n (n + 1) + 2n + 2 = = + n (n + 1 ) (n + 2 ) (n + 3 ) (n + 2) (n + 3) (n + 1 ) (n + 2) (n + 3) 2 + n (n + 1 ) (n + 2) (n + 3) ' The sum of n terms is, therefore, by the rule ( 271 ) , 2 1 2 (1 + 3 )+1 3 -n1 (1 − 3) + (2² 3 - ( +2 )( +3 ) + 3 ( 2.3 − (n + 2) + 3) + 11 )) (n 3) (« + 2) ((n + (n + 11 3n² +12n + 11 = 18 3 (n + 1) (n + 2) (n + 3)' If the form in (3787) is used, the total constant part C is determined finally 11 by making n = 0, which gives C = 18 =

The nth term is

n (n + 3)

3797

Theorem .

ƒ( E) a*p ( x) = a * f(aE) ¢ (x) ,

f being an algebraic function . PROOF.

Let ae , then the left

D+ f (E) em² e' p (x) = ƒ ( e") ex p (x) ( 3736 ) = exƒ (eP + m ) ø (x) (3475) =ƒ = a*f(aE) (x).

Class V.-If Ф (a) be a rational integral function , 3798

ar Za² + (x) == a " _" "

a² − A$ { • ( x) — a _ —_ — __1 ^p ( x) + (a— ~ 1)? A*• ( x) − &c. } .

The upper limit is understood to be a - 1 , and a constant is to be added , (3781-2) . -1 PROOF.— Zap (x) = ▲ ¯¹a*p (x) = ( E − 1 ) -¹ap (x ) = a* (aE − 1 ) ´¹ ¢ (x) -1 -1 ax ( 3797) = a* { a (1 + ▲) −1 } ˜¯`• (~) = a-1 _~_ ~1 (1 + _ — _ — _1 ) • (x). Then expand the binomial.

Zap (x) in successive derivatives of p (x) . A2 an 3799 £u*p(x) = a - · { 1 + 4 , $'( x) + 1.2 4.3 $¨ ( x ) + &c. } , -1 E - 1 -1 αΔ An On . where 4. = (aa-1 — 1 ) " 0" = ( 1 + a"- ) l "0". PROOF.-By (3757) , 4 (e¹ ) = ↓ (E) e°.D ; therefore (see last proof) a² (aE- 1 ) -¹ (x) (putting E = eº ) = a* (aE - 1 ) -¹ eº · D p ( x) a* 02 ~1 (1+ 1 + 0.0 0.D + = a-l ) { 1+ + 1 } ;# ( #). 1.2;D² + &c .· }

558

CALCULUS OF FINITE DIFFERENCES.

Ex .: To sum the series 2.1 + 4.8 + 8.27 + 16.64 + to n terms.

3800

We require 2

+ Σ2*x³ = 2 *x³ + 2² ( x³ −24x³ + 2²A²x³ — 2³A³µ³) = 2ºæ³ + 2º { x³ −2 (3x² + 3x + 1) +4 (6x + 6) −8.6 } = 2º { 2x³ — 6x² + 18x - 26

-n 3801 If A-" u, be known for all integral values of n , and if v be rational and integral, Σ κ Σu¸v¸ = Σ'u¸ • Vx -1 − Σ²ux . Avx - 2 + Σ³ux · A²vx - 3— &c . PROOF. Euv x = ( EE' — 1 ) −¹ u¸v¸ = (▲ E′ + ▲) −¹ u,v (3746) and (3736 ) . Expand the binomial operator, observing ( 3738) .

3802 Σ"u¸v¸ = u„Σ "v¸ - n ▲ un + ¹vx + 1 + Cn , 2 A² u₁₂

+ 2vx + 2- &c .

PROOF.

Σ"Uxvx = (A´ + AE ′) - " u¸vx, as in ( 3801 ) , -N -n -2 = ▲˜" vxx ― n (A - " - ¹E) v¸▲ u₂x + Сn,? (A˜ " -²E²) v „A²u , - &c .,

producing the above by (3735) and ( 3782) . Observe that, in (3801 ) and (3802 ) , two forms are obtainable in each case by expanding the binomial operator from either end of the series.

3803

Ex.: To sum the series sin a + 2² sin 2a + 3' sin 3a + to æ terms.

The sum is a² sin ax + x² sin ax. Taking u, = sin ax and v₂ = x², we know A-" sin ax, by (3729) ; therefore (3801 ) gives Σæ² sin ax = ( 2 sin §a) ¯¹ sin { ax— § (a + π ) } ( ≈ −1) ² -3 — (2 sin a) -² sin { ax − (a + π) } ( 2.x −3) + ( 2 sin la ) ¨³ sin { ax− (a + π) } 2.

APPROXIMATE

3820

SUMMATION.

The most useful formula is the following

B, du - B₁ d³u + &c. Σu 31 2u, = C + Su, dx + 1 2 + 21 de 4 / ! de

1 Su‚d = C+ c + Su , de + 2+ + 12 x+

dur dx de 1

――――

1

d³u

1

d'ur

30240

dx5

+ 720 dx³

ďux

1

d³u*

47900160

dax⁹

&c .

+ 1209600 da PROOF.- Eu , = (e" -1 ) -¹u . of x.

Expand by (1539) with D in the place

APPROXIMATE SUMMATION.

Ex. 1 : The value of

" at (2939) is given at once by the formula.

1 1 ... + imately, Ex . 2 : To sum the series 1+ 1242+ 1 3 x approx

3821

1 x + Put

559

- 1 - 1 + 1 1 = x 1 + C + logx — 2x 12x³ x 120x

= 10 to determine the constant ; thus 1+ 1 + 1 ... + 1 = C + log 10+ 1 2 3 10 20

from which C = 577215, and the required sum is 1 1 - 1 •577215 +log + + 2x 12x2 120x¹

3822

Ex. 3 :

1 + &c., 1200

&c .

1 1 1+ + 31/323+ 33 + 48 + &c . ,

1 =C Σ 12/ 1 / 143 2.x2

Σ113

&c.

1 2x³

1 1 = 14+ 124+ ე3 4

5B4 7B9B 3B2 + + 2.¹ 2x6 2x8 2x10' 1 -- 3 5 + 11/2+ 12 20 12

The convergent part of this series, consisting of the first five terms , is an approximation to the sum of all the terms. 3823 A much nearer approximation is obtained in this and analogous cases by starting with the summation formula at a more advanced term .

E.g .:

1+

1 1 1 1 + + + Σ39 23 33 43 x3

2035 + 1 + 1 + 3B₂ --- 5B + &c. 2.54 2.50 2.52 2.53 1728 1 1 1 2035 + + + 250 2500 50 1728

1 + &c. 187500

The converging part now consists of a far greater number ofterms than before, and the convergence at first is much more rapid.

3824 Ex. 4: The series for log F ( +1 ) at (2773) can be obtained by the above formula when a is an integer. For, in that case, log F (2 + 1 ) = log 1 + log 2 + log 3 ... +log x = log a +

loga,

and (3820) gives the expansion in question, the constant being determined by making a infinite.

3825

dx Formula (3820) may also be used to find fud e by the

process of summation , and Laplace's formula (3778) .

thus

answers the purpose

of

CALCULUS OF FINITE DIFFERENCES.

560

Zu, in a series of derivatives of u,. 3826

Lemma.— n — 1 ! (e² −1 ) − ” = ( —1 ) " − ¹ (d, + n — 1 ) (d, +n − 2) ... (d₂ + 1 ) { e' − 1 } -¹.

PROOF. -Put v, for n - 1 ! ( e - 1 ) -" . Then Vn+ 1 = − (d₂ + n) v₂ = (d, + n) (d, + n − 1) v „ - 1. "u may now be developed.

3827

Ex.-To develope

=

u,,

(Boole, p . 97)

-1 2 (e² − 1) −³ = («₂ + 2) ( d₂ + 1 ) { e' − 1 } -¹ -1 (d + 3d, + 2 )

+44 +&

with 420, and -

A2r + 1 = ( - 1 ) " B₂r + 1 ÷ ( 2r + 2 ) ! = ts

3 2 t2 + ²² + (24, + 34, −1 )

+ Σ° { ( r + 2) ( r + 1 ) Ar + 2 + 3 (r + 1 ) Ar.1 + 2A, } ť. Therefore, changing t into d,, we get 3 19 du- &c . Σu₂ = u₂dx Uz+ 2 dx Ju u₂ dx + , = 3 8 240 dx SS SSS

3828

¹u, in a series of derivatives of u̟ - ņ .

Let

a" cosec" x = 1 - C² + C

* - &c . , then

D 2 Σ"-ux

= D - " { 1+ C₂ (2 )² + C. ( 2 ) * + &c. } u₂ [Boole, p. 98. 3829 $ (0) − + (1 ) + 4 (2) - & c . =11/11 { 1 — — + 4

- & c . } $ (0 ) .

By this formula , a series of the given type may often be trans formed into one much more convergent .

PROOF.

The left =

1 1 1 1 $ (0) = 2 + 1 $ (0) = - (0), 21 + 14 1 +E

the expansion of which is the series on the right. 1 - 1 - 1 + &c. Summing the first six terms, Ex.-To sum 1 2 + 3 4 37 1 -1 it becomes + + &c. Taking $ (0 ) = ( 0 + 7) ´¹, 8 60 7 2 1 1 1 1 (1 2.3 + &c . = + + + 2.7.8 8.7.8.9.10 + &c. }. 4.7.8.9 7 2 8 7 3830

The sum after six terms converges rapidly by this formula, and more rapidly than if the formula had been applied to the series from its commencement .

PLANE

COORDINATE

SYSTEMS

OF

GEOMETRY.

COORDINATES .

CARTESIAN COORDINATES . 4001

In this system (Fig . 1 ) * the position of a point P in a

plane is determined by its distances from two fixed straight lines OX, OY, called axes of coordinates . These distances are measured parallel to the axes .

They are the abscissa PM

or ON denoted by a, and the ordinate PN denoted by y . The The abscissa a is axes may be rectangular or oblique. reckoned positive or negative according to the position of P to the right or left of the y axis , and the ordinate y is positive or negative according as P lies above or below the a axis conformably to the rules (607 , ' 8) . 4002 These coordinates are called rectangular or obliqué according as the axes of reference are or are not at right angles .

POLAR COORDINATES . 4003 The polar coordinates of P ( Fig. 1 ) are , the radius vector, and 0, the inclination of r to OX, the initial line , measured as in Plane Trigonometry (609) .

4004

To change rectangular into polar coordinates , employ

the equations

4005

x = r cos 0,

y = r sin 0.

To change polar into rectangular coordinates , employ

-1 r = √x² +y² ,

0

tan

See the end of the volume. 4 C

(봄)

PLANE COORDINATE GEOMETRY.

562

TRILINEAR

COORDINATES .

4006 The trilinear coordinates of a point P (Fig . 2) are a, ß, y, its perpendicular distances from three fixed lines which form the triangle of reference , ABC, hereafter called the trigon . These coordinates are always connected by the relation

4007 4008

aa + bB + cy = Σ , or

a sin A +B sin B +y sin C = constant,

where a , b, c are the sides of the trigon , and Z is twice its area.

4009 If æ, y are the Cartesian coordinates of the point aßy, the equations connecting them with the trilinear coordinates are, by (4094) ,

a = x cos a +y sin a -P₁ , cos ẞ +y sin ß —p2,

B:=

y= = x cos y + y sin y -P3. 4010 Here a has two significations . On the left, it is the length of the perpendicular from the point in question upon the side AB of the trigon . On the right, it is the inclination . of that perpendicular to the a axis of Cartesian coordinates. Similarly ẞ and y.

4011 The angles a , ẞ, y are A, B, C by the equations y — ß = π— A ,

connected with the angles

a ―y = π - B,

a - B = π + C,

only two of which are independent. 4012 P1, P2, P3 are the perpendiculars from the origin upon the sides of the triangle ABC.

AREAL COORDINATES . If A , B , C (Fig . 2) be the trigon as before , the areal co ordinates a' , ẞ ' , y' of the point P are a 4013

PBC

=

a' = P1

β B' =

ABC'

PCA =

P2

ABC'

PAB 2 y=Y := ABC P3

SYSTEMS OF COORDINATES.

563

The equation connecting the coordinates is now

4014

a' + B' + y' = 1 .

4015 To convert any homogeneous trilinear equation into the corresponding areal equation .

4016

Substitute

bß = ΣB' ,

aa = Σa' ,

= Σy'. cy = cy

Also any relation between the coefficients l, m , n in the equation of a right line in trilinears will be adapted to areals by substituting la, mb, nc for l , m, n. Similarly for a, b, c, f, g, h, in the general equation of a conic (4656), substitute aa², bb², cc², ƒbc , gca, hab. In either the trilinear or areal systems , a point is deter mined if the ratios only of the coordinates are known . ß :: y = P : Q : R, then , with trilinear co Thus , if a ẞ ordinates , P ΡΣ 4017 α= ; and, with areal , a = aP +bQ +cR ' P+ Q+ R®

TANGENTIAL COORDINATES . 4019 In this system the position of a straight line is deter mined by coordinates , and the position of a point by an equation. If la + mẞ + ny = 0 be the trilinear equation of a straight line EDF (Fig . 3) ; then, making a , ß, y constant, and l, m, n variable, the equation becomes the tangential equation of the point 0 (a , ẞ , y) ; whilst l , m , n are the co ordinates of some right line passing through that point. Let λ , μ, v (Fig. 3) be the perpendiculars from A, B, C upon EDF, and let P1 , P2, P3 be the perpendiculars from A , B , Cupon the opposite sides of the trigon ; then, by (4624) , we have

4020

Rλ = lp19

Rv = nps ,

= mp29 Rµ =

where R = √ ( l² +m² + n² — 2mn cos A - 2nl cos B - 2lm cos C') . Hence the equation of the point O becomes 4021 ra ta B + Pi P2

sin 0₁

2= 0 P3

or

λ

Pi

sin 0,3

sin 02

= 0,

+v

τμ

P2

P3

where p₁ = OA, 0, = ▲ BOC, &c. , and 2ABOC = P2P3 sin 0.

564

PLANE COORDINATE GEOMETRY.

Formula (4021 ) shows that, when the perpendiculars A, μ, v are taken for the coordinates of the line, the coefficients be come the areal coordinates of the point referred to the same trigon. 4023 Any homogeneous equation in l, m, n as tangential coordinates is expressed in terms of λ , μ , v by substituting for ‫ע‬ λ μ By (4020). l, m, n, " " respectively. P1 P2 P3 4024 An equation in A, μ, v of a degree higher than the first represents a curve such that λ , μ, v are always the perpen diculars upon the tangent. The curve must therefore be the envelope of the line (A, μ , v) .

TWO-POINT INTERCEPT COORDINATES .

X = AD, μ BE (Fig. 4) be variable distances from Let λ two fixed points A , B measured along two fixed parallel lines , then 4025

ad + bµ + c = 0

is the equation of a fixed point 0 through which the line DE always passes . This may easily be proved directly , but we shall show that it is a particular case of the system of three point tangential coordinates .

Let one of the vertices ( C) of the trigon in that system be at infinity (Fig. 3) . Then equation (4022) becomes A sin 01 μ sin 02 + sin COE sin 0, = 0. + P1 P2 For : P, sin COE always. Divide by sin COE ; then λ ÷ sin COE = AD, &c., and the equation becomes sin 01 AD + sin 02 AЕ + sin 03 = 0 . P1 Pa The only variables are AD and AE. Calling these A and μ, the equation may be written aλ + bµ + c = 0, the form taken by aλ + bµ + c'v = 0) when v∞

and c' vanishes.

ONE-POINT INTERCEPT COORDINATES . 4026

Let a, b be the Cartesian coordinates of the point O

(Fig. 5) ; and let the reciprocals of the intercepts on the axes

SYSTEMS OF COORDINATES.

565

1

1 of any line DOE passing through O be

=

η

AE'

AD

Then, by (4053 ) ,

a + bm = 1

4027

is the equation of the point 0, the variables being § , n.

This is a case of the system of three-point tangential coordinates in which two of the vertices (B, C) of the trigon are at infinity. Equation (4022) A sin 0, now becomes + sin BOD sin 0, + sin COE sin 0, =: 0, P1 sin 0, sin 0, + sin 0, or + = 0, AD AE P1 which is of the form a +bn = 1.

TANGENTIAL RECTANGULAR COORDINATES . 4028 This name has been given to the system last described when the two fixed lines are at right angles (Fig . 6) . The coordinates E, n, which are defined as the reciprocals of the intercepts of the line they determine, have now also the following values . 4029

Let x, y be the rectangular coordinates of the pole of

the line in question with respect to a circle whose centre is the origin and whose radius is k ; then

x & 8= 1/2 since x.OM = y.ON = it = = 0.

and

2;

η

= y k ??

for M, N are the poles of y = 0,

4030 The equation of a point P on NM whose rectangular coordinates are OR = a, OS = b, is

a +bm = 1 , by (4053) , this equation being satisfied by the coordinates of all lines passing through that point. In all these systems an equation of a higher degree represents a curve the coordinates of whose tangents in , satisfy the equation.

4031

ANALYTICAL

CONICS

IN

CARTESIAN

LENGTHS

COORDINATES.

AND

AREAS .

Coordinates of the point dividing in the ratio nn' the right line which joins the two points ay, ay' .

ntn Š= n + n'

4032

PROOF.

(Fig . 7. )

= x + AC = x +

§= x + x' x+x 2

η =

ny ' +n'y n + n'

n " ( -a). n+n

Similarly for n.

n = y +y 2

4033

If n = n' ,

4034

Length of the line joining the points ay, x'y' = √ (x − x')² + (y — y')².

The same with oblique axes 4035

√(x - x ) ² + (y —y')² + 2 (x − x') (y — y') cos w.

PROOF.-By (Fig. 7) , Euc. I. 47 , and ( 702) .

Area A of a triangle in terms of the coordinates of its angular points 11 , X2 Y2, x3Y3 . 4036

A =

{ x₁₂ − X¿Y₁ + X2Y3 − X3Y2 + X3Y1 — X₁Y3 } .

PROOF. (Fig. 8. ) By considering the three trapezoids formed by y₁ , Y , s and the sides of the triangle, we have A = } (y₁ + Y½) ( x, −x₁ ) + } (Y₂ + Y3 ) (X3 −X2) — § (Ys + Y1) (Xg — X1) .

LENGTHS AND AREAS.

567

y Area of the triangle contained by the

y = m₁x + c₁, (c₁ - c₂)2

A=

4037

2 (m₁ — m₂)

=

axis and the lines

Y = M 2x + C2 ,

(4052)

(B₁C₂ - B₂C₁ )² (4056) 2B¸В¸ (¸¸—АA¸B₁)'

PROOF. (Fig. 9.) Area = (c - c ) p, and p is found from pm2 —pm, = — C1 — Cg. The sign of the area is not regarded .

COR.-Area of the triangle contained by the lines Y = m₁x + C1,

y = M2X + C2,

Y = MzX +C3,

´ (C1 — C2 ) ² + (C2 — C3) ² + (C3 — Ci) ²´ A ^=} { mi-m2 Momz M3 -M1

4038

4039

= { c₁ (m, —m3) +c₂ (m¸ — m¸) + c¸ ( m, —m,) } ² -2 (m¸ — m²) (m, —m3) (m3 — m¸)

4040

=

(B₁C₂ - B₂C₁ )² + &c. 2B¸B¸ (¸¸—A,B₁) Square of Determinant (A 1, B, C3) 2 32 ( A¸В¿— ¸) ( A,B, —A¿B₂) ( A¸Â¸ —¸¸

4041

PROOF.

(Fig. 10. )

ABC = AEF+ CDE - BED. F

Employ (4037) .

Area of Polygon of n sides.

X1 Y1

First in terms of the coordinates of the angular points X2 Y2, ... xn Yn.

4042

24 = (x₁Y2 − X2Y1) + (X₂Y3 —− X3Y2) + ... + (Xn Y\ — X₁Yn) = X₁ (Y2 ― Yn) + x₂ (ys — y₁ ) + ..... ...... + x'n (Y1 — Yn − 1) .

Secondly, when the equations to the sides are given, as in (4037) .

4043

4044

2A = (C₁ - C2) ² + (C₂ - C3)2 + ... + (cn -c₁)2 mim₂ M2 M3 mn -m1 Also three values similar to ( 4039 , '40 , '41 ) .

PROOF.- By (4367) , adding the component triangles. Each expression for the area of a triangle or polygon will be adapted to oblique axes by multiplying by sinw .

4047

CARTESIAN ANALYTICAL CONICS.

568

TRANSFORMATION

4048

OF

COORDINATES .

To transform the origin to the point hk.

Put

x = x' +h,

y = y' +k.

To transform to rectangular axes inclined at an angle to the original axes .

4049

Put

x = x' cos ♦ -y' sin 0 , PROOF . Consider a' as cos y = sin (p + 0) (627, '9) .

y = y' cos 0 + x'sin

and y' as sin 4.

.

( Fig. 11. )

Then a = cos ( +0) and

Generally (Fig . 12) , let w be the angle between the original axes ; and let the new axes of x and y make angles a and ß respectively with the old axis of a. 4050

and

Put a sin ∞ = x' sin ( w − a) + y' sin (w − ß) y sin w = x' sin a +y' sin ß.

PROOF.- (Fig . 12. ) The coordinates of P referred to the old axes being OC = x, PC = y, and referred to the new axes , OM = a', PM = y', we have, by projecting OCP and OMP at right angles first to CP and then to OC, CD MF - ME, PN = ML + PK, which are equivalent to the above equations.

To change Rectangular coordinates into Polar, hk being the pole O, a the inclination of the initial line to the x axis (Fig. 13) , and ay the point P. 4051

Put x = h + r cos ( 0 + a) ,

THE

RIGHT

y = k + r sin ( 0 + a) .

LINE .

EQUATIONS OF THE RIGHT LINE . 4052

y = mx + c ......

4053

x y == 1 . 22 + 1/2 =

. (1 ) , .... (2) ,

THE RIGHT LINE.

569

4054

x cos a +y sin a = p

(3) .

4055

Ax + By + C = 0

(4) .

PROOF. (Fig . 14. ) Let AB be the line. Take any point P upon it, coordinates ON = x, PN = y. Then, in ( 1 ) ; m = tan 0, where 0 = BAX, − OC, and c is the intercept the inclination to the X axis ; therefore mx = OB. In (2) , a, b are the intercepts OA, OB. In (3) , p = OS , the per pendicular from O upon the line ; a = ▲ AOS. p = OR + LP = x cos a + y sin a . (4) is the general equation. b

A

-cot a.

=

m = tano =

4056

B

a

B

A

cos

sin@ =

4060

√A² + B²

4062

√ A² + B² с

C

√1 + m²

√A²+B²

p = c sin a =

Oblique Axes. Equations (4052 , 53 , '55 ) (4054) must be written

4065

hold for oblique

axes ,

(Fig. 14 )

a cos a +y cos B = p.

m sin w

A sin w =

tan 0 =

4066

but

A cos w
m, there are two ovals, as shown in the In that case, the last equation shows that if OPP'

meets the curve in P and P' , we have OP.OP ' = √ (a* —m¹) ; and therefore the curve is its own inverse with respect to a circle of radius = √ (a* —m¹) .

5315 O being the centre, the normal PG makes the same angle with PB that OP does with PA. PROOF. - From ( r + dr) (r' — dr′) = m² and rr' = m² ; therefore rdr = r'dr or r : r = dr : dr' = sin 0 : sin 0' , if 0, 0' be the angles between the normal and r, r'. But OP divides APB in a similar way in reverse order. 5316 Let OPR, then the normal PG, and the radius of curvature at P, are respectively equal to 2m²R$

m²R and R2+a2

3R* + a* + m*

THE LEMNISCATE . +

(Fig. 126)

5317 Characteristic. - This curve is what a Cassinian be comes when m — a . The above equations then reduce to (x² +y²)² = 2a² (x² - y²)

and

r² = 2a² cos 20.

5318 The lemniscate is the pedal of the rectangular hyper bola, the centre being the pole. 5319

The area of each loop = a².

(5206)

THE CONCHOID . ‡

(Fig. 127)

5320 Characteristic . - If a radiant from a fixed point 0 in tersects a fixed right line, the directrix , in R, and a constant length , RP = b , be measured in either direction along the radiant, the locus of P is a conchoid . If OB = a, be the per pendicular from O upon the directrix , the equation of the curve with B for the origin or O for the pole is 5321

x²y² = (a +y)² (b² - y²)

or

r = a seco ± b .

* B. Williamson, M.A. , Educ. Times Math. , Vol. xxv. , p . 81. + Bernoulli, Opera, p . 609. Nicomedes, about A.D. 100.

728

THEORY OF PLANE CURVES.

5323 When a < b , there is a loop ; when a = b , and when a > b, there are two points of inflexion .

a cusp ;

5324 To draw the normal at any point of the curve, erect perpendiculars , at R to the directrix , and at O to OP. They will meet in S the instantaneous centre , and SP will be the normal at P (5242) . 5325 To trisect a given angle BON by means of this curve, make AB - 20N, and draw the conchoid , thus determining Q ; then AON = 3AOQ . PROOF .- Bisect QT in S ; QT = AB = 20N, therefore SN = SQ = ON ; therefore NOS = NSO = 2NQO = 2AOQ.

The total area of the conchoid between two radiants each making 5326 an angle 0 with OA is

a² tan 0 +26³0 +3a√ (b² - a²)

or

a² tan 0 + 26¹0,

according as b is or is not > a. The area above the directrix ( between the same radiants } = 2ab log tan (

+ +

+620. )

The area of the loop which exists when b is > a is a + √ (b² — a³) +a√ (b²- a³). ¹³ cos -¹ - 2ab log a- √(b² - a²)

THE LIMAÇON. * 5327

(Fig. 128)

Characteristic. - As in the conchoid , if, instead of the

fixed line for directrix , we take a fixed circle upon OB as diameter. This curve is also the inverse of a conic with respect to the focus . line and axis of a is

5328

The equation, with OB for the initial

or

r = a cos 0 ± b

(x² +y² - ax) ² = b² (x² +y³) ,

where a = OB, b = PQ.

5330

With b > a, O is a conjugate point. With ba, O is a node.

5331

[ For ma, see (5332) .

The area = π ( } a² + b²) .

When a - 2b , the limaçon has been called the trisectrix.

* Blaise Pascal, 1643.

TRANSCENDENTAL AND OTHER CURVES.

THE VERSIERA.*

729

(Fig. 130)

(Or Witch of Agnesi.) 5335 Characteristic . - If upon a diameter OA of a circle as base a rectangle of variable altitude be drawn whose diagonal cuts the circle in B, the locus of P, the point in which the perpendicular from B meets the side parallel to OA, is the curve in question . Its equation is 5336

xy = 2a √ (2ax — x²) ,

where a

0C the radius .

5337 There are points of inflexion where a == 3a. The total area is four times the area of the circle .

THE QUADRATRIX .†

5338

(Fig. 131 )

Characteristic .-The curve is the locus of the inter

section , P, of the radius OD and the ordinate QN, when these move uniformly, so that xa :: 0 : , where x = ON, a = OA, and 0 = BOD. The equation is

a --x

π

a

2

y = x tan

5339 The curve effects the quadrature of the circle, for OC : OB :: OB : arc ADB. PROOF : OC : OB :: CP : BD. therefore CP : BD :: a : ADB. 5340

But CP = x in the limit when it is small,

The area enclosed above the x axis = 4a²¯¹log 2 .

PROOF . —In the integral ( ≈ tan ( a≈≈ 7 ) de substituto ≈ (a - x) = 2ay, and integrate Sy tan y dy by parts, using (1940) . The integrated terms , which vanishes though of at the limit - log cos produce log cos the form ∞∞ . The remaining integral is log cos y dy, and will be found at (2635) .

THE CARTESIAN OVAL.

5341

(Fig. 134)

Characteristic . -The sum or difference of certain fixed

multiples of the distances of a point P on the curve from two * Donna Maria Agnesi, Instituzioni Analitiche, 1748 , Art . 238. 5 A

† Dinostratus, 370 B.C.

THEORY OF PLANE CURVES.

730

fixed points A, B, called the foci, is constant. of the inner and outer ovals are respectively mr₁ +lr₂ = ncs, ncz,

5342

The equations

mr - lr₂ = nc3 ,

where r₁ = AP, r½ = BP, c¸ = AB, and n > m > l.

ι ncs To draw the curve, put m = μ and m = a ; therefore r₁ ± µr, = a, where a is > AB and µ < 1 ( 1 ) . Describe the circle centre A, and radius ᎪᎡ = a. Draw any radiant AQ, and let P, Q be the points in which it cuts the ovals, then, by ( 1) , 5343

5344

PR = µPB

and

QR = µQB ……………………… .

.. ( 2) .

Hence, by ( 932 ) , we can draw the circle which will cut AR in the required points P, Q. Thus any number of points on the oval may be found. 5345 By (2) and Eac. VI. 3, it follows that the chord RBr bisects the angle PBQ. Draw Ap through r, and let PB, QB produced meet Ar in p and and q. The = triangles PBR, qBr are similar, therefore grµqB ; therefore q is on the inner oval. Similarly p is on the outer oval . By Euc. vi. B., PB . QB = PR. QR + BR ; therefore, by ( 2 ) , ( 1 - µ³) PB.QB BR . Combining this with PB : Bq = BR : Br, from similar triangles, we get BR . Br a BQ.Bq = . (3) . 5346 2 = - 2 1- μ³ ·μ

5347

Draw QC to make 4BQC = BAq ; therefore , A, Q, C, q

being concyclic , we have, by (3 ) ,

a² --c3 BQ.Bq = AB.BC =

(4) .

1 - μ² Hence C can be found if a, μ, and the points A, B are given . C is the third focus of the ovals , and the equation of either oval may be referred to any two of the three foci . Putting BC C₁, AC = C₁, AB = c,, 2the equation between l, m, n is obtained from (4) thus : cc₁ (1 - µ³) = a² - c ; therefore c; (c, + c, ) = a² +µ³ñ‚ © • Nc3 μ = and the result is But C + = C₂, a = " m m 5348

F°c₁ + n°c, = m²c₂

where CA

or

l²BC + m²CA + n²AB = 0 ….. (5) ,

- AC.

Putting 1 , 72, 73 for PA, PB, PC, the equations of the curves are as follows

TRANSCENDENTAL AND OTHER CURVES.

Inner Oval.

731

Outer Oval.

5349

mr. +lr₂ = nc3 ... (6) ,

2 mr₁ - lr₂

= nc3 ... (7) ,

5351

+ lrs = mc2 ... (8), nr₁ +lr₂

nri - lrs

= mc2 ...

5353

mrs - nr₂ = lc₁

nr₂ —mr₂ = lc

... ( 10) ,

(9) ,

... (11 ) .

That (6) and ( 7) are equations of the curve has been shown . To deduce the other four, we have APB = AqB = ACQ (5347) ; therefore ACQ, APB are similar triangles. But, by (6), mAP + IBP = nAB, therefore mAC + ICQ = nAQ or nAQ - 1CQ = mAC, which is equation ( 9 ) . Again, ABQ, APC are similar. But, by (7) , mAQ - IBQ = nAB ; therefore mAC - ICP = nAP or nAP + 1CP = mAC, which is equation (8) . Equations ( 10) and ( 11) are obtained by taking (6 ) from ( 8) and ( 7) from (9), and employing (5).

5355

AP.AQ

AB . AC

constant.

PROOF. Since A, Q, C, q are concyclic, QCA = QqA = ABB ; there fore P, Q, C, B are concyclic ; therefore AP.AQ AB . AC constant ( 12) . CP.CP' = CA. CB

5356

constant.

PROOF: PCB = PQB = Bpq = BCq. Hence, if CP meets the inner oval again in P' , CBq, CBP' are similar triangles . Again, because ▲ BPC = BQC = BAq = BAP', the points A, B, P , P are concyclic ; therefore CP.CP'CA . CB constant. Q. E. D. Hence, by making P, P' coincide, we have the theorem : 5357

The tangent from the external focus to a series of tri

confocal Cartesians is of constant length , and = √ (CB.CA) .

5358

To draw the tangents to the ovals at P and Q.

De

scribe the circle round PQCB, and produce BR to meet the circumference in T ; then TP, TQ are the normals at P and Q. The proof is obtained from the similar triangles TQR, TBQ, which show that sinTQA sin TQB = 1 : m, by (2) , and from differentiating equation ( 7), which produces dr . dra = l : m .* ds ds

5359 The Semi- cubical parabola y = ax is the evolute of a parabola (4549) . The length of its arc measured from the 8

origin is

9

s= 29a 7a { (1 +2 ax) ' -1 }. 2

* For the length of an arc of a Cartesian oval expressed by Elliptic Functions, see a paper by S. Roberts, M.A. , in Proc. Lond. Math. Soc. , Vol . v., p. 6.

732

THEORY OF PLANE CURVES.

5360 The Folium of Descartes , x³ -3axy +y³ = 0, has two infinite branches , and the asymptote x +y + a = 0.

For the lengths of arcs and for areas of conics, see (6015) , et seq.

LINKAGES AND LINKWORK .

5400 A plane linkage, in its extended sense , consists of a series of triangles in the same plane connected by hinges , so as to have but one degree of freedom of motion ; that is , if any two points of the figure be fixed , and a third point be made to move in some path, every other point of the figure will , in general , also describe a definite path. With two points actually fixed, the linkage is commonly called a piece-work, and if straight bars take the place of the triangles , it is called a link-work.

THE FIVE-BAR LINKAGE. 5401 Mr. Kempe's fundamental five-bar linkage is shown in Figure ( 135) . A, B, D' are fixed pivots indicated by small circles . C, D, B' , C' , in the same plane, are moveable pivots indicated by dots .

The lengths of the bars AB, BC, CD, DA

are denoted by a, b , c , d.

The lengths of AB′ , B'C' , C'D ' , D'A

are proportional to the former, and are equal to ka , kb, kc , kd, Hence ABCD, AB'C'D respectively. ' are similar quadri laterals , and LAD'C' = ADC. P being any assigned point on BC and BP = λ , P' must be taken on DC so that cd D'R' = λ Draw PN, P'N' perpendiculars to AB. Then, ab throughout the motion of the linkage in one plane , NN' is a constant length . PROOF : NN' = BD' - (BN + N'D') .

But BD'

a - kd, and A cd cos D = BN+N'D' = λ cos B-λ (2ab cos B- 2cd cos D) 2ub ab

=

λ ·(a² + b² — c² —d³) (702) . Hence 2ab

LINKAGES AND LINKWORK.

NN '

5402

a- kd

733

λ Qab (a² +b² — c² —d³).

(a - kd) ab then — c² — d² ' a² + b² ' BD NN' = = 2 ; consequently, if the bars PO BB and P′0 =

5403

CASE I.- (Fig. 136.)

If λ =

P'D' be added , the point O will move in the line AB. If, in this case , dka and b = c, then λ = b and P coin cides with C, P' with C', and B' with D, O as before moving in the line AB.

5404

CASE II .- (Fig. 137. )

If, in Case I. , kd = a and

a² + b² = c² + d², A is indeterminate ; that is , P may then be taken anywhere on BC. D ' coincides with B, and ÑN' = 0 . PP' is now P'O be added, O will move in POP'B and

always perpendicular to AB. If the bars PO, of lengths such that PO² - P'O² =· P.B² — P'B², the line AB. If, on the other side of PP', bars P'O' = PB be attached, then O ' will move in a

perpendicular to AB through B.

d 5405 CASE III .— ( Fig. 138. ) If, in Case I. , kd = a, b = s and ca , the figure ABCD is termed a contra- parallelogram . a² and BP' BPA is indeterminate, BC ' = kc = — — λ. d

Hence BC and BP' are measured in a reversed direction ; PP ' is always perpendicular to AB, and if any two equal bars PO, P'O are added, O will move in the line AB. 5406

If three

or

more similar contra-parallelograms be

added to the linkage in this way, as in Figure ( 139) , having the common pivot B and the bars BA, BC, BE, BG in geo metrical progression ; then, if the bars BA, BG are set to any angle , the other bars will divide that angle into three or more equal parts . 5407

If, in Figure ( 138 ) , AD be fixed and DC describe an

angle ADC, then B'C' describes an equal angle in the opposite direction. Mr. Kempe terms such an arrangement a reversor, and the linkage in Figure ( 139 ) a multiplicator. With the aid

UNIV

THEORY OF PLANE CURVES.

734

of these, and with a translator ( Fig. 140) , for moving a bar AB anywhere parallel to itself, he shows that any plane curve of the nth degree may, theoretically, be constructed by link work.*

5408 CASE IV .-- ( Fig . 141. ) If, in the original linkage (Fig. 135) kda, D' coincides with B. Then, if the bars RPO, RP'O ' be added by pivots at P, P ', and R ; and if OPPR = BP' and O'P'P'R = BP ; the points 0, 0 will move in perpendiculars to AB. For by projecting the equal lines upon AB, we get NL = BN ' and BN = N'L ', therefore BL = BL = NN' = a constant, by (5402 ) .

5409 CASE V.- (Fig. 142. ) Make kad and λ = b. Then Replace D'C B' coincides with D, P with C, and P ' with C. ', CD by the bars DK, KD ' equal and parallel to the former. Also add the bars CO = DK and OK = CD. Draw the per Then by projection , pendiculars from O, C and C' to AB. = NL N'D' ; therefore BL = BN+ NL = BN+ N'D = BD ' Hence the point 0 will move perpendi -NN' constant . cularly to AB. 5410 CASE VI .— ( Fig. 143. ) In the last case take k = 1 . Therefore da , D ' coincides with B, BK = BC, and CDKO This is Peaucellier's linkage. is a rhombus .

5411

CASE VII.- ( Fig. 144. )

In the fundamental linkage

(Fig. 135 ) , transfer the fixed pivots from A, B to P, S, adding the bar SA, so that PBSA shall be a parallelogram. Then, since NN' is constant (5402 ) , the point P' will move perpen dicularly to the fixed line PS. 5412 Then

Join AC cutting PS in U, and draw UV parallel to AD. UV : AD = PU : AB = CP : CB = constant ; there

fore PU and UV are constant lengths .

Hence it follows that

the parallelism of AB to itself may be secured by a fixed pivot at U and a bar UV instead of the pivot S and bar SA. 5413

In Case VII . ( Fig . 144 ) , with fixed pivots P and S

Proc. of the Lond. Math. Soc. , Vol. vII., p. 213.

735

LINKAGES AND LINKWORK.

and bar SA, make b = a , dc , kad, λ = b .

Then B' coin

cides with D, N' with N, P with C and L, and P' with C′ ; and we have Figure 145. DC, DC' are equal, and they are equally inclined to AB or CS ; because, in similar quadri laterals , it is obvious that AB and CD and the homologous sides DC and AD ' include equal angles . Therefore CC' is perpendicular to CS, and C' moves in that perpendicular only.

5414 If two equal linkages like that in (5413 ) , Figure ( 145) , but with the bars AS, CS removed , be joined at D ( Fig. 146 ) and constructed so that CDy ' , yDC ' form two rigid bars , then AB, aẞ will always be in one straight line . Let A, B be made fixed pivots, then , while C describes a circle , the motion of the bar aß will be that of a carpenter's plane. On the other hand , if the linkage of Figure ( 145) , with AS and CS removed as before, be united to a similar inverted ' common , then , with fixed linkage ( Fig . 147) , with DC, DC ' , the motion of the bar aß will be that of a lift , pivots A, B, D directly to and from AB. 5415

5416 The crossing of the links may be obviated by the arrangement in Figure ( 148 ) . Here the bars C'B, C'D, C'D' are removed, and the bars FD, FE, FG added in parallel ruler fashion.

5417 CASE VIII.— ( Fig . 149. ) In Case VII . , substitute the pivot U and the bar UV for S and SA. Make da , and therefore 1. Then b' = b and c c , making BCDC ' a contra-parallelogram ; D' coincides with B, and B' with D. The bars AB, AD are now superfluous .

Take BP = λ ; then

BP ′ = \ { ; therefore PP ' is parallel to CC' , therefore to BD, therefore to PV ( 5412 ) ; therefore V, P, P ' are always in one right line . and AB.

P' , as in Case VII . , moves perpendicularly to PU This arrangement is Hart's five-bar linkage.

5418 When a point P (Fig . 152) moves in a right line PS, it is easy to connect to P a linkage which will make another point move in any other given line we please in the same

THEORY OF PLANE CURVES.

736

plane.

Let QR be such a line cutting PS in Q.

Make Qa

fixed pivot , and let OQ , OP, OR be equal bars on a free pivot O. Then, if the angle POR be kept constant by the tie-bar PR, PQR, being one half of POR (Euc. III. 21 ) , will also be constant, and therefore , while P describes one line , R describes the other. If the bar PO carries a plane along with it, every point in that plane on the circumference of the circle PQR will move in a right line passing through Q.

THE SIX-BAR INVERTOR .* 5419

If in the linkwork ( 5410 , Fig. 143) the bar AD be

removed , and D be made to describe any curve , O will describe the inverse curve , just as, when D described a circle , O moved in a right line which is the inverse of a circle. PROOF. Let BOD and CK intersect in E. Then BO.OD = BE - OE = BC² - 00² = a constant called the modulus of the cell.

THE EIGHT- BAR DOUBLE INVERTOR. 5420 Two jointed rhombi ( Fig. 150) having a common diameter AB form a double Peaucellier cell termed positive We or negative according as P or Q is made the fulcrum. = have PQ.PR PQ.QS = AP2 -AQ , the constant modulus of the cell.

THE FOUR -BAR DOUBLE INVERTOR . 5421

If,

on the bars

of a contra - parallelogram ABCD

(Fig. 151 ) four points p, q, r, s be taken in a line parallel to AC or BD, then in every deformation of the linkage, the points p , q , r , s will lie in a right line parallel to AC ; and Pq⋅ pr = pq . qs a constant modulus . Thus , if p be a ful crum and curve.

describes a curve, q will describe the inverse

If q be the fulcrum, p will describe the inverse curve .

PROOF. Let Ap = mAB, therefore pq = mBD, and pr = ( 1 − m) AC , therefore pq.pr = m ( 1 — m) AC.BD = m ( 1 — m) ( AD²—AB²) = constant. * Since the curve described is the inverse and not the polar reciprocal of the guiding curve, it seems better to call this linkage an invertor rather than a reciprocator.

LINKAGES AND LINKWORK.

737

THE QUADRUPLANE, OR VERSOR INVERTOR. 5422 Let the bars of the contra-parallelogram invertor ( 5421 , Fig. 151 ) carry planes , and let P, Q, R, S be points in the planes similarly situated with respect to the bars which contain p, q, r , s respectively, so that PAp = QAq and = AP: Ap AQ : Aq ; and similarly at C. Then , if P be the fulcrum and R traces a curve , Q will trace the inverse curve and the angle QPR will be constant .

PROOF.- Let PA = nAB and PB = n'AB, therefore, by similar triangles, PAQ, BAD, PQ = nBD. Also, by the triangles PBR, ABC, PR = n'AC ; PA.PB therefore PQ.PR = nn´AC.BD = ·(AD - AB²) , a constant. AB2 Again, the inclination of PQ to BD = that of AP to AB, which is con stant. Similarly, by the triangles PBR, ABC, the inclination of PR to AC = that of BR to BC, which is also constant ; therefore QPR, the sum of these two inclinations , is a constant angle.

THE PENTOGRAPH , OR PROPORTIONATOR.

5423 Let ABCD ( Fig . 153) be a jointed parallelogram, A, B fixed pivots, q a tracer placed at any assigned point in BC produced ; then a pencil at p will evidently reproduce any figure traced by q diminished in linear proportions in the ratio of Bq to BC.

THE PLAGIOGRAPH, OR VERSOR PROPORTIONATOR. 5424 In the same figure, make an angle qBQ = pDP, = BQ Bq, and DP = Dp, and let a tracer Q and pencil P be rigidly connected to the arms BC and DC. Then P will pro duce a similar reduced figure as before, but no longer similarly situated . It will be turned round through an angle QBq. This is Prof. Sylvester's Plagiograph . PROOF. -Let BC = k. Bq ; therefore AD = kBQ, DP = kAB, and ABQ = PDA ; therefore ( Euc. vi . 6) AP = kAQ. Also PAQ is a constant angle, for PAQ = BAD- BAQ- PAD = BAD- BAQ - BQABAD- (π - ABQ) = BAD- + ABC + QBq = QBq.

THE ISOKLINOSTAT,

OR ANGLE-DIVIDER .

5425 This linkage ( Fig. 154) accomplishes the division of an angle into any desired number of equal parts . The dia * Invented and so named by Prof. Sylvester. 5 B

738

THEORY OF PLANE CURVES.

gram shows the trisection of an angle by it. A number of equal bars are hinged together end to end, and also pivoted on their centres to the same number of equal bars which radiate , fan- like, from a common pivot. bars make equal angles with each other.

The alternate radial

The same thing is accomplished in a different way by Kempe's Multiplicator ( 5406 , Fig . 139) .

A LINKAGE FOR DRAWING AN ELLIPSE.

5426 In the arrangement of (5413 , Fig . 145 ) the locus of any point P, on DC' , excepting D and C' , is an ellipse. PROOF. -Take CS, CC' for x and y axes ; P the point xy ; SCD = 0, Then we have x = (c - h) cos 0, and therefore CDC' = 20 ; PD = h. x2 y³ + y = (c +h) sin 0, therefore 1 is the equation of the locus. (c - h)² ' (c + h)² Any point on a plane carried by DC also describes an ellipse round C ; but if the point lies on a circle whose centre is D and radius DC, the ellipse be comes a right line passing through C, as appears from (5418) .

A LINKAGE FOR DRAWING A LIMAÇON, AND ALSO A BICIRCULAR QUARTIC.*

5427 (Fig. 155. ) Let four bars AP' , AQ ' , BC, CD be pivoted at A, B, C, D, and let AB = BC = BQ '= a AD = DC DP'b. Take a fulcrum F on BC, a tracer at P, and a follower at Q, so that PQ is parallel to BD.

Let FP = p,

FQ ; then, if P traces out a circle passing through F, Q will describe a limaçon . PROOF.- Let BQ = ma, therefore PD = mb ; r = 2m . BN, Р =(m + 1 ).DN + ( 1 − m) BN. Also BN - DN² = a² - b². Eliminate BN and DN, and the equation between r and pρ is ² + (1 ― m) rp - mp² = m (m + 1) ² (a² — b²) = k³. If P describes the circle pc cos 0, Q describes the locus r² + (1 − m) cr cos 0 - mc² cos² 0 = k², which is the inverse of a conic, that is, a limaçon (5327) . If C be made the fulcrum, the equation reduces to

-p² = 4 (a² —b²).

5428 With the same fulcrum F, drawing FH parallel to AC, if a tracer at H describes the circle , then a follower at K on CD will trace out a bicircular quartic . * W. Woolsey Johnson, Mess. of Math., Vol. v., p. 159.

LINKAGES AND LINKWORK.

739

PROOF. - Draw FL, LK parallel to BA, AD. Let FH = p, FK = r, CK = B, CFanFB, and therefore CL = np. Now 2 (a² +(²) = r² + n²p² +

(a²-32)2 7.2

Therefore, if I moves on the circle p = c cos 0, K will describe the curve +n²²² cos20−2 ( a² + ẞ²) r² + (a²²)² = 0, or (x² + y²) ² + (n²c² — 2a² — 2ẞ²) x² ---—2 ( a² + ß³) y² + (a²— (3²) ² = 0 .

A LINKAGE FOR SOLVING A CUBIC EQUATION. * 5429

Let the three-bar linkwork (Fig . 156 ) have the bars

AB, DC produced to cross each other.

Let AB = AD = a,

BC = b, CD = c ; and let b and c be adjustable lengths . Suppose a -qx +r = 0 a given cubic equation.

Make c = } √ (2 + 2 ), 6 = + √ ( 2− 2 ) ; then deform the quadrilateral until EC = CD ; DE will then be equal to a real root of the cubic . PROOF :

- _ (x + a) ³ +4c² —a² cos E = x² +c² — b³ = 2cx 4c (x + a)

from which

x³-2 (c² +b²) ∞ + 2a (c² — b²) = 0 . Equate coefficients with the given cubic.t

ON THREE -BAR MOTION IN A PLANE. 5430 If a triangle ABC (Fig. 157) be connected by the bars AO, BO ' to the fulcra O, O ' , the locus of C is called a three -bar curve . OA, O'B meet in Q, the instantaneous centre of rotation of the triangle, since QA, QB are perpendicular to the movements of A and B respectively. Therefore CQ is the normal to the locus of C. 5431 If a triangle similar to ABC be placed upon 00 ' (homologous to AB) , the circum- circle of the triangle will pass through the node, and the vertices of the triangle are called the foci of the curve.

* M. Saint Loup , Comptes Rendus, 1874. + The foregoing account of linkages is taken chiefly from a paper by A. B. Kempe, F.R.S. , in the Proc. of the Royal Soc . for 1875 , Vol. XXIII . Other results by the same author will be found in the Proc. of the Lond. Math . Soc. , Vol . Ix. , p . 133 ; and by H. Hart, M.A. , ibid., Vol. vI. , p. 137, and Vol . VIII. , p. 286. See also The Messenger of Mathematics, Vol. v .

740

THEORY OF PLANE CURVES.

Figures ( 158 ) and ( 159 ) exhibit different varieties of the curve according to the relative proportions between the lengths of the bars . *

MECHANICAL CALCULATORS .

The Mechanical Integrator .+ 5450 This instrument computes not only the area of any closed plane curve , but the moment and also the moment of inertia of the area about a fixed line.

The principle of its

action is shown in Figure ( 160 ) . OP is a bar carrying a tracer at P, and a roller 4 at some point of its length. The end is constrained to move in the fixed line ON. When the tracer P moves round a closed curve, the length OP mul tiplied by the entire advance recorded by the roller is equal to the area of the curve. PROOF. - Let the motion of the tracer from P to a consecutive point Q be decomposed into PP' and P'Q parallel and perpendicular to ON. Let OP = a and PON = 0. When the pointer moves from P to P', the roll accomplished is PP' sin 0. The roll due to the motion from P' to Q will be neutralized by the exactly equal and opposite roll in the motion of the pointer from q to p', since the bar will there have again the same inclination . Con sequently the product of the entire roll and the length a is equal to the sum of such terms as aPP' sin 0. But this is the area ÕPP′0' = NPP'N' . The algebraic addition of such rectangles gives the entire area, and the instru ment effects this, for the area SN is subtracted, by the motion of the roller, from the area QN which is added.

5451

The instrument itself is shown in Figure ( 161 ) .

A

frame moving parallel to OX by means of the guide BB carries two equal horizontal wheels geared to a central wheel which has two circumferences , such that its rate of angular motion is half that of the lower wheel and one third of that of the upper. The latter wheels carry two rollers , M and I, on horizontal axles ; and the middle wheel carries an arm OP, a pointer at P, and a roller A.

In the initial position , the

The curve is a tricircular trinodal sextic, and is completely discussed by S. Roberts, F.R.S. , and Prof. Cayley, in the Proc. ofthe Lond. Math. Soc. , Vol. vii . , pp. 14, 136 . † Invented and manufactured by Mr. J. Amsler-Laffon, of Schaffhausen. The demon strations (which in clearness and elegance cannot be surpassed) of the action of this instru ment, and of the Planimeter which follows, were communicated to the author by Mr. J. Macfarlane Gray, of the Board of Trade.

MECHANICAL CALCULATORS.

741

rollers A and I are parallel, while M is at right angles to A. The frame is thus supported above the paper on the three rollers ; and if the arm OP be moved through an angle AOA', the axles of the rollers M and I will describe twice and three times that angle respectively . Putting OP - a as above, and A, M, and I for the linear circumferential advances recorded by the three equal rollers respectively , we have the following results I.—The area traced out by the pointer P = Aa . II .—The moment of the area about OX

=M Ma² 4 as

III . - The moment of inertia about OX

= (3A +I):

12 PROOF. - I. Since O moves in the line OX, while the pointer P moves round a curve, the roller A will, as shown above, make the rolling Eh sin 0, where h = PP' in Figure ( 160 ) , and the area of the curve = aΣh sin or a × roll . II. The moment of the area about OX

= Σ ( ah sin 0 ×

a sin 0 a² = ( h - Σh cos 20) . 2 9 4 )=

Now Σh vanishes when P returns to the starting point, and -Σh cos 20 is the roll recorded by M. For, when OP makes an angle 0 with OX, the axis of M will make an angle -( 90° + 20) with OX. In this position, while P makes a parallel movement h, the roll produced thereby in M will be -h sin ( 90° +20) a² =-h cos 20. Therefore x roll of M = moment of area. 4 III. Lastly, the moment of inertia of the area about OX as = Σah sin 8 x a² sin²0 ) = (ah 12 (3h sin 0 - h sin 30) . Now, when OP makes an angle with OX, the axis of I makes –30 ; there fore --Eh sin 30 is the entire roll of I. Hence the moment of inertia as as X roll of I. = x roll of 4+ 12

The Planimeter. 5452

(Fig. 162)

This instrument * is a simpler form of area computer .

O is a fixed pivot ; OA, AP are two rods having a free pivot at A ; C is the roller , and P the pointer. The area of a closed curve traced by the pointer is equal to the total roll multiplied by the length AP. * Like The Integrator, the invention of Mr. Amsler.

THEORY OF PLANE CURVES.

742

PROOF. Decompose the elementary motion PQ of the pointer into PP', effected with a constant radius OP, and P'Q along the radius OP' , and so all round the curve . The roll of C accomplished while P moves from P' to Q will be neutralized by the equal contrary roll when P moves from q to p' on the radius Op' = OP. Thus the total roll recorded will be the sum of the rolls due to the movements PP', QQ', &c. Draw OB perpendicular to AP, and, when P comes to R, let B' be the altered position of B. The area PQSR = } ( OP² - OR²) w, where w = POQ. But OP² = 04² + PA² - 2PA . AC - 2PA.BC (Euc. II . 13) ; therefore, since BCis the only varying length on the right, we have PQSR = PA(BC ~ BC ) w. But BC is the roll of C due to the angular motion w of the rigid frame OAP, and the subtraction of the area OSR from OPQ is effected by the instrument, since when the pointer moves from S to R the direction of the roll must be reversed . Hence the total area = PA × the total recorded roll.

APPENDIX ON BIANGULAR COORDINATES . *

5453

In the figure of (1178) , the biangular coordinates of a

point P are defined to be a = cote and ẞcot p. 5454

= PSS ' and Φ = PS'S, or

The equation of a right line YY' is aa + bB = 1,

where a

cot SYS ' and b = cot SY'S'.

PROOF.- Supplying the ordinate PN in the figure and denoting the angle S'SY by 4, the equation is obtained from CN cos +PN sin = p the per pendicular on the tangent, SS' sin & YY' and SS' cos = SY- SY'.

cota - b.

5455 5456

Equation of a line through C :

a- ẞ ß = const.

5457

Equation of the line at infinity :

a +ß = 0 .

5458

Let SS'c, then the distance between two points

αιβι , α , β , is

1 = c² { ( a + B.

1

2

a , + B₂ 8.) ² + (a, 48, -a , 78.) ). a₂+ B₂ "}

* Quarterly Journal of Mathematics, Vols. 9 and 13 ; W. Walton, M.A.

APPENDIX ON BIANGULAR COORDINATES.

5459

743

The equation of a line through the two points is

α1 -α₂

a -a1 =

B₁ - B₂

B-B₁

The length of the perpendicular from a'ß' upon the 5460 line aa +bẞ = 1 is aa' +bB - 1 P=

' ` √ { (a − b)² + 1 } ' a ' +ß 5461 COR. - The perpendiculars from the poles S, S' are therefore bc ac S'Y' = SY =

√{(a− b)² +1 } ' 5463

√{ (a− b)² + 1}'

When the point a'ß' is on SS' at a distance h from S, (a - b) h + bc p = √ { (a − b)² + 1 } '

With two lines aa + bẞ = 1 , a'a + b'ß = 1 , the condition

5464

of parallelism is

5465

of perpendicularity (a - b) (a' —b') + 1 = 0 .

a - ba - b' ,

5466 The equation of the line bisecting the angle between the same lines is

aa + bB- 1

=

a'a + b'ß - 1

✓ { (a −b)² + 1 } = √{ (a' — b')² + 1 } ' 5467 The equation of the tangent at a point a'ß' on the curve F (a , ẞ) = 0 is (a - a') F + (B - B ') F = 0. 5468

And the equation of the normal is a - a'

B- B =

(a'B ' −1 ) Fp + ( 1 +a²) F。 5469

(a’'B ' −1) F. + (1 +ß³) F„ ;

The equation of a circle through S, S' is aß− 1 = m (a +B),

where m

cot SPS' the angle of the segment.

744

5470

THEORY OF PLANE CURVES.

If C be the centre, the equation becomes αβ = 1 .

5471 And , in this case, the equations of the tangent and normal at a'ẞ' are respectively

å +B = 2

and

a - ẞ = a' - 8 '.

5472 The equation of the radical axis of two circles whose centres are S, S' , and radii a , b , is

(c² — a² + b²) a = (c²+a² —b³) B. PROOF. By equating the tangents from aß to the two circles, their lengths being respectively c² (1 + a²) -a² and c² (1 + ß³) -b', by (5458 ) . (a + B) (a +B)¹

5473 To find the equation of the asymptotes of a curve when they exist , Eliminate a and ẞ between the equations of the line at a +B = 0,

infinity

the curve and the tangent

F (a, B) = 0, (a - a') F + ( B — B') Fg = 0.

Ex. The hyperbola a' +6 a− B = ± m√2.

m³ has, for the equation of its asymptotes,

SOLID

COORDINATE

SYSTEMS

GEOMETRY.

OF COORDINATES .

CARTESIAN OR THREE - PLANE COORDINATES .

5501 The position of a point P in this system (Fig . 168 ) is determined by its distances , a = PA, y = PB, z = PC, from three fixed planes YOZ, ZOX, XOY, the distances being measured parallel to the mutual intersections OX, OY, OZ of the planes , which intersections constitute the axes of coordi nates . The point P is referred to as the point xyz , and in the drawing x, y , z are all reckoned positive, ZOX being the plane of the paper and P being situated in front of it, to the right of YOZ and above XOY. If P be taken on the other side of any of these planes , its coordinate distance from that plane is reckoned negative .

FOUR-PLANE COORDINATES . 5502 In this system the position of a point is determined by four coordinates a, ß, y, 8 , which are its perpendicular distances from four fixed planes constituting a tetrahedron of reference . The system is in Solid Geometry precisely what trilinear coordinates are in Plane . The relation between the coordinates of a point corresponding to (4007 ) in trilinears is 5503

Aa + BB + Cy + D8 = 3V,

where A, B, C, D are the areas of the faces of the tetrahedron of reference, and V is its volume.

TETRAHEDRAL COORDINATES . 5504 In this system the coordinates of a point are the volumes of the pyramids of which the point is the vertex and 5 C

746

SOLID GEOMETRY.

the faces of the tetrahedron of reference the bases : viz. , The relation between them is BB, C , D. 5505

Aa,

' + y + 8 = V. a²+ ß

POLAR COORDINATES . 5506 Let O be the origin ( Fig. 168 ) , XOZ the plane of reference in rectangular coordinates , then the polar coordi nates of a point P are r, 0, p , such that r = OP, 0 = ▲ POZ, and XOC between the planes of XOZ and POZ.

THE

5507

RIGHT

LINE .

The coordinates of the point dividing in a given ratio

the distance between two given points are as in (4032 ) , with a similar value for the third coordinate Z.

5508

The distance P, Q between the two points xyz , x'y'z' is PQ = √ / { (x − x') ² + (y — y') ² + ( ≈ — ≈') } . ( Euc. I. 47) .

5509 The same with oblique axes , the angles between the axes being A, µ, v. PQ = √ / { (x − x') ² + (y — y ' ) ² + (≈ — ≈' ) ² + 2 (y — y' ) (≈ — ≈') cos à +2 (≈ —≈') (x − x') cos µ +2 (x − x') (y — y') cos v } . (By702).

5510

The same in polar coordinates , the given points being

rop, r'e'p', PQ = √ [r² + r'² −2rr { cos

cos

+ sin

sin & cos (4 — ☀') } ] .

PROOF.- Let P, Q be the points, O the origin. Describe a sphere cutting OP, OQ in B, C and the z axis in A ; then, by (702), PQ = OP² + OQ³ -20P.OQ cos POQ and cos POQ, or cos a in the spherical triangle ABC, is given by formula (882) , since b = 0, c = 0', and  = ¢ — ¢´.

DIRECTION RATIOS. 5511 Through any point Q on a right line QP (Fig. 169) , draw lines QL, QM, QN parallel to the axes, and through any other point P on the line draw planes parallel to the coordi

747

THE RIGHT LINE.

nate planes cutting the lines just drawn in L, M, N ; then the direction ratios of the line OP are

QL 5512

QM

1 ι =

m = QP'

QN

n= QP

QP'

The angles PQL, PQM, PQN are denoted by a , ß , y ; and the angles YOZ, ZOX, XOY between the axes by λ, µ, v. 5513 When λ, μ, v are right angles , the axes are called rectangular , and the direction-ratios are called direction cosines , being in that case severally equal to cosa, cosß , cos y. 5514 When L, M, N are the direction-ratios (or numbers proportional to them) of a line which passes through a point abc, the line may be referred to as the line (LMN, abc) , or, if direction only is concerned , merely the line LMN.

EQUATIONS BETWEEN THE CONSTANTS OF A LINE. 5515

The relation

between the constants of a line with

rectangular axes is

l² + m² + n² = 1 ; and with oblique axes , it is 5516

I cos a + m cos B + n cos y = 1.

PROOF. The first by (Euc. 1. 47) . The second by projecting the bent line QLCP ( Fig. 169) upon PQ, thus PQ = QL cos a +LC cos ẞ + CP cos y, and QL = PQ.1, &c. , by ( 5512) .

5517

Also , when the axes are oblique , cos a =

1 + m cos v + n cos μ,

cos B = m + n cos λ + cos y =

cos v,

n + l cos μ + m cos λ.

PROOF. By projecting QP in figure ( 169) and the bent line QLCP upon each axis in turn, and equating results ; thus PQ cos a = QL + LC cos B + CP cos y, applying ( 5512) .

5518

The relation between l , m, n and λ , µ , v is l² + m² + n² + 2mn cos λ +2nl cos µ + 2lm cos v = 1 .

PROOF.-By eliminating cos a, cos ß, cos y between (5516) and ( 5517) .

SOLID GEOMETRY.

748

The relation between cos a, cos ẞ, cos y and λ, μ, v is

5519

cos² a sin²

+ cos² ß sin² μ +cos y sin² v

+2 cos ẞ cos y (cos μ cos v- cos λ) +2 cos y cos a (cos v cos λ - cos μ) +2 cos a cos ẞ ( cos à cos μ - cos v)

= 1 - cos²λ - cos² μ - cos² v + 2 cos à cos μ cos v. PROOF. ( 5517) .

By eliminating l, m, n between the four equations in (5516) and

5520 The angle between two right lines lmn, l'm'n' , the axes being rectangular :

cos

= ll' + mm' + nn'.

PROOF. -In Figure ( 169) , let QP be a segment of the line lmn. The projection of QP upon the line l'm'n' will be QP cos 0. And this will also be equal to the projection of the bent line QLCP, upon I'm'n' , for, if planes be drawn through Q, L, C, and P, at right angles to the second line I'm'n' , the segment on that line intercepted between the first and last plane will be QP cos 0, and the three segments which compose this will be severally equal to QL . , LC.m', CP.n', the projections of QL, LC, CP. Then, by ( 5512 ) , QL = QP.l, &c.

5521

sin² 0 = (mn' —m'n) ² + (nl' — n'l)² + (lm' — l'm)³.

PROOF.-From

1 — cos³ 0 = (l² +m² + n³ ) ( l'² +m²² + n'²) — ( ll' + mm' + nn')' (5515 , '20) .

5522

With oblique axes ,

cos 0 = ll' + mm' + nn' + (mn' + m'n) cosλ + (nl' +n'l) cos µ + ( lm' + l'm) cos v. As in ( 5520 ) , substituting from ( 5517) the values of cosa, &c.

PROOF.

EQUATIONS OF THE RIGHT LINE.

-

x-a =

5523 Ꮮ

2--C =

M

or N

- a y-b x7ª = m

2- c n

Here abc is a datum point on the line, and if r be put for the value of each of the fractions , r is the distance to a variable point xyz.

L, M, N are proportional to the direction ratios of

749

THE RIGHT LINE.

the line, which ratios must therefore have the values M

L

5524

m =

1= ' + M² + N ')' √ (L

V (E +M +N N

n= √(L* +M³ +N®)' 5525 NOTE. - Instead of a, b, c in the equation we may use kL + a, kM+ b , kl + c, where k is an arbitrary constant.

5526

The equations of a line may also be written in the

forms

5527

x = λz + a,

y = μz + B.

These are the equations of the traces on the planes of

xz and yz, and are equivalent to

x -a

y- B

≈-0 =

=

ī

μ

If the line is determined as the intersection of the two ' , we may D and A'x + B'y + C'z = D planes Ax + By + Cz 5528

write equations (5523 ) by taking L = BC ' — B'C,

M = CA' - C'A ,

DA' -D'A

DB' -D'B

b =

@ = N

N = AB ′ - A'B,

"

c = 0.

N

PROOF. Eliminate z between the equations of the planes, then the reciprocals of the coefficients of x and y will be L and M.

5529 The projection of the line joining the points xyz and abc upon the line Imn is 1 (x− a) + m (y — b) +n (z − c) .

Hence , when the line passes through abc , the square perpendicu lar from xyz upon it is equal to of the 5530

(x — a)² +(y —b)² + (≈ — c)³ — { l (x − a) + m (y — b) + n (≈— c) } *. 5531

Condition of parallelism of two lines LMN, L'M'N' : L : LM : M'N: N'.

750

5532

SOLID GEOMETRY.

Condition of perpendicularity : LL +MM' +NN' = 0.

5533

(5520)

Condition of the intersection of the lines (LMN, abc)

and (L'M'N', a'b'c') (5514 ) : (a - a') (MN' - M'N) + (b − b') (NL ' — N'L) + (c - c') (LM' - L'M) = 0 . PROOF. - Eliminate x, y, z between the equations y —b′ — z — c′ = r ', a = y =b = 2 - c x -a' x-α = = = r and = L' M N M' L N'

by subtracting in pairs, and then eliminate r and r'.

5534

The shortest distance between the same lines is

-(a - a') (MN' - M'N) + (b − b') ( NL' — N'L) + ( c— c' ) (LM' — L'M) √ { (MN' — M'N)² + (NL ' — N'L)² + (LM ' — L'M)² } PROOF. -Assume λ, μ, for the dir- cos. of the shortest distance . Then, by projecting the line joining abc, a'b'c' upon the shortest distance, we get p = (a − a ') x + ( b − b ') µ + (c − c ) v. Also, by (5520) , LX + Mμ + Nv = 0 and V = O, giving the ratios μ : v = MN' - M'N : NL' — N'L L'λ+ M'µ + N'v : LM' - L'M ; and ( 5524 ) then gives the values of λ, µ, v.

5535

The equation of the line of shortest distance between

the lines (Imn, abc) and ( l'm'n' , a'b'c') is given by the inter section of the two planes u + u' cos 1ι (x − a) + m (y — b) +n (≈ — y) =

....... (i .) , sin² 0 u' + u cos 0

l' (x − a') + m' (y — b') + n' (≈ — y') =

... (ii .) , sin

where

u = 1 (a' — a) + m ( b' — b) + n ( c' — c) , u' = l ' (a — a') + m' ( b − b' ) + n' ( c - c') ,

and

cos 0 = ll' + mm' +nn' .

PROOF. (Fig. 170. ) Let O be the point xyz on the line of shortest dis tance AB ; P, Q the points abc, a'b'c' on the given lines AP, BQ. Draw BR and PR parallel to AP and AB ; RT perpendicular to BQ ; and QN, TM per pendicular to BR. Then RBQ = 0, RN = u, QT = u' , therefore NM = u' cos 0 and RM = RN + NM = u + u' cos 0, and in the right-angled tri angle RTB, RM cosec² = RB, the projection of OP upon AP, that is, the

THE RIGHT LINE.

751

left member of equation (i.) . Similarly for equation (ii.) . Itshould be observed that (i. ) and (ii. ) represent planes through AB respectively per pendicular to the given lines AP and BQ. 5536

Otherwise , the line of shortest distance is the inter section of the two planes whose equations are l' (x − a) + m' (y — b) + n' (z − c) = cos 0 1 (x− a) + m (y — b) + n (≈ − c)

= 1 (x - a) + m (y — b') + n (z — c') l' (x − a') + m' (y — b') + n' (≈ — c')' For these equations state that cos is the ratio of the projections of OP or of OQ upon the given lines, and this fact is apparent from the figure.

5537 Equations of the line passing through the two points abc, a'b'c' : x -a y - b = ལ་ 7= a b-b ' C 5538 A line passing through the point abc and intersecting at right angles the line lmn :

--a y- b = 2 - c a N' = "M Ꮮ where_L = lm (b — b') + nl (c — c′) — (m² + n²) (a — a'), and symmetrical values exist for M and N. PROOF.-The condition of perpendicularity to Imn is Ll +Mm + Nn = 0 ;

(5520)

and the condition of intersecting the line is (a - a') (Mn - mN) + ( b − b′ ) ( Nl — nL) + (c — c′) ( Lm — IM) = 0 . These equations determine the ratios L : M : N.

5539 Equations of the line passing through the point abc , parallel to the plane La + My + Nz = D, and intersecting the line ( l'm'n' , a'b'c') : x-a y-b 2- c = = m n ι

where l, m , n are found , as in the last, from Ll +Mm +Nn = 0,

and

(a — a') (mn' — m'n) + (b − b') ( nl' — n'l) + ( c — c') (lm' —l'm) = 0 .

752

SOLID GEOMETRY.

5540 Equations of the bisector of the angle between the two lines 1m , n , lmn :

X

≈

y

=

= l₁ +l₂

m₂ + m²

n₁ + n₂

PROOF.-Taking the intersection of the lines for origin, let x, y,,, yz, be points on the given lines equidistant from the origin ; then, if xyz be a point on the bisector midway between the former points, x = { ( x, + x9) , &c. (4033) ; and the direction-cosines of a line through the origin are propor tional to the coordinates.

5541

are

The equations of a right line in four plane coordinates 8-8 - BB-B a-á B = 2y ==R aa² N Ꮮ M Ꭱ

(i.) ,

where aßyd is a variable point , and a'ß'y's' a fixed point on the line. The relation between L, M, N, R is

5542

AL + BM+ CN+ DR = 0 .....

..... (ii .).

PROOF.- For, since equation (5503) holds for aßyd and also for a’ß'y'd', we have A (a - a') + B (B − ß ') + C (y — y') + D ( d — d′) = 0. Substitute from (i.) a - a' rL, 6 - ẞ' = rM, &c. 5543 In tetrahedral coordinates the same equation (i . ) sub sists , but the relation between L, M, N, R becomes , by changing Aa into a , &c . ,

5544

L + M+ N+ R = 0 .

THE

5545

PLANE.

General equation of a plane : Ax +By + C + D = 0.

5546

Equation in terms of the intercepts on the axes : x a +3 /

+ 32 = 1. с

5547 Equation in terms of p, the perpendicular from the origin upon the plane, and l , m, n , the direction- cosines of p :

lx+my + nz = p.

THE PLANE.

753

PROOF. If P be any point xyz upon the plane, and O the origin, the pro jection of OP upon the normal through O is p itself ; but this projection is lx + my + nz, as in (5520) . 5548

The values of l , m, n, p for the general equation

(5545) are

A

–D

1=

&c . , √(A²+B²+ C²)'

p = √ (A²+B² + C³)*

PROOF. - Similar to that for (4060-2) : by equating coefficients in (5545) and (5547) and employing ' +m² + n² = 1 .

5550

The equation of a plane in four-plane coordinates is la + mẞ+ ny + rd = 0 ,

l : m : nr = αι . βι . γ . δι Pi P2 P3 P4

with

where a₁, B1 , 71, & ar th pe e e rpendiculars upon the plane from A, B, C, D, the vertices of the tetrahedron of reference , and P1, P2, P3, P₁ are the perpendiculars from the same points upon the opposite faces of the tetrahedron . PROOF.-Put y= = 80 for the point where the plane cuts an edge of the tetrahedron, and then determine the ratio : m by proportion. See Frost and Wolstenholme, Art. 81 . 5551 The equation of a plane in tetrahedral coordinates is also of the form in ( 5550) , but the ratios are, in that case , l : m : n : r = а₁ : ß₁ : Y₁ : §₁ .

The relation between the three-plane and four- plane coor dinates is

5552

a = p ― lx - my— nz.

The equation of a plane in polar coordinates is { cos e cose +sine sine cos (4-4) } = p .

PROOF. Here p is the perpendicular from the origin on the plane, and p, 0', o' the polar coordinates of the foot of the perpendicular. Then, if ↓ is the angle between p and r, we have pr cosy and cosy from ( 882) .

5553

The angle

between two planes

lx +my +nz = p

l'x + my + n'z = p'′ and 5 D

SOLID GEOMETRY.

754

is given by formula ( 5520) , and the conditions of parallelism and perpendicularity by (5531 ) and ( 5532) , since the mutual inclination of the planes is the same as that of their normals .

5554 The length of the perpendicular from the point x'y' upon the plane Ax + By + Cz +D = 0 is

Ax +By + C + D ±

= p - lx' — my ' — nz'.

√(A + B + C )

PROOF .

5556

As in (4094) .

The same in oblique coordinates

(Ax + By + Cz + D) = p - x cos a - y cos B- z cos y,

P where p is found from ( 5519) by putting A , B, C for p cos a , ρ cos ẞ, p cos y. p

5558

p² =

This gives

{ A² sin³ \ + B² sin μ +C sin v +2BC (cos μ cos v -cos λ) +2CA (cos v cos λ — cos µ ) + 2AB (cos à cos µ — cos v) 1 - cosλ- cos² μ - cos² V + 2 cos è cos μ cos

5559 The distancer of the point from the plane ' y' Ax + By + Cz + D = 0 , measured in the direction Imn, the axes being oblique : ' + By + C + D r = Ax Al + Bm + Cn PROOF. - By determining r from the simultaneous equations of the line and the plane, viz.,

x•— -x' = r x' = Y — Y = = =2' = n m ι

and

Ax +By + Cz + D = 0.

Otherwise, by dividing the perpendicular from x'y'z' (5554) by the cosine of Al +Bm + Cn its inclination to lmn, viz., √(A² +B²+ C²)

EQUATIONS OF PLANES UNDER GIVEN CONDITIONS. 5560 A plane passing through the point abc and perpen dicular to the direction Imn : l (x − a) +m (y— b) +n (z − c) = 0.

THE PLANE.

5561

A plane passing through two points abc, a'b'c' : x -a

+

λ

a- a 5562

755

with

y b μ b-b' + == c

= 0,

λ+ μ + v = 0.

PROOF. By eliminating n between the equations 1 (x - a) + m (y - b ) + n (z − c) = 0, 1 ( a − a') + m (b − b ′) + n ( c − c') = 0, and altering the arbitrary constant.

5563 A plane passing through the point of intersection of the three planes u = 0, v = 0, w = 0 :

lu + mv + nw = 0. 5564 A plane passing through the line of intersection of the two planes u = 0 , v = 0 : lu + mv = 0. 5565 A plane passing through the two points given by u == 0 , v = 0 , w = 0 and u = a , v = b , w = c : lu +mv +nw = 0

with

la + mb + nc = 0.

5566 The equation of a plane passing through the three points a₁y11 , X2Y2Z2, X3Y 3 Z 3

x y z X1 Y1

1

1 1

= 0. or A, B, C, is given by the determinant annexed , in which the coefficients of x, y , z

X2 Y2 Z2 1 X's Y3 Z3 1

represent twice the projections of the area ABC upon the coordinate planes . PROOF. The determinant is the eliminant of Ax + By + Cz = 1 , and three similar equations. Expanded it becomes ∞ (YqZ3 —YsZg + Y3Z1 — Y13 + Y₁₂ — Y₂1) + y ( &c. ) + z ( &c . ) + x1Yq%3− &c . = 0 . Hence, by (4036 ) , we see that the coefficients are twice the projections of ABC, as stated .

5567 The sum of squares of the coefficients is equal to four times the square of the area ABC. PROOF.- For, if l, m, n are the dir- cos . of the plane, and ABC = S, the coefficients are = 281, 28m, 2Sn, by projection .

5568 The determinant ( ~1 , 2, 3) , that is, the absolute term in equation (5566) , represents six times the volume of the tetrahedron OABC, where O is the origin.

SOLID GEOMETRY.

756

PROOF.-Writing the equation of the plane ABC, Ax +By + Cz + D = 0, we have for the perpendicular from the origin, disregarding sign, D D = (5567) , p= √ (A² + B³ +C²) 28 therefore D = 2pS = 6x the tetrahedron OABC. 5569

If xyz be a fourth point, P, not in the plane of ABC,

the determinant in (5566) represents six times the volume of the tetrahedron PABC.

PROOF. By the last theorem the four component determinants represent six times (OBCP + OCAP + OABP + 0ABC) for an origin O within the tetrahedron.

5570 A plane passing through the points abc, a'b'c' , and parallel to the direction Imn :

x

∙a

y-b

z- c

a --·a'

b- b '

c-c

ι

m

n

= 0. b-b' PROOF. Eliminate A, μ, between the equations ( 5561-2) and nv ἰλ = ημ + = 0, the condition of perpendicularity between lmn a- a b- b' c - c' and the normal of the plane (5561) .

5571 A plane passing through the point abc and parallel to the lines lmn , I'm'n' : x -a

ι

ľ

y- b

m

m'

2- c

n

n'

: 0.

PROOF. The equation is of the form ( -a) + µ (y — b ) + v (z — c) = 0, and the conditions of perpendicularity between the normal of the plane and the given lines are lλ + mµ + nv = 0, l'λ + m'µ + n'v = 0. Form the eliminant of the three equations .

5572 A plane equidistant from the two right lines (abc, lmn) and (a'b'c' , I'm'n') : x-

(a + a')

y = 1 (b + b') x − 1 (c +c')

l

ľ

m

m'

n

n'

= 0.

By (5571) .

TRANSFORMATION OF COORDINATES.

757

5573 A plane passing through the line (abc, lmn) and per pendicular to the plane l'x +m'y + n'z = p : The equation is that in (5571 ) . For proof, assume λ, μ, v for dir- cos . of the normal of the required plane, and write the conditions that the plane may pass through abc and that the normal may be perpendicular to the given line and to the normal of the given plane.

TRANSFORMATION OF

COORDINATES .

5574 To change any axes of reference to new axes parallel to the old ones : Let the coordinates of the new origin referred to the old axes be a, b, c ; xyz and x'y'z', the same point referred to the old and new axes respectively ; then x = x' + a,

5575

To change

y = y' + b,

z = 2 ′ +c.

rectangular axes

of

reference to new

rectangular axes with the same origin : Let OX, OY, OZ be the original axes , and OX' , OY' , OZ' the new ones , l₁ m₁n₁ the dir- cos . of OX' referred to OX, OY, OZ, do . do . OY' do.

lg Mg ng

l my ns

do .

OZ'

do.

do.

xyz, En

the same point referred to the old and new axes respectively. Then the equations of transformation are

5576

x = l₁ &+ l₂ n +ls s

......

(i.) ,

y = m₁§ + m₂n + mb

(ii .) ,

z = n₁ š + n₂ n + ns b

(iii .) .

And the nine constants are connected by the six equations 5577

li + mi +·n₁ ni = 1 ... (iv. ) , 2 l₁₂+ m² + n² = 1 ... ( v .) , 2 l} +m} + n} = 1 ... (vi.) ,

ll +m¿m¸ + n‚nç = 0 ... (vii .) , ll₁ +mm₁ +n¸n¸

... (viii.) ,

ll₂ +m¸m¸ + n¸n₂ = 0 ...

so that three constants are independent.

(ix. ) ,

SOLID GEOMETRY.

758

PROOF.- By ( 5515 ) and ( 5532) , since OX', OY', OZ' are mutually at right angles.

5578

The relations (iv. to ix . ) may also be expressed thus

4 =

mənz―msn

m1 = nglz―nzlą Me

l2 =

mgN1-myns

ngl1—nils

13

Ms n¸lɔ— nål

m₁ng-mqn1

ni = ± 1 ...... (x.), l₂m — l¿m² Նջ = ± 1 ...... (xi. ) , Içm₁ — l₁m³

Ng = ±1 ....... (xii .).

Im₂ - l₂m₁

Obtained by eliminating the third term from any two of equations (VII.-IX. ) . Also, by squaring each fraction in (x. ) and adding numerators and denominators , we get l² + m²1 + n² = 1, by (5577) . ( l² + m² + n²) ( l² + m² + n²) − (lls + m«m« + NgNg)

5579

If the transformation above is rotational , that is , if it

can be effected by a rotation about a fixed axis , the position of that axis and the angle of rotation @ are found from the 2 cos 0 = l₂ +m₂ + n3−1 ,

equations

cos² a

cos² Y

cos² B =

=

5580 - m₂+nç − l₁ − 1

n +h −m − 1

l₁ + m¿ — nç— l '

where a, B,, y are the angles which the axis makes with the original coordinate axes . PROOF. (Fig. 171. ) Let the original rectangular axes and the axis of rotation cut the surface of a sphere, whose centre is the origin O, in the points x, y , z, and I respectively. Then , if the altered axes cut the sphere in E, n,, we shall have 0 = 4 x in the spherical triangle ; Ix = I = a ; Iy = In = ß ; Iz = I = y, and by ( 882) applied to the isosceles spherical triangles IE, &c., l₁ = cos x = cos' a +sin a cos 0, m₁ = cos yn = cos* ß + sin² ß cos 0, ng = cos z = cos² y + sin² y cos 0. From these cos 0, cos a, cos ẞ, and cos y are found.

5581

Transformation of rectangular coordinates to oblique :

Equations (i . to vi . ) apply as before , but (vii . to ix . ) no longer hold, so that there are now six independent constants.

759

THE SPHERE.

THE

SPHERE .

5582 The equation of a sphere when the point abc is the centre and r is the radius , (x — a)² + (y — b)² + (≈ — c)² = r². 5583

The general equation is x² +y² +z² + Ax + By + Cz + D = 0 . A

B

2'

2

The coordinates of the centre are then

C , -N

and the radius = } √ ( A² + B² + C² —4D) . PROOF.-By equating coefficients with ( 5582) . 5584 If xyz be a point not on the sphere , the value of (x − a)² + (y — b) ² + ( - c)2-2 is the product of the segments of any right line drawn through xyz to cut the sphere.

PROOF. -From Euc. III . , 35 , 36.

THE RADICAL PLANE. The radical planes of the two spheres whose equations 5585 are u = 0, u' = 0 , is u- u' = 0. 5586 The radical planes of three spheres have a common section , and the radical planes of four spheres intersect in the same point. PROOF.- By adding their equations, and by the principle of (4608) extended to the equations of planes.

POLES OF SIMILITUDE . 5587

DEF. -A pole of similitude is a point such that the

tangents from it to two spheres are proportional to the radii. 5588 The external and internal poles of similitude are the vertices of the common enveloping cones .

SOLID GEOMETRY.

760

5589

The locus of the pole of similitude of two spheres is a

sphere whose diameter contains the centres and is divided harmonically by them .

CYLINDRICAL AND

CONICAL

SURFACES .

5590 DEF. -A conical surface is generated by a right line which passes through a fixed point called the vertex and moves in any manner. If the point be at infinity, the line moves always parallel to itself and generates a cylindrical surface. 5591

5592

Any section of the surface by a plane may be taken

for the guiding curve.

5593

To find the equation of a cylindrical or conical surface.

RULE . -Eliminate xyz from the equations of the guiding X -a Ꮓ = y-ß curve and the equations n of any generating m 1

- =

line ; and in the result put for the variable parameters of the line their values in terms of x, y, and z.

5594 Ex. 1.- To find the equation of the cylindrical surface whose generating lines have the direction Imn, and whose guiding curve is given by b'x +a'y' ab and z = 0. y-B x -α 2 At the point where the line ==== n meets the ellipse, z = 0, m ι x = a , y = ß. Therefore b'a² +a²ßª = a²b². Substitute in this, for the lz mz variable parameters, a, ß, a = x—-la , ẞ = y - m² ; and we get, for the n n cylindrical surface b' (nx - lz)2+ a² (ny - mz)² = a²b³n².

5595

A conical

surface whose vertex is the origin and

guiding curve the ellipse ba +a²y² = a²b² , z = c , is x2

/3

a² +21/2

PROOF.--Here the generating line is

0.

= У = 2. At the point of inter m N

761

CONICOIDS.

lc b212c2 a2m²c² = —, y = mc ; = a²b². + n² n n n2 Substitute for the variable parameters l m n the values ay : z , and the result is obtained.

section of the line and curve z = c,

CIRCULAR SECTIONS.

5596 RULE .- To find the circular sections of a quadric curve, express the equation in the form A (x² + y² + z² + c²) + & c . = 0. If the remaining terms can be resolved into two factors, the circular sections are defined by the intersection of a sphere and two planes. Generally the two quadrics ax² + by² + cz² + 2fyz + 2gzx + 2hxy

=

5597

:1

and (a + λ) x² + (b + λ) y² + ( c + λ ) z² + 2fyz + 2gzx + 2hxy = 1 have the same circular sections .

PROOF.-Let r, p be coincident radii of the two surfaces having lmn for a 1 Then = al² + bm² + cn² + 2fmn + 2gnl + 2hlm and 171 =10 ρ

common direction .

= the same + λ. Therefore, if r has a constant value throughout any section, p is also constant throughout that section .

5598 Ex.- An oblique circular cone whose vertex is the point a , 0, b, + y² = c² ; z = 0 ; is and guiding curve the circle (az- bx)² + b²y² = c² (z — b) ². The equation may be written b³ (x² + y² + z² — c³) = z≈ { 2abx + ( b² + c² — a²) z — 2bc² } , and therefore the cone has two series of parallel circular sections, z = k and 2abx +(b +c - a ) z - 2bc2 = p² (5583). (Frost and Wolstenholme. )

CONICOIDS .

5599 DEFS. - A conicoid is a surface every plane section of which is a conic . The varieties are the ellipsoid, the one-fold and two-fold hyperboloids, the elliptic and hyperbolic paraboloids , spheroid of revolution , the cone, and the cylinder . 5 E

the

762

SOLID GEOMETRY.

In any of the following equations of a conicoid, by making one of the variables constant, the equation of a section parallel to a coordinate plane is obtained, and the equation of the surface is by that means verified . Thus, in the equations of (5600 ) or (5617) , Figs . ( 172) and ( 173 ) , if z be put = ON, we get the equation of the elliptic section RPQ, the semi- axes of b a = which are NQ = C √ ( c² — ON¹ ) and NR = с √ (c² — ON³) , a , b , c being the principal semi-axes of the conicoid ; that is, OA , OB, OC in the figure .

THE ELLIPSOID . 5600 The equation referred to the principal axes of the figure is = 1.

(Fig. 172 )

Z a² + 4 b* + 2 5601 There equations are

are

two planes

of

circular

section

whose

- ~2 №2 ~ (1 1 ) — ~ ( 1 − 1 ) = 0, b2 − a²

1

with a > b > c. PROOF :

x²

1 = 0 ( 2 − 1 ) + v'² ( B −- 1 ) + ²² ( 1 − 1 ) =

is a cone having a common section with the conicoid and a sphere of radius r. If the common section be plane, one of the three terms must vanish in order that the rest may be resolved into two factors . Since a > b > c, the only possible solution for real factors is got by making r = b. 5602

Sections by planes

parallel to the above are also

circles , and any other sections are ellipses .

5603 The umbilici of the ellipsoid (see 5777 ) are the points whose coordinates are

a²-b² x = ±a

b2

y = 0, a²

C²'

~ = ± cv a² —c²*

PROOF . - The points of intersection of the planes ( 5601 ) and the ellipsoid b2-c² a² -b² z =±c (5600 ) on the az plane are given by a' = ± a a² -C2 a²-c² Since, by (5602) the vanishing circular sections are at the points in the s a с plane conjugate to x' and z ', we have, by (4352) , x = - 응 2 , z = a x' .

CONICOIDS.

763

5604 If a = b , in (5600 ) , the figure becomes a spheroid , and every plane parallel to ay makes a circular section . Hence the spheroid is a surface of revolution . It is called prolate or oblate according as the ellipse is made to revolve about its major or minor axis .

THE HYPERBOLOID. 5605 The equation of a one-fold hyperboloid referred to its principal axes is x2 ༧ y² = 1. (Fig. 173) +4-3 b2 a² 5606 The planes of circular section , when a > b > c , are all parallel to one or other of the planes whose equations are

- · ཆུ a² — ~ ( -1)

y v³ (

PROOF.

= 0. a² ) = +1

As in ( 5601 ) , putting r = a.

5607 The generating lines of this surface belong to two parallel systems (i . ) and (ii . ) below, with all values of 0. 5608

x

2

x

= cos 0+

≈

sin

= cos 0

sin 0 a

a

C

... (i .) , y = sin 0.-92/4 b

….. (ii .) . y Y = sin 0 + / COS 0 b

cos C

For the coordinates which satisfy either pair of equations , (i . ) or (ii . ) , satisfy also the equation of the surface . The equations may also be put in the forms -_____ a cos e

a sin 0 5612

y-b sin o =

=

5610

If z = 0,

-b cos 0

응.

x = a cos 0 and y = b sin 0.

Hence 0 is

the eccentric angle of the point in which the lines ( i . ) and (ii. ) intersect in the ay plane.

5613

Any two generating lines of opposite systems intersect ,

but no two of the same system do,

764

SOLID GEOMETRY.

5614 If two generating lines of opposite systems be drawn through the two points in the principal elliptic section whose eccentric angles are 0 + a , 0 — a , a being constant , the coordi nates of the point of intersection will be y = b sin 0 sec a,

x = a cos 0 sec a,

z = ± ctana,

and the locus of the point , as 0 varies , will be the ellipse №2 y² = 1 ; 21 + c tan a . 5615 + b² sec² a a² sec² a PROOF . - From (i . ) and ( ii. ) , putting

a for 0. *

The asymptotic cone is the surface given in (5595) . PROOF.-Any plane through the z axis whose equation is y = mx cuts the hyperboloid and this cone in an hyperbola and its asymptotes respectively.

5616

5617

The equation of a two- fold hyperboloid is 32 ~2 y = 1. ² a b2 c²

(Fig. 174)

and the equation of its asymptotic cone is 5618

a²

-b?

= 0.

PROOF. -Any plane through the x axis, whose equation is y = mz, cuts the hyperboloid and this cone in an hyperbola and its asymptotes respec tively. There are two surfaces, one the image of the other with regard to the plane of yz. One only of these is shown in the diagram.

5619

The planes of circular section when b is > c are all

parallel to one or other of the planes whose joint equation is

•22 0. X2 100( 10+10 1-0) b2)-23( b2 = 1 G -b². PROOF. -As in ( 5601 ) , putting r² = — 5620 If b = c , the figure becomes an hyperboloid of revo lution.

THE PARABOLOID . This surface is generated by a parabola which moves with its vertex always on another parabola ; the axes of the two curves being parallel and their planes at right angles. 5621

* The surface of a one-fold hyperboloid, as generated by right lines, may frequently be seen in the foot-stool or work-basket constructed entirely of straight rods of cane or wicker.

765

CENTRAL QUADRIC SURFACE.

The paraboloid is elliptic or hyperbolic according as the axes of the two parabolas extend in the same or opposite directions .

5622

The equation of the elliptic paraboloid is y² = x, 1/2+==== b

(Fig. 175)

b and c being the latera recta of the two parabolas . PROOF : QM = b.OM ; PN² = c.QN ; ..

QM² PN2 = OM+ QN = x. + b с

If bc , the figure becomes the paraboloid of revolution .

5623

Similarly the equation of the hyperbolic paraboloid is 2 y ≈2 x. (Fig. 176)* b2 c2

5624

The equations of the generating lines of this surface

are

y =m 3/6 ± 7 /c

and

y // F

x = с

"

m

the upper signs giving one system of generators and the lower signs another system .

5625

The equations of the asymptotic planes are y

ལྔ

= 0. 1/2 ± 老

CENTRAL

QUADRIC

SURFACE .

TANGENT AND DIAMETRAL PLANES .

x²

y2 + + b? a² oids l d o i b e o r d s e u and the two hyp both the ellip = 1 to incl

5626

Taking the equation of a central quadric

The curvature of this surface is anticlastic, a sort of curvature which may be seen in the saddle of a mountain ; for instance, on the smooth sward of some parts of the Malvern Hills, Worcestershire,

766

SOLID GEOMETRY.

according to the signs of b and c², the equation of the tangent plane at xyz is

ny = 1. ² + 2 +3+

By ( 5679).

5627 If p be the length of the perpendicular from the origin upon the tangent plane at xyz ,

56

2 1 = = = +2 + 3 b¹ p² PROOF.

From ( 5549 ) applied to (5626 ) . 56

5628 The length of the perpendicular let fall from any point n upon the tangent plane at ayz is

Ex (5554 & 5627)

p

. a² ++ 警 -1)

5629

Direction cosines of the normal of the tangent plane ι = = pay, a²

at xyz,

m = py b?

n = pz c²

PROOF.- By ( 5548 ) applied to ( 5626) and the value in (5627) . 5630

If l , m, n are the direction cosines of P,

p = lx + my + n≈

and

p² = u²l² + b²m² + c²n².

PROOF.- (5630 ) By projecting the three coordinates x, y , z upon p. (5631 ) By substituting the values of x, y, z , obtained from ( 5629) , in (5630 ) .

5632

The equation of the normal at xyz is

a² b2 = (n− y) —— = (8— =) —, (E — x) — x y since the dir- cos . are the same as those of the tangent plane at ( 5626 ) . 5633

Each term of the above equations = p √ { (§ − x)² + (n − y) ² + ( 5 — ≈)² }

or p multiplied into the length of the normal .

767

CENTRAL QUADRIC SURFACE.

(n ---− y)² = (5-2)² y² b₁ # Add numerators and denominators, and employ ( 5627) .

PROOF.- Each term squared

=

( -x)³ x a¹

5634 Equation (5631 ) is the condition that the plane lx + my + nz = p may touch the conicoid ; and if p = 0, we have for the condition of the plane la + my + nz = 0 touching 72 æ² the cone = 0, + + c² b2 a2 a²l² + b²m² + c²n² = 0 .

5635

5636

The section of the quadric made by a diametral plane

conjugate to the diameter through the point ayz has for its

equation

5637

Ex a² + 7 +2= 0.

By (5688) .

Hence the relation between the direction cosines of

two conjugate diameters is

W

nn'

mm'

+

+ ="

b2

= 0. c²

ECCENTRIC VALUES OF THE COORDINATES . 5638

These are defined to be

x = aλ,

ybp,

≈ = cv,

with

λ² +μ² + v² = 1 .

5640 , μ, are the dir-cos . of a line called the eccentric = rλ , n = 1µ , & line ; and rv are the coordinates of the corresponding point upon an auxiliary sphere of radius r . 5641 The eccentric lines of two conjugate semi-diameters are at right angles . By (5637) .

5642 The sum of the squares of three conjugate semi diameters is constant and = a² + b² + c². PROOF. - Let a' , b' , c' be the semi- diameters , and ₁₁₁ , yZ2, 333 their +y + = a , &c. , extremities. Put the eccentric values in the equations and add. By ( 5641 ) , λ; + λ² + x² = 1 , & c.

768

SOLID GEOMETRY.

5643 The sum of the squares of the reciprocals of the same is also constant . PROOF. - Put r, cos a₁, r₁ cos ₁, r₁ cos y₁ for x₁ , y₁, %, in the equation of the quadric . So for x , y , z, and 3, 3, 23. Divide by r1 , 72, 73, and add the results.

5644

The sum of squares of reciprocals of perpendiculars

on three conjugate tangent planes is constant. PROOF. For each perpendicular take ( 5627) , and substitute the eccentric values as in (5642) . 5645 The sum of the squares of the areas of three con jugate parallelograms is constant.

PROOF. By the constant volume of the parallelopiped p,4, = P₂A₂ = P3A3 (5648) and by ( 5644) .

The sum of the squares of the projections of three conjugate semi-diameters upon a fixed line or plane is constant . 5646

PROOF. With the same notation as in ( 5642) , let ( Imn) be the given line. Substitute the eccentric values ( 5638 ) in ( la, + my₁ + nz₁ ) ² + ( læ₂ + my₂ + nz₂) ² + (lxz + myz + nzz) . In the case of the plane we shall have a²² - (lx₁ + my + nz₁ ) ² + &c . ________ COR COR .. The extremities of three conjugate semi ters diame being ay₁₁ , 2 , 333, it follows that , by pro

5647

jecting upon each axis in turn , x²+x² 2 +x² = a² ;

y} +y + y = b² ;

~ + ~ + ~ = c².

5648 The parallelopiped contained by three conjugate semi diameters is of constant volume = abc. PROOF. -By ( 5568 ) , the volume

by the eccentric values (5638 ) . (584, I.) . 5649

COR .- If a' ,

X1 Y1 Z1 = abcμy X2 Y2 Z2 με νε λ X3 Y3 Z3 Аз μз 1'3 But the last determinant = 1 by =

' , c' are the semi- conjugate diameters ,

w the angle between a' and ' , and p the perpendicular from the origin upon the tangent plane parallel to a'b' , the volume of the parallelopiped is pa'b' sin w = abc.

769

CENTRAL QUADRIC SURFACE.

5650

Hence the area of a central section in the plane of a'b' abc

Ta'b' sin ∞ = T p

5651

Quadratic for the semi-axis of a central section of the ‫دارم‬ x² y³ quadric + 2 + ² = 1 made by the plane la + my + nz = 0 : a² c b

b2m²

a²12 + a² -p²

c²n²

= 0. -b² — p² + c² —

PROOF. The equation is the condition, by (5635), that the plane lx + my + nz = 0 may touch the cone

x²

+y ( 1-1 ) +00 (10-1) + a

as in the Proof of ( 5600 ) .

5652 form

(1-1 ) = 0,

For another method , see ( 1863) .

When the equation of the quadric is presented in the

ax² +by² + cz² +2fyz + 2gzx + 2hxy = 1 ,

takes the quadratic for the form of the determi

1 a

h

h

b

nant equation annexed . Or, by expanding, and for the same writing determinant , with the 1 erased, the fraction 7.2 equation becomes

g

ι

f

m

1

2.2

= 0. 1

g

ι

f m

n

C

202 n

A'r' + { (b + c) l² + (c + a) m² + (a + b) n² - 2fmn - 2gnl - 2hlm } r² -12 - m² - n² = 0. The equation of the cone of intersection of the sphere and PROOF. quadric now becomes ' (a − 1 ) x² + ( b − 1 ) y² + ( c− 1 ) ² + 2fyz + 2gzx + 2hay = 0, and the condition of touching (5700) produces the determinant equation . 5 F

770

SOLID GEOMETRY.

5654

To

find the axes of a non- central quadric x² + y² + 22 = 1 . a² b2 c²

section

of the

Let PNQ (Fig. 177) be the cutting plane. Take a parallel central section BOC, axes OB, OC, and draw NP, NQ parallel to them.

These will be the axes of the section PNQ, and NQ ON² , NQ² will be found from the equation + =: 1 . OA2 OC2 5655

The area of the same section πabc -2). р (1

where p' and p are the perpendiculars from 0 upon the cutting plane and the parallel tangent plane.

PROOF .-The area

NQ² = TNP.NQ = π .OB.OC 00²

=π * ( 1— ON³) OB.OC =

(1–2 ) , by ( 5650) . abe (1-7), P

SPHERO-CONICS . DEF.-A sphero -conic is the curve of intersection of the surface of a sphere with any conical surface of the second degree whose vertex is the centre of the sphere . Properties of cones of the second degree may be investi gated by sphero- conics , and are analogous to the properties of conics .

A collection of formule will be found at page 562 of Routh's Rigid Dynamics, 3rd edition .

CONFOCAL QUADRICS . 5656

DEFINITION .--The two quadrics whose equations are Xx2

x2 ++

a² are confocal.

= 1

and

1, a² + λ+++ + b² + λ + c² +λ = 1

We shall assume a >b > c.

decreases from being large and positive , the 5657 As third axis of the confocal ellipsoid diminishes relatively to the

4

CENTRAL QUADRIC SURFACE.

until λ == = -c², when the surface merges focal ellipse on the xy plane,

others

22

--

771

into the

y2 = 1. + 2 b -c²

A still diminishing , a series of one-fold hyperboloids appear until λ = -b² , when the surface coincides with the focal hyperbola on the za plane ,

X2

≈2 -

a² -b²

1. b² - c²

The surface afterwards developes into a series of two -fold hyperboloids until λ = -a² , when it becomes an imaginary focal ellipse on the yz plane.

5658 Through any point xyz three confocal quadrics can be drawn according to the three values of furnished by the second equation in ( 5656) . tions , becomes 5659

That equation , cleared of frac

λ³+ (a² +b² +c²− x² — y² — ~²) λ²

{b°c² +c²a² + a²b²— (b² +c²) x² — ( c² + a²) y° — ( a² +b²) z² } λ + a²b²c² —b²c²x² - c²a²y² — a²b²~² = 0 .

These three confocals are respectively an ellipsoid, a one fold hyperboloid , and a two-fold hyperboloid . See Figure (178 ) ; P is the point ayz ; the lines of intersection of the ellipsoid with the two hyperboloids are DPE and FPG, and the two hyperboloids themselves intersect in HPK. PROOF. --- Substitute for A successively in (5659) a², b² , c², -∞ ; and the 2 +₂ left member of the equation will be found to take the signs + , accordingly. Consequently there are real roots between a² and b², b² and c², c² and -∞ .

5660 Two confocal quadrics of different species cut each other everywhere at right angles . PROOF. Let a, b, c ; a', b', c' be the semi-axes of the two quadrics ; then, at the line of intersection of the surfaces, we shall have

1 1 -72 (1-1 ) = 0, ~ () / − 1 ) + y² (0 (} − ) + ( − 1 ) = 0, )

SOLID GEOMETRY.

772

which , since a² — a² = b²² — b² = c²² - c² = λ , becomes the condition of per pendicularity of the normals by the values in ( 5629) . Thus, in ( Fig. 178 ) , the tangents at P to the three lines of intersection of the surfaces are mutually at right angles.

5661

If P be the point of intersection of three quadrics

a₁₁₁, abc , аžbзes confocal with the quadric abc ; the squares of the semi -axes , d₂, d , of the diametral section conjugate to P in the first quadric are (considering a, > a, > ag, and writing the suffixes in circular order) d₁ =: a₁ ~ a In the second ,

d₁ = a² ~ až

in the third ,

d² = až ~ aj

2 d₁ = aj ~ az, d₁ = a; ~ a₁ , 2 2 di = až ~ a². d₁

Or, if for a , a , as we put a² +λ₁, a² + λ², a² + λ3, the above values may be read with A in the place of a and the same suffixes .

PROOF .- Put a - a² = µ ; then x² + |

1/2 z3 = 1 and + b2 a²-

+

1

+ b2-μ

c²- μ

are confocal quadrics .

Take the difference of the two equations, and we x2 + &c. 0. Comparing this with . a² (a² -μ) (5651 ) , the quadratic for the axes of the section of the quadric by the plane æ' lx + my + nz = 0, we see that, if l, m, n have the values &c., μ is identical a²' e with ; the plane is the diametral plane of P ; and the two values of µ are the squares of its axes. Let d , d be these values ; then, since there are but three confocals, the two values of μ must give the remaining confocals, i.e., = a - da and a — d = a . obtain, at a common point x'y'z',

The six axes of the sections are situated as shown in the diagram ( Fig. 179 ) . Either axis of any of the three sections is equal to one of the axes in one of the other sections , but the equal axes are not those which coincide . O is supposed to be the centre of the conicoids , and the three lines are drawn from O parallel to the three tangents at P to the lines of in tersection .

5662 Coordinates of the point of intersection of three con focal quadrics in terms of the semi-axes :

773

CENTRAL QUADRIC SURFACE.

2 bibb

2 2 2 ajaa y² =

x² =

(a₁ — bi) (bi— ci) '

(a - b₁) (a₁ -ci) '

2 2 2 C1 C2 C3

~2 =

(ai - ci) (bi - ci) The denominators may be in terms of any of the confocals 2 = since a -ba — b₂ = a3 — b3, &c.

x2 PROOF. -The equation of a confocal may be written a² + 1 = 1 , producing a cubic in a2, the product of whose roots a², a

z3 ya 2 -h2 + a² -k² 1 1 , a²3 gives a².

5663 The perpendiculars from the origin upon the tangent planes of the three confocal quadrics being P1 , P2, P3 : 2 22 2 a bc abe P₁ = P₁₂ = (a — a²) (a — a³) ' (a₁—a²) (až— a ) ' 2 2 2 a3b3c3 p3 = p₁ (a — až) ( a — a³) PROOF.- By (5649) , p₁d₂d¸ = a¸b₁₁ ; then by the values in (5661 ) .

RECIPROCAL AND ENVELOPING CONES .

5664

DEF .-A right line drawn through a fixed point always

perpendicular to the tangent plane of a cone generates the reciprocal cone. The enveloping cone of a quadric is the locus of all tangents to the surface which pass through a fixed point called the vertex.

5665

The equations of a cone and its reciprocal are respec

tively ·Ax² + By² + C≈² = 0 ...... (i .) ,

and

x2 4 A

= 0 …………… . (ii .) . +2/3+ C

PROOF. - The equations of the tangent plane of ( i . ) at any point xyz, and of the perpendicular to it from the origin, are ૐ = η = Հ Ax + Byn + Cz¿ = 0 ......... ………………… .. ( iii . ) , and (iv.) . Ax Cz By Eliminate x, y, z between (i . ) , ( iii . ) , and (iv. ) .

5667

The reciprocals of confocal cones are concyclic ; that

774

SOLID GEOMETRY.

is , have the same circular section ; and the reciprocals of con cyclic cones are confocal .

PROOF.-A series of concyclic cones is given by Ax² + By² + Cz² + λ ( x² + y² + z² ) = 0 by varying λ ; and the reciprocal cone is 23 y² x² = 0. + + C +λ B +λ A+λ

(5665)

5668

The reciprocals of the enveloping cones of the series 22 x2 y2 of confocal quadrics + + = 1 , with fgh for a² + λ b²+ λ c² + λ the common vertex , P, of the cones , are given by the equation a²x² +b²y² + c²≈²− (fx +gy + hz)² + λ ( x² +y² +z²) = 0 . PROOF. - Let Imn be the direction of the perpendicular p from the origin upon the tangent plane drawn from P to the quadric. Equate the ordinary value of p² at (5631 ) with that found by projecting OP upon p ; thus (a² + λ) l² + ( b² + λ ) m² + (c² + λ ) n² = (fl + gm + hn)². Now p generates with vertex ( a cone similar and similarly situated to the reciprocal cone with vertex P, and l, m, n are proportional to x, y, z, the coordinates of any point on the former cone. Therefore, by transferring the origin to P, the equation of the reciprocal cone is as stated. 5669

COR .

These reciprocal

cones

are

concyclic ;

and

therefore the enveloping cones are confocal ( 5667) .

5670

The reciprocal cones in ( 5668) are all coaxal.

PROOF. - Transform the cone given by the terms in ( 5668 ) without A to its principal axes ; and its equation becomes A + By² + Cz² = 0. Now, if the whole equation, including terms in A, be so transformed, a + y² + z² will not be altered . Therefore we shall obtain (A + λ) x² + (B + λ) y² + ( C + λ ) z² = 0, a series of coaxal cones.

5671 The axes of the enveloping cone are the three normals to the three confocals passing through its vertex. PROOF. The enveloping cone becomes the tangent plane at P for a con focal through P, and one axis in this case is the normal through P. Also this axis is common to all the enveloping cones with the same vertex, by (5670 ) . But there are three confocals through P ( 5658) , and therefore three normals which must be the three axes of the enveloping cone.

5672

The equation of the enveloping cone of the quadric

THE GENERAL QUADRIC.

x2

22

y² +

+ a² + λ

775

b² + λ

1

is , when transformed to its prin

c² + λ

cipal axes , y² x² + λαλι πλ

≈2 =0 -λg

~2 y2 = 0, + Χ +X ' + dX + d

x2 or

where A1 , A2, A, are the values of

for the three confocals

through P, the vertex , and d₂, d, are the semi - axes of the diametral section of P in the first confocal (5661 ) . PROOF. -Transform equation (5668) of the reciprocal of the enveloping cone to its principal axes, as in ( 5670 ) . Let A , A , A be the values of X which make the quadric become in turn the three confocal quadrics through P. Then the reciprocal (4+ ) x² + ( B + λ) y² + ( C + λ ) z² = 0 must become a right line in each case because the enveloping cone becomes a plane. Therefore one coefficient of x , y , or z must vanish. Hence A + ₁ = 0, B +λ = 0, C + λ = 0. Therefore the reciprocal cone becomes (^—λ₁) œ² + (^ —λ₂) y² + (^ —λg) z² = 0, and therefore the enveloping cone is x2 y³ + = 0. + λ- λ λ- λα λα- λ

THE

GENERAL EQUATION

OF A QUADRIC .

This equation will be referred to as f(x, y , z ) = 0 or 5673 U = 0 , and , written in full , is ax² +by² +cz² +2fyz + 2gzx + 2hxy + 2px + 2qy + 2rz + d = 0. By introducing a fourth quasi variable t = 1 , the equation may be put in the homogeneous form 5674

ax² +by² + cz² + dw² + 2fyz + 2gzx + 2hxy +2pxt +2qyt +2rzt = 0,

abbreviated into (a, b, c, d, f, g, h, p , q, r(x, y , z , t) ² = 0 , as in (1620) . Transforming to an origin a''z' and coordinate axes parallel to the original ones , by substituting a' + E , y' + n , % + ¿ for x, y, and z , the equation becomes , by (1514) ,

776

SOLID GEOMETRY.

5675

a§² + bm² + c5² + 2ƒn $ + 2g$$ + 2h&n + §U₂ +ŋU„ + ¿ U₂ + U = 0 ,

where Uf(x , y , z) (omitting the accents ) .

The quadratic for r, the intercept between the point the quadric surface measured on a right line drawn and a'y' from 'y' in the direction Imn, is 5676

r² (al² + bm² + cn² + 2fmn +2gnl +2hlm ) +r (l₂ +mU, + nU₂) + U = 0 . Obtained by putting & = rl, n = rm , L = rn in ( 5674) . 5677 The tangents from any external point to a quadric are proportional to the diameters parallel to them. PROOF .

From ( 5676) , as in ( 1215 ) and (4317) .

The equation of the tangent plane at a point ryz on the quadric is 5678

(§− x) U¿ + (n −y) U, + (¿ −≈) U, = 0

5679

or

§U¿ +nʊ„ + ¿U₂ ++U ; = 0,

with T and t made equal to unity after differentiating. PROOF. From ( 5676 ) . Since xyz is a point on the surface, one root of the quadratic vanishes. In order that the line may now touch the surface, the other root must also vanish ; therefore IU₂ + mU„ + n U₂ = 0 . Put rl = ¿ −x, rm = n − y, rn = 6-2 ; En being now a variable point on the line, and therefore on the tangent plane.

5680 Again, therefore

x Uz + y Uy + z U₂ + tU, 2U, by ( 1624) , æUx + y U₁y + zU₂ = − t Ut which establishes the second form ( 5679) .

5681 Equation (5679) also represents the polar plane of any point xyz not lying on the quadric surface. Written in full it becomes

(ax + hy + g + p)

x (a + hn +g +p)

or

+n (hx + by+fz +q)

+y (h§+ bn +ƒ$ + q)

+ $ (gx +fy + cz + r )

+ ≈ (g§+fn + c$+ r)

+

px + qy + rz + d = 0,

+

pš+ qn + r$ + d = 0.

THE GENERAL QUADRIC.

5683

777

That is , the forms

§U₂ +nʊ, + ¿ U₂ + U = 0

and

U¿ + yU„ + zUç + U = 0

are convertible, U standing for f(x, y, z) in the first, and for f( , n, ) in the second . The intersection of the polar planes of two points is called the polar line of the points .

5685

5686 The polar plane of the vertex is the plane of contact of the tangent cone.

PROOF. If En be the vertex and xyz the point of contact, equation (5683 ) is satisfied . If x, y , z be the variables and En constant , the second form of that equation shows that the points of contact all lie on the polar plane of the point in .

5687 Every line through the vertex is divided harmonically by the quadric and the polar plane. PROOF. In equation ( 5684) put a = & + Rl, y = n + Rm , z = 5 + Rn_to determine R, the distance from the vertex to the polar plane. This gives -20 R= 9 employing (5680) . IU¿ + mU₂ + nU Now, if r, r' are the roots of the quadratic ( 5676) , with §, n , written for 2rr' x, y, z, it appears that = R, which proves the theorem. r+ r'

Every line ( mn) drawn through a point ayz parallel 5688 to the polar plane of that point is bisected at the point, and the condition of bisection is

IU₂ + mU¸ + nU₂ = 0 . PROOF. -The equation is the condition for equal roots of opposite signs in the quadratic (5676) . Since l, m, n are the dir. cos . of the line and Ur, Uy, U₂ those of the normal of the polar plane (5683 ) , the equation shows that the line and the normal are at right angles (5532) .

The last , when x, y , z are the variables , is also the equation of the diametral plane conjugate to the direction Imn . Expanded it becomes 5689

(al +hm +gn ) x + (hl + bm +fn ) y + (gl +fm + cn ) z

+pl+ qm + rn = 0. 5 G

778

SOLID GEOMETRY.

For the point xyz moves, when x, y , z are variable, so that every diameter drawn through it parallel to lmn is bisected by it, and the locus is, by the form of the equation , a plane. If the origin be at the centre of the quadric , p, q, and r of course vanish.

5690 The coordinates of the centre of the general quadric U = 0 (5673) are

x=

Δ ', 2A

Δ ལྤ =

y = 2A

Δ 2A

PROOF. -Every line through xyz, the centre, is bisected by it. The condi tion for this, in ( 5688 ) , is U₂ = 0 , U₁ = 0 , and U₂ = 0, in order to be inde pendent of Imn. The three equations in full are a h g P a h g ax + hy + gz + p = 0 ) h b f q A = hb f hx + by + fz + q = 0 ; and A' = c gf gf c gx + fy + cz + r = 0 d p q r⠀ -Solve by (582).

5691

The quadric transformed to the centre becomes

a5ª + bn³ + c8ª + 2ƒn6+ 2g65+ 2h5n + ÷ = 0. PROOF. By the last theorem, the terms involving E, n , in (5675) vanish. The value of U or ƒ (x, y , z) , when xyz is the centre, appears as follows :

′ (1647) . U = §U₂ (5680) = px + qy + rz + d = p²p + q¹ + rA; + d (5690 ) = = 2A The last equation , being again transformed by turning the axes so as to remove the terms involving products of coordi dinates, becomes

5692

5693

ax² + By² +yz² + — ' = 0,

where a, ẞ, y are the roots of the discriminating cubic

R³ — R² (a +b + c) + R (bc + ca + ab —ƒ² — g² — h²) — A = 0, or ( R — a) (R — b) ( R — c) — ( R — a) ƒª² — (R — b ) g²— (R — c) h -2fgh = 0. PROOF.

It has been shown , in ( 1847-9 ) , that the roots of the discrimi

nating cubic ( multiplied in this case by -maximum aud minimum values of

are the reciprocals of the

+y +z . But such values are evidently

779

THE GENERAL QUADRIC.

the squares of the axes 2of the quadric surface. Let the central equation of x² y³ 1 Δα the surface be + + = 1. Therefore 2 = -producing a² A" &c., bi the equation above.

5694 The equations of the new axis of a referred to the old axes of E , n , are (F + af) x = (G + ag) y = (H+ah) ≈ ; and similar equations with ẞ and y for the y and z axes . PROOF. -When Imn, in ( 5689 ) , is a principal diameter of the quadric, the diametral plane becomes perpendicular to it, and therefore the coefficients of x, y, z must be proportional to l, m, n . Putting them equal to Rl, Rm, Rn respectively, we have the equations The eliminant of these equations is (a − R) 1 + hm + gn = 0 ...... ( 1) the discriminating cubic in R al hl+ (b - R) m +fn = 0 ...... ( 2) ready obtained in (5693) . gl+fm + (c - R) n = 0 ..... (3)

From (1 ) and ( 2) , and from (2) and (3) ,

l : m = hf— g (b − R) : gh —ƒ (a − R), m : n = fg - h (c - R) : hf- g (b - R) ;

therefore (gh - af + Rf ) l = (hf- bg + Rg) m = (fg ― ch + Rh) n, which establish the equations, since xyz = 1 : m : n and F = gh - af, &c., as in (4665) .

5695

The direction cosines of the axes of the quadric .

If the discriminating cubic be denoted by $ ( R) = 0 , and its roots by a, ß, y ; the direction cosines of the first axis are

-

Φαa ( α) " Pa (a)

- Φι(α ) P.(a)'

- pcC (a) Φα ( α )

For the second and third axes write ẞ and y in the place of a . ...... (i.) , F + af= L, G+ ag = M, H + ah = N ...... PROOF . Let (a − b) ( a — c) —f² = λ , ( a − c) ( a− a) — g² = µ, (a — a ) (a − b) — h³ = v... (ii. ) . ( a) = 0 may be put in either of the forms Then the equation ...... L2 = µv, M³ = vλ, N³ = λµ .... (iii.) .

Now the dir. cos. of the first axis are, by (5694) , proportional to 1 1 1 = √λ : √µ : √v, by (iii. ) . : L M N Their values are, therefore, √ √ ( +µ + v)'

νομ √ (^ +µ + v)'

√v √ (x + µ + v)`

But λ = do (a) and λ+ µ + v =- do (a) , by actual differentiation of the da da cubic in (5693) .

780

SOLID GEOMETRY.

5696 Cauchy's proof that the roots of the discriminating cubic ( 5693) are all real will be found at (1850 ) .

5697 The equation of the enveloping cone, vertex xyz, of the general quadric surface U = 0 (5673 ) is 4 (abcfghlmn) U = (IU¸ + mU„ + nU₂)², with

-x, n - y,

-z substituted for l , m , n.

PROOF. -The generating line through xyz moves so as to touch the quadric. Hence the quadratic in r (5676) must have equal roots . The equa tion admits of some reduction . 5698

When U takes the form ax + by + cz² = 1 ,

equation

(5697) becomes (al² + bm² + cn²) (ax² + by² + c≈² - 1) = (alx + bmy + cnz)².

5699 The condition that the general quadric equation may represent a cone is A' = 0 ; that is , the discriminant of the quaternary quadric, (5674) or ( 1644) , must vanish. PROOF. - By (5692 ) . Otherwise A' = 0 is the eliminant of the four equations U = 0, U, = 0, U₂ = 0, U = 0, the condition that equation (5675) may represent a cone.

The condition that the plane lx + my + nz = 0 may touch the cone (abcfghXxyz) = 0 is the determinant

a

h

g

ι

h

bf fc m n

m n

equation on the right .

g ι

5700

= 0.

PROOF.-Equate the coefficients l, m, n to those of the tangent plane (5681) , p, q, r being zero, and xyz the point of contact . A fourth equation is lx + my + nz = 0, which holds at the point of contact. The eliminant of the four equations is the determinant above.

The

condition

that

the

a

hg

may

h

b

gf (abcdfghpqrXxyz1)² = 0 (5673) is the determinant equation on the right.

PROOF. As in (5700).

p ι

9 m

f c r

P

Z

q

m

d t

12

plane lx + my + nz + t = 0 touch the quadric

P4T2 * 1 2

5701

= 0. t

RECIPROCAL POLARS.

781

5702 If the origin is at the centre, p = q = r = 0. In that case, transposing the last two rows and last two columns , the determinant becomes

a h g l 0

= 0,

or,

d

a

h g

ι

= t²

a h g

h bfm 0

h b fm

h b f

gfcn0 l m n 0 t

g fc n I m n

gƒ c

0 0 0 td

5703

The condition that the line of intersection of the

planes lx + my + nz + t = 0 ... (i . )

and

l'a + m'y + n'z + ť′ = 0 ... (ii . )

may touch the general quadric (abcdfghpqrXxyz1 )² = 0 , is the determinant equation deduced below. Multiply equation (i . ) by E and (ii. ) by ʼn to obtain the plane (lE + l'n) x + (me + m'n) y + (në + n'n ) z + t + t'n = 0 ......... (iii . ) ,

RECIPROCAL

220

294 སྣ་ ༤ོང

passing through the intersection of (i. ) and (ii. ) . The line of intersection will touch the quadric if (iii. ) coincides with the tangent plane at a point xyz, and if xyz be also on ( i . ) and ( ii . ) . Therefore, equating coefficients of (iii.) and the tangent plane at ryz ( 5681 ) , we get the six following equa tions, the eliminant of which furnishes the required condition , a h 9 P ι ι ax + hy + gz + pw = 15+ l'n b f q m' hx + by + fz + qw = m² + m'n n' с 2° gx + fy + cz + rw = në + n'n g f = 0. ť px + qy + rz + dw = t¿ + t'n q r d m n t 1x + my + nz + tw = 0 m' n' t' l'x + my + n'z + t'w = 0

POLARS .

5704 The method of reciprocal polars explained at page 665 is equally applicable to geometry of three dimensions. Taking poles and polar planes with respect to a sphere of reciprocation , we have the following rules analogous to those on page 666.

SOLID GEOMETRY.

782

RULES FOR RECIPROCATING . 5705

A plane becomes a point.

5706

A plane at infinity becomes the origin.

5707 Several points on a straight line become as many planes passing through another straight line. These lines are called reciprocal lines. 5708

Points lying on a plane become planes passing through

a point, the pole of the plane. 5709 Points lying on a surface become planes enveloping the reciprocal surface. 5710 Therefore, by rules (5708) and ( 5709) , the points in the intersection of the plane and a surface become planes passing through the pole of the plane and enveloped both by the reciprocal surface and by its tangent cone. 5711 When the intersecting plane is at infinity, the vertex of the tangent cone is the origin. 5712 Therefore the asymptotic cone of any surface is orthogonal to the tangent cone drawn from the origin to the reciprocal surface.

The cones are therefore reciprocal.

5713 The reciprocal surface of the quadric is a hyperboloid, an ellipsoid, or a paraboloid, according as the origin is without, within, or upon the quadric surface. 5714 The angle subtended at the origin by two points is equal to the angle between their corresponding planes. 5715 The reciprocal of a sphere is a surface of revolution of the second order. 5716 The shortest distance between passes through the origin.

The

reciprocal

(abcdfghpqrXxyz1 ) x² + y² + z² = k² , is

a

h

g

hb

f

gf p q

c r

= 0 (5674) ,

p

ૐ

զ 1°

η

d

surface

r

of

the

η

C - k²

quadric

the auxiliary sphere being

dah g = 0,

h b fn

-k² g fc l

ξ

general

or, if p = q = r = 0,

3 ~

5717

two reciprocal lines

§ η ८०

| -k* | a h g h b f

1g f c

THEORY OF TORTUOUS curves.

783

PROOF. The polar plane of the point n with respect to the sphere is Ex + ny + 5z — k² = 0. This must touch the given surface, and the condition is given in (5701) .

5718 The reciprocal surface of the central quadric x² y² 22 + + = 1 , when the origin of reciprocation is the a³ b2 c²

point x'y'z' , is (§x' +ny ' + ¿z' — k²) ² = a²§² + b²n² + c²5³ ,

or, with the origin at the centre , 5719

a²§² + b²n² +c²¿² = k¹.

PROOF. -Let p be the perpendicular from a'y'z' upon a tangent plane of the quadric, and in the point where p produced, intersects the reciprocal surface at a distance p from x'y'z' . Then (5630) k²p - ¹ = p = lx' + my' + nz′ — √✓ (a³l³ + b³m³ + c²n³). Multiplying by p produces the desired equation .

THEORY OF TORTUOUS CURVES .

DEFINITIONS. - The osculating plane at any point of a 5721 curve of double curvature, or tortuous curve, * is the plane containing either two consecutive tangents or three consecu tive points . 5722 The principal normal is the normal in the osculating plane. The radius of circular curvature coincides with this normal in direction. 5723 The binormal is the normal perpendicular both to the tangent and principal normal at the point. 5724 The osculating circle is the circle of curvature in the osculating plane, and its centre, which is the centre of circular curvature, is the point in which the osculating plane intersects two consecutive normal planes of the curve . 5725

The angle of contingence , dy, is the angle between two

consecutive tangents or principal normals . The angle oftorsion, dr, is the angle between two consecutive osculating planes.

* Otherwise named "' space curve."

784

SOLID GEOMETRY.

5726 The rectifying plane at any point on the curve is per pendicular to the principal normal ; and the intersection of two consecutive rectifying planes is the rectifying line and axis of the osculating cone.

5727

The osculating cone is a circular cone touching three

consecutive osculating planes and having its vertex at their point of intersection . The rectifying developable is the envelope of the rectifying planes , and is so named because the curve, being a geodesic on this surface, would become a straight line if the surface were developed into a plane . 5728 The polar developable is the envelope of the normal planes , being the locus of the line of intersection of two con secutive normal planes . Three consecutive normal planes intersect in a point which is the centre of spherical curvature : for a sphere having that centre may be described passing through four consecutive points of the curve . 5729

The edge of regression is the locus of the centre of

spherical curvature. 5730 The rectifying surface is the surface of centres ( 5773 ) of the polar developable . 5731 An evolute of a curve is a geodesic line on the polar developable. It is the line in which a free string would lie if stretched between two points , one on the curve and one any where on the smooth surface of the polar developable .

5732 In Figure ( 180 ) A , A' , A ”, A"" are consecutive points on a curve. The normal planes drawn through A and A' intersect in CE ; those through A' and A" in C'E', and those through A" and A"" in C" E". CE meets C'E' in E, and C'E' meets C" E" in E' . The principal normals in the normal planes are AC, A'C', A" C" , and these are also the radii of curvature at A, A' , A", while C, C' , C" are the centres of curvature. LACA' =du and CA'C' dr. The surface ECC'C"E' is the polar developable, CC'C" being the locus of the centres of curvature, and EE'E" is the edge of regression . EA is the radius and E the centre of spherical curvature for the point A. hн, ÍÍ', H'H " are elemental chords of an evolute of the curve, AhH being a normal at A, and A'HH' a normal at A' , and so on . The first normal drawn is arbitrary, but it determines the position of all the rest.

THEORY OF TORTUOUS CURVES.

785

PROPERTIES OF A TORTUOUS CURVE. 5733 The equation of the osculating plane at a point ayz on the curve is (§− x) λ + (n − y) µ + ( 5 — ≈) v = 0 . 5734

λ, μ, v are the direction cosines of the binormal, and

their complete values are P (Ys28 - Y2ss) ,

5735

p (≈s X25 - Z28 X3) ,

P (XsY2s — X25Ys) .

The angle of contingence

d¥ = √ { (Ys≈2s —Y2s≈5) ² + (≈5X28 − ≈2, X¸) ² + ( X¸Y23 — X25Ys)²} ds . PROOF. - Let the direction of a tangent be lmn, and that of a consecutive tangent l + dl, m + dm, n + dn . Since the normal of the plane must be per pendicular to both these lines , we shall have, by ( 5532) , lλ + mµ + nv = 0 and ( l + dl) λ + ( m + dm) µ + (n + dn) v = 0 , λ :: µ : v = mdn — ndm : ndl ― ldn : ldm ― mdl, therefore and the denominator in the complete values of λ, μ, v is √ { (mdn − ndm) ² +&c. } = sin dų, by (5521 ) ; that is, = dy . Also l , m, n = x , y , z, and dl = x²ds, &c. ds Therefore λ = (y₁ =28 - Y27 ) Similarly, μ and v ; and s = p, by ( 5146) . dy

5736

The radius of curvature p at a point xyz .

2 = x² +y² +≈28 = I B PROOF : therefore

X22t + Y₂t 2t + ≈2t•

$2 2t

st

d¥ = √ { (Ys7qs— Y2878)² + &c. } ds , in ( 5735) , = √ { ( x² + y² + z² ) (x² + y² + 223) − (í „X28 + YoY 28 + Zg%28) ³ } = √ (x²28 + y² + z2 ) ; since

+y +

= 1 ;

and differentiating this equation makes a, 2, + &c. = 0. Otherwise, geometrically, precisely as in the proof of ( 5141 ) , we find the direction cosines of the principal normal to be

5737

cos a = px287

cos B = py281

cos y = p2s.

Therefore p² ( x² + y²28 + z2 ) = cos² a + cos² ß + cos³ y = 1 . The change to the independent variable t is made by ( 1762) .

5738

The angle of torsion, in terms of λ , μ , v of ( 5734) , is

35 + µY3s + v≈38) pds dr = √ (x² +µ² + v² ) ds = (λX38 ·√ { (µv¸ — µ‚v) ² + (vλ¸ — v¸λ)² + (dµ, —d¸µ)2} . 5 H

SOLID GEOMETRY.

786

PROOF. -By (5745 ) , we have (dr) ³ = (dλ) ³ + (dµ ) ² + (dv ) ³ ............ (i .) , which gives the first form. The third reduces to this by the method in (5736) . For the second form put u = Y , Z28 - Y287 , &c. , then du ds udK บ 1 λ μ = = = = &c. (5734), dλ = u w K dự K ย K Substitute in (i. ) , reducing by K² = u² + v² + w² and KdKudu + v dv + wdw.

CURVATURE AND TORTUOSITY.

ds 5739

1

du

P

ds

1

dr

Curvature =

Radius of curv. , p = dy ds

Radius of torsion , σ =

Tortuosity =

; dr

If T, changes zero

sign while

or infinity, there is

ds passing

a point

of

through the inflected

values

torsion

or

a cuspidal point, respectively. If 7 , without changing sign, passes through zero or infinity, there is a point of suspended torsion or infinite torsion respectively.

Ts is zero , identically, the curve is plane. If 7,

5740

The radius of spherical curvature, R = √ (p²+ pr).

PROOF.- In Fig. ( 180) R = p² + EC and EC = p, by analogy with q =P n a plane curve (see proof of 5147) . 5741 The element of arc of the locus of centres of circular curvature is ds'

Rdr , and therefore R = 87 .

PROOF. - In Fig. ( 180 ) ds' = CC′ = pdr sec

5742

= Rdr.

The radius of curvature of the edge of regression

= S4 = RR, = p + p₂r 9 S" being the arc of the edge of regression . PROOF. An inspection of Figure ( 180) shows that R and p stand in the same relation to the edge of regression that r and p occupy with regard to a curve in the standard formula. In fact we may substitute R for r, p for p,

THEORY OF TORTUOUS CURVES.

787

for 0,7 for 4, and remains . The chosen line of reference AB being always parallel to the tangent EC, then AEC = BAE = $. Also the angle of contingence CEC' = CAC' - dr, by the right angles at C and C' . Ac cordingly, we have the formula p = s = rrp = p + P24 from (5146-8) , and the values above corresponding to them .

5743 A method of estimating the variation in direction of a right line whose position is given as depending upon the form of a tortuous curve at every point. Let x, y, z be the direction cosines of the line referred to a fixed principal normal , tangent , and binormal of the curve [x, y, z may either be constants with respect to the varying principal normal , tangent, and binormal, or they may be func tions of the angle between the binormal and the spherical radius ] . 5744 The complete changes in x, y , z, with respect to the fixed origin and axes , will be

Sx = dx +yd↓ —zdr , Sy = dy - xdy , Ez = dz + xdr .

Let a PROOF.- In Figure ( 180) AC , AB are the fixed axes of a and z. line AL of unit length be drawn always parallel to the line in question ; then , if x, y, z be the coordinates of L, x, y, z will also be the direction cosines of AL, and therefore of the given line. Now, suppose A to move to A' , and consequently AL to take the position A'L'. Then the changes in æ, y, z will be the changes da, dy , dz relatively to the moving axes, plus the changes due to the rotations du round the binormal and dr round the tangent. With the usual notation , we shall have dx = dx + w₂z— w₂y, dy = dy + wzx - w₁z , dz = dz + w₁y — wzx, with w₁ = 0 , w₂ = −dr, w₂ = —dy.

5745

If dx be the angular change in the direction of the

right line ,

dx = √ { (8x) ² + (dy)² + (d≈) ? } . For dx = LL' since AL is a unit length.

EXAMPLES . 5746

The angle between two consecutive radii of circular

curvature being dɛ, (de)² = (d¥)² + (dr)².

SOLID GEOMETRY.

788

dz

PROOF. Here, in ( 5744) , x = 1 , y = 0 , z = 0 , therefore dx = 0 , dy = —d¥, dr. Substitute these values in (5745 ) .

5747 The angle , dn, between two consecutive radii of spherical curvature, being the inclination to the binormal , (dn) ² = (dy.sin 4)² + (do — dr)². PROOF. - In (5744 ) the direction cosines of R (Fig. 180 ) are x = sin ¢, y = 0, z = cos &, therefore da cos o (do - dr) , dydy sin p, dzsin ø (do - dr). Substitute in (5745).

5748 The angle of contingence of the locus of the centres of circular curvature , (dx) = (dy.cos

PROOF.

) ² + (do + dr )².

The dir. cos. of the tangent at C to the locus (Fig. 177) are x = cos p, y = 0 , z = sin & ;

therefore desin ø (do + dr), Substitute in (5745) .

dydy cos p,

5749 The osculating plane direction cosines in the ratios

dy

sino cos

-

:

dx

of the

dr dx + dx

do

dz = cos ¢ (do +dr) .

same

curve has

its

- dy cosp . dx

PROOF. As in the Proof of ( 5735) , the dir. cos . of the normal to this Substitute the plane are proportional to yez - zdy, zòx - xồz , xòy — yềx. values in last proof. 5750

The angle of torsion of the same curve is found from

(5745) and (5744) as above , x, y , z being in this case the dir. cos. of the normal of the osculating plane as given in (5749).

5751

The direction cosines of the rectifying line are

dr 0, de

dy de

PROOF. The rectifying plane at A' ( Fig. 180 ) is perpendicular to the normal A'C'. Therefore its equation is a − yd¥ +zdr = 0. The ultimate intersection of this plane with the rectifying plane at A (that is, the plane of yz ) is the rectifying line. Hence the equation of the latter is ydy = zdr ; and the dir. cosines reduce to the above by ( 5746) .

THEORY OF TORTUOUS CURVES.

789

COR. - The vertical angle of the osculating cone

5752

-1 dy = 2 tan dr

5753

The angle of torsion of the involute of the curve is

= = √ (4² + r² ) de. S PROOF . This angle is also the angle between two consecutive rectifying lines. Therefore, taking the dir. cosines from (5751 ) , we must put in (5744) dr x = 0, y = z = dy ;. de' de dr d y dx = dy therefore de dr = 0; by T₂de ; dz = 42de. de 5754

The angle of torsion of an evolute of the curve

= dy sin (a —T). PROOF. (Fig. 180.) Let HH'H" be an evolute of the curve, AH the tangent to it in the normal plane of the original curve at A, and let a = CAH, the inclination of AH to the principal normal. At any other point H" of the evolute, where its tangent is A″H'H ", let the corresponding angle be 0 = C" A"H". Then α -7, 7 being the sum of the angles of torsion between A and A", or the total amount of twist of the osculating plane. Now the normal of the osculating plane of the evolute at H", is perpendicular to HH' and H'H ", two consecutive tangents . Therefore its dir. cosines in (5744) must be x = -- sin (a - T) , y = 0, z = cos (α - T) ; therefore

dx = cos (a - 7) dr + 0 — cos ( a − 7) dr = 0, dy = sin (a −7) d↓ ; dz = sin (a −− 7) dr — sin ( a -− r) dr = 0.

Hence the angle required = dy = d↓ sin ( a — 7) .

5755 Approximate values of the coordinates of a point on a tortuous curve near to the origin in terms of the arc , the axes of x, y , z being the principal normal , tangent, and binormal , and the arc s being measured from the origin : 3 - S "Ps 6p² 2p

$2

x0

53

y = s

Ps + &c., + 6p² + 8p³

S4 (1-2p + PP2 ) + & c. , 24, p σs

Upo - 2/4 po + 2p.) + &c. ,

ρ and σ being respectively the radii of circular curvature and torsion .

SOLID GEOMETRY.

790

PROOF. - By Taylor's theorem ( 1500 ) , since x, y, z, s are the same as dæ, dy, dz, ds initially, we have a = x« s + } x2,8² + Z3 + &c . , and similar ex pansions for y and z. The dir. cosines of the principal normal at the point ayz will be, from (5737) ,

cos (-4) = P289

COS

COS

+ 4 ) = P! 289 (+

= p228 ; ( -)

↓ = dy and 7 = dr being estimated positive as drawn in Figure ( 180 ) for positive values of x, y , z. Differentiate these equations for s, and in the results put the initial values 1 2,2,0, y , = 1 , = r = 0, 4, = " T8=11 / & c. , ρ to determine the derivatives in the above expansions.

THE HELIX . 5756 The helix is a curve traced on a cylinder of radius a , so that its tangent preserves a constant inclination , = 17 - a, to the axis . Taking the axis of the cylinder for the z axis of coordinates , the equations of the helix are

x = a cos 0,

y = a sin 0,

≈ = a0 tan a.

5757

The radius of curvature p = a sec²a.

5758

The radius of torsion

= 2a cosec 2a .

PROOF. -p from (5806) ; since p₁ = a , P₂ = ∞ , and 0 = a at every point. By (5739) , σ = s . But dz = ds sin a and adr = dz cos a.

5759 The helix of closest contact with a given curve may be found as follows . Determine the constants a and a from equations (5757-8) , with the known values of p and for the given curve ; then place the helix to have a common tangent with the curve at the point, and make the osculating planes coincide.

GENERAL

THEORY

OF

SURFACES .

5770 DEFINITIONS . -A tangent plane passes through three consecutive points on a surface which are not in the same right line. 5771 The normal at any point of a surface is perpendicular to the tangent plane .

THEORY OF SURFACES.

5772

791

A normal plane is any plane through the normal.

5773 A line of curvature on a surface is a line along which consecutive normals to the surface intersect. At every point of a surface there are usually two lines of curvature at right angles to each other ; and to these correspond two principal radii of curvature . The two lines of curvature coincide with the principal axes of the indicatrix at the point.

See (5778) .

5774 The surface of centres is the locus of the centres. of principal curvature . There are two such surfaces , for there are two centres on each normal , and the normal is a tangent to both surfaces . Either surface may be regarded as generated by the evolutes of the lines of principal curvature. 5775 A geodesic is a line traced on a surface along which the osculating plane at every point contains the normal to the surface. See (5779) . 5776

The radius of geodesic curvature

of a curve traced

on a surface is measured by the ratio of the element of arc of the curve to the angle between consecutive normal sections of the surface drawn through consecutive tangents of the Geodesic curvature, being the reciprocal of this , is curve. therefore the rate of angular deviation of the normal section per unit length of the curve. 5777

An umbilicus is a point on a surface where a section

parallel to and close to the tangent plane is a circle ; in other words, the indicatrix is a circle. For a definition of Indicatrix , see (5795) . 5778 In Figure ( 182 ) OCD is the normal at O to a curved surface ; AOA', BOB' are the lines of curvature, therefore the normals to the surface at A and O intersect in the centre of curvature radius p₁ (5773 ) , and the normals at B and O, in the centre, radius 2. The normals to the line of curvature BOB' at B and O, drawn in the osculating plane BOB', intersect in K, and those at B' and O intersect in H. HOD is the angle between the osculating plane of the line of curvature and the plane of normal section . Similarly for the line of curvature AOA'. 5779 If POP' be a geodesic, its osculating plane POP' contains OD the normal to the surface at O, and therefore p = OD, the radius of curvature of this section at 0 ; but PE, the normal to the surface at P, does not inter sect OD, the consecutive normal at O, unless the geodesic coincides with one of the lines of curvature , OA or OB. The angle DPE is the angle of torsion which vanishes in the latter case.

* Not to be confounded with the radius of curvature of a geodesic.

792

SOLID GEOMETRY.

GENERAL EQUATION OF A SURFACE . 5780 Let the general equation of a surface be represented by (x, y, z ) = 0.

The equations of any tangent at a point xyz are

5781

x

n- y =

=

m

with

lo + mo +n &₂ = 0 .

n

PROOF. -At an adjacent point x +rl, y + rm, z + rn , we have (x + rl, y + rm, z + rn) = 0 , therefore, by ( 1514 ) , p (x, y , z ) + r ( lox + mp, + noz) = 0, the rest vanishing in the limit . But p (x , y , z ) = 0 , therefore 19x + mpy + noz = 0. But l, m, n are the direction cosines of the line joining the two points, which becomes a tangent in the limit ; and if ¿ŋ be any point on this line distant p from xyz, x = pl, n − y = pm, 5 — z = pn , &c.

5782

The equation of the tangent plane at xyz is (§− x) Þx +(n − y) 4, + ( 5 − ≈) 4; = 0 .

PROOF. -Eliminate l, m, n from lo̟₂ + mp, + ng₂ = 0 by -x = pl, &c. , as above.

TANGENT LINE AND CONE AT A SINGULAR POINT. 5783 tives of

If, in the expansion in ( 5781 ) by Taylor's theorem, all the deriva (x, y, z ) of an order up to n inclusive vanish , we have JN +1 (x + rl, y + rm , z + rn) = (x, y, z) + (ld₂ + md + nd₂) n + ¹ (x, y, z) = 0 . n+ 1

There are in this case n + 2 coincident points at xyz in the and since the equation (ld, + md, + nd₂ ) n + ¹ (x, y, z ) = 0 is degree in l, m , n ; n + 1 tangents to the surface at xyz can, drawn in any given plane through that point. This equation place of the conditional equation in (5781 ) . 5784

direction Imn, of the n + 1th in general, be now takes the

Equation ( 5782 ) is now replaced by { (§ − x) d¸ + (n − y) d, + ( ¿ —≈) d. } " $ ( x, y , z) = 0 ,

the equation of the locus of all tangents at the point xyz , and representing a conical surface generated by the motion of those tangents .

5785

The equation of the normal at xyz is ε—x Px

= n- y = y Pz

(5782)

THEORY OF SURFACES.

793

' on the 5786 The equation of the tangent at a point a'y'z curve of intersection of the tangent plane at ayz with the surface is

___ ? — ~ _ n —y' = _ §— x' = μ with the two conditions λφ + μφ ,ย + νφ = 0 ,

λφ

+ μφ +νφ

= 0.

For these are the conditions of perpendicularity to the normals of the tangent planes at xyz and x'y'z' respectively. There are three exceptional cases in which the ratios λ Aμ have more than one set of values ; namely—

— 5787 P , tangent cone at xyz .

y,

z vanish simultaneously, there is a

5788 II.- When ', ', ' vanish simultaneously, x'y'z' is a singular point on the surface . Ф ቀ III ._ _ _ _ _ _ _ _When Pr = у = P. In this case the point Py' Px Pz, x'y'z' coincides with xyz , and the tangent there meets the curve in more than two coincident points , the condition for which is

5789

with

(\d + µd, + vd )² p (x, y, z) = 0

(i .) ,

λφ . + μφ + νφ . = 0

(ii .) .

These equations furnish two sets of values of the ratios Aμv, giving thereby the directions of two inflexional tangents (tangents to the curve of intersection ) at xyz , each meeting the surface in three coincident points . If all the derivatives of an order less than n vanish at xyz , equation (i .) will be replaced by (Ad +ud + vd. ) " p (x, y , z ) = 0 , which , together with (ii . ) , will determine n inflexional tangents at the point.

5790

The polar equation of the tangent plane at the point

r0p, r' , ' ,

' being the variables , is , writing u for r¹,

u ':= = ( u cos 0 —u, sinė) cos✪ + ( u sin @ + u, cos ) cos (p ' — 6) sinơ ′ + u , cosec 0 sin ( 5 I

− ¢) sin @ .

SOLID GEOMETRY.

794

-PROOF. Write the polar equation of the plane through paß, the foot of the perpendicular on the plane from the origin ; thus pu = cos 0 cos a + sin sin a cos (4 — ẞ) . Differentiate for 0 and to find puo and риф , and eliminate p, a, and B. This elimination is troublesome.

5791 The length of the perpendicular from the origin upon the tangent plane at xyz, xox + y

+ zz

p=

ne or

√ {& + & + 6 }

, {

(5782, 5549)

+6 +6 }

the second form being the value of p when the equation of the surface is 4 ( x , y , z ) = c , a constant , and when homogeneous function of the nth degree ( 1624) . 5793

is a

In polar coordinates , 1

r²+r +r cosec² 0 = u² +u² +u¿ cosec² 0 = 204

p

PROOF.-Add together the values of the squares of pu, puo and риф found in (5790 ) . For a geometrical proof, see Frost and Wolstenholme, Art. ( 314) .

1 1

THE INDICATRIX CONIC . 5795 DEF . -The indicatrix at any point of a surface is the curve in which the surface is intersected by a plane drawn parallel to the tangent plane at that point and infinitely near to it . 5796 The following abbreviations will be employed— The derivatives of p Φ (x, y , z), Par , Pay, Paz, Pyz , Pzx, Pxy , Pz› Py, Pz› will be denoted by

5797

a,

b,

c,

f,

g,

h,

l , m, n.

PROP.- The indicatrix at a point xyz of a surface

p (x, y , z ) = 0 is the conic in which the elementary quadric surface 5798

I. R2

a² +bn² + c5² + 2ƒn ( + 2g¢§ + 2h§n =

√l²+m² +n² = N

P is intersected by the tangent plane at xyz , whose equation is 5799

II .

l§ + mn + n { + } N = 0 .

1

795

THEORY OF SURFACES.

The origin of coordinates is the point wyz in both equa R is an indefinitely small radius from the centre of the quadric (I. ) to a point n on the indicatrix , and p is the

tions .

radius of curvature of the section of the surface & by a normal plane drawn through R ; the ratio Rp being constant for all such planes . PROOF. - Let O, in Fig. ( 181 ) , be the point xyz on the surface ; x + ³, y+n, z + an adjacent point P. Then • (x + E, y + n, z + 5) = 4 ( x, y , z ) + lĘ + mn + n5 + 1 ( a§² + ... + 2h &ŋ ) + &c. With xyz for origin, draw the quadric surface a² +bn² + c + 2ƒn % + 2g5€ + 2h&n = N.. . (i. ) and the plane ... l& + mn + n $ + 1 N = 0 …………........ .. (ii. ) . Since §, n , are very small, N is likewise . Also the unwritten terms in the above expansion may be neglected in the limit. Hence, any point En lying on the intersection of the quadric (i . ) and the plane (ii. ) will also lie on the original surface ( x + E, y + n , z + 5 ) = 0 . To determine N, we have the perpendicular from xyz upon the plane (ii.) , -N (5549) . The radius of curvature of the section of the p= 2 √ (1² + m² + n³) surface made by a normal plane at O drawn through P being p, we have R2 R2 p= " and therefore N :—— √ (l² +m² +n³) . 2p ρ In the Figure, R = OP, p = OL, and the intersection of (i . ) and (ii. ) is the conic PSQ. Since p is indefinitely small, we may put N = 0 in equa tion (ii. ) . This amounts to taking the parallel section of the quadric by the tangent plane at O instead of the section PSQ. But these two will be equal in all respects , since the section of the quadric is a central one.

5800

If m = 0, equation ( II . ) becomes l + n

= 0 , and if

the inclination of the indicatrix plane to the plane of ay be a, ι tan a = To obtain, in this case , the equation of the n indicatrix in its own plane, put n = n', in equation ( I. ) .

= E' cos a ,

=

' sina , and

When none of the three constants l , m , n are zero , 5801 the quadric (I. ) simplifies as follows From (II . ) we have l + mn - n and two similar equa tions. Square these, and by the results eliminate the terms in nl, ¿E , Eʼn from (I. ) , which then becomes

5802 where

III.

H&² + Kn² + LC² = N,

ι H = a+ K = b + m² mn (lf-mg- nh) , nl (mg− nh — lf), n L = c + Im 2 (nh ― lf— mg).

SOLID GEOMETRY.

796

This is the equation of another quadric intersecting the plane (II . ) in the indicatrix.

5803 The equation of a surface for points near an origin ( (Fig. 182) , the normal at O being taken for z axis, is

X2

y² = 2%,

+ P2

Pi

where P1, P2 are the radii of curvature of the normal sections through the x and y axes , and those sections will be proved to be the lines of curvature at 0. PROOF. Let AC = a and BC = b be the semi-axes of the indicatrix conic at a small distance z from 0 ( 5795) . The equation of the conic will there. x² a³ b2 fore be + = 1 ; but = 2p₁ and - = 20 giving the equation a² b2 Z 2 required. Secondly, on a line of curvature, the normal to the surface at a point ryz will intersect the z axis ( 5773) . The condition for this , by ( 5533 ) [ with xyz for abc, the origin for a'b'c', L, M, N = $x, $y, 4z (5785) = 2 , 24 , -2, and L′ , M ′, N' = 0,0,1 ] , Pi Pa gives xy ( 1, − 1) = 0, therefore x = 0 or y = 0 on a line of curvature. Q. E. D.

5804

If the equation of the surface with the same axes be

z = ax² + 2hxy + by² + 2fyz + 2gzx +cz² +higher powers ,

1

1

then

P2 =

P₁ = 2a'

26*

PROOF. - Put y = 0 and divide by z, therefore 1 = a 2² + 2gx + cz + &c. When a and z vanish, we have 1 := 2api. 5805

For a normal section making an angle 0 with AC,

= 11 -= - 2 (a cos²0 +2h sin P

cos 0 +b sin³ 0) .

PROOF.-Turning the axes in ( 5804) through the angle ℗ by ( 4049) , the coefficient of "2 becomes a cos²0 + as above.

5806

Euler's Theorem .— If Р be the radius of curvature of

797

THEORY OF SURFACES.

any other normal section at O, making an angle ACP = 0 with AČ ( Fig . 182 ) , 1 cos² 0 sin² 0 + P2 P1 P PROOF. - Let = CP ; then x = r cos 0, y = r sin 0, and r² = 2pz, which substitute in ( 5793) . 5807

COR. - The sum of the curvatures of two normal

sections at right angles to each other is constant ; or, if p, p' be the radii of curvature for those sections, and Pa, Pb Po the radii for the principal sections ,

1 1 1 = • + + РР Pa Pb 1

5808 The radius of curvature of a normal section varies as the square of the radius of the indicatrix in that section .

PROOF. From

5809

= 2pz, in Figure (182).

Meunier's Theorem. -The radius of curvature of an

oblique section of a surface is equal to the radius of curvature of the normal section through the same tangent multiplied by the cosine of the inclination of the planes . PN2 PN2 NO ρ p' = NO' p = NO' therefore ρ == NO = cos p,

PROOF. (Fig. 183. ) when NO and NC vanish.

5810

Quadratic for y, at a point on the surface z = p (x , y)

giving the direction of the principal normal sections , and , therefore, of the lines of curvature (notation 1815) . { pqt— (1 + q²) s } y¿ + { ( 1 +p²) t− (1 + q²) r } Yx + { (1 +p²) s −pqr } = 0 . PROOF.- (i .) The equations of the normals at the consecutive points xyz and x + dx, y + dy, z + dz of the surface (x, y, z) = 0 are n- y -x 5-2 E = = and E- (x + dx) = n− (y + dy) = 2 (Fer mat's last th .) : A.26,58 : An.64 : C.24,89,90,98 : J.17. æ² —y² = 2œ” : L.40. am' .+by” = cz" : N.79 . " y" +1, impossible : N.50,70,71 . x² - ay² = " : C.99. simultaneous : x = u² ; x + 1 = 2v2 ; 2x + 1 = 3w2 : N.78. x² + a = y² ; y² -a = z2 : An.55 : C.78. x² + x + 2 = y² ; x2 - x - 2 = 22 : N.76. Ayz + Bzx + Сxy ax +by +cz = 0 ; = 0 : A.28. +z² -z2 z² = 0 ; x² 2 +y2 — x² - y² + x2 —y² z2 = 0 ; −x² + y² + z² = □ : E.20 . x²+ axy + y = 0 ; y² + ayz + z² = 0 ; z²+azx +x² = □ : E.20,21 . x² + y² + z² = 0 ; x+ y + z = 0 ; w3 + y³ +z3 = 0 : E.17. six eqs. in nine unknowns : N.50 . exponential, ay* : A.6 : Z.23. a - by = 1 : N.57. *Indeterminate forms : 1580-93 : A.26 : AJ.1 exponential : J.1 : Me.75 : 00 0 N.48,77 : 8, A.21 ; Z.1 : ∞0 ; L.41 , *

42 : N.46 :.ƒ (∞) when x = ∞ , 1592: x 0º, J.11,12 : 0 exp 0º, J.6 . * with two variables : 1592a. *Indeterminate multipliers : 213 , 1862 , 3346 : N.47. *Index law : 1490. Indian arithmetic, th : L.57 : calculation of sines, N.54. *Indicatrix : 5795 : C.92 : Mc.72 : N.74. a rect. hyperbola, condition : 5824. * two coinciding lines : 5825. an umbilicus : 5819. to determine its axes : L.78,82. determination of a surface from the indicatrix : A.59. * Indices : 29 : N.765,779,78 . in relation to conics : N.722. of functions , calculus of : JP.15. *Induction : 233 : C.39 : G.15 : L.48. *Inequalities : 330-41 : A.1,24. in integrals : Mél.3 : f.d.c. Mém.59. ( n) >n" : N.60. 5 X

*

Indeterminate equations-(continued) : 1,1,5 ; 2,3,6 ; 5,5,5 ; L.64 : 1,5,5 ; 1,6,6 ; 1,9,9 ; 1 ,n, n ; 2, n,2n ; L.65, 59 : with cab, C.42 and L.56. ax² + by² + cz² + dt2 = u, with the follow ingvalues of a, b, c, d : 2,2,3,4 ; 2,3, 3,6 ; 3,3,3,4 ; 1,2,6,6 ; 2,3,4,4 ; L.66 : 2,2,3,3 ; L.65 : 3,4,4,4 ; 3,4,12,48 ; L.63. x² + y² + z² + t² = 4u : L.56. ax² + by² + cz² + dt² + exy +fzt = u, with the following values of a, b, c, d, e, f: 1,2,-2,2,1,2 ; L.63 : 1,2,3,3, 1,3 ; 2,2,3,3,2,3 ; 1,1,6,6,1,6 ; L.64 : 2,3,2,3,2,2 ; 2,5,2,5,2,2 ; L.64 : 1,1 , 2,2,0,2 ; 1,1,1,1,0,1 ; 1,1,1,1,1,1 ; 2,2,3,3,2,0 ; 1,1,3,3,1,0 ; L.63 ; 3,5 , 10,10,0,10 ; 2,3,15,15,2,0 ; 2,3,3,3, 2,0 ; L.66. x² +2y² + 2x² + 3t² + 2yz := u : L.64. 2x² + 3y² + 3z² + 3t² + 2yz = u : L.66. x² + y² + z² + Qu² + 2uv + 2v² + t² = w : L.64. x² + y² + 2z² + 2zt + 2t² + 3u² + 3v² = w : L.64. 2(x² + xy + y²) + 3 (z² + t² + u² + v²) = w : L.64. xy + yz + zt + tuv : C.622 : L.67. 2 = x + x² + ... + x : G.7. y² quadratics in seven unknown inte gers : x² + ay² + bz2 + ct² + du² + ev² = w, with the following values of a, b, c, d, e : 4,4,4,4,4; 1,4,4,4,4 ; 2,2,4,4,4 ; 1,1 , 4,4,4 ; 1,2,2,4,4 ; 1,1,1,4,4 ; 1,1,2 , 2,4 ; 1,1,1,1,4 ; 4,4,4,4,16 ; L.65 : 1,1,1,2,2 ; 1,1,1,1,1 ; 1,1,1,1,2 ; 1,2,2,2,2 ; 2,2,2,2,2 ; 2,2,2,2,4 ; 3,3, 3,3,3 : L.64. higher degrees : cubic : AJ.2 . = +a : N.782,83 : with a = = 17, a3y² N.77. a3y3uz3: N.78,80. x3 + y³ + z3 + u³ = 0 : A.49 . ax¹ + by+ = z2 : C.87,91,94 : N.79 : a = 7, b = -5, L.79. x+ 2y = z2 and similar eqs : L.53. z2 : Mém.20. x++ ax²y² + y+ ax¹ +bæ²y² + cy¹ + dx³y + exy³ = fz² : C.883. x5+y5 = az5 : L.43. x7 +y7 = z², impossible : C.823 : L.404 .

889

J 890

INDEX.

Inequalities-(continued) : a > : A.14. if æ² + y² = z², æ™ + ym zm² : A.20. geo. mean of n numbers < arith . mean : 332 : N.42. Infinite -equalities : M.10 : prG.22. functions : An.71 : J.54 . from gnomonic projection : Me.66. linear point-manifoldness : M.15,17 , 20,212,23. point-mass : M.23. products : J.27 : N.69. value expressed by r functions : Ac.3. exhibiting circular arcs, logarithms and elliptic functions of the 1st kind : J.73 : Ac.4. use of in mathematics : C.73 . *Infinitesimal calculus : 1407 : M.11,18 . Infinitesimal geometry : An.59 : C.82. of a surface, formula : G.13. Infinity, points at on alg. surfaces : C. 59. * Inflexional tangents : 5789 : A.35. of a cubic curve : E.30 : J.38,58. Inflexion curves : Z.10. *Inflexion points : 5175 : CM.4 : J.41 . of cubic curves : J.28 : axis , E.31 . Herse's equation : N.81 . Inscribed figures : In-circle - of a quadrilateral, locus of centre: A.52. * of a triangle : 709, 953, 4747—50 : tg.e 4889 : CM.1 . in-centre : 709 , t.c4629 , tg.e4882 . In-conic of a circle : thJ.91 . four of a conic : prJ.39 . of a developable quartic : An.59. of a polygon : M.25. of a quadric : J.41 . of a quadrilateral : tg.e4907 : N.63 : four, N.56. of a triangle : 4739-46 , tg.e4887 : A.2 : N.50 ; max, A.8 : Q.2 : lat. rect ., E.34. In-cubic of a pencil of six lines : Q.9. In-hexagon :- -of a circle : A.22. of a conic : 4781 : N.57,82. In-parabola of a triangle : CD.7. In-pentahedron of a cubic : Ac.5 : M.5 . * In-polygons :--of a circle : 746 : CM.1 : J.35 : N.50. regular of 15,30,60,120, &c. , sides : A.62. do. 9 and 11 sides : LM.10. do. 17 sides : TI.13 . do. four of 30 sides : N.78. do. 5, 6 and 10 sides , relation : A.40, 43,45,48.

In-polygons-(continued) : two stars , one double the other : A.61 . of a circle and conic (Poncelet) : G.1 . of a conic : 4822 : thsN.47 : cnTN.69. with sides through given points, cn : 4823 : An.512. semi-regular : N.63. of a cubic (Steiner) : M.24. of a curve: Q.7. of a polygon , th : CD.5 . of a quadric with sides through given points : LM.22. In-quadrics --of a developable : Q.10 : quartic, An.59. 6 of a quadric, 2 touching 4 : An.69. of a circle : 733 : *In-quadrilateral : A.5 : cnE.21 area, N.44 : P.14. * of a conic : 4709. of a cubic : N.84. In-sphere of a tetrahedron : A.61 . In-spherical quadrilateral : N.49 . In-square of a circle : J.32 . of a quadrilateral : A.6. In-triangles of a circle : P.71 . with sides through given points : J.45 N.44. of a conic : J.7 : Maccullagh's th, N.65. with given centroid : G.23. similar : A.9. of a triangle : thsQ.21. two ( Steiner's " Gegenpunkte ") : J.62. In- and circum-circles of a poly gon : N.45. distance of centres : A.32. of a quadrilateral : Fuss's prMél.3 . of a triangle : 935 : A.38 . distance of centres : 936, 4972 : eq, 4644. In- and circum-conics : of a pentagon : N.782. of a polygon : J.64,70 : regular, Z.14. of a quadrilateral : 60 theorems , N. 453 : N.76. of a self-conjugate triangle : Me.81 . of a triangle: 4724, 4739 : An.52 : G.222. In- and circum-heptagons of a conic : A.3. In- and circum-pentagon : of a circle : A.22,43. In and circum- polygons (see also " Regular polygons ") :- of a circle : L.16 : N.80 : P.11 : Q.11. sum of squares of perps., &c.: ths 1099. difference of perimeters, ths : N.433. of two circles, respectively : C.53 : G.21 : L.78.

U

·

3

I

INDEX.

891

In- and circum-polygons-(continued) : Integrals or Integration- (continued) : of a conic : A.4 : ellipse An.52 : An. from orthogonal surfaces' theory : 57 : J.64 : N.57,84. L.38. of two conics : C.90. by Pfaff's method : A.47. of a curve: C.78. by series : Me.71 . by substitution : A.18. of a homonymous polygon : A.50. by Tchebychef's method : L.74 : M.5 . In- and circum-quadrics of a tetra hedron : eqsN.65 . comparison of transcendents : Me.79 : Pr.8. In- and circum-quadrilaterals :-of a circle : A.48. complex, representing products and of a conic : 4709 and pentagon ths, powers of a definite integral : J. N.48. 48. In- and circum - spheres :-of a tetra connected with trinomial integrals : L.55. hedron : N.73. convergency of : M.13. of a regular polygon : A.32. * definite :- -applied to Euler's, & c .: of a regular polyhedron : 910. In- and circum- triangles : -of a circle J.16. with finite differences : J.12. (Castillon's pr) : Q.3. from indefinite : J.41,51,52. equilateral, of another triangle : Me. 74. whose derivatives involve explicit functions of the same variable : and square of an ellipse : A.30. of two conics : 4970 : N.80. C.12. envelope of base : 4997. determination of functions under the respectively of two conics having a sign : JP.15. common pole and axis : CD.4. difference between a sum and an *Instantaneous centre : 5243. *Interest : 296-301 : and insurance, A. integral : 2230 : G.9. 26. division into others of smaller inter vals : A.4. Integrability of functions : An.50,73 : C.28 : J.59,79 : JP.17 : L.492. * eight rules for definite integration : criterion for max. and min . values of 2245. equations for obtaining functions as a primitive : An.52 . integrals J.79. *Integral calculus : 1900-2997 : A.ext expressible only by logarithms : An. 18,26 Euler's, A.20 : C.14,42 : 76. Newton, CD.8 : G.19 : L.47 : Me. 72,74,75 : Mém.18,36. extended independently of the con ception of differentials : A.61 . paradoxes : C.44. formulæ of: A.1 : J.18,19 : M.4 : Me. theorems , &c.: 2700-42 : A.45 : C. 76 : Mo.85 : N.85 : failure of f, 13: L.geo.ap50,56 : Me.77 : Mém. CM.2 . prs15,30. and gamma-function : LM.12 : Z.9,12. Integral functions : C.88,89,98 : G.4,22 : h.c.f of G.2 . higher, of composite functions : A.20 . with binomial divisors : J.70. with imaginary limits : C.23 : J.37. and continued fractions : An.77. use of imaginaries in : M.14. inverse method : CM.4 : CP.4,5 : L.78. reciprocal relation of: A.67. involving elliptic functions : Pr.29 : * Integrals or Integration : 1908 : A.1,2, 4,5,6,10,23 : Ac.1,32,44 : C.90 : Q.19. limits of: 2233-44 : L.74. CD9 : CP3 : J.2,4,8,17,25,36,39 , 61,92 : JP.9,10,11 : L.39 : M.6,16 , multiplication of : Pr.23. 73,75 : Mém.31 : P.14,36,37 : Pr. number of linear independent of 1st kind : An.82. 7,39 : Q.11,13 : Z.7,11,15,18,22,23. of alg. differentials by means of approximation to : 2127, 2262, 2991 : logarithms : Mo.57 : An.75 : C. A.9,14 : C.97 : CM.2 : G.3 : J.1 , 902 : J.12,24,78,79 : Mo.84 : N.81 16,18,37,48 : L.80 . Gauss's : 2997 : C.84 : M.25. (see " Integrals "). of algebraic surfaces : C.993 : J.26 : by the principle of Abel's derivative : octic, An.52 : cubature, C.80. J.23. of circular functions : 1938-97 : 2453 by differentiating under the sign of -2522 : No.1799 : LM.4 : M.6 : integration : 2258. Mém.9. by elliptic functions : G.11 : L.46 .

892

INDEX.

Integrals or Integration-(continued) : sine and cosine : G.6 : M.11 . * of exponential and logarithmic func tions : 2391-2431 : E.17,18. ** of circular, logarithmic and expo nential functions : 2571 - 2643 (see " Integrals " ) . of a complex function : A.66 : th of Cauchy, Ac.84. * of a closed curve : 5204 : C.23 : E.28 : 2.17. of differentials containing the square root of a cubic or biquadratic : Me.57. of discontinuous functions : 2252 : C.23 LM.6. of dynamics : L.52,55,58. of explicit functions , determination of algebraic part of result : An.61 . of functions which become infinite between the limits : 2240 : J.20 : JP.11 : Q.6. of infinite relations : M.14. of irrational alg. curves by loga rithms : An.61 . ** of irrational functions : 2110-20 : AJ.2 : An.56 : C.322,89 : L.53,64 : Mém.30. limits of 1903, -6 , 2233 , 2775. * for quadrature of curves : 5205—11 : C.68,70 circle, J.21,23 : JP.27. triple integrals : J.59. of rational functions : 2021-32 , 2071 -2103 : L.27 : N.73. * of rational fractions : 1915 : CD.3 : Mém.332 : N.72. of total differentials : C.992. of transcendental functions : JP.26 . of two-membered complete differen tials J.54. periods of: C.36,38,752 : G.753 : JP.274 . principal values of infinite definite : 2240 : A.68 . properties by elliptic coordinates : L.51. quotient of two d.i of the form

Jdx dy... dz : J.67. reduction to elliptic functions : An. 60 : LM.12 : Me.77,78 . residues of : JP.27. Riemann's of first kind : An.79 . singular values of : A.11 . * successive : 2148 : L.62 : 2nd order, M.20 and Z.11. * summation of : 2250 : J.47 : JP.12,21 . tables of definite, by B. de Haan, note on : C.47. and Taylor's theorem : Me.84. theorems L.48 : P.55 : Q.10,12.

Integrals or Integration- (continued) : * transformation of : 2245-52 : A.10 : CM.4 : J.f15,22,36 : L.36 : Mél.3 : Q.1. variation of arbitrary constant : 2247 : J.33 . whose values are algebraic : J.10 : JP.14 : L.38.

*

** **

**

*

ALGEBRAIC FUNCTIONS . Indefinite : unclassified : An.75 : C.90 : J.12,24, 78,79 : Mo.84 : N.81. simple functions of a² + a2 : 1926-37 : 1 x", A.4 : √ (a² — x²) , A.38 : ວ2 + aº” am N.82 : An.68. √ ( 1 -x ) ' fractions involving a binomial surd : 2008-19 . 1 : Mém.13. (1 +x) / (2x - 1 ) æm :* (x" -a) √(x" —b) Q.18. √(1 +24) : 2015 : L.80 : Z.8. 1 - x4 (1 + 2)2 : Mém.10. (1x ) ( 1 + bx +x+)? x : A.3 . (x + 8) √ (x3—1) ' xm and deductions : 2021-8. " ±1 x"(a + bx") : 2035-60 : A.36 : Mém.11 . 1 : 2061-5 : J.36. x²+a2 1 1 : 2007. : A.40 ; (x-a)P (x — b)я' x" (x - 1)" functions of (a + bx +cx²) : 2071-80 : 2103-9. xm : A.55. (a + be +ca²)" ' functions of (a + bx²+cæ¹) : 2081—5 . functions of (a +bx" +cx²" ) : 2086 2102. rational algebraic functions of irra tional ones : integrated by rationalizing : 2110 20. reducible to elliptic integrals, viz. : rational functions of X4, X, and ✓X6: where X is a quartic in æ : 2121 -41 . F(x, √X ) : 2121 : LM.8.: L.57. F (x) :, 2133-6 : J.36 : LM.14. √X₁

INDEX.

ALGEBRAIC— ( continued) : Integrals : x+ a : L.64 : Mél.3. √X₁ 1 : C.59 : CD.1 : E.36 : J.10. √x : C.51. f(x) F (x) X 1 : 2141 : J.17 : Man.79 . √X3 F ( X ): L.57. 1 : Me.82 . (1 — œ³) ³ 1 &c., reduced to (æ³ — 1 ) .√/ (æ³ —b³)' Jacobi's functions : Q.18. f(x) : J.32. √ ( 1-28) sundry : CM.2 : L.47 . Limits 0 to 1 : Euler's see " Euler's integrals.” * x² - ¹ ( 1 - x)m-1 : 2280 : A.40 : C.16 : J.11,17.3. x²-1+xm - 1 : 2341 : C.55 . (1+2) +m similar forms : 2342-4, 2352 , 2356 -67. nana -1 a -1 x : 2367 : A.10 . ** 1 - xn -x 1 am (1- x)" : L.59. (1 + ax) x² (1 -x) {1 + √ (1 + ax)} ² + 29 +2 : L.57. 2g+2

xm + (1 − x) m - } (a +bx+cæ²) m+1 : L.56—7. 1 : J.42 . (a − bx)" (1 - x)¹- " æ” Limits 0 to ∞ : æm -, &c.: 2309-12 , 2345-55 : A.38 : 1 ±x J.24 : L.41 : Q.12 : Z.19. 1 : 2364 . (x²+a²)" Xnx-r: Me.83 . xa-1 2n > a > 0 : N.48. 1 + æ" +æ² ' xa -1 &c.: A.16 . 1 + x² + ... + æ² (N −1) '

-1(1 + x2) and 12 similar : A.35. An+2 .

893

·ALGEBRAIC— (continued) : Integrals * x²n " F { ( ax− 1 ) * } (Cauchy) : 2712 : ∞™ A.9. xn :: A.12 : E.41 . 1 +2x cos +x² ° dedu ctions from this involving in te grals of the forms man sin 1-æn de and n : A.35. SH •æ sin ar Other limits : b F (x) dæ : 2368-9 . F (x) da * -1 √(1 -x ) ' S ( -c)" D ////// xm-1dx : A.35. (a -bx) x ‫م‬0 ‫ہ‬ sin a dx =f(a) : geoN.85. -2x cos a + x2

CIRCULAR FUNCTIONS . Indefinite : sin æ, sin- ¹æ, &c .: 1938-49 : sin" x, &c., 1954-7 : cos" , N.74. 1 1 : 1951 : J.9; (a +bcos x)" a+b cos x 1958 : Me.80. products and quotients of sine and cosine and their powers : 1959 80, 2066-70 : A.49 . * binom . funcs . of sin and cos : 1982-92 . * ditto of tangent : 1983, 1991 . F (sin x, cos x) : 1994-7 : A.12 : a cos x +b sin x +c J.19,32. * (a + 2b cos x +c cos 2a) -1 : 1993 ; xm 2029. x²2n -2x" cos no +1 ' F (cos x) : 1996. (a +b₁ cos x)(a + b₂ cos x) ... &c . a function of sine or cosine in a ra pidly converging series, and suc cessive integration of it : J.4,15 . sin" : A.11 . (1 —k sin þ ) √ (1 —k² cos²p) (sin amx) : J.81 . {1-k2 sin (a +x) sinº (a- x) } : J.39. sin" x : A.17 : with m = = 1 , G.7. xm

894 INDEX.

Integrals : -CIRCULAR. Indef.—(cont.): sin kæ æ or sin æ f(x) (Fourier) : 2726-42 : A.382.

Limits 0 to

: 2 : 2453-5, 2458 , 2472 : E.29 : n = 1, E.28. tan22m-1 : 2457. sin" a cosa : 2459-65 . * COSP sin n : 2481 , 248492 : L.43 . COS cos" -1 sin ux π = and similar : A.59. sin a 2 a cos"-2 sin næ : 2494. 1 ** and similar : 2496 a2 cos2a + b² sin² sin"

-2501 , 2344. -n- ] æ sin" -1 æ sin Cosp-n COS px ; and sin -2 sin pe : 2585-8. a cot ax : A.34 : No.19 ; æ tan (1 — bæ) : 5340.

Limits 0 to : * sin" cosa : 2459 . * sin na sin COS px : 2467-9 . sin " sin COS x COS nx : 2474-82, 2493.

**

sin " dnrf(cos x) : 2495. cos n (x -a sin x) : L.41 : 0 to 2π, C.39. cos (a +na) cos (b +px) cos (c +qx) : C. 54. æ sin æ : Pr.25 : with a = 1 , 2506 1 +a2 cos2 x and Q.11 . a sina or sina sin ræ or cos rx : 2623--9. 1-2a cosx + a² sin " or cos pr Q.11 : LM.11 : n = 1, (1—2a cos x + a²)" * L.74. F (sin x cos x ) : JP.27. about 250 integrals with limits chiefly from 0 to π, some from 0 to ∞ : Pr.252,26,27,29,30,31,322,33.

Limits 0 to ∞ : COS X cos x2, cos (ax)2 : 2507-9 : Q. √x 12 : 2602. sin (sin)%:: Q.13 ; COS x" : Me.72.

Integrals : ∙ CIRCULAR. 0 to ∞ —(cont.): + m sin ba 2579-8 x± 1. : COS cos bæ COS & 9,also da : A.10. x+ a ∞D æ sin" x ap : 2510 : A.30 : E.26 : Z.5 : n = 2 and 3, 2511-2. cos qx -cos p ; 2513-5. x or x2 sin ax sin bæ and similar : 2514-22. x cos 2ax sin" x : 2722-5. Xm (sin ax, cos bæ), reduction of : J.15. • (sin ax, cos bæ) , : J.23,25. æ sin ræ æ : 2572 : L.49 : Z.7,8. * COS re 2573 : A.10,11,59 : J.33 : Me. a²+x2;: 72,76 : Z.7,8. æ sin ræ : 2575 : A.10 : J.33. a²+ x² COS ræ : L.40 : Q.18 . (a²+x²)" sin rx (n) COS : A.11 : with $ (x) = x”, a2 +x2 L.46 : (x) = tan -¹ce, A.11. sin (ax)2 COS and similar : Z.13. a2 +x2 sin ax and related integrals : (1 +x2) sin be Ac.7. 1 - cos" a : Z.7,8. x2 1 or a sin ca : 2630-2 . (1 +x2) ( 1—2a cos cx +a²) tan -¹ax 1 * tan -1 x tan -1 x : 2503 ; x2 x ( 1 + x²) a b' 2504. tan‍¹ax- tan -¹ bæ : 2505. æ Other limits : da 2π: So a+beosetc sin æ lim . 0 to A.55. tan -1 x dæ, deduced from ( 2416) : Star 1+x L.69.

895

INDEX.

—(continued) : Integrals : CIRCULARtan -1 ax dæ * : 2502 . S x√(1 - x²)

° F(a) da, where F (x) is a rational 00 x" integral circular function : CM.3. sine-integral, &c. : see 66 Functions." EXPONENTIAL FUNCTIONS. Indefinite : * e*, a* : 1924 ; xm + x, 2004. * e* { (x) + ' (x)} : 1998. e exp ( 2) : E.34. X being a rational integ. funct . of a : eX : J.13 ; ex, Mém.83 ; e exp (-x ) , Q.1 . e exp (-x2) or the same X x2 or √(a +bx2) + (c + dx²) : L.52.

Limits 0 to ∞ : * e-kzx" : 2284-91 : see " Gamma function." * e exp (-kx2) : 2425 : evaluation by a continued fraction, J.12. * other forms : 2426-31 , 2595, '8, 2601 . * e exp : 2604-5 : L.56. ( x²-02) .nz e-n Z.6 ; ditto xx", L.46 . 1 +x2 -nx e and the same xx : A.10 . x2-a2 e -ax ei (x² +bx) : Mél.2. e exp (ax") . Ei ( ± ax") : Q.18. Other limits : expon-integral : see " Functions." e exp (-2) = Erfc & (Glaisher)] : See Me. 76. e-k F (x) dx : C.774.

LOGARITHMIC. Indef. Integrals : (continued) : xm log type of several : Mém.18. (1- ") '

Limits 0 to 1 : 1 1 2291 . : 2284 ; at ( log * log ) , 25 x involving loga or log (1 + x) : 2391 2403, 2416-22. log ( 1+ ) , 2416 : C.59 : J.6 : L.43,44. 1 +x2 2-1logæ log & " : 2636 : Q.12 ; 1 +xm √( 1-2) and many cases , J.34. *

1log 1+ : 2403 : L.73 . 1-x xm-1 A.3 x" (x - 1)" Z.3 7. ; : 9 log æ log x about 540 expressions chiefly formed from log ( 1 ) or log (1 ± xn) orlog (1 +x +x2) or log ( 1 + x² +x¹ ), joined to a single factor of the 1 ± æm form amor (1 ± æ”) or 1 + xm with integral values of m and n : A.39,40 . about 280 expressions, nearly all comprised in the form a (1 -x")" (loga)', with integral values of m , n, p, q, and t: A.40. about 130 expressions of the forms , a (1 -x )" (log x)2 or 3,

2m (1 - x ) loga, and ∞ am ( n -1) loga , with integral x -1 values of m , n, p, q, and r : A.40 . Limits 0 to ∞ : 2423-4.

S 02" eе exp(-21

a2 : C.12. )

F(resi) e - noi : A.15. Oa LOGARITHMIC FUNCTIONS.

Indefinite :

loga : 1950 ; am (log ) ", 2003-6. am log (1 +x) , &c.: A.39. F (x) log : G.12. (log x)" and a similarform : 2030-2. ∞ ±1

Other limits : log (1 - x) : 2408-12 : L.73 : Z1. x 1+x limits 0 to 2-1 : 2415. 1 log 1-x' x log -integral : see " Functions ." CIRCULAR-EXPONENTIAL FUNCTIONS. In definite : ear COS )"ba : 1999 ; e sin" x cos" æ, (sin 2000.

INDEX.

896

Integrals - CIRCULAR- EXPONENTIAL (continued): (eiz sin x) -1 : L.74. Limits, 0 to 0 : e-ar sin ræ : 2571 , 2591. x ** e-ax sin be : 2583 : 0 to 1, Mém.30. ** e-arm sin bx: 2577, 2589 : J.33 : Z.7. COS 772 e-ax a: 2608-11 : A.7 : (sin ) with limits -

to

, 2612 .

* ear +e-ax sin ma, & c.: 2593-2600. e** +e-** * e- kx cos ma sin" x : 2717-20 . * e exp (-a-x2) COS 2be, and similar : cosh 2614-8 : Z.1,10 . a2 x & c. : e exp { - (x² + x2 a² ) cos 0 }

Integrals :-CIRCULAR LOGARITHMIC— (continued): sin log COS e and log (1- k sin x), each with the above denominator : J.92.

Limits 0 to π : * log (1-2a cosx +a²) : 2620—2 : L.38 : Q.11. cos re log (1-2a cos x +a2) : 2625 : A. 13. a log sin æ : 2637 ; a log sin² æ, 2638. Limits 0 to : cos ax log 2cx : A.11. b2 +x2 log (1-2 a cos cx+a²) : 2631 . 1 +x2 Other limits : a +b cos de: A.53 . (0 a6-b00320

F (cos nx) log sind : Z.10. 0 cos ( -1)x log tan æ dæ : A.16 .

2606 : Q.12 . COS x-e-ax : 2619. æ CIRCULAR-LOGARITHMIC FUNCTIONS : x" (log x)™ : 2033. 2-2x" cos no +1 Limits 0 to 1 : sin COS (m log x) / log æ, and similar : 2641 -3. log ( 1—2x cos +x²) , &c . , &c .: A.34. x

log 1—2x cos +x², & 13 similar : L.732. x ( 1 - x2) log sina log cosa, &c. ( limits 0 to #): E.22.

0

x log (1-2 a cos x +x²) dx, and 0

similar : J.403. EXPONENTIAL- LOGARITHMIC FUNCTIONS : 00 -ar ear. - log (x² + u²) dx, &c . : J.38. 0 21 e- log ( 1-2a cos x +a²) de : J.40. 0 MISCELLANEOUS THEOREMS : * formulæ of Frullani, Poisson, Abel, Kummer, Cauchy, &c.: 2700—13.

[" ƒ ′ ( x) =ƒ(b) —ƒ(a) : 1901—3 : A.16 . [º º (ax)XN —• (bæ) dx : Me.81 : n = 0, 2700. * f (x) {√ (x)} ±" de : 2001-2. de (0 of Taylor's th) : LM.13 . S*£(0ær.0)

Limits 0 to 7: F (x) de, approx. to : C.97.

* log sina : 2635 : CM.2 : E.23 : Q.12 : O to no and similar integrals , A. 16 . * log (1 +ccosa) : 2633 : 0 to π, 2634. COS & tan a log cosecx : E.27. log (1 +n² sin² x) :, L.46 . ✓ (1 - k² sin² x)

[ƒ(x−R) dx = [ƒ(y) dy : CD.4 . √

汁 1 )f (x) dx , and similar : (x+

Me.75,76,77. [f(x) (x) dx : L.49.

Lwyde Sude [' vde : AJ.7.

INDEX.

f. (u), du : Z.25. S', (u + k)k+2 du = ( −1)¢ [ ', (u)2+2du : A.38.

[ Pm (x) Pn (x) dx (Pm ∞ = Legendre's coeff. ) : Pr.23. du + uU4 (u) = log ( u) C : J.19. (x +p) (x +q) do am + 1 (x +p + q)" +1 ? C.78.

= pos integer :

(sin 2a) cos a da S

2 = [0 L.532.

(cos2x) cos x dæ :

A.21 :

f (x ) sin wred (log ) : Mo.85 . sin2 I' COS xy sin cos2 x, transf. of: r2+x2

J.36. integrals deduced from z" + Pz' + Qz = 0, and y" + (P + 2R) y' + { Q + R (P + R) + R'} y = 0 : J.18. f (x, y) da : J.61 : Mo.61 : Q.7. if f(x +iy) = P + iQ, th : 2710. [ Upwards of 8,000 definite integrals have been collected and arranged in a 4to volume by D. Bierens de Haan ; Leyden, 1867 (B.M.C.: 8532. ff.) ] *Integrating factors of d.e : pp.468–471 , 3394. *Integrator mechanical : 5450. Intercalation : CM.3. *Intercepts, to find : 4115. *Interest : 296 : N.48,61,64. Interpolating functions : C.11 . *Interpolation : 3762 : A.32,61,62,70 : AJ.2 : C.19,48,68,92 : J.5 : L.37, 46 : Me.78 : N.59,76 : Q.7,8 . of algebraic functions, Abel's th : J.28. Cauchy's method : A.2 : C.373 : L.53. by circular functions : N.85. by cubic and quintic equations : C.25 . formulæ : J.2 : C.99, Mo.65 . Lagrange's : 3768 : J.1,84 : N.57,61 . Newton's : N.57,61,71 . for odd and even functions : C.99. * and mechanical quadrature : 3772 : A.20.

Interpolation-(continued) : by method of least squares : C.373 : Mém.59. * by a parabolic curve : 2992 : C.37. Stirling's series : Me.68. and summation : I.11,12,14. tables : I.113. of values from observation : Mél.2 : tr, Mém.59. Intersection :of circles and spheres : L.38. * of 2 conics : 4916 : CD.5,6 : N.66. of 2 curves : 4116, 4133 : CM.3 : J.15 : L.54 by rt. lines , Me.80. of 2 planes : 5528. of 2 quadrics : C.62 : N.684 . of right line and conic : see " Right line." of successive loci, ths : N.42. of surfaces : J.15 : L.54 : by rt. lines, Me.80. *Invariable line, plane, conic,and quadric : 5856-66. * Invariants : 1628 : An.54 : C.853 : E.42 : G.1,2,15 : J.62,68,69 : L.55,61,76 : M.3,5,17,19 : Me.81 : N.58,592,69, 70 : P.82 : Pr.7 : Q.12 : Z.22. of binary cubics : An.65 . of binary forms of 8th deg. , C.84 : G.2 M.5 : simultaneous, M.1 . of higher transformations : J.71 . superior limit to number of irre ducible : C.86 . of a binary quadric : M.3 . of a binary quantic : 1648 : E.40 : Me.79 of two, 1650. of a binary quartic : M.3 : Q.10 . of a binary quintic, table of irreduci ble : AJ.1 . of a bi-ternary quadric : J.57. of a conic : 4417 : 4936-5030. * of two conics : 4936 : N.75. of three conics : Q.10 . of particular conics : 4945. of a correspondence : G.20. differential : M.24 : of given order and degree belonging to a binary 10-ic, C.89. of d.e linear : C.88 : of 4th order, Ac. 3. and covariants of f(x2, y2) relative to linear transformation : G.17. of linear transformations : M.20. mutual relation of derived invariants : J.85. of an orthogonal transformation : J. 65 : LM.13 . of a pair of homog. functions : Q.1 . partial : LM.2 . of points, lines, and surfaces : Q.4. 5 Y

**

TH.

*

Integrals : MISCELLANEOUS (continued):

897

898

INDEX.

Invariants- (continued) : of a quadric : J.86 : oftwo, M.24 : Q.6 . of a quintic of 12th order, Q.1 : of 18th order, C.92 : J.59 . related to linear equations : C.94. of sixth order : G.19. skew, of binary quintics, sextics, and nonics , relations : AJ.1. of ternary forms : G.19 . transformation theorem : M.8 : Me.85. Inverse calculus of differences : N.51 . * Inverse equation of a curve or inverse method of tangents : 5160 : No. 1780 : J.26 : Mém.9,26 . *Inversion : 1000, 5212 : An.59 : C.94,pr 90 : LM.5 : Me.66 : thsN.61 : Pr. 34: problems by Jacobi, J.89 : geo.ths, Q.7. formula: An.53 : Lagrange's , J.42,54. of arithmetical identities : G.23. of a curve : 5212 : G.4 : J.14 : Pr.14. angle between radius and tangent : 5212 , 5219 : E.30 . of 2 non-intersecting circles into con centric circles : E.39. ** of a plane curve : 5212 : G.4 : Pr.14. of a quadric : J.52,76 : Q.11. of a system of functions : An.71 . and stereographic projection : E.35. *Involute : 5149 , -53, -66. * of a circle : 5306 : C.26 : successive , E.34. and evolute in space : CD.6. integrals of oblique : C.85. * of a tortuous curve : 5753. * Involution : 1066 : A.55,63 : gzAn.59 : At.63 : CD.2 : thCP.11 : thE.33 : G.10,20 : J.63 : N.53,64,65. and application to conics : A.4,5 . of a circle : Me.66. of a cubic space and the resulting complex : Z.24. of higher degrees : C.99, JM.72 : of 3rd and 4th , An.84 : Z.19. of numbers, machine for : P.15 . of n-tic curves : C.87. relation between a curve and an n-tic, the latter having a multiple point of the n- 1th order : C.96. pencils with problems in conics and cubics : N.85. of points on a conic : N.82 . of 6 lines in space : C.52. of right lines considered as axes of rotation : C.52. * systems of points in : 4826, 4828. ditto , marked on a surface : C.99. Irrational fractions : decomposition of, J.19 : irreducibility of, Mém.41 : rationalization of, A.18.

Irrational functions : M.4 : of the 2nd degree, C.98 . Irreducible functions with respect to a prime modulus : C.70,90,93 : L. 732. calculus, N.85 : homog. Isobaric : functions, G.22 : algorithm , N.84. Isogonal relations : A.60 : Z.18 , 20. do. represented by a fractional func tion of the 2nd degree : M.18 . X and representation of

*Vax"+b : Z.26. transformation of plane figures : N.69. Isomerism , pr : AJ.İ. Isoperimeters, method of : N.47,74,82. problems : J.18 : M.13 : Mél. 5. triangle with one side constant, and a vertex at a fixed point : C.84. Isoptic loci : Pr.37. Isosceles figures : C.87 : JP.30. Isotherms , families of: Z.26. Isotropic functions : C.262,272. Iterative functions : L.84. *Jacobian : ths 1600-9 : AJ.3 : thZ.10. of 3 conics : 5023 : LM.4. formulæ d.c.1471 : J.84 : Mo.84. function : one argument, G.2. of several variables : Mo.822. modular eq. of 8th degree : M.15. and polar opposites : Me.64. sextic equation : Q.18. system, multiplicator : M.12 . Jacobi- Bernoulli function : J.42. Kinematics : A.61 : AJ.3 : G.23 : L. 53,80 : LM.th17 : N.82, ths 83 and 84. of plane curves : A.55 : N.82 : caus tics, Z.23. paradox of Sylvester : Me.78 .. of plane figures : Mél.26 : N.78,80 : of a triangle, N.53 . of a point : N.49,82 : barycentric method, Mél. 5. of sliding and rolling solids : TA.2. Kinematic geometry : of space : J. 90. of similar plane figures : Z.19,20,23. Knight's move at chess : C.31,52,74 : CD.7 : CM.3 : E.41 : N.54 : Q.14. Knots : TE.28 : with 8 crossings , E.33. *Kummer's equation, i.c.: 2706. rational integrals of: M.24. an analogous eq.: C.99. Kummer's 16-nodal quartic surface : C.92 : J.84. figures of : M.18. lines of curv. of: J.98.

INDEX.

899

Lamé's equation : An.79 : C.86,90,91 , | * Limaçon : 5327 : C.98 : N.81 . ths92. Limited derivation and ap. thereof : Z.12. Lamé's functions : C.87 : J.56,60,62 : M.18. Limiting coefficients : C.37. (d.c) : 1552 : *Lagrange's theorem Limits theory of, Me.68. of functions of two variables : M.11 . C.60,77 CD.6 : CM.3 : Mél.2 : x gzC.96 and Me.85 : gzQ.2. of when ∞ L.40 : *Lambert's th. of elliptic sector : 6114 : ( 1+ 12 /) of a parabolic sector, A.16,33,48 : N.85 : Q.5. Me.78 : Q.15. Life annuities : A.42 : cn of tables , P. *Landen's th. of hyperbolic arc : 6117 : 59. E.21 . Linear associative algebra : AJ.4 . Laplace's 66 coefficients or functions : see construction : Man.51 . Spherical harmonics." coordinates in space : M.1. Laplace's equation : and its analogues, dependency of a function of one vari CD.7 : and quaternions, Q.1. able : J.55. Laplace's th. (d.c) : 1556. dependent point systems : J.88. Last multiplier, Jacobi's th . of : L.45. forms : L.84 with integral coeffi Lateral curves : A.58 . cients , J.86,88. *Latus rectum : 1160. function of n variables : G.14. *Law of reciprocity : N.72 : d.e,3446. U² = V² where U, V are products of n ext. to numbers not prime : C.90. linear functions of two variables : *Least divisors , table of, from 1 to 99000 : CD.5. page 7. geometry , th: M.22. Least remainder (absolute ) of real identities between square binary quantities : Mo.85. forms : M.21 . Least squares , method of : A.11,18,19 : systems, calculus of : JP.25. AJ.1 : C.34,37,40 : CP.8,11 : G.18 : L.52,53,67,75 : J.26 : Linear equations : A.512,70 : Ac.4 : C. 81 ,th942 : G.14 : J.30 : JP.29 : L.f Me.80,81 : Mél.1,4 : gzZ.18. 39,66 Mo.842 : N.51,75,802 : Z. Legal algebra (heredity) : N.63. 152,22. *Legendre's coefficient or function , X : analogous to Lamé's : C.98. 2936 : C.47,91 : L.76, gz79 : with real coefficients : M.6. Me.80 : Pr.27 . similar: N.452. and complete elliptic integral of 1st solution by roots of unity : C.25. kind : Me.85. * systems of : 582 : A.10,22,52,57 : G. rth integral of and log integral of: 15 : C.81,96 : L.58 : N.469. Me.83. in one unknown : C.90 : G.9. product of any two expressed by a of nth order : J.15 . series of the same functions : standard solution : 582 : Q.19 : gen. Pr.27. th, A.122. symbolic solution in connexion with Legendre's symbol ( ~ ) : Mél.4,5 . the theory of permutations : C.21 . Leibnitz's th . in d.c : 1460 : N.69 : a whose number exceeds the number of formula, Mo.68. variables : N.46 . * Lemniscate : 5317 : A.55 , cn3 : At.51 : Linesalg. representation of : C.76. thsE.4 : L.46,47 : Me.68 : N.45. " de faîts et de thalweg " in topo chord of contact, en : Z.12. graphy : L.77. division of perimeter : C.17 : L.43 : generated by a moving plane figure : into 17 parts, J.75 : irreducibility C.86. &c. of the partition equation , of greatest slope : A.29 : and with J.394. vertical osculating planes, C.73. tangents of: J.14 : cnZ.12. loxodromic : J.11 . Lemniscatic geometry : Z.212 : coordi six coordinates of : CP.11. nates, Z.12 ; of nth order, J.83 . * Lines of curvature : 5773 : A.34,37 : biquadratic function : Lemniscatic An.53 : C.74 : CD.5 : L.46 : M.2, multiplication and theorem, 32,76 : N.79 : Q.5. transformation of formulæ, J.30 . of alg. surfaces : Z.24. ** and conics, analogy : 5854 : Me.62 . Lexell's problem : LM.2.

900

INDEX .

Lines of curvature- ( continued) : dividing a surface into squares : C. 74: LM.4 : Mo.83 . of an equilateral paraboloid : N.84. of an ellipsoid : A.38,48 : An.70 : ths CD.3 : CM.2,3,4 : JP.1 : N.81 . comparison of arcs of, by Abel's th : An.69. and of its pedal surfaces : Q.12 . projection of : Z.2. rectification of : An.73. generation of surfaces by : J.98 : N. 77. and geodesics of developables : L.59. * near an umbilic : 5822 : A.70 : Q.10 . osculating plane of : 5835. plane, condition for : 5843 : An.68 : C.363,422,96 : G.22 : Me.64. plane or spherical : An.57 : C.46 : JP. 20 : L.53. of a quadric : 5833-4 : C.222,49,51 : G.11 : J.26 : Me.1 : N.632 : Pr.32 : TI.14. pd constant along it : 5836. projected from an umbilic into con focal Cartesians : E.19 . quadratic for y , giving the direc tions : 5810. of two homofocal quadrics : L.45. of quartic surfaces : C.59 : L.76. of ruled surfaces : C 78. spherical : C.362,422. and shortest distance of 2 normals one of which passes through an umbilic : L.55. of surface of the 4th class , correlatives of cyclides which have the circle at infinity for a double line : C.92. of the tetrahedral surfaces of Lamé, & c.: C.84. and triple orthogonal systems : M.3. *Linkage and linkwork : 5400—31 : E. 28,30 : Me.75 : N.75,78. * 3-bar : 5430 , E.34 ; 4 -bar, Me.76. conjugate 4- piece : LM.9. * for constructing :an ellipse : 5426. a lemniscate : AJ.1 . * a limaçon : 5427 : Me.76. 2 and x : AJ.1 and 3. *** root of a cubic equation : 5429. * Hart's : 5417 : LM.6,8%. * Kempe's : 5401 Pr.23. * Peaucellier's : 5410 : E.21 : LM.6 : N. 82. the Fan of Sylvester : E.33. * the Invertor : 5419. * the Multiplicator : 5407. * the Pentograph : 5423. N the Plagiograph : 5424. * the Planimeter : 5452 .

Linkage and linkwork—(continued) : the Proportionator : 5423. * the Quadruplane : 5422. * the Reciprocator : 5419. * the Reversor : 5407. * the Translator : 5407. * the Versor-invertor : 5422. * the Versor-proportionator : 5424. Lissajons' curves : A.70 : M.8. *Lituus : 5305. Loci, classification of : C.83.85 : P.78 : Pr.27. Locus of a point :—the centre of a circle cutting 3 circles in equal angles : N.53. the centre of collineation between a quadric surface and a system of spherical surfaces : A.65. dividing a variable line in a constant ratio : gzAJ.3. of intersection of common tangents to a conic and circle : N.63,79. of intersection of curves : CM.2. ditto of two revolving curves : N.64. on a moving right line : L.49. on a moving curve : Mém.18. the product of2 tangents from which, to 2 equal circles is constant : An.64. at which 2 given lengths subtend equal angles : A.68. whose sum of distances from 3 lines is constant : A.17,46 : from 2 lines, N.64 from lines or planes, A.19 ,prs and ths31 . whose distances from 2 curves have a constant ratio : An.58 : or satisfy a given relation, A.33 symbolic f. of pole of one conic with Locus respect to another : N.42. of remarkable points in a plane tri angle : A.43. of vertex of constant angle touching a given curve : N.61. of vertex of quadric cone passing through 6 points : N.63. *Logarithmic -curve, 5284 : quadra ture, N.45 . integral : A.9 , , 19 : J.17 : Z.6. numerical determination : A.11 . of a rational differential : J.3. parabola : CD.7. potentials : M.3,4,8,13,16 : Z.20. rational fractions : A.6. systems : A.14. transcendents : P.14. waves : LM.22. *Logarithms : 142 : P.1792 , 1787, 1806 , 17 : TE.26 . and anti-logarithms, cn : 1.12,,24.

INDEX.

901

Logarithms- (continued): Malfatti's problem, to inscribe 3 circles * calculation of : 688 : A.24,27,42 : LM. in a triangle touching each other : A.15,16,20,55 : J.10,452,76, 1,5 : Me.74 N.51 : Pr.31,32 : TE.6,14 : TI.6,8. 77,84,89 LM.7 : M.6 : P.52 : Pr. 6 : Q.1 : TE.24 : Z.21. Huyghen's method : C.662,682. and circular functions from definite Malfatti's resolvents of quintic eqs : A. 4.5. integrals : A.65 . common or Briggean : A.24. Malm's surfaces, th : J.84,88. constants in integral : 60. Mannheim, two theorems : G.8. of different orders of numbers : L.45. Martin's measure of distance : A.19 . higher theory of : trA.15. Matrices : E.42 : LM.4,16 : thMe.85 : impossible : CM.1. P.58,66 : Pr.9,14. natural, or Napierian, or hyperbolic : a, b A.25,26,57. = ax+b a; d and function f(x) = ca + d : c, of commensurable numbers or of algebraic irrationals : C.95. Me.804. base of: see 66 e." Cayley's th : LM.16 : Me.85 . modulus of : 148 : A.3. equation, pa = xq : C.992. of negative numbers : No.1784. of 2nd order : linear eq., C.992 : quad new kind of : J.70. ratic, Q.20 . powers of: CM.2 . of sum and difference of 2 numbers : of any order : linear eq., C.994. notation of : J.50. A.45. persymmetrical , th : E.34. with many decimal places : N.67. product of : G.5,11 . of 2 , 3, 5, 7, 10 and e all to 260 decimal roots of a unit matrix : C.94. places : Pr.27. whose terms are linear functions of e : of 2, 3, 5, 10 and e to 205 places : Pr. J.50. 6,20. of primes from 2 to 109 : table viii.,p.6. | * Maximum or minimum : 58, 1830 : A.4, 13,22,35,49,53,602,70 : C.17,24 : J. tables of log sines, &c. cn : Q.7. Logic, algebra or calculus of : A.6 : AJ. 48 : JP.25 Me.1, geo5,72,76,81 , 83 : N.43 : Z.13. 3,72 : CD.3 : M.12 : Man.71,76,823 : problems on : 1835-40, 1847 : A.2, Q.11. of equivalent statements : LM.11 . geo19,38,39 geoL.42 : Mém.11 , a paradox, N.63. Logic of numbers : AJ.4. of an arc as a function of the abscissa : Logocyclic curve : Pr.9 : Q.3. J.17. Longimetry applied to planimetry : J.52. continuous : 1866. Loto, game of : L.42. of a definite integral : 2.21. Loxodrome : eqA.21 : N.61,: Z.5. of a surface of revolution : N.74. discontinuity in : CD.3. distances between points, lines , and of cylinder and sphere : A.2. of ellipsoid and sphere : A.32. surfaces, geo : At.65. duplication of results : Me.80. of paraboloid of rotation : A.13. Loxodromic triangle upon an oblate ellipse which can pass through 2 spheroid : A.27. points and touch 2 right lines : A.14. Lucas's th : G.14 : analogous f., G.13. Ludolphian number : Mo.82 . elliptic function method : Mél.5. Lunes : J.21 . of figures in plane and in space : CM. * Maclaurin’s th . (d.c ) : 1507 : A.12 : 3: L.41 : J.242 : Z.11 . * functions of one variable : 1830 : ditto. CM.32: J.84 : N.70. symbolic form : CM.4. with an infinity of max. and min. values : J.63. Maclaurin -sum -formula : J.12. functions of 2 variables : 1841 : Mém. Magical equation of tangent : Q.6. Magic : cubes , Q.7 : CD.1 . 31 : Q.5,6 : Lagrange's condition, CM.2 . cyclovolute : TA.5,9. * functions of 3 variables : 1852 : CD.1 : parallelopiped : A.67. prs 1860-5 . rectangles : A.65,66. * functions of n variables : 1862 : L.43 : squares : A.21,57,66 : CD.1 : CM.4 : Mém.59 : Q.12 symmetrical, E.8 J.44 Me.73 : Pr.15,16 : Q. Mél.2. 6,10,11 TA.5,9.

902

INDEX.

Maximum or minimum-(continued) : of in- and circum-polygon of a circle : A.29,30 : do. of ellipse, and analo gous th. for ellipsoid, An.50. indeterminates : CM.4. in-polygon (with given sides) of an ellipse: A.30. by interpolation , f : A.25. method of substitution : A.23. of multiple definite integrals : Mél.4. planimetrical groups of : A.2 . of single integrals between fixed limits J.54,69 M.25. of the sum of the distances of a point from given points , lines, or planes : J.62 . of the sum of the values of an integral function and of its derivatives : L.68. solids of max. vol. with given surface and of min. surface with given vol. C.63. * of f F(x, y) ds, &c., to find 8 : 3070-2. * of SS F (x, y, z) dS, &c . , to find S : 3078 -80 . Maximum : ellipse touching 4 lines : A.12 . ellipsoid in a tetrahedron : Z.14. of a factorial function : Me.73. polyhedron in ellipsoid : A.32. of a product : N.44. of a sphere, th : N.53. solid of revolution : 3074. tetrahedron :-in ellipsoid, A.32 : whose faces have given areas, C. 54,66 : N.623. * volume with a given surface : 3082 . sin of : A.362 : of /x, &c., A.42. x of ax +by + &c., when x2 +y2 + & c. = 1 : N.46. Mean centre of segments of a line cross ing three others : A.40 . Mean distance of lines from a point : Z. 11. Mean error of observations : A.25 : C. 377 : TI.22. in trigonometrical and chain measure ments : A.46 : Z.6. Mean proportionals between two lines : A.31,34. Mean values : C.18,20,23,26,272 : L.67 : LM.8: M.6,7 : Z.3. of a function of one variable : G.16 : of 3 variables , C.29 . and probabilities, geo : C.87 : L.79.

*Measures of length &c.: page 4. exactitude of: Z.6 : do. with chain, Z.1. Mechanical calculators : C.28 : I.16 : P.85 . for " least squares ” : Mél.2. Mechanical construction of :-curves : M.6 : N.56. Cartesian oval : AJ.1 . conics : An.52 : three, N.43. ellipse : A.65 : Z.1 . lemniscate : A.3. conformable figures : AJ.2. cubic parabola : N.58. curves for duplication of roots : A.48. (a2 -x )/y : E.18. surfaces of 2nd order and class : J.34. Mechanical :-division of angles : Q.4. measurement of angles : A.61. integrators : 5450 : C.92,94,95 : Pr.244. for Xdx +Ydy : Me.78. involution : AJ.4 . quadrature : 3772 : A.58,59 : J.6,63 gzA.66 and C.99. solution of equations : Me.73 : N.67. linear simultaneous : Pr.28. cubic and biquadratic, graphically : A.1 . Mensuration of casks : A.20. Metamorphic method by reciprocal radii N.54. Metamorphic transformation : N.46. Metrical - system : E.30. properties of figures, transf. of : N. 58,59,60 : J.4. properties of surfaces : AJ.4 . * Meunier's theorem : 5809 : gzC.74. Minding's theorem : Quaternion proof : LM.10 . Minimum : -theory of : L.56 : prM.20. angle between two conj . tangents on a positive curved surface : A.69. area : J.67. of circum-polygon : CD.3. of a hexagonal " alvéole," pr : N. 43. circum-conic of a quadrilateral : A. 13 : An.54. circum-tetrahedron of an ellipsoid : Z.25. circum-triangle of a conic : Z.28 : of an ellipse, Z.25. curves on surfaces : J.5 : see " Geo desics." distance of 2 right lines : G.4 ; of a point, ths : A.8. ellipse through 3 points and ellipsoid through 4 : L.42. ellipsoid, th : Mo.72. N. G. F. of a binary septic : AJ.2.

INDEX.

Minimum : —theory of―(continued) : numerical value of a linear function with integral coefficients of an irrational quantity : C.53,54. perimeter enclosing a given area on a curved surface : J.86. questions relating to approximation : Mél.2 : Mém.59. sum of distances from two points : 920-1 . sum of squares of distances of a point from three right lines : Z.12. sum of squares of functions : N.79. eqA.38 : G.14, Minimum surfaces : 22 : J.81,85,87,ext78 : Mo.67,72 : projective, M.14 : metric, M.15. algebraic M.3 : lowest class-number, An.79. not algebraic and containing a succes sion of algebraic curves : C.87. arbitrary functions of the integral eq. of : C.40. between 2 right lines in space : C.40. generation of : L.63. representation of by elliptic functions : J.99. of a twisted quadric : At.52. limits of (Calc. of Var.) : J.80 on a catenoid, M.2 : determined by one of the edges of a twisted quadrilateral, Mo.65. variation of surface, capacity of : Mo. 72. Minimum value of (_¸ (4+Bæ + Cx² + &c.) dæ : N.73 . of √ √ (dx² +dy² + ... ) when the varia bles are connected by a quadric equation : J.43. Models : LM.39 : of ruled surfaces , Me.74. Modular -equations : An.79 : of 8th degree, 59 : C.483,493,66 : M.1,2 : Mo.65 : see also under " Elliptic functions." degradation of : M.14. factors of integral functions : C.24 . functions and integrals : An.51 : J. 184,19,20,21,23,25 : M.20. indices of polynomials which furnish the powers and products of a binomial eq: C.25. relations : At.65. Modulus of functions, principal : C.20. of series : C.17. * of transformations : 1604 : A.17. *Momental ellipse : 5953.

903

*Momental ellipsoid : 5925, 5934 ; for a plane, 5936. *Moment of inertia : 5903 : An.63 : At . 43 : M.23. * of ellipsoid : 6150 : CD.8 : J.16. by geometry of 4 dimensions : Q.16. * principal axes : 5926, 5967, 5972 : At . 43. * of a quadrilateral : 5951 : Q.11. of solid rings of revolution : Q.16 . * of a tetrahedron : 5957. * of a triangle : 5944 : Me.4 : Q.6 : polar, N.83. * of various laminæ and solids : 6015— 6165. Monge's theory "des Déblais et des Remblais " : LM.14. Monocyclic systems and related ones : J.98. Monodrome functions : C.43 : G.18. Monogenous functions (Laurent's th) : Ac.42 C.32,43. Monotypical functions : C.32 . Monothetic equations : C.99. Mortality : A.39. " Mouse-trap " at cards : Q.152. Movements : JP.15. elliptic and parabolic : JP.30. groups of : An.692. of a plane figure : thAn.68 : JP.20 , 28 : LM.3. of an invariable system : C.43. of a point on an ellipsoid : AJ.1 : J.54. relative : JP.19 : of a point, L.63. of a right line : C.89 : N.63. of a solid : JP.21 . transmission of and the curves result ing : JP.3 . of "ahnlich veränderlicher " and " affin-veränderlicher 99 systems : Z.24 and 19. *Multinomial theorem : 137 : Me.62 . Multiple-centres, geo. theory : L.45. Multiple curves of alg. surfaces : An.73. Multiple Gauss sums : J.74. * Multiple integrals : 1905, 2825 : A.64 : An.52 C.8,11 : thsCD.1 : thE. 36 : J.69 L.39,433,452,46,th48,56 : LM.82 : Me.762 : Z.13,3. double : 2710, 2734-42 : A.13 : Ac.5 : An.70 J.272: G.10 : L.58 : Mém . 30. approximation to : J.6. Cauchy's theory, ext. of : C.752. change of order of integration : 2775 : A.19. expressing an arbitrary function : J.43. reduction of : J.45 : Z.9. residues of : C.752.

904

INDEX.

Multiple integrals , double- (continued) : ( 2-2) dxdy SvON{(12 SS — b²) ( c² —œ²) (b² —y²) (c²—y²) } ' { (x² — π = : L.38. 2 same with log of numerator : L.50. (x' —x) ( dy dz' —dzdy') + sym SJ( x = x)( ✓ { (x - x)² + ... }3 = 4mπ : C.66.

do d↓ transf. of (sin² -sin² cos²↓) J.20. 1 2x cos ix cosje de dy -pp T²Jo Jo √ { 1 + a² +2a (µcosa +vcosy) } * C.96.

Multiple integrals-(continued) : volume integral of

: 2.14 . a )² + ( t)²+ (~C )' ·· (~ SSSeе exp (-x² -y2 - z²), æ”yª z dx dy dz : N.54. SSS ... $ (ax• + by® + &c.) œ” y¹ ... dx dy ... with limits 0 to ∞ in each case (Pfaff) : J.28. dx dy ... ... by dis SS {(a - x)² + (b -y)² + ... } " continuous functions : TI.21 . m and with a numerator do. with n = 2 (a—x) F ( 2 y2 h2 + k2 + .) : CM.3.

SSF (a + be +cy) dx dy : A.37. SSF' (x + iy) dx dy : J.42. F SS (w, t, z) dt du : C.96. G (u , t, z) 18-1 ၂၂ မှ (ax +by") x - ya - ¹ de dy : J. 37. evaluation : A.5 : by Fourier's th, CM.4. expansion of Q.8. Frullanian : LM.15. limits of LM.16. reduction of : An.57 : L.41,39. by transf. of coordinates : C.13.

S” F (x² +y² + ...) (ax + by+ ... ) dxdy ...: L.57. of theory of attraction : CD.7. * transformation of : 2774 : A.10 : An . 53 : No.47 : CM.4 : Mél.2 : Mém . 38 : Q.4,12. an indef. double : J.8,10 . a triple integral : 2774-9 : J.45. y₁dæ₁ + ... + ynda, LM.11 . ** triple : 2774 : A.30 : J.45. which are unaltered in form by trans formation of the variables : J.15, 91. SSS ... dx dy dz ...

Q.23.

* SSS ... x² - ¹y - 12 - ¹ ... dx dy dz ... with different limiting equations : 2825 CM.2 : L.51 . some other integrals evaluated by r functions : 2826-34. SSSF (ax +by +cz, a'x + b'y + c'z , a" x + by + c" z) dx dy dz, limits ± ∞ : A.30.

SS ... F (x, y, z ...) PQ dx dy dz … where P = (1 − a )^ -1 (1− y) Q ya za + b fa +b + c... : L.59. arising from (2604) , viz :

-¢ — dx : Pr.42. Se exp (−22—3) dz *Multiple points : 5178 : CM.2 : thG.15 : Me.2 : Q.2,6. on algebraic curves : An.52 : L.42 : N.51,59,81 , at ∞ 643. on two curves having branches in contact : C.77. on a surface : J.28. * Multiplication : 28 : J.49 ; abridged, N. 79. by fractions : Me.68. Multiplicator equations : M.15. Multiplicity or manifoldness : J.842,86 : thAc.5: Z.20. Music : E.273,28 : Pr.37. Nasik squares and cubes : Q.8,15 . Navigation, geo. prs. of use in : A.38. Negative in geometry : No.1792 . Negative quantities : At.55 : N.443,67 : TE.1788. Nephroid : LM.10. Net surfaces : J.1,2 : M.1 : any order, An.64. quadric : J.70,82 : M.11 . quartic : M.7. and series : C.62. trigonometrical : Z.14. having a 3-point contact with the intersection of two algebraic sur faces : G.9. Newton, autograph m.s.s of : TE.12. *Nine-point circle : 954, 4754 : A.41 : E. 7,30,th35,pr39 : G.1,ths4 : Me.64, 68 : Q.5-8.

INDEX.

Nine-point circle-(continued) : an analogous circle : A.51. * contact with in- and ex-circles : 959 : Me.82 : Q.13. and 12-point sphere, analogy : N.63. Nine - point conic of a tetrahedron : Me.71 . Nonions (analogous to Quaternions) : C.97,98. Non-uniform functions : C.88. Nodal cones of quadrinodal cubics : Q. 10. Node cusps : Q.6. Nodes, two-plane and one-plane : M.22. *Normals : 1160 : 4122-3, 5122 : A.13, 53 : LM.9 : p.eMe.66 : Z.cn2 and 3. of envelopes : Me.80. * plane of a surface : 5772. * principal : 5722 : condition for being normals of a second curve, C.85. of rational space curves : J.74. section of ellipsoid (geodesy) : A.40. * of a surface : 5771 , 5785 : C.52 : CD.3 : CM.2 : L.39,47,72 : M.7. coincident : L.48. transformation of a pencil of : C.882. *Notation (see also " Functions ") : A,B, C,F,G,H : 1642 . * A.P., G.P. , H.P.: 79,83,87. * (α₁becз) : 554.

* * * *

(n) (2 ) , Jacobi's function (see " Functions "). ab or a" : 2451 . 1 1 : 160. a+ b+c + Ban, Bernoulli's nos.: 1539. C (n ,r) or Cnr : 96. Otherwise C (n,3) = number of triads of n things, & c.

(2) = rth coeff . of nth power of (1+«) : also Jacobi's function (see " Func tions " ). * D: 3489 : dr,dne, &c.: 1405. * d³y , dæ³, dy, &c.: 1407. ' de' * d (uvw) : 1600. d (xyz ) *A: 582, 1641 , 3701 ; ▲', 1645. *E : 902, 3735. *e : 150, 1151. 1 1 = e= == e exp + or ea + 1= x x *f(x): 400, 1400 ; f-1 (x) , 430. *f ' (x)f (x) : 424, 1405.

905

Notation-(continued ) : * (aẞy) = u : 4656 ; (λµv) or U, 4665. *Gux: 3732. *H (n , r) : 98. *J : 1600 . N.G.F = numerical generating func tion. N= b (mod r) signifies that N- b is divisible by r. n = n(") = n ! : 94, 3713. Tas operator : 3500 . * P (n, r) or Pn,, = n " ) : 95. Also , P (n, r) number of triplets of n things , &c. * ¥ (x) or Z' (x) = d, log г (x) : 2743. * R, r, ra : 909-13. * Sm, Sm, p : 534 ; Su, 2940. * sin - 1, &e.: 626 ; sinh,đc., 2180. * Σ : 3781-3. * Un : 3499. Si (n)= sum of divisors of n. In or f 23 d or E (2 ) = integer next d < ጎ d' n n = integer next > d' Ta n = rth coeff. of (1+ )". (2/2) not greater 1 = not less than ; than. (÷) = denominator to be stated after wards. ( X ) and ( X ) : 1620. algebraic : CP.3 . for some developments : C.98. continuant contd. fraction determi nant. median = bisector of side of a triangle drawn from the opposite vertex. subfactorial : Me.78 . suggestions : Me.73. *Numbers (see also " Partition of," and "Indeterminate equations") : 349 : A.2,16,26,58,59 : Ac.2 : AJ.4,6 : C.f12,43,442,454,f60 : CM.1 : G.16, 32 : J.93,39,404,48,77 ; tr,273,28 and 292 : JP.9 : L.37-39,41,45,58,59, 60: LM.4 : Mém.22,24 ; tr (Euler), 30 : N.443,62,79 : Q.4 : TE.23. ap. of algebra, JP.11 ; of r function , No.81 ; of infinitesimal analysis, J.19,21 . formulæ L.642,652. relation of the theory to i.c : C.82. 5 Z

906

INDEX.

Numbers-(continued) : approximation : to N, E.17 ; to functions of large numbers , C.82. binomial eqs. with a prime mod : C.62. cube : Q.4. cubic binomials : ³ ± ³ : C.612. determined by continued fractions : LM.29. digits, calculus of, th : J.30. digits terminating a power : A.58 : N.46. Σ Dirichlet's th. (m ) = $ (m ) : L.57. Dirichlet's f. for class numbers as positive determinants : L.57. division of : A.26 : J.13 : Mél.3 : Pr. 7,10 ; by 7 and 13, A.25,26 ; by me2 +ny2, Mém.15 : P.17,88 : Q. 19,20. divisors of y2 + Az2 when A = 4n + 3 a prime : J.9. divisors arising from the division of the circle : L.60. 4m +1 and 4m +3 divisors of a num ber : LM.15. factors of: Mém.41 . Gauss's form : L.56. integral quotients and remainders : An.52. large, analysis of : A.2 : C.2,29. method with continuous variables : J.41 . multiples of: C.2. non-pentagonal th : J.31 . number of integers prime to n in n ! = ¢ (n) : L.57 . odd : A.1, : and prime to all squares, C.67. Pellian equation : prA.49 : LM.15 : sol. by ell. functions, Mo.63. perfect : C.81 : N.79. polygonal , Fermat's th . of : P.61 . polynomials havingdeterminate prime divisors : C.98. powers of, 12 theorems : N.46. prime to and < N: A.3,29 : E.28 : J.31 : N.45. prime to and the product of the first n primes : A.66 . prime with respect to a given ar.p : C.54. prime to the radix having multiples made up of repeating digits : Me.76. products of divisors of : Q.20. quadratic forms of : Mém.53 . rational linear functions taken with respect to a prime modulus, and connected substitutions : C.483.

Numbers-(continued) : representation of by forms : C.92 ; by infinite products , A.1 . square having prime factors of the form 4n +1 : N.78. squares of : J.84 : M.13 : Pr.63,7. three in ar.p : N.62. sums depending upon E (x) : L.602. sums of digits : Me.66 : TE.16 . sum and difference of two squares : thsN.63 . sums of divisors : 377 : Ac.6 : G.7 : L.632 : Mél.2 : Mém.50. sums of powers of (see also " Series ") : 276, 2939 : An.61,65 : thCD.5 : Me.75 : N.42,56,70 : Q.8. of cubes : An.65 : L.66 ; of the odd nos . , A.64. of n primes : N.792 ; 4th powers, A.54. of squares : A.67 . of uneven orders : Mo.57. symmetrical functions of : Q.7. systems : 2.14 ; history of, by Hum boldt , J.4 . theorems : A.7,10,20,49 : An.70 : C. 252,43,83 : CM.2 : G.8 : L.48,52 : N.75 ; Cauchy's , gzC.53 ; Eisen stein's, J.27,50,83 : LM.7 : Q.5,6. Gauss's on XP -y❞ ; C.98 . x-y Lagrange's arithmetical : A.47. p"+q" in terms of pq : N.75. on 21 : C.85,86 : Me.78. 2, biquadratic character of : C.57, 66 : L.59. on (n + 1 )" ― n " ; N.44 ; p. (n) , L.69. on n' ― n (n - 1) + , &c. : 285 : A.30. on 2m positive numbers : N.43.

on P (m) +E (

1 ) P (1) + : Q.3.

on the greatest product in whole numbers of given sums : J.57. on an odd sum of 12 squares : L.60. on products of sums of squares : G.2. on 4 squares : N.57. on 2, 4, 8, and 16 squares : Q.17. on (a) + (a ) + , &c. = n, where a, a ' , & c., are the divisors of n: Cm.3. Numeration, ancient decimal : C.6,8 ,. Numerical approximations : N.423,53. Numerical functions : L.57 : Me.62. simply periodic : AJ.1 . sums of, approximately : C.96. which express for a negative deter minant the number of classes of a quadratic form, one at least of whose extreme coefficients is odd : C.622.

INDEX.

Obelisks : A.9,11. Oblique :-bevilled wheels , cn : J.2. coordinates : 4050, 5511-9 : fN.54. cyclic surface : TI.9. and osculating circle of a conic : G.22. Octahedron function : Q.16. Octahedron, centroid of : LM.9. Octic equations : G.7,10 ; and curves, M.152. Octic surface : G.13: M.4 : Q.14. * Operative or symbolic calculus : 1483, 3470-3628 : AJ.4 : C.17 : G.202 : J.5,59 :: LM.123 : Me.82,85 : P.37, 44,60-63 : Pr.10,11,12,13 , : Q. 4,5,8. applications : G.19 : Me.82 . algebraic : TE.14 ; ap. to geometry, CM.1,2 ;, CM.3. expansions : Pr.142. formula : C.393. * index symbol : 1485 : CD.6 . integration : CD.3 : exMe.76. representation of functions : C.43 : CD.2 . seminvariant operators : Q.20. on the symbols a", log , sin æ, cos æ, sin- , cos a : A.9,11 . theorems : A.57 : CD.8 : LM.11 : Q. 32,15,163 ; from Lagrange's series, Q.16; from рu -pwu = pu, CD.5. * Operators : de, 1405 ; endr, 1520—1 , Q.92. * (d. - m) -1, &c. : 3470-85 ; gz of 3474, C.43. * D (D- 1) ... (D- n + 1) : 3489. * = adt ,+ & c.: 3500. expansions and formulæ for : * F (xd ) U, where U = ƒ (x) = a + bx + cx² + : 3486. * ƒ(D) uv, 3494 : uf (D) v , 3495, with f as above.

Daf(xD) U: E.36 . { (D) er " Q: 3491 . e*+DF (x), &c.: E.34. D+ 1 x +1 e x ** F (x ) = " x + ¹ F (x + 1 ) : E.36 . QF (a, x) : Q.13. +1 2 +/{ a√v ± b√(wi)} , &c.: C.96. f(x +hD) .1 : E.39. фо = $ (d, +x) ↓y : 3498. * (d, + y) $x * reduction of F (π₁ ) : 3503. * F (T) U and F -¹ (π) U : 3509—10 . * C (u, m) u,/m !: 3514. transformation of Vdr . Ud ..., &c : G. 21 . Orthocycle : Q.17.

907

Orthogonals, algebraic system of : C.69. *Orthogonal :-circles : 4170, 4182-4 ; of in- and circum-circles of a tri angle, Q.18. circle and conic : E.7. coefficient system : A.2,61 . coordinates : C.60 ; curvilinear, JP. 26. conics : N.84 ; families of, A.63. curves : J.35 : N.52,81. system from logarithmic repre sentation : Z.16. * lines of a triangle : 4633. lines and conics : C.72. * projection : 1087. in metrical projective geometry : GM.14. of a circle into an ellipse : A.2. of a triangle : E.30,31,36,37. substitution : J.67 : M.13 : Z.24. surfaces C.29, spheres 36,542,59,72, 79,87,ths17 and 21 : J.84 : JP.17 : L.43,44,46,47,63 : N.51 : P.73 : Pr.21 : Q.19. cubic eq. for : C.762. with elliptic coordinates : C.53 : J.62. and isothermal : C.84 : JP.18 : L. 43,49. systems : C.67,754 : M.7 ; condition , J.83 ; quadric, J.76 ; parallel, M.24. triple : A.55 - 58 : An.63,77,85 : C.alg58,67 ; cyclic, G.21,22 : J.84 ; quartic, 82 : Ž.23. trajectories : L.45 : Me.80 : Z.17. of circles : Me.85. of circular sections of an ellipsoid : L.47. of a moveable plane : Pr.41 . of a moveable sphere : C.42. of a surface : Mém.20. Orthomorphic projection of an ellipsoid on a sphere : AJ.3. Orthomorphosis of a circle into a para bola : Q.20. Orthoptic :- -lines of a conic : A.57. loci of : LM.13 : Pr.37 ; of 3 tangs. to a quadric, E.40. surface of a quadric : J.50. Orthotomic circles : Me.64,66 : Q.2. *Osculating circle : 5724 : L.39. of conics : A.70 : N.60. of a family of curves : N.70. of a parabola, ths : N.66. of quadric curves N.43. of tortuous curves : N.81 . ** cone : 5727 : angle of, 5752. conic : L.39 : triply, A.69 : Z.19. of a cubic curve : J.68 :: 2.17.

908

INDEX.

Osculating-(continued) : curves : Q.11 . helix : N.71 . line of a surface : C.82 : J.81 . parabola N.81 . paraboloid : JP.15 : N.82 : of a quadric, L.382. * plane : 5721 , eq 5733 : and radii of curv . at a multiple point of a gauche curve : An.71 : C.68. of a tortuous curve : C.96 : J.41,63. sphere - Mém.20 : N.70 : of curve of intersection of two surfaces , cn : C.83. of two curves having a common principal normal : LM.16. surfaces C.792, degree of 98 : L.41 , 80 : of quadrics , N.60 . Oval of Cassini : see 66 Cassinian oval." Oval of Descartes : see " Cartesian oval." Pangeometry : G.5,15. *Pantograph : 5423 : Mém.31 : TE.13 . Paper currency : A.42. Pappus , prs in plane geometry : A.38 : Z.5. *Parabola : geo. 1220-44 : anal . 4200 39 : eqCM.2 : N.42, , 54,70 : geo CM.4 : Me.71 : cn1249 : thsN.60 , 63,71,76,802. circum-hexagon and triangle : CM.1 . circle of curvature of : 1260 : A.61. chords of : 1239, 4224. two intersecting : 1242. determination of vertex and axis : N.58. eq. deduced from eq. of ellipse : 1219. focus and directrix : N.49. * focal chord : 4235-9. ** latus rectum : 1222 : Me.75. normal, length of : 4233-4. plane and spherical : A.60 . quadrature of : 1244 : A.32. and right line : see 66 Right line." segment of: 6078 : A.26,29. solid generated by it : N.42 . sector : E.30 : N.57 : Lambert's th , J.16. in space : A.3. ** tangents : see under " Conics." * through 4 points, cn : 4837 : J.26 . * triangle of 3 tangents : 1237, —68 : A. 47 : Me.75. trigonometry of : CD.8 . *Paraboloid : 5621 , 6126-41 : N.61 : Q. 13. * generating lines of : 5624. of eight lines : C.84. * elliptic : 5622 : A.11 : L.58 : P.96 .

Paraboloid-(continued) : quadrature of : 6127 : An.55 . segment of : 6127-33 : A.29. hyperbolic : 5623 : A.11. of revolution : 6134. Paradoxes of De Morgan : J.113,12 , 13, 16 . Parallels : A.8,47 : At.51 : J.11,73 : Mél. 1,3 : Mém.502 : Z.21,22 : Thibaut's proof, A.15 . in analytic geometry : A.44. Parallel curves : J.55,ths32 : LM.3 : Q. 11 : Z.5. closed : A.66. of ellipse : 4960 : A.39 : An.60 : N.442: Q.12. Parallel surface : C.54 : LM.12. of surface of elasticity : An.57. of ellipsoid : A.39 : Ăn.50,60 : E.17 : J.93 . Parallelogram with sides through four given points : A.39. Parallelogram of Watt : A.8 : L.80. *Parallelopipeds : on conjugate diame ters : 5648. diagonals, &c.: CM.1 . equality of: A.4. analogues of parallelograms : LM.2 : Me.68. on a spherical base : N.45. system of: LM.8 . Partial differences : question in analysis : J.16. *Partial differential equations ( P.D.E. ) : 3380-3445 : C.3,11,16,78,95,96 : thsCD.3 : J.58,80,prs26 : JP.7,10, 11 : L.36,80,83 : M.11 : Z.6,8,18. *P.D.E. , first order : 3399-3410 : A.33 , tr50 : An.55,69 : C.146,533,545 : CD.7 : CM.1 : J.2,17 : trJP.22 : L.75: M.9,11 ,th20 : Z.22. complete primitive connected with any solution : 3405. derivation of the general primitive and singular solution from the complete primitive : 3401 . derivation of a singular solution from the differential equation : 3403. with a general first integral : Me.78. integration by : - Abelian func tions : C.94. Cauchy's method : C.81 . Charpit's method : 3399. Jacobi's first method : C.79,82 : and ap. to Pfaff's pr, J.59. Jacobi-Hamilton method : M.3. Lie's method : M.6,8. Weiler's method : M.9. law of reciprocity : 3446. and Poisson's function : C.912.

909

INDEX.

P.D.E., first order— (continued) : simultaneous : C.68,76 : L.79 : M.4,5. * singular solution : 3401-3 : J.66. systems of: A.56 : M.11,17. theorem of Jacobi : C.45. 3 variables : J.64. * n variables : 3409 : A.22 : J.60 : LM. 10,11. with constant coefficients : Mél.5. integration by calc . of variations : C.14. z =px +qy + F (p , q) : Z.5. x wa y z p m q" A : CM.1 . (1 + P₁ + ... + P (dz , dy )" Z = Q: Z.13 . *P.D.E . , first order, linear : 3381-95 : reduction to , C.15 : J.81 : Me.78 . = R : 3383 : extension to * PZx + Qzy n variables , 3384. L(px +qy-z)-Mp - Nq+R = 0 : C.83. a(yu. - zu ,) + b (zu, -xu, ) + c( xu, —yu,) = 1 : Q.8. * Ptx+ Qty + + RtS: 3387. Y ; xy = x² +y2 : 3390-1 . √ (y²—x2) * (x−α) %x + (y — b) %y = c — z : 3392. * x² + y2 + z2 = 2ax : 3393 . * simultaneous : 3396-7 ; ex. 3398 : J.81 . u = vy and u₁ = — Vx : J.70 . *P.D.E . , second order : 3420-45 : A.33 : C.544,702,78,98 : JP.tr22 : L.72 : M.15 : Me.762,77 : Mél.3 : N.83 : P.46 transf . of, C.97. in two independent variables : trA. 54 : trNo.81 : C.92 ; transf . of, 97 : M.24. in ,4 and 5 variables : Mém.13 . * Rr + Ss + Tt = V, Monge's method : 3423 CM.3 : N.76 : Q.6. * Rr + S8 + Tt + U (rt — s² ) = V : 3424, 3434-40 : J.61 . * Rr + Ss + Tt + Pp + Qq + Zz = U : 3442. r +t = 0 : A.2 : CM.1 : J.59,73,74 : L.43 . r+t+ k²z = 0 : M.1 . * r-ast0 : 3433. * r—a²t = $ (x , y) , &c.: 3565. (r-a2t) = 2np : E.13. r-a2y2t = 0 : E.27,28 . r = q²mt : C.98. r = q:: J.72 .

* 2x=

P.D.E., second order-(continued): da (p sin x) + t +n (n + 1 ) z sin² = 0 : L.46. r (1 +q²) = t ( 1 +p²) : J.58 . 42 q²r—A²t + ny q =0: E.22. η dr (px) + dy (qx) = 0 : Z.28. construction of explicitly integrable equations ofthe form 8 = zλ (x,y) : JP.28. 2f (x)F " (y) z : C.81 . {ƒ (x) + F (y)}2 a dry log A +λ = 0 : L.53. 8+ Pp + Qq + Z = 0 : Me.76 . * 8 + ap + by + abz = V: 3444. (ax +by+ c) s + aλq + bµp = 0 : A.33. (x+y)²s + a (x +y) p + b (x + y ) q +cz = 0: A.33,38. z2 (28 -pq)² +q = F (y) : A.70.

8=

rt-82 : geoQ.2 . * P = (rt—s2)" Q, Poisson's eq.: 3441 . (1 +r) t+(1 + t) r— 2pqs = 0 : An.532 . * q (1 +q) r+p (1 +p) t− (p + q +2pq) 80: 3432 . 48² +(r −t)² = 4k2 : approx. integn ., C.74. As +Bq + (r , p , q , x, y , z) = 0 : C.93 .

*

** * * **

(log )xy + az = 0 : C.36. r dr U U2t = Rr where t = √(2 Rr² +42) r L.38. U2x +U2y + U23 = 0 (see also " Spherical harmonics ") : 3551 , 3626 , J.36 : Mo. 78. =yz : 3552 . Ustu tu = aux + buy + c uz = xyz, &c.: 3554. xuzx +aux - qxu = 0 : 3618 . a² (Uzx + U2y + U2z) = U21 : 3629 : C.7 :

LM.72. integration by change of variables : C.742. P.D.E. , third order, two independent variables : LM.8 : N.83. P.D.E. , fourth order : ▲▲u = 0 : C.69. P.D.E., any order : No.73 : C.80,89 : M.11,13. 28 2 x² %nx = Zny : Z.7. two independent variables : C.75 : CP.8.

910

INDEX.

P.D.E. , any order (continued) : any number of functions and inde pendent variables : C.80. and ap. to physics : JP.13. of cylinders : Me.77. and elliptic functions : J.36 : hyper ell. , J.99. integration by definite integrals : An. 59 : C.94 : L.54. of dynamics : C.5 : J.47. Hamiltonian : M.23 : Z.11. of heat: L.48 ; of sound, L.38. integrated in series : C.15,16. of Laplace : G.23 . linear : An.77 : C.13,90 : CD.92 : CM. 2 : J.69 : JP.122 : L.39 . of orthogonal systems of surfaces : Ac.4 : C.77. with periodical coefficients : C.293. of physics : L.72,47. P.D.E. , simultaneous : C.92,th78 : LM.9 : M.23 : Z.20. linear : J.65. P.D.E. , system of : C.13,74,81 . am %mt = x²mzmx : A.30,31 , by z = eªtf(x). Znx = x*%(m + n) y + F₁ ( y ) + xF2 (y) + ... + x - ¹Fm(y) : A.51 . Aznt + (d2x + day + ….. ) " z = 0 : C.94. dz Hdx + Kdy + Ldp + Mdq + Ndr + & c.: J.14. Partial differentials of x : J.11 . x²+y² *Partial fractions : 235, 1915 : A.30,66 : C.46,492,783 : CM.1 : G.2 : J.1,5,9, 10,11,22,32,50 : JP.3 : L.46 : Mém. 9: N.452,64,69 : Q.5 . Partition of numbers ( see also " Num bers " and " Indeterminate eqs ." ) : AJ.2,5,6 : An.57,59 : At.65 : C.80, 86,90,91 : CP8 : J.13,61,85 : M. 14 : Man.55 : Me.78,79 : Mém.13, geo.ap20,44 : Mél.1 : N.69,85 : P. 50,56,58 : Pr.7,8 : Q.12,2,7,15 : Z. 20,24. by Arbogast's derivatives : L.82 . of complex numbers in Jacobi's th : C.96. by elliptic and hyper-elliptic func tions : J.13. formula of verification : Pr.24. into 2 squares : An.50,52,54 : C.87 : J. 49 : LM.8,9 : N.54,78,alg65 : odd squares , Q.19. into 3 squares : J.40 : L.59,60 . into 4 squares : C.99 : L.68 : Pr.9 : Q.1. 4 odd, or 2 even and 2 odd : Q.19,20. into 5 squares : C.97,98 : J.35.

Partition of numbers-(continued ) : into ten squares : C.60 : L.66. into p squares : C.39,90 : L.61 : N. 54. and an integer : L.57. into the product of two sums of sqs.: L.57. into parts, the sum of any two to be a sq.: Mém.9. into 2 cubes : L.70. into sum or difference of 2 cubes : AJ.2. into 4 cubes : N.79. into maximum nth powers : C.95. into 10 triangular numbers : C.62. formation of numbers out of cubes : J.14. 2 squares whose sum is a sq.: E.20 : N.50. 3 squares, the sum of every two being a sq.: E.17. 4 squares, the sum of every two being a sq.: E.16. 3 nos., the sum or diff. of two to be a sq.: Mém.18. 2 sums of 8 sqs. into 8 sqs.: Me. 78. a sum of 4 sqs . into the product of 2 sums of 4 sqs.: TI.21 . n nos . whose sum is a sq. and sum of sqs. a biquadrate : É.18,22,24. a quadric into a sum of squares : N. 80,81. ns-m³ into 3p² +q² : N.49. a square into a sum of cubes : N.67. a cube into a sum of cubes : E.22, 23. into 4 cubes : N.77 ; into 3 or 4 cubes, A.60. n12-9m12 or its double into 2 cubes : N.81 . 3 nos. whose sum is a cube, sum of sqs. a cube, and sum of cubes a sq.: E.26. 5 biquadrates whose sum is a square E.20. of n into 1 , 2, 3, &c. different num bers : E.34. of pentagonal numbers : C.96. a series for the : AJ.6. tables , non- unitary : AJ.7. theorems : AJ.6 : C.40,96 : Me.76,80, 83 : pr. symm. functions , G.10. Partitions :- -in theory of alg. forms : G.19. number of for n things : E.10. in planes and in space : J.1. Sylvester's theorem : Q.4. trihedral of the X-ace and triangular of the X-gon : Man.58.

INDEX.

*Pascal's theorem : 4781 : AJ.2 : CD.3, 4 : CM.4 : J.34,41,69,84,93 : LM. 8 : Me.72 : N.44,52,82 : Q.1,4,5,9 : Z.6,10. extension of and analogues in space : C.82,98 : CD.4,5,6 : G.11 : J.37,75 : M.22 : Me.85 . ap. to geo. analysis : A.18. on a sphere : A.60. Steiner's " Gegenpunkte " : J.58. Pascal lines : E.30 . Pedal curve : A.35,36,52 : J.48,50 : M.6 : Me.80,81 : Q.11 : Z.52,21. circle and radius of curvature : C.84 : Z.14. of a cissoid, vertex for pole : E.1 . of a conic : A.20 : LM.3 : Z.3. central : A.9 : Me.83 : N.71 . foci and vector eq.: LM.13. negative central : E.20,29 : TI.26. negative focal : E.16,17,20 . nth and n - 1th : E.18. of evolute of lemniscate : E.30. inversion and reciprocation of : E.21 . of a parabola, focal and vector eqs.: LM.13. rectification of difference of arcs of : Z.3. which is its own pedal : L.66. Pedal surfaces : A.22,35,36 : M.6 : J.50 : Z.8. counter : AJ.5. volumes of: C.55 : A.34 : An.63 : J. 62 : Pr.12. Pentagon, ths : A.4 : J.5,56 : N.53. diagonals of: A.57. Pentagonal dedecahedron : A.25. Pentahedron of given volume and mini mum surface : L.57. Periodic continued fractions : A.62,68 : C.96, J.20,33,53 : N.68 : Z.22. closed form of : A.62. representing quadratic roots : A.43. with numerators not unity : C.96. Periodic functions : A.5 : J.48 : N.67 : gzC.89 : Mo.84. cosx- cos 3x + cos 5a : CD.32. doubly : C.32,40,70,90 : J.882 : L.54. of 2nd kind : C.90,98 : gzL.83. of 3rd kind : C.97. monodromic and monogenous : C.40. with essential singular points : C.89. expansion in trig. series : N.78. 4- ply, of 2 variables : J.13. 2n-ply, of n variables : Mo.69 . multiply: C.57,58 . integrals between imaginary limits : A.23. real kind of: Mo.66,84.

911

Periodic functions-(continued) : of 2nd species : M.20 . of several variables : C.43 : J.71. in theory of transcendents : J.11 . of 2 variables with 3 or 4 pairs of periods : C.90. with non-periodic in def. integrals : C.18. Periodicity theory : M.18. Periods : cyclic, of the quadrature of an algebraic curve : C.80,84. of the exponential e* : C.83. of integrals : see " Integrals." law of: C.963. in reciprocals of primes : Me.733. *Permutations : 94 : Al .: C.22 : CD.7 : L.39,61 LM.15 : Me.64,66,79 : N. 44,71,763,81 : Q.1 : Z.10 . alternate : L.81 . ap. to differentiation and integration A.21 . of n things : C.95 : N.83 ; in groups, L.65 . of 3q and 2q letters , 2 and 2 alike : N.74,753. number of values of a function through the permutations of its letters : C.20,21,46,47 : L.65. successive (" battement de Monge " ) : L.82. with star arrangements : Z.23 . upon *Perpendicular from a point : a line length of : 4094, t.c4624 : eq4086, t.c4625 : sd5530. upon tangent of a conic : 4366-73. * upon a plane : 5554. upon tangent plane of aquadric : 5627. * ditto for any surface : 5791-3. Perpetuants : AJ.72. * Perspective : 1083 : A.692 : G.3 : thsL. 37 : Me.75,81 . analytical : A.11. of coordinate planes : CM.2 . drawing : 1083-6. figures of circle and sphere : A.57. isometrical : CP.1 . oblique parallel : Z.16. projection : A.16,70. relief : A.36,70 : N.57. * triangles : 974 : E.29 : J.89 : M.2,16 : in a conic, A1. Petersburg problem : A.67. Pfaff's problem : A.60 : C.94 : J.61,82 : M.17 th of Jacobi, J.57. Pfaffians, ths on : Me.79,81 . (see also " Expansion of ") : C.95 : E. 30 : N.42,45. calculation of : A.6,18 : E.27 : G.2 : cnJ.3 : Me.73,74 : N.50,56,66. by equivalent surfaces : N.48.

INDEX.

912

π:

- calculation of―(continued) : by isoperimeters : N.46 . by logarithms : N.56. to 200 decimal places : J.27 ; to 208, P.41 ; to 333, A.212 ; to 400, A.22 ; to 500, A.25 ; to 607, Pr.6,11,22. formulæ for, or values of : A.12 : J.17 : L.46 : M.20. √2 approx .: Me.66 ; T=3+ 10 = 2 log i, J.9. 2

functions of p.6 : A.1 : C.56,74. ¹ : E.27 ; to 140 places, LM.4 . hyperbolic logarithm of : LM.14. 75 -¹ : LM.8 . powers of and of π incommensurable : 795. series for : Q.12 : TE.14. theorem on and e : Q.15 . II ( ) =1 (1 + ) ... { 1+ ( n - 1 ) x } : A.12 : J.43,67. II (a) and imaginary triangles and quadrangles : A.51 . Piles of balls and shells : N.72. Pinseux's theorem : Mo.84. Plane : J.20,45 : Mém.22 . * equations of : 5545, q.c 5550, p.c 5552 : 2.1. * under given conditions : 5560-73. ** condition for touching a cone : 5700 . * ditto for a quadric : 5635, 5701. cond. for intersection of two planes touching a quadric : 5703. figures, relation between : A.55 : J.52 : M.3. kinematics of : Q.16. and line, problems : CD.2 : J.14. motion of JP.2 : LM.7. point-systems : J.77 ; perspective, Z.17. representing a quadric : N.71 . * Plane coordinate geometry : 4001-5473. Planimeter A.58 : Mél.2,3 ; Amsler's, 5452, C.77 ; Trunk's, A.44 ; polar, A.51 : N.80. Planimetrical theorems : A.37,60 . Plücker's complex surfaces : M.7. Plückerian characteristics of a curve discriminant : Q.12. Plückerian numbers of envelopes : C.782. Point-pair, absolute on a conic : Q.8. harmonic to two such : Z.13. Point-plane system : M.232. Points - in a plane, relation between four : A.2,26. tg.eq of two : 4669, 4913. on a circle and on a sphere : N.82. of equal parallel transversals : A.61 .

Points-(continued) : four, or lines, ths : CD.8. at infinity on a quadric : N.65. roots in a closed curve : N.68. in space, represented by triplets of points on a line : LM.2. systems : JP.9 : M.6,25 : N.58. of cubic curves : Z.15. three : coordinates of, N.42 ; pr, A.8. *Polar : 1016 , 4124 : A.28 : J.58 : gzLM.2 : Me.64,66 : N.72,79 . * of conics : 4762 : thsN.58. of cubic curves : J.89 : L.57 : Mél.5 : Q.2. curves, tangents of : N.43. * developable : 5728. inclined : N.59. line of two points with respect to a quadric : 5685. plane : Q2 : Z.22. of a quadric : 5678, 5687 : An.71 ; of four, LM.13. of a quartic : L.57. of 3 right lines : A.1 . * subtangent : 5133. Polar surface :-of a cubic : J.89 ; twisted , Z.232,24. of a plane : C.60 : N.66. of a point : N.65. tetrahedron : J.78 : N.65 : Z.13. of a triangle : A.59 ; perspective, J. 65. Pole : of chords joining feet of nor mals of conic drawn from points on the evolute : N.60. * of the line λa +µß + vy : 4671 . * of similitude : 5587. *Pole and Polar : 1016, 4124. Political arithmetic : trA.36-38. Pollock's geo. theorems : Q.1 . Poloids of Poinsot : CD.3. Polyacrons, A-faced : Man.62. Polydrometry : A.38,39. * Polygonal numbers : 287 : Pr.10,11,12 , 13. Polygonometry : thsAn.52 and J.2,47 : Mém.30. Polygons (see also " Regular poly gons ") : An.cn53,63 : JP.4,9 : Ñ. 74 : Z.11 ; theorems : A.1,2 : C. 262 : prsCD.52 : Mél.2 : N.58. area of: 748, 4042 : J.24 : N.48,52. articulated and pr . of configuration, tr : An.84. centroid : N.77. of circular arcs, cn : A.3 : J.76. classification : Q.2. division into triangles : A.1,8 : L.38 , 39 : LM.13 : Pr.8. of even number of sides : LM.1.

INDEX.

Polygons-(continued) : family of: N.83. maximum with given sides : J.26. of n +2n sides , numbers related to : A.62 . of Poncelet, metrical properties : L.79. semi-regular : JP.24 ; star, A.59 and L.79,80. sum of angles of : A.52 : N.50 . Polyhedral function (Prepotentials) : CP.132. *Polyhedrons : 906 : C.462,60-62 : J.3 : JP.4,92,15,24 : L.66 : Man.55 : N. 83 : P.56,57 : Pr.8,9,112,12 : Q.7 : Z.11,14 ; theorems : E.39-42 : J. 18 : N.43 ; F+ S = E + 2 : 906 : A.24 : E.20,272. classification of : C.51,52 . convex : angles of, C.74 ; regular, A. 59. diagonals, number of : N.63. Euler's theorem : J.8,14 : Mém.13 : Z.9. minimum surfaces of : A.58. maximum : regular, C.612 ; for a given surface, M.2 : Mél.4. regular ellipsoidal, C.27 ; self-conj . , A.62 ; star, A.62 : thsC.26 . symmetrical : J.4 : L.49. surface of: A.53 ; volume, J.24 : N.52. Polynomials : geometry of, JP.28 : th Q.14. determined from its partial differen tial : A.4. product of two : N.44. system of : L.56 . oftwo variables analogous to Jacobi's : A.16. value when the variable varies be tween given limits : C.98. Polyzonal curves, √U + √V + = 0 : TE . 25. Porisms : L.59 : P.1798 : Q.11 : TE.3,4,9 . of Euclid : C.29,48,56-59 : L.55. of two circles : Me.84. Fermat's fourth : A.46. of in- and circum-polygon : Me.83 : P.61 . of in- and circum -triangle : LM.6,9 : Q.1. Poristic equations : LM.4,5. Poristic relations between two conics : LM.8. Position, pr. relating to theory of num bers : Mél.2. Potential : ths An.82 : C.88 : G.15 : J. 20,32,63,70,th81,85 : M.2,3 : N.702 : Z.17. of a circle : J.76.

913

Potential-(continued) : of cyclides : C.83 : J.61 ; elliptic , Me. 78. cyclic-hyperbolic, tables : J.4,63,72,83 , 94. of ellipsoids : J.98 : Me.84 : Q.14 ; two homog., J.63,70. elliptic : J.472. of elliptic disc, law , r -³ : Q.14. Gauss's f. and theory : Z.84. gz. of first and second : L.79. history of : J.86. of homogeneous polyhedra : J.69. Jellett's eq. and ap .: Q.16. Newton's : M.11,13,16. one-valued : J.64 . p.d.e of : C.90. Poncelet's ths : Z.3. a related integral : L.45. of a right solid : J.58 . of a sphere : Me.81 ; surface of, Me. 83: Q.12 : Z.7. surfaces : J.54 ; conicoids, &c ., 79. vector : Me.80 . Pothenot's problem of the sphere : A. 44,47,542. Powers : angular functions, &c .: J.72. and determinants, relation : Z.24. of negative quantics : Me.73. of polynomials : JP.15. Power remainders : M.20 . Prepotentials : P.75. Prime divisors of quartics : J.3. Prime factors ofnumbers : J.51 : N.71,75. Prime-pairs : Me.79 . *Primes : 349-378 : A.2,19 : AJ.7 : fAn . 69 C.ths13,49,50,f63,962 : G.5: J.th12 ,th14,20 : L.52,54, th79 : LM.2 M.21 : Me.41 : N.46,56 : Pr. 5: Q.5 : up to 109, M.3 . in ar.p : Z.6 . calculation of : J.10 : in 1st million, M.25. in a composite number : C.32 . distribution of, ap . of recurring series : C.82. division of a prime 4n + 1 into sum of squares : J.50 : ditto of 8n +3, 7n +2, and 7n +4, J.37. even number = sum of 2 odd primes (P) : E.10 : N.79 . Fibonacci's problem : LM.11 . of the following forms and theorems respecting them : 4k + 1 , 4k + 3 ; L.60 : 6k + 1 ; J.12 : 8k + 1 ; L.613, 62 : 8k +3 ; L.58,60,614,622 : 8k + 5 or 7 ; L.60,61 : 12k + 5 ; L.61,63 : 16k +3 ; L.61 : 16k +7 ; L.60,61 : 16k + 11 ; L.60-62 : 16k + 13 ; L. 61 : 20k +3, 20k + 7 ; L.63,64 : 6 A

914

INDEX.

Primes (continued) : 24k + 1, 5, or 7 ; L.614 : 24k + 11 or 19 ; L.60 : 24k + 13 ; L.61 : 40% +3 ; L.61 : 40k + 7 or 23 ; L. 603,613 : 40k + 11 or 19 ; L.60 : 40k +27 ; L.613 : 120k + 31 , 61 , 79 , or 109 ; L.61 : 168k + 43, 67, or 163 ; L.63 : m² +kn2 with k = 20,36,44, 56, or 116, and n an odd number ; L.65 : 4m² + 5n2 with m odd ; L. 66. general formula for C.63. of a geo. form, limit of : C.74. irreducibility of 1 + a + ... + - ' , when pis a prime : J.29,67 : L.50 : N.49. law of reciprocity between two, ana logue : J.9 . * logarithms of (2 to 109) : page 6. * number infinite : 357 : Mo.78. number within given limits : A.64 : C.953 : M.2 : Z.5 . number of digits in their reciprocals : Pr.222,23 . number in a given quantity : Mo.59. product of n, th : N.74. relative : 349-50 , 355, —8, 373 : J.70 : L.49. tables of from 108 to 100001699 : of 153 of the 10th million : of cube roots of to 31 places : Me.78 . of sums of reciprocals and their powers : Pr.33. Mab for a prime or composite modulus : Q.9. totality of within given limits : AJ4 : L.52 . transformation of linear forms of into quadratic forms : C.87. on that prime number λ for which the class -number formed from the Ath roots of unity is divisible by λ : Mo.74. Primitive numbers : thC.74 . Primitive roots : C.64 : JP.11 : fL.54 : taJ.45. of binomial eqs .: thsL.40 : N.52. of primes : J.49. product of, for an odd modulus : J.31 . of unity : C.92. Abel's theorem : An.56. divisors of functions of periods of : C.92. period, Jacobi's method : C.70. of primes J.49 : and their residues , Me.85. sum of, for an odd modulus : J.31 . table, for primes below 200 : Mém. 38. table of, for primes from 3 to 101 : J.9.

*Principal axes of a body : 5926 ,'60 , 72,77 : J.5 : JP.15 : L.47 : N.46 . Prismatoids : A.39 : volume of, Z.23. Prismoids : A.39. Prism : volume of, A.6. *Probability : 309 : A.12,19,47 : C.65,97 : CP.9 : E.274,30 : G.17 : I.15 : J. 262,302,33,34,36,42,50 : L.79 : N. 51,73 : Z.2. theorems Bernoulli's : LM.5 ; P.62 ; TE.9,21 . problems : A.61,64 : CD.6 : E. all the volumes G.16 : L.37 : Me.4,6 : Q.9. de l'aiguille, &c.: L.60. in analysis : J.6. on decisions of majorities : L.38,42. duration of life : CM.4. rouge et noir : J.67. drawing black and white : I.14 : L.41 . errors in Laplace, p. 279, and Poisson, p. 209: CP.6 games : A.11 : head and tail ; C.94. of hypothesis after the event : 324 : Mél.3. local : P.68 : exE.7. notation : LM.12. position of double stars : Pr.10. principal term in the expansion of a factorial formed from a large number of factors : C.19. random lines : Pr.16 : E. frequently. repeated trials : 317-21 : C.94. statistics : L.38. testimony and judgment : TE.21 . continued : Me.772. Products : of differences : thQ.15. of 4 consecutive integers : N.62 . of inertia : 5906. infinite, convergency of : A.21 : transf. of, C.17. of linear factors : C.9. of n quantities in terms of sums of powers : Me.71 . of 2 sums of 4 squares ( Euler) : Q. 16,17. systems of: L.56. *Progressions : 79-93 : An.64 : C.20 : G.62,7,11 : of higher order, G.122. with n = a fraction : N.42. *Projection 1075, 4921 : A.3,6,12 : prsJ. 70 : Q.21 , pr13 . and new geometry : A.1 . central : G.13 : derivation from or thogonal, A.62. central and parallel, of quadrics into circles : A.52. * of conics : 4921-35 : A.66 : J.37,86. of a cubic surface : M.5.

INDEX.

*Projection- (continued) : of curves : J.66 : Z.132 : loci of cen tres, A.6. on spheres : Q.14. tangents to : cnL.37. of a curved surface on a plane : J.67. of an ellipsoid on a plane : J.59. of figures in one plane : A.1. gauche : N.66 . of Gauss : Mél.2 . map : Mercator's, 1093 : A.50 : G.18 : JP.24: N.793. * of one line on another : 5529 : on a plane, A.6. * orthogonal : 1087. plagiographic : A.8 . of ruled quartics : M.2. of shadows : N.56. of skew hyperboloid : Me.75 . of solids : Man.84. of the sphere : 1090 : AJ.2. stereographic : 1090 : A.30-32 : L. 42,54. of surfaces on a plane : M.5. of surface of tetrahedron on a sphere : J.70. of two rectangular lines : 4934. correspondence between Projective : two planes and two spaces : G.22. equations of a surface, relation to tg.e : E.2. figures on a quadric : A.18. generation of alg. surfaces : A.1,2 : of curve forms, An.14 : J.54 : M.25 : Z.18. geometry : A.8 : An.75 : At.752,78 : G.12-14,17 : J.84 : M.17,18 : ths N.77 : ths of Cremona, G.132,14 : thZ.11. loci and envelopes : G.19. and P.D. eqs.: A.1. point series : J.91 . and perspectivity of higher degree in planes : J.42 . *Proportion : 68 : A.8 : G.6,13 : TE.4. Pseudosfera : G.10. Ptolemy's theorem : A.2,67 : At.19 : J. 13 : LM.12 . ext. to ellipse : A.30 : inverse of, A.5. Pure mathematics, address by H. J. S. Smith, F.R.S.: LM.8. Pyramid triangular : A.1,3,21,28,32, 36: J.3 : volA.14. vertices of : A.2. and higher n-drons : prs A.9. and prism sections, collineation, & c .: A.9. Pythagorean theorem : A.11,17,20,24 : gzJ.26 and N.62. spherical analogue : A.44 : N.52 .

915

Pythagorean triangles : taA.1 : E.20 . *Quadratic equations : 45 : A.24 : with imaginary coefficients , A.8. graphic solution : Me.76 . real roots of : J.61. solution by continued fractions : L.40. by successive approximation : N.74. 66 Quadric Quadratic forms (see also functions " ) : trA.15 : Ac.7 : C. 85 : J.27,f39,54,56,76,86 : L.512, 73 : M.6,23 : Mo.682,74 : thsAn. 54 : J.53 : M.20 and Z.16,19. Dirichlet's method : Mo.64. having one at least of the extreme coefficients odd : L.67,69. Kronecher's : L.64. multiplication of : An.60. number of the genera of : J.56. number which belong to a real deter minant in the theory of complex numbers : J.27. odd powers of sq. root of 1-2nU+n² : Mél.5. positive : A.112. reduction of : J.39 : L.48,56,57 . relation, anal. investigation : Z.14. Quadratic loci, intersection of : AJ.6. *Quadratrix : 5338. tangent, cn : N.76. of a curve : C.763. Quadrangle : prA.55 : C.95. of chords and tangents : A.2. differential relation of sides : Me.77. dualism in the metric relations on the sphere and in the plane : Z.6. and groups of conics : A.1 . of two intersecting conics, area : LM.8. metrical and kinematical properties : C.95. *Quadrature : 5871-83 : A.26 : An.502 : CD.ths5,9 : E.6 : J.34 : L.54 : Mém.24,41 : N.42, f55,54 : N.75 : TI.1 . approximate : C.95 : N.58. of the circle : Me.74 : Pr.7,20. Cote's method : N.56. with equal coefficients, f.: C.90. Gauss's method : A.32 : C.84,90 : J.gn55,56. from integrals of differentials in two variables : M.4 . * Laplace's formula, f.d.c : 3778. of a small geodesic triangle, Gauss's th : J.16,58 . by lines of equal slope : 5881 . quadrics : CD.1 : L.63 : formed by intersecting cylinders, An.65. sphero-conic : J.14.

916

INDEX.

Quadratures (continued) : which depend upon an extended class of d.e with rational coeffi cients : C.92. *Quadric cones : 5590,5618 ,'54 ,'97 : N.66. locus of vertex : C.52 . through six points : C.52. Quadric functions or forms : A.13,38 : C.44,55,78,892,95 : J.472 : JP.28, arith 32, positive 99 : algL.74 : Mo.58 : see also " Quadratic forms. " bipartite : P.58. in coefficients and in indeterminate complexes : J.24. equivalence of: C.93. in n variables : Me.2 : disappearance of products, N.55. quaternary : An.59 : L.54 : M.5,13 : whose det. < 0 , C.96 ; and corres ponding groups of hyperabelians, C.98. reduction of : C.91,93,96 : G.5 : to sum of squares , Mél.5 . represented by others : C.93. transf. of: C.86 : CD.4 : G.1 : LM.16 : N.66 : Q.17 . reciprocal : J.50. invariability in number of pos. and neg. squares : J.53. with two series of variables : C.94. *Quadric surfaces : 5582-5703 : A.32,4, 12,16,45,56 : An.52, geo60 : At.51 : C.76 : CD.3 : G.13,14 : J.1,18,38, f42,63,69,89 : JP.6 : L.39,43,50 : M.2.23 geoMe.74 : N.56,57,582, 593,60,61,77 : Z.5, cn13. theorems : An.54 : CD.5 : J.54,85 : L.43 : Mo.79 : N.632,642 : Q.2,4. problems : An.61 : C.60 : J.73 : N. 58 : Q.10. analogy with conics : Me.72. anharmonic section of : G.12. bifocal chords of : CD.5. * central equations : 5599-5672 . central sections : 5650 : area, 5650 : axes, 5651. locus of focus : N.66. * non-central sections : 5654. * centre : N.75 : area, 5655. centre, coordinates of : 5690 : A.16. circular sections : 5596, 5601 , ' 6,'19 : C.43 : CD.1 : CM.1 : E.30 : J.47, 65,71,85 : N.51. common enveloping cone : G.6. condition for a cone : 5699. conjugate diameters : 5637 : N.42,61 ; parallel, G.1 ; rectangular sys tem , L.58.

Quadric surfaces : -conjugate diame ters-(continued) : parallelopiped on them : vol. 5648 : sum of squares of areas of its faces, 5645: do. of reciprocals of the perpendiculars on its faces, 5644. sum of squares of their reciprocals : 5643 : ditto of their projections on a line or plane, 5646 . construction and classification by projective figures : A.9. correlation of points and planes on : CD.5. * cubature of : 6126-65 : A.14. diameters : 5677, -88. diametral plane : 5636 ; gn.eq, 5689. of constant sectional area : N.43. director-sphere of : Q.8. duplicate CD.6. enveloping cones of : 5664-72, 5697. equation between the coeffs .: J.45. without foci : L.36 . focales, a property from the theory of : L.45. * general eq.: 5673 : CD.5 : CM.4 : LM. 12,13 : M.1 : Me.64. condition for a cone : 5699. condition for a sphere : Q.2. coefficients : A.1 . generation : 5607-24, N.47,75 ; Ja cobi's, J.73 . homofocal : L.60. indices of points, lines, and planes , theory : N.704. * intersection of two : 5660 , C.64 : G. 6 : M.15 : Q.10 ; ruled , N.83 ; with a sphere, ths , N.64. parameter representation of: M.15. tangent to : LM.13. ** intersection of three : 5661 , J.73 : TN.69. loci from : A.27. normals of : 5629-32 : C.78 : J.73,83 : N.63,78 : Q.8. oblique coordinates : N.82 . polar plane of : 5681-8. Plücher's method : L.38. of revolution : N.72,81 ; through 5 points, 66 and 79. self-reciprocal : M.25 : Z.22 . sections of: J.74. similarity of two : CD.8. with a " Symptosen-axe ": A.60,61 . system of: Q.15 ; reduc. and transf. , L.74: Z.6. * tangents : 5677 : G.12. tangent planes : 5626, -78 : CM.1 : cnJ.42: LM.11 : N.46.

*

INDEX.

Quadric surfaces- (continued) : locus of intersect. of three : LM.15 . through 9 points or under 9 condi tions : A.17 : CD.4 : J.24,62,68 : L.58,59 : Q.8 : Z.25 . under 8 conditions : C.62. through 12 points : G.17,22. through a twisted quintic curve : E. 38. transformation of two, by linear sub stitution into two others , in which the squares only of the variables remain J.12. principal axes : see " Axes." principal planes of: L.36 : N.67,71 : Z.24. and their umbilics : C.54. volume of segment : A.27 ; of oblique frustum, E.19. Quadricuspidals : L.70. Quadrilateral : A.6,48 : Me.66 : thG.5 : prsN.43. area: N.48 ; as a determinant, N.74. between two tangents to a conic and the radii to the points of contact : A.532. bisectors, a property : N.75. * and circle, geo.: 733 : Q.5. * complete : 4652 : A.24,69 : At.63 ; mid points of 3 diagonals, collinear, Me.73. * and conic : 4697 : N.75,76. and in-conic : tg.c 4907 : Q.11. cn. with given sides and equal diags .: A.5. convex : A.66 ; area, Q.19. Desargues' theorem : Z.24. plane and spherical : A.2 : Z.6. and quadrangle : A.1. right-angled : A.2,3. with sines of angles in given ratio : A.2 . sum of sqs. of sides : th 924. th. extended to 3 dimensions : J.56. Quadruplane : LM.14. *Quantics : 1620 : C.47 : J.56 : LM.6 : N.48 Cayley P. 54,56,58,59,61 , 67,71 and 78 : Pr.72,8,9,11,15,17, 18,23 : thsCD.6 : J.53 : N.53 : Q.14 ; Cauchy, Pr.42. derivatives , relation between 1st and 2nd : E.39. derivation from another by linear substitution : C.42. of differentials : J.70,71 : Mo.69. index symbol of : CD.5. integration of a rational : C.97. in linear factors : C.50 : L.ths61 : Q.6. transf. of ax2 + by² + cz2 + dw2, by linear subst.: J.45.

917

Quantitative function, transf. of : CD.3 . 66 Quartic equations : see Biquadratic.' Quartic curves : An.76,79 : At.52 : C.37, ths64,65,77,98 : CD.5 : G.14,16 : J.59 M.th1,4,7,12 : prN.56. aU+ 63H = 0 : M.1 . and Abel's integrals : M.11. binodal, mechanical cn . of : E.18. characteristics of a system : C.75. chord of contact, eq. M.17. classification by inf. branches : L.36. with cuspidal conics : M.19. with 3 cusps of 1st kind : An.52 . degenerate forms : LM.2. developable reciprocated : Q.7. with a double line : A.2 : C.75 . with a double point : M.19 ; two, C.97.; three, Q.18 ; three of inflexion, N.78. with double tangents : J.66. and elliptic functions : J.57,59. of 1st kind and intersections with a quadric : An.692. generation of C.45 : J.44 ; 3rd class , J.66 and Z.18. 16 inflexion points of 1st species of : trZ.28 : E.32. and in-pentagon, th : M.13. oblate : C.74. parameter representation of : M.13. penultimate : Me.72. with quadruple foci : Q.18 . rectification : C.872. and residual points : E.342. and secants : M.12. singularities of : L.75 : M.14. synthetic treatment of : Z.23. through which one quadric surface only can pass : An.61 . trinodal : thE.30. unicursal twisted : LM.14. Quartic surfaces : A.12 : C.70 : G.11,12 : LM.3,19 : M.1,7,13,18,20 : Mo.66 , 72 : N.70,873 : Q.10,11 . containing a series of conics : J.64. with a cusp at infinity : LM.14. with double conic : A.2 : M.1,2,4 : Mo.68. with eq. Sym. det. := 0 : Q.14. generated by motion of a conic : J.61 . Hessian of : Q.15. and 2 intersecting right lines : M.3. with 12 nodes : Q.14. with 16 nodes : J.65,73,83,84,85,86,87, 88 : Mo.64 ; principal tangent curves of, M.23 and Mo.64. Steiner's : C.86 : J.64 . with a tacnode at infinity at which the line at infinity is a multiple tangent : LM.13.

918

INDEX.

Quartic surfaces-(continued) : with triple points : M.24. Quaternions : C.86,98,: CD.4 : G.20 : M.11,22 Me.623,64,81 : P.52 : Q. 6 : TE.27,28 : TI.21 . ap. to linear complexes and congru ences : Me.83. ap. to tangent of parabola : AJ.12. elimination of aßy from the conditions of integrability of Suadp, &c.: TE.27. equations : C.98 ; linear, 99, : of sur faces, Joachimstahl's method, E.43. vp op = 0 : TE.28 : qQ - qQ' = 0, Me.852. finite groups : AJ.4. f. for quantification of curves , sur faces , and solids : AJ.2. geometry of: CD.9 . integration ths : Me.85 . transformations : Man.82. *Quetelet's curve : 5249. Quintic curves : cnM.25 . Quintic equations : AJ.6,7 : An.65,68 : C.462,48,50,61,629,73,802,85 : J.59 : M.13,14,15 : P.64 : Pr.11 : Q.3 : TI.19 Z.4. auxiliary eq. of : Man.152 : P.61 : Q.3. condition of transformability into a recurrent form : E.35. irreducible : AJ.7 : J.34. functions of difference of roots : An. 69. reduction of : Q.6. resolvents of: C.632. whose roots are functions of a varia ble : Q.5. solution of : An.79 : J.59,87 : N.42 : Q. 2,18. Descartes' method : A.27. Malfatti's : An.63. in the form of a symmetrical deter minant of four lines : An.70 . Quintic surfaces : LM.3. having a quintic curve : An.76. Quintics, resolution of : Q.4. Radial curves : LM.1 : of ellipse, Q.18 . of conics , catenary, lemniscate, & c.: E.24. Radiants and diameters of a conic : C. 26. *Radical axis (see also " Coaxal Cir cles " ) : 958, 984-99 , 4161 : LM. 2 : Me.66. of symm. circle of a triangle : A.63 . of two conics : Q.15. *Radical centre : 997. *Radical plane : 5585.

Radii of curvature of a surface : A.11 , 55 : Q.12 : Z.8. principal ones : L.47,82 : M.3 : Me.80 : N.55. *Radii of curvature of a surface : 5795 5817 : A.11,55 : Q.12 : Z.8 . ellipsoid : 5831 . flexible surface : L.482. principal : 5814-6 : L.47,82 : M.3 : Me.80 : N.55. constant : Me.64. for an ellipsoid : 5832. equal and of constant sign : C.41 : JP.21 : L.462,50. Euler's theorem : 5806. one a function of the other : An.65 : C.84 : J.62 . product constant : An.57. reciprocal of product : An.52. sum constant : An.65 . sum = twice the normal : C.42 . * Radius of curvature of a curve : 5134 : A.cn4,9,31,33 : CD.7 : J.2,45 : ths M.17 : N.62,74 : q.c and t.c Q.12 : Z.3. absolute : CM.1 . circular : 5736 ; ang. deviation, 5746. of conics : 1259 : A.9 : CM.1 : J.30 : L.36 : Me.66 : Mél.2 : N.45,682. at a cusp or inflexion point : N.54. in dipolar coordinates : Me.81 . of evolutes in succession : N.63. of gauche curves : N.66. of a geodesic : L.44 : on an ellipsoid, An.51 . and normal in constant ratio : N.44. of normal section of (x, y, z ) = 0 : 5817. of a parabola : 1261 , 4542. of polar curves : A.51 : CM.2. of polar reciprocal : N.67. of projection of a curve : N.61 ; of contour of orthogonal projection of a surface, C.78. of a roulette : 5235 : N.73 . transf. of properties by polar recipro cals : L.66 . of tortuous curves : Mém.10. of circum-sphere of a tetrahedron in terms of the edges : N.74. * Radius of gyration : 5904. Radii of two circles which touch three touching two and two : A.55. Radius vector of conic : J.30 : N.479. Ramifications : E.30,33,40, pr 37, th27. sol. by a diophantine eq.: Mo.82 . Randintegral : J.71 . * Ratio and proportion : 68 ; compound , 74. of ab : geo.cnN.44.

INDEX.

Ratio and Proportion- (continued) : of differences of geo. quantities : C.40. limits of: 753. * of segments of lines and triangles, geo.: 929-32. of two distances, geo.: 926-8. Rational ::- derivation, cubic curve : AJ.33. divisors of 2nd and 3rd degrees : N. 45. functions, development of : AJ.5. Rationalisation of : alg. fractions : A.13,33,35. alg. equations : A.13 : CD.8 : J.14 : P.14 : TI.6. alg. functions : A.69. a series of surds ( Fermat's pr.) : A.35. Rational functions :-of n elements : M.14. infinite form systems in : M.18. Ray systems (see also " Congruences ") : J.57 : L.60,74 : Me.66 : N.60,613, 622 : Z.16. of 1st order and class and linear pen cils : J.67,692 : Z.20. 1st and 2nd order : Mo.65 : M.15,17. 2nd order and class : J.92. 3rd order and 2nd class : J.91 . 6th order and 2nd class : J.93. 2nd class and 16 nodal quartics : J.86 : Mo.64. complex of 2nd degree and system of 2 surfaces : M.21 . and refraction theory : Q.14 : TI.15 -17. forming a group of tangents to a sur face : Z.18 . infinite geometry of : Z.17. *Reciprocal polars : 4844, 5704 : A.36 : gzE.24 : J.77 : LM.2 : N.48,49 : num.fG.21 . * Reciprocal of a circle : 4845. * cones : 5664, 5670. ** of a conic : 4866-8. * of a quadric surface : 5717-8. radii: M.13. relations : J.48,79,90 : M.19,20. * spiral : 5302 . * surfaces : 5704-19 : J.79 : M.4,10 : P. 69. curvature of: L.77. degree of: CD.2 : TI.23. of Monge: C.42. of quadrics : gn 5705 ; central, 5706. of surface of centres o a quadric : Q.13. of the same degree as their primi tives : Mo.78. theorems on conics and quadrics : L. 61.

919

Reciprocal (continued) : transformation, geo.: L.71 . triangle : th Q.1 ; and tetrahedron, Q.1. Reciprocants : LM.172. Reciprocity : anal., A.7 : geo. , CD.3 . * Reciprocity_law : 3446 : AJ.1 ,th 2 : C. 902 : J.282,39 : LM.2 : Mo.58 : d.e, 3446 and A.33. in cubics : M.12. history of: Mo.75. for power residues : C.84 ; quadratic, C.24,88 : J.47 : L.472 and Mo.80, 84,85 ; cubic, in complex numbers from the cube roots of unity, J. 27,28. quadratic F2 system of 8th degree : J.82. supplementary theorem to : J.44,56. *Rectangle : M. I. of, 6015 . * Rectangular hyperbola : 4392 : Me.62 , 66,72 : N.42,65. *Rectification of curves : 5196 : A.26 : Ac.5 : An.69 : CR.95 : CD.9 : G. 11 : J.14: L.47 : N.ths 53,64. approximate : M.4 : Mél.4. by circular arcs : C.77,85 : L.50 : Mém . 302. by elliptic arcs or functions : J.79 : Mém.30. by Poncelet's theorem : C.94. mechanical : Z.16. on a surface : Mém.22. * Rectifying - developable : 5727. * line : 5726 , -51 : N.73. * plane : 5726. surface : 5730. Reflexion : from a revolving line : TE.28. from plane surfaces : A.60 . from quadric surfaces : J.35 . Refraction curve : A.51 .

Règle à calcul : C.58 : N.69. *Regular polygons : 746 : A.21 ,cn24,39 : L.38 : M.cn6,13 : N.42,44,47. convex : Me.74. eqs. of and division into eqs . of lower degrees, tr. : A.46. in and circum : N.45 : Q.2. in space : Me.75 . spherical : N.60,67. star : J.65 : Me.74 : N.49. funicular : N.49 . 5-gon : M.83. 7-gon : A.17. 7-gon and 13-gon : M.6. 8-gon : A.6. 12-gon : complete, C.95.

920

INDEX.

Regular Polygons- (continued) : 17-gon : A.6 : J.cn24,75 : N.74 ; and division of the circle, A.42. eqs . for sides and diagonals : A.40. * Regular polyhedrons : 907 : A.112,47 : Pr.34 : Me.66 : Q.15. relation of angles : 909 : Me.74. volumes by determinants : A.57. Related functions : M.25. Relationship problems : E.35,382. Relative motion : N.66 . Rents : A.40. Representative functions : M.18 . Representative notation : Q.6 . Reproduction of forms : C.97. Reptation : N.54. Residues : A.26 : C.122,13,32, ap32,41,44, 49 : CM.1 : J.252,31,89 : L.38 : Mél. 4: N.462, Mém 70. ap. to infinite products : C.17. ap. to integrals whose derivatives in volve the roots of alg. eqs .: C.23 . ap. to reciprocity law of two primes and asymptotes : C.76. of complex numbers : Mo.80. primes of 5th, 8th , and 12th powers : J.19 . biquadratic : C.64 : J.28,39 : L.67. cubic : A.43,63 : C.79 : J.28,32 : L.76. quadratic : Ac.1 : J.28,71 : Q.1. ext. of Gauss's criticism : Mo.76. of primes, also non-residues : L.42 : Mél.4. and partition of numbers : J.612. quintic : C.76 : Z.27. septic : C.80. of 9exp. (9exp.9) by division by primes A.35. Residuation in a cubic curve : Me.74. Resultant alg.: M.16 : ext C.583. and discriminants and product of dif ferences of roots of eqs ., relation : Me.80. of two equations : J.30,50,53,54 : M.3 : P.57,68. of two integral functions : Z.17. of n equations : An.56. of covariants : M.4. of 3 ternary quadrics : J.57, N.69 . Reversible symbolic factors : Q.9. Reversion of angles : LM.6. Rhizic curves : Q.11 . Rhombus quadrisection by two rect angular lines : Mém.11 . circumscribing an equil. triangle : A. 45. Riemann's surface : LM.8 : M.6,18,: thsZ.12 . of 3rd species : M.17. new kind of : M.7,10.

Riemann's surface-(continued) : irrationality of : M.17. Riemann's function : A.68 : J.83 : M.21. ext. to hyper-geo .-functions of 2 vari ables C.952. e-formula, gz : Ac.3. *Right-angled triangles : 718 : prs A.2. with commensurable sides : E.33. Right cone : Me.72,73,75,76. Right line : A.49,57 : fQ.15 : t.c Me.62. and circle : ths N.56. coordinates of: G.10. and conic : Q.7 ; cn. for points of sec tion, A.59,66 : N.85. quadratic for abscissæ of the points : 4319. tg.e of the points : 4903 . condition of touching : 4315, 4323, t.c 5017. * drawn from x'y' across a conic : quad ratic for the segments in an el lipse, 4314 ; parabola, 4221 ; gen. eq., 4494 ; method , 4134. joining two points, coordinates of point dividing the distance in a given ratio : 4032 , t.c 4603 , 5507. tg. eq. of the point : 4879. joining two points and crossing a conic : quadratic for ratio of seg ments in an ellipse, 4310 ; para bola, 4214 ; gn. eq. , 4487 , t.c 4678 ; method, 4131 . constants, relations between : sd. 5515. coordinates of, relation between the : 4897. and curve : 4131-5 : ths. in which pairs of segments have a constant length, C.836 ; a constant product, C.822,832 ; a constant ratio, C.83 ; ths. in which systems of 3 seg ments have a constant product, C.839. crystallography : A.34. equations of : 4052-66, p.c 4107, t.c 4605-8 ; sd 5523, q.c 5541 . geometry of: A.64 : thsJ.8. at infinity : 4612-4, tg.c 4898. cond. for touching a curve : 4900 . pencils of : C.70 : L.72 ; quadruple, J.67. and plane : prs CD.1 and CM.2 : t.c and q.c Q.5. pole of: t.c 4671 : tg.e 4674. and quadric : 5676 ; harmonic divi sion , 5687. and quadric of revolution : N.82. six coordinates of : CP.11.

INDEX.

Right line-(continued) : system of : At.68 : G.9,10,16 . of 1st degree, G.6 ; of 2nd, G.7. in space : G.113,12 : L.46 . and planes, geo. of 2nd kind : At. 65. * three, condition of intersection : 4097 : t.c 4617. * three points lying on, cond.: 4036, t.c 4615 . * through a point : 4073, 4088-9, 4099, t.c4608. * condition : 4101 . and perp. or paral. to a given line : sd5538-9. * through two points : 4083 , sd5537 ; t.c4616, 4789 ; p.c4109 : on a conic, equation of, ellipse 4324, parab. 4225. through four lines in space : A.1 : CM.3 : Gergonne's pr. J.2 and N.17. touching a surface : condition , 5786. quadric : 5703. planes or points through or on given points, lines, or planes , number of such : Z.6. * two :-angle between them : 4112 ; sd5520, 5553 : CP.2 : N.66 . ** bisector of the angle : 4113 , sd5540 , q.c5543. cond . of parallelism : 4076 ; t.c4618 ; sd5531 . cond. of perp.: 4078 ; t.c4620 : sd 5532. cond. of intersecting on a conic, gn. eq : 4962. cond. of either touching the conic : 4964. cond . of intersection : sd5533. coordinates : 4090, t.c4611 . shortest distance : sd5534-6. drawn to the points of section of a right line and conic, eq. of : A.69. through origin, eq. of : 4111 . under given conditions : C.73,74 : under four, C.68. * Right solid : M. I. of, 6018 : thA.34. Rodrique's th .: Me.80,842. Rolling cones : L.53. Rolling and sliding solids : geo thsC.46 . Rosettes : N.48. Rotation : CM.3 : LM.32 : infinitesimal, C.78. of system of lines drawn through points on a directrix, modulus of : C.21 . Roots of algebraic fractions : N.46 .

921

* Roots of an equation (see also " Equa tions ") : 50,402 A.14 : CM.2 : CP.8 : E.36 : J.20,31 : N.42,56 : P.1798,37,64 : Q.1,5,6. of a biquadratic , en : N.44. by parab. and circle : N.87. * commensurable : 502 N.45 ,th57. limits to the number : N.59. common : 462 : C.80,88 : N.55,69. as continued fractions : CM.3. continuity of : N.76. in a converging series : C.23,38. of cubic : L.44 : N.42. of cubic and biquadratic : An.55 : L. 55. as definite integrals : J.2. y" —xy" -¹— 1 = 0 : Me.81 : P.64. as determinants of the coefficients : A.59,61 . * discrimination of : 409 : A.46. * equal : 432-47 : CD.5 : E.33 : Mél.1 : P.1782 Q.9,18. with equal differences : G.152. existence of : A.15 : CD.2 : CP.10 : E.36 : G.2 : J.5,44,88 : LM.1 : Q. 11 : TI.26 . expanded in power series : J.48 . of the form a + √b + √c + : N.45 . forms for quadrics, cubics, and quar tics : Z.24. functions of the roots of another eq. : 425,430 ; products in pairs, Q.13. squares of differences : 541 : ap,N. 50 : E.40. as functions of a variable parameter : C.30. functions of : similar, L.54 ; relation to coefficients , TE.28. geo. cn of: JP.10 . in g.p : N.89. in a given ratio : J.10. * imaginary : 408 : geo.cn , A.15,45 : C.24, 86–88 : JP.11 : L50 : N.46,47 , 682 approx. N.45,53 : Q.9. between given limits : A.21 : L.44. Newton's rule : Me.80 : N.67 : Pr. 13 . Newton-Fourier rule : Q.16. Newton- Sylvester rule : 530 : C. 994 : LM.1 : Me.66 : Pr.14 : Q.9 : TI.24. incommensurable : 506 (see " Sturm's th.") infinite : N.442,45. in infinite series : A.69. integral, by Newton's method of divisors : 459. least : M.9 : TE.28. * limits of 448 : C.58,60,93 : geo CP. 12 : N.43,45,592,72,802,81 . 6 B

922

INDEX.

Roots of an equation— (continued) : number between given limits : A.1 : G.9 : J.52 : L.40. the eq. containing only odd powers of a : N.63. Rolle's th.: 454 : AJ.4 : N.44 : ext L.64. Rolle, Fourier, and Descartes : A.1. number satisfying a given condition : C.40. product of differences : Me.80 : P.61 . of a quadratic : 50-3. of a quartic and of a Hessian, rela tion : E.34. of quintics : C.59,60 : LM.14 : TI.18. rationalization of : P.1798,14. real : A.36,58 : C.61 : J.50 : JP.10 : N.50 : P.57. of a cubic : N.72 : Z.2. Fourier's th.: N.44. developed in a series : L.78 : N.56. limits of: J.1 : N.53,79. series which give the number of : Z.2. to find four : An.55. rule of signs : 416-23 : A.34 : C.92, 982,992 N.43,46,47,67,69,79. separation of : A.28,70 : J.20 : N.682, 72,74,75,809. by differences : N.54. for biquadratics : A.47. for numerical : C.89,92 : G.6. simultaneous eqs.: C.5. squares of differences : 541 : CM.1 : N.42,44 : Q.4. sums of powers : 534 : E.38 : thJ.9 : N.53,75 : gzMe.85 : Q.19 . in sums of rational functions of the coefficients : Ac.6. surd forms of : CM.3. symmetrical functions of : 534 : A.16 : AJ.1 : An.54,55,60 : C.44,45 : G. 5,11 : J.19,54,81 : Me.81 : N.48, 50,55,66,84 : P.57 : Pr.8 : Q.4 : TI.25. do. of the common roots of two eqs. : N.60 : Z.15. do. of differences of roots : C.98. which are the binary products of the roots of two eqs .: An.79. with a variable parameter : C.12. which satisfies a linear d.e of 2nd order : C.94. *Roots of numbers : 108 : A.17,26,35 : C.58 : E.36 : Me.75 : N.61,70. * square root : 35 : C.93 : N.452,46,61 , 70. as a continued fraction : 195 : A.6, 12,49 CM.22 : L.47 : Me.85 : Mém.10 : TE.5 : Z.17.

Roots of numbers-(continued ) : to 25 decimal places : Me.77. cube root : Horner's method : 37: A.67. of 2 to 28 decimal places : Me.76,78. and sq. root, limit of error : N.48. fourth root : A.30. nth root as a fraction : A.46. *Roots of unity : 475-81 : C.38 : J.40 : L.38,54,59 : Me.75 : N.432 : TE. 21 : Z.22. cubic roots, alg. and geo. deductions : C.84. function theory : Z.22. 23 roots, composition of number 47 : J.55,56. *Roulettes : 5229 : Ac.63 : C.70 : CP.7 : J.65 : L.80,81 : N.56 : TE.16 : Z.28. areas of, and Steiner's transf.: E.35. generated by a circle rolling on a circle : JP.21. by focus of ellipse rolling on a right line : A.48. by centre of curvature of rolling curve : L.59. Ruled surfaces : An.68 : CD.8 : G.3 : J.8: L.78 : N.61 . areas of parallel sections : Z.20. and guiding curve : A.18. of minimum area : L.42. octic with 4 double conics : C.60. P. D. eq. of: Me.77. quadric : Me.68. quartic : A.65 ; with 2 double lines, A.65. quintic : J.67. represented on a plane : C.853. of species , p = 0 : M.5. symm. tetrahedral : C.62. torsal line : M.17. transformation of : C.88. *Scales of notation : 342 : J.1 : L.48,5ry, 10ry,20ry: Phil. Soc. of Glasgow, vol. 8. Screws : TI.25. Scrolls : A.53 : CD.7 : CP.11 : J.20,67 : M.8 : cubic, M.1 : P.63,64,69 : Pr.12,13,16 : Z.cn28. condensation of: LM.13. cubic on a quadric surface : Me.85. flexure and equilib. of : LM.12. ruled : A.68 : z = mxy², A.55. tangent curves of : M.12. *Sections of the cone : 1150. *Sectors and segments of conics and conicoids : 6019-6162 : G.1 : thsZ.1 . Secular eq. has real roots : J.88. *Self-conjugate triangle : 4755, 4967 : G.8 : N.67 : Q.5,10.

I L

INDEX.

22/02/2

Self- conjugate triangle (continued) : of 2 conics : Me.77 ; of 3 conics , 5025. and tetrahedron in conics and quad rics : Z.6. Self - enveloping curves and surfaces : Z.22. Self-reciprocal surface : Mo.78. * Self-reciprocal triangle : 1020. Seminvariants : AJ.7 : E.th42,6 : Q. 19-21 . critical and Spencian functions : Q.4,6. and symm. functions : AJ.6. Septic equations : Mo.58. 99 *Series : (see also " Summation and 66 Expansions "): 125-9, 149 59, 248-95, 756-817 , 1460 , 1471 -2, 1500-73, 2708-9, 2743—60, 2852-64, 2880 , 2911-68 , 3781 , 3820 : A.4,52,9,14,18,23,60 : No.39, 472 : C.29, pr92 : CP.9 : G.10 : J. 3,17,34,38,th53 : L.th56,81 ,: Me. 64 : N.59, th62,70 : Q.3 : Z.15,16, 23.

x+

༄ }༅ ༅「8 }8 r༠ c/「c

Useful summations : 1 + ... = log: -x : 156.

+

3

x +

=log (1 +x) : 155.

+

+20

+

... =

x

+

e -1 : 150.

...

+

+

23 3! 23 3!

:

T

tan-¹ : 791 .

x2 x+ 2!

= =1 1-1 :

T

+

x

+

+

x+

1 +æ log 1 - x : 157.

―

第一

= 2

5!

+

212283

sina : 764. 3! x2 x4 = e*+e-* : + + + 2! 4! 6! 2 x2 x4 x6 ― .. = 1 - cosa : 765. + 2 ! 4! 6! * 1 +2 + +n : 2939 : A.65 : Me.78. p = 1,2,3, or 4 : 276 : A.64 : E.34. ―― ... & 1P -3P + 5P — ... : 1P 2P +3P J.7. Enx" : A.27. (a +n) " : N.56. Σ (ɑn + b„x²) x¹¬n : Z.15. n-3 + (n - 4) (n - 5) -...: J.20. 2.3

923

Series ( continued) : a ±nb + C (n , 2) c + &c.: J.31. nr- n (n - 1) + C (n , 2) (n - 2) -... : 285 ; rn, CM.1 .

1+ 1x+ a (a +1)x2+ ... : J.37 ; with b (b + 1) x = 1 , J.2. 1 Σ : N.59 . (a + nd)* deductions from (13) x2n Az x =Σ Σ = 171 ) == 2"n !* 1)," + (a a(n) LM.9 . 1 1 + 1 1 ± 2n 21/0 + 3n ± and 1 ± t. : 3n 5n 2940-4 : A.41 : LM.8 : Q.7 : with n = 1,2... 8; 2945 ; E.32,39 ; G.10 ; N.79 ; Z.3. Note that by (2391 ), 1 1 + 1 - &c. =flogæ dæ 32 52 +x2 an : Mém.11 : with a = 1 , J.5. n2 nx n : A.61 . Σ Σ A.41 n! an " x13 X- x7 + ... E.44 ; with x = 1 , 7! 13! J.5. Σ (n- 1) x" : A.50 . n ! a" -1 xn : Q.6. Σ (-1)" (a +2n) (1 − x)" 1 Σ : E.36. x(")

* Σ anx(") : 2709. x n) æ( : A.26 . n+1! 1.3 ... 2n - 1 : L.60. n ! 2" (2n + 1 )

Σ ( -1 )"

1+ (qª— 1) (q³— 1) x + (q - 1) (q - 1) +1 (g - 1)(q1-1)(q - 1) (q +1-1 )x22 (q — 1) (q² — 1) (q'— 1) (q¹¹¹ — 1) + ... : J.32,70.

Σ { ( " + 1 ) ( 53) + ... } : G.11 . ε x Σ : A.35 . 1 + am an + a₁nm - ¹+

INDEX.

924

Series

(continued) : Xha +B Σ Κ B a pos. integer < a, na +B' K, = the general term of some recurring series : C.86. a Σ ni +a : A.34.

N= 2 n : C.50. Σ .# = 1 E (NL) :

* Σ

sin ¢ )) / n * : J.54 ; with k = 2, COS(2n + 110

2960-1 , J.8. Σ sin³ (2n + 1 ) / (2n + 1 )+ : E.39. or 4 Σ (sin no)20 / nt and sin Σ COS no )³ / n³ : L.73 .

sin * Σ ε (a" COS no) In : 2922—3 . sin n2 21π Σ : L.40. COS P n sin no Σ : 2962 : M.5. a²+n² COS Σ An cos" sin no : Z.1 . * Σ sin (a +nẞ) : 800 : Q.3 ; COS * Σ c" sin (a +nẞ), 783 ; COS sin Σ n!cos (a + nẞ), 788. sin Σf (n ) no: J.42 : L.52. COS sin Σ Η (m ,n ) æn COS no : J.41 . Σ 1 tan Φ . A.44. 2n 2n

Σf(nx) : L.51 . ΣΑ , φ (n ) an, J.25,28. n m f (x + m 1 ) : A.22 . from

(1-x)" da : A.47.

2 1+ sin x da : G.10. cos2 x log 1 1 - sin of Abel : C.93 : N.85. application : thA.48 ; to arith , G.7. in a.p : see " Arithmetical progres sion." analogous series : N.69. a.p and g.p combined : A.9. with Bernoulli's nos.: An.53 ; and bi nomial coefficients , A.23.

from

Series (continued) : binomial (see " Binomial theorem ") : analogous series : E.35 : J.32 : N. 82 ; with inverse binomial co efficients, Me.80 . coefficients independently deter mined : A.18. whose coefficients are the sums of divisors of the exponents , sq. of this series : Me.85. combination : A.26. convergent : 239 : A.2,6,8,14,26,41,67 , 69 : No.44 : C.10,11,28,40,432 : J. 2,32,11,13,16,22,42,45,76 : L.39 42 : M.10,20-22 : Me.64 : N.45, 46,67,69,70, : P.87 : Z.10,11 . and of d.i with a periodic factor : L.53. power -series : A.25. representing integrals of d.e : C.403. representing functions : M.5,22. in Kepler's problems : Ac.1799 . multiplication of : M.24 . and products , condition : M.22. whose terms are continuous func tions of the same variable : C.36. with constant ultimate differences : Pr.52. converted into continued fractions : J.32,33 : Mém.9 ; into products of an infinite no. of factors, J.12 : L.57 : N.47. in cosines of multiple angles : C.44 : Mém.15. and definite integrals : L.82 : Man. 46. derived : A.22 ; from tan- ¹0, A.16. developed in elliptic integrals of 1st and 2nd kind : An.69. difference : 264 : A.23,24. differential transf. and reversal of : Pr.7. and differentiations : A.10 : J.36.

: C.21 : L. Dirichlet's f. for Σ$ (1) n 46. discontinuous : CP.6 : L.54 : Me.78, 82 : N.85. A.64 : No.68 : C.17.20 : divergent CP.8,10 : J.11,13,41 : M.10 : 2.10. division of : AJ.5. double : C.63. doubly infinite : CD.6 : M.24. ext. of by any parameter : A.48. factorial : 268 : Mém.20 : N.67 : TE. 20. of fractions : L.40. Fourier's : A.39 : C.91,92,96 : CM.2 : Z.27. of Gauss and Heine : C.73 : G.9.

INDEX.

Series (continued) : Gregory's : 791. harmonic periodic : J.23,25. of Hermite, a th.: E.29. from infinite products : Me.733. integration of infinite : A.3. irrationality of some : J.37. involving two angles : L.74. Klein's higher : An.71. of Lagrange: C.23,343,52 : L.57 : N. 76,gz85 : Q.2 ; remainder, C.53. an analogous series : C.99. xn : A.10 : An.68 : of Lambert ; & 1 - xn J.9 : Z.6. Laplace's (d.c) : C.68. of Laplace's functions : EY , C.882 ; En Yn, C.44. in Legendre's function X : An.76 : C.44 . of Leibnitz : J.89. limits of : A.20 : Me.76 ; remainders , C.34 ; by the method of means , J.13. from log (1 +x) , (1 +x)" and e* by in termitting terms in the expan sions : A.21 . modular : C.19 . 99 of, multiple : C.192 ; " regulateur C.44 . neutral : CP.11. obtained by inversion from Taylor's series : Mém.11 . of odd numbers : A.64. a paradox : Me.72. periodic, critical values of : CP.8. of polynomials : C.96. of posterns : G.6. of powers (see also " Numbers ") : 277 : C.87 G.2 : cubes, L.64 and 65 : M.23 : Mo.78 : Q.8 : Z.1. approximate fractions : J.90. of a binomial : Mém.13. in a convergent cycle, constants in : M.25. like numbers : N.71,77. or multiples of 3 : A.272. of terms in ar.p : L.46. products of contiguous terms of: Mém.18. of reciprocals : Q.8. recurring : 251 : doubly, An.57 : J.33, 38 : Me.66 : Mém.24,26 : N.84 : of circles and spheres , N.62 : f, Z.14. represented by rational fractional functions : J.30 . * reversion of : 551 : J.52,54 : LM.2 : TI.7. of Schwab : N.59.

925

Series (continued) : self-repeating : CP.9. of spherical functions : An.75. of Sterling, for transformation : J.59 . with terms alternately positive and negative : C.64. whose terms are the coefficients of the same power of a single vari able in a multiple integral : C.20. in theory of numbers : C.892. transformation of : C.59 : J.7,9 : into a continued fraction, Mém.20, Z.7 ; of Σ fƒ (x, t) dt and others, C.13. in a triangle problem : A.64. trigonometrical (see above) : A.63 : Ac.2 : C.95,97 : M.4-6,162,17,22 , 24 J.71,722 : representing an arbitrary function between given limits, J.4 ; conversion in mul tiples of arc, L.51 ; symbolic transf. of, Q.3. triple : G.9. two infinite, multiplication rule : J.79 . * Seven-point circle : 4754c. Seven planes problem : N.56. Sextactic points of plane curves : Pr. 13,14. Sextic curves : ax + by + c = 0, Q.15 ; mech.cn, LM.2. bicursal : LM.7. and ellipse, pr : J.33. Sextic developable : Q.7,9. Sextic equation : C.64 : M.20. irreducible : J.37. solution when the roots are connected by (a-3) (by) (c−a) + (a— b) (B — c) (y — a) = 0 : J.41 . Sextic torse : An.699. Sextinvariant to a quartic and quart invariant to a sextic : AJ.1 . between two *Shortest distance : lines : 5534 : A.46 : G.5 : N.49,66. between two points on a sphere : A. 14 : N.14,67,68. from the centre of a surface : A.63. of a point from a line or plane : N.44. Shortest line on a surface : A.23,37,64 ; in spheroidal trigonometry, A.40. Signs : CP.2,11 : J.12 : Me.73 ; ( = ) , Me.75 ; ( + ) , CD.6,7 : Me.85 : N. 48,49. Similarity :- -of curves and solids : A.13. Similarly varying figures : LM.16 . Simson line of a triangle : E.29. *Simpson's f. in areas : 2992 : C.78. Sines of higher orders : C.914,922 ; ap. to d.e, C.903,91 .

926

INDEX.

Sines, natural, limit of error : N.433. -- 03 : geo.N.75. Sin 0-0 4 Sin-1 (x +iy): Q.15. Sine and cosine :-extension of mean ing : A.31 : C.863. * in factors : 807 : A.27 : J.27 : L.54. of infinity : CP.8 : Me.71 : Q.11. of multiple arcs (see also “ Expan sions ") : CM.4 : TI.7. * of particular angles : 690 ; 3°, 6°,...to 90°, N.53. sums of powers : An.1 . tables, formation of : 688 : A.66 : N. 422. values near 0 and 90° : G.9. of (a +b) : 627 ; A.6,21,36. Six-point circle of a triangle : Me.82 , 832. or sphere : LM.2. Six points on a plane Skew surfaces : see 66 Scrolls." Sliding rule : LM.6. *Small quantities of second order : 1410. Smith's Prize questions , solutions : Me. 71,723, -4, 77. Solid angle : A.42 ; section of, No.19. Solid harmonics : Me.80. Solid of revolution : A.60,67. between two ellipsoids : A.2. cubature and quadrature of : 5877 80 : A.68 : N.42. Space homology : G.20. Space theories : An.70 : LM.14 : P.70 : Z.17,18. absolutely real space : G.6. continuous manifoldness of two di mensions : LM.8. space of constant curvature : An.69, 73 : J.86 : M.12. Plücker's " New geometry of" : G.8: L.66 : P.11 : Z.11,12. Grassmann's " Ausdehnungslehre " : AJ.1 : CP.13 : M.7,12 : Z.24 ; ap. to mechanics , M.12. non Euclidean or n-dimensional : A. 6,29,58 : ths64 : AJ.4,5 : An.71 : C.75 : G.6,10,12,23 : M.4,52-7 : Me.ths68,72 : Pr.37. 3-dim. , J.83 ; 4-dim., J.83, M.24 ; 6 dim . , G.12. angles (4-dimen . ) : A.69. areas and volumes : A.69 : CD.7. bibliography of: AJ.1,2. circle : G.12,16,18. conics AJ.5. curves C.79 : M.18. Feuerbach's points : G.16. hyperboloid : Z.13.

Space theories- (continued) : imaginary quantities : Z.23. loci (anal.) : C.24. planes (4-dimen . ) : A.68 . plane triangle : A.70 . point groups : ths Ac.7. polars and alg. forms : J.84. potential function : An.82,83. projection ; M.19 ; 4-dim. into 3-dim., AJ.2 . quadric, super lines of ( 5-dim. ) : Q.12. quaternions : CP.13. regular figures : AJ.3. reversion of a closed surface : AJ.1. representation by correlative figures : C.812. simplicissimum of nth order : E.44. screws, theory in elliptic space : LM. 15,16. 21 coordinates of : LM.10. Sphere : geo, C.92 : ths and prs M.4 : q.c Me.62. and circle : geo, A.57. cn . from 4 conditions : JP.9. cutting 4 spheres at given angles : An.51 : N.83. cutting a sphere orthogonally and touching a quadric, locus of cen tre : TI.26. diameters, no. of all imaginable : A.24. equation of: 5582. 5 points of: J.23 : N.84. illumination of: Z.27. kinematics on a : LM.12. sector of (eccentric) : A.65. small circle of: Me.85. touching an equal sphere : E.31,32 ; as many as possible, A.56. 4 spheres, pr.: L.46 . 4 touching a 5th : At.19. 8 touching 4 planes : E.19 : N.50. 16 touching 4 spheres : J.37 : JP.10 : Me.cn82 : Mém.10 : N.44,47,65, 66,84 : Z.142. volume, &c. of segment and zone : 6050 : A.3,32,39 : An.57 : P.1. *Spherical :- -areas : 902. catenaries : J.33. class cubics with double foci and cyclic arcs : Q.15. conics : thQ.3 ; and quadrangle, Q.13 ; homofocal, L.60. coordinates : CD.1 : CM.1 ,ap2 ; ho mogeneous , G.6. curvature : 5728, '40,'47 : thE.34. curves : A.35,36 : Mém.10. of 3rd class with 3 single foci : Q. 17. of 4th class with quadruple foci : Q.18.

INDEX.

* Spherical :-curves-(continued) : of 4th order : J.43. with elliptic function coordinates : J.93. equidistant : An. 71 : J.25. and polars : No.63. rectification of: An.54. ellipse : t.cQ.8. quadrature, &c.: L.45 : N.484,54. epicycloid : G.12. excess : Mél.2 : cn , N.46 : f, Z.6. of a quadrilateral : Me.75 . figures, division of : J.22. geometry : G.4 : J.6,8,132,ths15 and 22 : M.15 : N.48,585,59 : Q.4 : TI.8. harmonics or 66 Laplace's functions 99 : An.682 : C.86,99 : CD.1 : CM.2 : J.26,56,60, geo 68,70,80,82,90 : L. 45,48 LM.9 : Me.77,782,85 : P.57 : Pr.8,18 : Q.72 : Z.24. analogues of : J.66 : LM.11. and connected d.i : Q.19 . as determinants : Me.77. and homogeneous functions : CM.2. and potential of ellipse and ellip soid : P.79. and ultra-spherical functions : Z.12 . P" (cos y), n = ∞ : J.90. S¹¸ -1 P¡µ™ dµ, &c. : Q.17. bby continued fractions : JP.28 ;

Qn Qn : P.70. S loci in spherical coordinates : TE.122. oblong : An.52 ; area, J.42 . polygonometry : J.2. polygons in- and circum-scribed to small circles of the sphere, by elliptic transcendents : J.5. quadrilateral : A.4,40 : th, E.28 : N.45. surface of: A.342,352. quartics : foci, Q.21 ; 4-cyclic and 3-focal, LM.12. representation of surfaces : C.68,75, 944,96 : M.13. surface represented on a plane : Me.73. triangle : A.9,11,20,ths50,65 : E.f30 : J.10,pr28 : JP.2 . ambiguous case : Me.77,852. angles of, calculated from sides : A.51 . by small circles , area : N.53. and circle : 898 : A.29,33. cos (A + B + C), f.: Me.72. and differentials of sides and angles : A.10. and ex-circles : 898 : E.30. graphic solution : AJ.6.

927

Spherical triangle- (continued) : and plane triangle : A.1 ; of the chords, A.33 : An.54 : Z.1. right angled : 881 : A.51 ; solution by a pentagon, A.11 . of very small sides : N.62. two, relations of sides and angles : A.2 . trigonometry : 876 : A.ths2,11,13,28, 37 : J.prs6,132 : LM.11 : N.42 : Z.16. d.e of circles : Q.20. cot a sin b : Me.64 ; mnemonic, 895 : CM.3 . derived from plane : A.26,27. formulæ 882 : A.5,16,24,26 : N.45, 46,53 : graphically, A.25 : ap. in elliptic functions, A.40. geodetic reduction of a spherical angle : A.512. Cagnoli's th.: 904. Gauss's eqs .: 897 : A.13,17 : J.7,12 LM.3,13. Legendre's th.: C.96 : J.44 : L.41 : M.1 : N.56 : Z.20. * Llhuillier's th.: 905 : A.20. * Napier's eqs. 896 : A.3,17 : CM.3 : LM.3,13 . Napier's rules : 881 . supplement to, and geodesy : A.36. * Sphero-conics : 5655a : tg.c, Q.8,99 : Me.3: Z:6,23. homofocal : C.50. mechanical cn : LM.6. Sphero-conjugate tangents : An.55. Sphero-cyclides : LM.16. Spheroidal trigonometry : J.43 : M.22 . Spheroidic transformation f. of Bessel : A.53. *Spheroids : 5604, 6152 ; cubature , 6158 , A.2 . Spieker's point : A.58. Spiral : A.28 : L.692 : N.79 : Z.14. * of Archimedes : 5296 : A.65,66. conical : N.45. * equiangular : 5288. * hyperbolic : 5302 . * involute : of circle, 5306 ; of 4th order, C.662. Squares : J.22. whose diagonals are chords of given circles : A.64. maximum with given sides : J.25. maximum in a triangle : J.15. whose sides and diagonals are rational : J.37. whose sides pass through 4 points : A.43. 64 transformed into 65, geo.: Me.77. sum of three : G.15 ; of four, L.57.

928

INDEX.

Standards of length : Pr.8,21 . Statistics : Z.26. Steiner's ths. and prs.: A.53 : J.13,14,16, 18,23,25,36,71,73 : N.48,562,593,62 : Hexagon, A.6 : Ray · systems, CD.6. Stereographic projection : A.39 : JP . 16 : L.46 . Stereograms of surfaces : LM.2 . Stereometry : JP.14 : thsA.10,31,43,57 ; J.1,5. multiplication : J.49. quadric and cubic eqs. and surfaces : J.492. Sternpolygon and sternpolyhedron : A. 13. Stigmatics see 66 Clinant." Stewart's geo.ths .: CM.2 : L.48 : TE . 2,15. Striction lines of conicoids : Z.28. Strophoids : AJ.7 : N.752. *Sturm's functions : 506 : A.62 : C.36 , 62,68 : G.1,20 : J.48 : L.46,48 , 67: Mo.58,78 : N.43,46,52,54,66, 67,81 : Q.3. and addition th . for elliptic functions of 1st kind : Z.17. ext. to simultaneous eqs .: C.35. and H. C. F.: Pr.6. and a quartic equation : A.34. and their reciprocal relations : Mo. 73. remainders J.43,48. tables : P.57 : Pr.8. ap. to transcendental eqs.: J.33. ap. to transf. of binomial eqs .: L.422. Subdeterminants of a symm . system : J.93 : M.82. Subfactorial n : Me.78. Subinvariants = seminvariants to bin ary quantics of unlimited order : AJ.5. *Subnormal : 1160. *Subsidiary angles : 726, 749. Substitutions : A.62 : C.ths66 and 67, 74,76,79 G.9,10,11,14,19 : L.65, 72 : M.13. ap. to functions of six or fewer vari ables C.21. ap. to linear d.e : C.78. by approximation , of the ratio of the variables of a binary quantic to another function of the same degree : C.80 . canonical forms of: L.72. and conjugate substitutions : C.215,22. n-3 -2 of the form (r) = e (r” −² +ar ² ) : M.2. linear : C.98 : J.84 : M.19,202.

Substitutions : —linear—(continued) : of a determinant : An.84. and integral : M.24. powers and roots of : C.94. reduction of : C.90 : JP.29. for reduction of elliptic functions of 1st kind : An.58. successive : A.38. which transform quadric functions into others which contain onlythe squares of the variables : J.57. of n letters : geo. for n = 3,4,5,6, and mystic hexagram : An.83. no. of in a given no. of cycles : A.68. permutable amongst themselves : C.21 . of equidistant numbers in an integral function of a variable : N.51 . of systems of equations : N.81 . of six letters : C.63. a th. of Sylvester : AJ.1 . which admit of a real inversion : J.73. which do not alter the value of the function : C.212. which lower the degree of an eq. in two variables and their use in Abelian integrals : C.15 . Sum and difference calculus : A.24 . Sum of squares of lines drawn from a point to cut a curve in a given angle : J.11 . *Summation of series (see also "Series " and " Expansions ") : 3781 A.10-13,262,30,55,62 : No. 84 : C.87,88 : J.10,31,33 : LM.4, 72: Mém.20,30 : P.1782,1784-7,— 91 ,—98 , 1802, —6,—7,—11 , —19 : Pr.14. approximate : 3820 : J.5 : 1.24. of arcs : A.63. Bernoulli's method : J.31 . Cauchy's th.: M.4. C (n, r) products of a, a + b, ... a + (n- 1) b : Q.18. by definite integrals : A.4,6,38 : J.17, 38,42,46,74 : Man.46 : Mém.11 . of derivatives and integrals : C.442. * by differences : 264 : Mém.30. by differential formulæ : L.31 : Mém. 15. by (x) : 2757. formula : A.47 : An.55 : J.30. Maclaurin’s , C.86 : f (A, D) DF ( n) , Q.8: Lagrange's, J.34 : P.60 : Vandermonde's, L.41 . of terms of a high order : C.32. Kummer's method : C.64. Lejeune Dirichlet's : CD.9 . periodic : J.15 . selected terms : Me.75.

929

INDEX.

*Summation of series- (continued) : of sines and cosines : J.4.

* Surface or surfaces-(continued) : ++=== 1 : A.35 ; с b a = b =c = 1 , A.21 . areas of: G.22 : N.52. argument of points on : LM.16. complex : M.5 ; of 4th order and class, M.2 . whose coordinates are Abelian func tions of two parameters : C.92 : M.19. of corresponding points : M.4. * cut orthogonally by spheres : 3393 : C.36. deficiency of : M.3. * definitions : 5770. determined from two surfaces of cen tres : A.68. Dirichlet's problem : An.71 . Dupin's theorem : C.74 : CM.4 : Q.12 . doubly circumscribing an n-tic sur face : J.54. of elliptic cone : An.51 . of equal slope : C.98 : N.65. equation of: gn5780 : A.3 ; for points. near origin, 5803. of even order : A.70. families of : C.70 : Me.72 . flexure of: J.18 . Gaussian theory of : LM.12 : N.52. generation of : C.94,97 : G.9 : J.492 : L.56,83 : M.18. implexes of : C.79,82. of minimum area : C.57 : J.8,13 : L.59 : Q.14. of nth order : cnM.23 ; 2nd , 3rd and 4th, Mél.6. of normals : N.59. whose normals all touch a sphere or conical surface : JP.4 ; do. for surface of revolution , JP.5. number under 9 conditions : C.62. octic of zero kind : G.12. order determined : Me.83. one-sided : A.57. parabolic points of : CD.2 . and p. d. e: C.13 : CD.2 : Z.7. and plane curves : J.54,72 : M.7. and point moving on it : L.76,77. and point at ∞ on it : J.65. relation of in Riemann's sense : M.7. representation of: J.83. on a plane : An.68,71,76. one upon another : An.77. of revolution : C.86 : G.4. areas and volumes : 5877,-9 : A.48. of a conic about any axis in space : JP.23 : L.63. of constant mean curvature : L.412. 6 с

a

theorem reXƒ(x, l) da : 2708.

of transcendents in alg. differentials : J.19. trigonometrical and infinite : Me.64. Superposition : TE.21,23. * of small quantities : 1515. *Supplementary :-angles : 620. * chords : 1201 . cones : thN.48 . * curves : 4917-20 : Man.54. Supplement integrals : J.98. *Surds : 108 : N.47. quadratic (see " Roots of numbers.") ↓ (a²+b²), √ (a2 -b2) approximately : J.13. ✓ (a ± √b) : 121 : A.3,13 : J.17,20 : Mém.10 N.46,48. * 3/ (a ± √b) : 122 ; / ( a ± √b ) , 124. *Surface of centres : 5774 : A.682 : An . 57: C.70,71 ,: E.30 : L.41,58 : LM.4: M.5,16. curvatures of the two ; relation : C. 74,79. the two focal conics of a system of homofocal quadrics : C.53 . of a quadric : Q.2. bitangents of: Q.13. model of : Mo.62. principal axes of : N.48. Surface curves : A.39,60 : An.54 : C.21 , 802 L.66 G.19 : J.2 : N.54,84. an an alg. surface : An.63. on cubic surfaces : M.21 . curvature of : N.65. on a developable : An.57 ; the oscu lating plane making a constant angle with it, L.47. on an ellipsoid : An.51 : CD.3 : geoL.50. groups of rational : M.3. of intersection : M.2 : N.68 ; of 2 quadrics, A.16. multiple: G.142 ; singularities of, M.3. on a one-fold hyperbola : CD.4. on quadrics : N.70. rectification of : A.36. relation to their tangents : C.82. on surfaces of revolution : Z.18,28. and osculating sphere : C.73. *Surface or surfaces : 5770 : A.14,f32,41 , 59,60,62 : An.51 — 3,55,60,61,65 , 71 : At.57 : C.17,33,37,49 ,ths58, 61,64,67,69,80,86,992 : G.3,21 : J. 9,ths13,58,63,64,85,98 : JP.19,24, 25,33 : L.44,47,512,60 : M.2,4,7,92, cn19 : Mo.82,833,84 : N.53,652,68, 72 : TI.14 : Z.74,8,20.

930

INDEX.

* Surface or surfaces- (continued) : meridian of a lemniscate, G.21 ; generation of, C.85. meridian and contour curves of: Z.21 . oblique : Q.6. passing through a given line, tan gent plane of : N.84. in perspective : JP.20. of reg. polygon about a side, vol . : A.57. quadric A.55 : L.60. shortest line on : A.38. superposable : C.80 : N.81 . Riemann's symmetrical ; and perio dicity modulus of the related Abelian integral of the 1st kind : Z.28. ruled, with generators part of a linear complex : C.84. ruled octic with 5 quartic curves : C.61. screw, parallel projection of : cnZ.18. section of, homogeneous eq .: LM.15. self-reciprocal : LM.2 ; quadric, E.36. sextic of first species : M.21 . singularities of: A.25 : An.79 : CD.7 : CM.2 : J.72 : M.9 : N.64 : Q.9. cubics : M.4 ; and quadrics, A.17. 16 singular points : C.922. solutions by infinites of 3rd order : geoC.802. Steiner's : LM.5,14 : M.5. touching a plane along a curve : CD.3 : CM.2. touchinga fixed surface always : G.20. transformation of: M.19. and transversal, th.: N.49. trapezoidal problem : Z.14. two series of : prsAn.73. web-system : J.82. which cuts the curve of intersection of two alg. surfaces in the point of contact of the stationary oscu lating planes : L.63. Surveying N.50, : geo. th.A.37. Symbolic geometry : (Hamilton) , CD . 1-4. Symbolical language : E.28 . Symmedian line : E.42 : N.83-5 : Q.20 . * Symmedian point : 4754c. Symmetrical: -conics of triangles : A.59. connections by generating functions : J.53 . * expressions : 219. figures : J.44. functions : C.76 : J.69,93,98 : LM.13 : Q.20 : Z.4 . Brioschi's th.: C.98.

Symmetrical- (continued) : of the common sol. of several eqs.: An.58. multiplication of : Me.85. of any number of variables : C.82. simple and complete : M.18,20. tables of : AJ.5 ; of 12-ic , AJ.5. points of 1st order : A.60 : cnA.64. of tetrahedrons : A.60. of a triangle : A.583. products : P.61 : Pr.11. with prime roots of unity : Me.85. tetrahedral surfaces : Z.11 . Symmetry ; plane and in space : N.47. Synthesis : C.18. *Synthetic division : 28 ; evolution , Me.68. *Syntractrix : 5282 . Tables : mathematical ; Sect . I., con tents , p . xi . of Bernoulli's nos., logarithms, & c ., calculation of: J.2. for empirical formulæ : AJ.5. in theory of numbers : Q.1. : CP.13. of e*, e-*, log100 , log10 Tac-loci : Me.83. Tamisage : LM.14. *Tangencies (circles , points, and lines) : 937 : Pr.9 : Q.2,8. Tangential eq. with the intercept and angle of inclination for coor dinates of the line : P.77. *Tangent cone at a singular point : 5783. *Tangents : 1160 : cnA.4,33 : J.562,73 : N.42 : Z.23. and contact-point in loci and en velopes : C.85 : cnN.57. conjugate (and Dupin's th. ) : An.60 . construction of : M.22 : N.80. double : Q.3. eq. of; to find it : 4120, -32 ; Me.66 . faisceaux of : N.50. cut by lines at a constant angle : A.43. locus of intersections of three : LM. 15. at a multiple point, cn .: JP.13. and normals : 5100 ; relations, A.51 . parallel : N.45. segments of : 4307 ; equality of, C.814 . of a surface : 5781 ; at singular points : 5783 : M.11,15. *Tangent planes : 5770, -82 ; p.c5790 : CM.4: N.45. and surface ; intersection of : 5786 -9 : L.58. to equidistant surfaces : cnZ.28. triple : C.77 . tan 120° : P.8,18. tan a as a continued fraction : Z.16 ; do. tan na, Me.74.

INDEX.

tan na : f. in N.53. tan - 1 (x +iy): Z.14. * tanh S : 2213 . tantochrone : L.44 : An.53. *Taylor's theorem : 1500 , -20 , -23 : A. 8,13 : AJ.4, ext1 : C.58 : CM.1 : J.11 : L.37,38,45,58 ,gz64 : M.212, 23 : Me.724,73,75 : Mém.20 : N. 52,63,70,743,79 : TA.7,8 : Z.gz 2 and 25. analogues of J.6. convergence of : C.60,74-6 : J.28 : L.73 . Cox's proof : CD.6 . deductions : G.12. for an imaginary variable : Me.79. kinematic meaning of J.36. reduced forms of: C.84. * remainder : 1503 : An.59 : C.13 : J.17 : N.60,63 : Z.4. a transformation of : C.78. Terminology : LM.14. Ternary bilinear forms : G.21 . Ternary cubics : AJ.2,3 : C.56,90 : CD.1 : J.39,55 JP.31,32 : L.58 : M.1,4,9. in factors : An.76 : Q.7. as four cubes : LM.10 . parameter of the canonical transf.: LM.12. reduction to canonical form : C.81 . transformation of : J.63. Ternary forms : At.68 : G.1,9,18 . order of discriminant : LM.3. with vanishing functional deter minants : M.18 . Ternary quadric forms : At.652 : C.92 , 94,96 : G.5,8 : J.20,40,70,77 ; transf. of, 71,78 and 79 : L.59,77 : P.67 . and corresponding hyperfuchsian functions : Ac.5 . indefinite J.47 ; with 2 conjug. inde terminates, C.68. representation by a square : An.75. simultaneous system : J.80. table of reduced positive : J.41 . Ternary quartic forms : C.563 : An.82 : M.172,20. Terquem's th.: Q.4. Tesselation pr.: LM.2. Tesseral harmonic analogues : CP.13. Tetradrometry, differential f.: A.34 : Z.5. Tetragon, analogue in space : CD.7. Tetrahedroid : J.87. *Tetrahedron : 907 : A.3,16,232,51,56 : J.65 : LM.4 : Me.62,662,82 : Mél . 2 : N.pr74,80,81 : Q.5: geo Z.27. determined from coordinates of ver tices : J.73.

931

Tetrahedron (continued) : of given surface : J.83. with opposite edges ; equal, N.79 ; at right angles , Me.82. with edges touching a sphere : N.74. and four spheres : LM.122. homologous J.56. and quadric : CD.8 : thN.71 ; 2 quad rics, Q.8. 6 dihedral angles of, eq.: N.46,67. theorems : A.9,10,31 : N.612,662 : Q.3,5. two : M.19 . volume : 5569 : A.452,57 : LM.2 : N. 67 : Z.11 . volume and surface in c.c and g.c : A.53 : CD.8 : N.68. volume and normal, relation : N.54. Tetratops : A.692. Theoretical value function : J.55. Theta-functions : Ac.3 : C.ths90 : J.61 , 66,74 : M.17 : Me.ap78 : P.80,82 : ths2.12. analogues of : C.93. addition theory : J.88,89 : LM.13. ap. to right line and triangle : A.3. argument : J.73 : 4thM.14 : M.16. characteristic of : complex AJ.6 : C. 882 : J.28. th. of Riemann : J.88. as a definite integral : Me.76. double : LM.9 : Q. transf. 21 . and 16 nodal quartic surface : J.83, 85,87,88. formula of Riemann : J.93. Jacobian, num. value : M.7,11. modular integrals : trAn.52,54 : J.71 . multiplication of : LM.1 : M.17. quadruple : AJ.6 : J.83. reduction of, from two variables to one : C.94. representation of : M.6 : Z.11. transf. of: A.1 : An.79 : J.32 : L.80 : M.252. linear Me.84 : Q.21 . triple : J.87. in two variables : C.922 : J.84 : M. 14,24. Theta - series : constant factors of, J.98: Me.81. n-tuple : J.48. Sea (c- x)/ { c (c - x) -b2} : Q.15. *Three - bar curves : 5430 : LM.7,9 : Me.76. triple generation of : Me.83. Toothed wheels : cnTE.28. Topology with tables : M.19,24 . Tore section of : G.10 : N.59,61,642,652 circular, 56 and 65. and bi-tangent sphere : N.74.

932

INDEX.

Toroid : A.8 : rectif. & c ., A.9 : N.44. Toroidal functions : P.81 : Pr.31 . Torse sextic : CP.112 : Q.14. circumscribing two quadrics : Me.72. depending on elliptic functions : Q.14. and curve : Q.11 ; and sphere, G.12 . *Torsion A.19,62,65 : J.60 : angle of, 5725 ; of involute, 5753 ; of evo lute, 5754. inflected, infinite and suspended : 5739 : N.45. constant : L.422. *Tortuous curves : 5721 : Ac.2 : C.19,43, 58,cn62 G.4,5,21 : J.16,59,93 : JP.2,18 : L.502, th51,522 : M.5,19 , th25 : Mo.82 : N.60 : No.79 : Q. 4,5,7. approximate coordinates of a point near the origin in terms of the arc : 5755. * circular curvature of : 5722 : An.60 : CD.9 : J.60 : JP.152 : Q.6. locus of centre of do.: 5741 , 5748 -50. with circular and spherical curvature in a constant ratio : L.51 . cn. upon ruled cubics and quartics : C.53 . with coordinates rational functions of a parameter : M.3. cubic J.27,80,: M.20 : Z.27. definitions : 5721 . determined from relation of curva ture and torsion : A.65 . cubics : An.58,59 : C.45 : L.57 : of 3rd class, J.56 ; four tangents , M.13. and developable surfaces : CD.5 : L.45 : M.13. elements of arc : N.733. generation of : C.94 : L.83 ; by two pencils of corresponding right lines, G.23. with homog. coordinates : Q.3. the loci of similar osculating ellip soids ; d.e of same : C.78. with loops : M.18. with a max. or min. property: Mém.13. on a one-fold hyperboloid : C.53. with the same polar surface, d.e : C.783. quadrics : J.20. quartics C.543 ; 1st species , J.93. cn. of two : C.53. intersection of quadrics : C.54. quintics C.54,58 . * radii of curvature and torsion : 5739 : L.48 Mém.18 . and right line method : 5743. sextics, classification of : C.76.

*Tortuous curves—(continued) : singularities of: C.67 : J.42. spherical curvature : 5728, -40, —47 : A.19 . triple, and their parallels : A.65. *Tractrix : 5279 ; area, E.35 . *Trajectory : 5246 : Mo.80 . of a displaced line ; oscul . plane, &c.: C.70,76. of 3 homofocal conics : An.64. of meridian of surface of revolution : L.46 . surface of points of an invariable figure whose displacement is subject to 4 conditions : C.76,77. of a tortuous curve : L.43. Transcendental : -arithmetic : J.29. curves : 5250 : An.76 : M.22. equations : C.5.59,72,94 : G.6,10 : J. 22 : L.38 : N.55,56. mechanical solution : CP.4. 6. separation of roots by compteurs s logarithmique " : C.44. without a root : J.73. *Transcendental functions (see also 66 ' Functions ") : 1401 : A.37 : C. 86 : J.3,th9,20 : L.51. of alg. differentials : J.23 : L.44. arithmetical properties : M.22 . classification of : L.37. connected with elliptic : C.17. decomposition into factors by calcu lus of residues : C.32. and definite integrals : P.57. expansion of : J.16. integral : C.94,95 : G.23 : J.98. reduction of: Me.72 . squares of: Me.72 . theorem of Sturm : L.36 . whose derivatives are determined by cubic eqs.; summation of the same : J.11. which result from the repeated integ ration of rational fractions : J.30. Transformation : -bilinear : M.2. birazionale : G.7 ; of 6th deg. in 3 dimensions, LM.15. contact : M.8. of coordinates : pl.4048 ; cb.5574-81 : A.13 : A.26 : CM.1 ,: J.2 : JP.7 : N.63. in 3 dimensions : A.13 : Q.2 : J.8. rectangular into elliptic : Mél.4. Cremona's : M.4. of curves : L.49,50. of differential equations : Me.82. of equations : A.40 : N.642 ; 3 vari ables, G.5 . of a characteristic eq. by a discri minant : Au.56.

INDEX.

Transformation—(continued) : quadric : CM.2 : N.75 : M.23. simultaneous : Q.11 . of figures in a plane : A.4 : N.64,774. in space : C.94,952,96 : N.79,802. double reciprocal by normals to a sphere : C.56. formulæ J.32. of functions : An.50 ; in an inf. series , A.3. by substitution : Mém.31 . quadric : Q.2 ; quadric differential, J.70. (ay- ba)2+ (bz - cy) ² + (cx — az )2 : CM.1,2 . two variables : Mém.11 . homographic plane : L.61. in geometry: C.71 : J.67 : JP.25. descriptive : G.13 : Z.9. * linear : 582, 1794 : CD.1,6 : CM.3,4 : M.2 : Mo.84. groups of: M.12,16 . methods of : C.92 ; which preserve an invariable relation between derivatives of the same order, C.82 : L.76. which preserve the lines of curva ture : C.92. * modulus of : 1604. orthogonal : 584, 1799. for equations of dynamics : C.67. plane : M.5 ; and in space, G.16. of powers into binomial coefficients : No.75. of product of n factors : An.51 . quadratic, of an elliptic differential : M.7. in rational space : An.73 : LM.3. rectangular LM.14. reciprocal : C.92 : G.17 : N.64,82,83. of rectilineal space coordinates : J.63. by series doubly infinite : An.56. space, for representation of alg. sur faces : M.3. of surfaces : M.21 : N.69 : Q.12. of symbolic functions into isotropic means : C.43. -c of tan -1 + symmet . y, z : c+α CD.9. Tschirnhausen's : An.58 : P.62,65 : Pr.11,14 . ext. to quintics and higher : E.12,4 : J.582. unimodular : 1605 . of variables : G.5 : No.78. Transitive function : of 24 quantities : L.73. doubly, of 5 or 6 variables : C.21,22.

933

Transitive function-(continued) : reduction to intransitive : C.21. *Transversals : 967-74 : A.13,18,27,30, 56 : CD.5 : CM.1 : J.84 : G.22 : Me.t.c68,75 : N.432,48 : TE.t.c24. orthogonal : AJ.3. of plane alg. curves : Z.19. of two points : A.66. parallel : A.13,57. of spherical triangle and quadrangle : A.45. Trees, analytical : AJ.4. Triads of once- paired elements : Q.9 . *Triangle 700 : A.17,19,22,29,33,36,43, cn46,61 : G.21 : J.50 : M.geo17 : Me.q.c62 : N.42,,43. angles of: 677 : 738–45 : A.65 : P.28 : division, A.51,58 : sum, geo960 : C.69,70 difference , Q.8. area : 707, 4036-41 : A.45,57 : CM.2 Mém.13 . bisectors : of angles : 709, 742, 932, 4628 , -30. * of sides : 738, 922 , 951 , 4631 : Mém.pr10,13. * central line : 957,4644. and circle, ths.: A.9,40,47,60 : Q.7,8, 10. and 3 concentric circles : Me.85. of 3 intersecting circles : Q.21 . circle and parabola : Q.15. and conic : N.70. ** of constant species : 977. construction of : 960. * equilateral, sum of sqs. of distances of any point from its vertices : 923 : A.69 : gz1094. formed by joining the feet of bisec tors of a triangle : A.64 . Gauss's equations for a plane tri angle : A.5. * notation : 4629 : E.44. orthocentre : 952, 4634. pedal line of : Me.83. perpendicular bisectors of sides : 713, 4639. * perpendiculars on sides : 952 , 4633 : Mém.13. and point : Q.5. and polars : circle, G.11 ; conic, Q.7. of mid-points of sides , t.cQ.8. quadrisection of : Mém.9. rational : A.51,56. * remarkable points of (see also " In centre,"&c.) : 955-9 : A. four, 47 ; two, 48 ; five, 52 ; 66,67 : E.28, 30,40 : LM. nine, 14 : N.70,73,83 : Z.11,15. of reference in t.c : 4006 . and right line : A.59.

934

INDEX.

Triangle-(continued) : scalenity of: Me.66,68. sides of: bisectors : A.17. division of : A.63 : N.83. containing conjugate poles with respect to four conics : An.69. cubic eq. for in terms of ▲, R, and r: J.20. similar : C.792 ; under 3 or 4 condi tions, C.782. solution of : 718, 859 : A.3,51 : J.44 : TE.10. six-point circle : Q4.5 . symmetrical properties : A.57 : Pr.11 . theorems : A.9,43,45,55,57,60,613,63 , 68 J.68 71 ,pr28 : Q.9, t.c7 and 8, geo1,2, and 4. relating to triangles of same peri meter under four other condi tions : C.84. *Triangular - numbers : 287 : E.30 : J.69 L.ths63. prism : thN.42. pyramid : At.19 : No.pr32 . Tricircular and tetraspheric geometry : An.77. *Trigon. in t.c : 4006. *Trigonometry : 600 : A.12,2,8,11,13,30 : G.13 : extM.35. formulæ A.27,65 : Me.81 , gz1 : N.77 : geoQ.5,10. * formulæ : 627,700,823 : A.33 : N.46,80. functions : P.1796 ; in factorials , A.43. in binomial factors : L.43. in partial fractions and products : Z.13 . functions closely allied to : A.27. tables, cn : A.1,232,25. theorems : A.2,50,51 : Q.pr7. on product of 4 sines : Me.81 . ix on ,16 . b ) ..... : Q.15 a ) (1 + i (1 + in cos²a sin¹ +2 sin a cos a sin cos + sin² a cost = 0 : Trihedron : about a parabola, locus of vertex : N.63. and quadric : N.71 . and tetrahedron : A.57. Trilinear forms : C.92,93. Trilinear relation of plane systems : J.98. *Triplicate-ratio circle : 4754b : E.40,42, 435,44 . Twisted surface : see 66 Scrolls." Twist of a bar : Me.80. * Umbilics : 5777, 5819-23 : J.65 : JP . 18 M.9 : Q.3. * of quadrics : 5603, 5834 : C.96.

Underdeterminants of a symm. deter minant (i.e., successive first minors) : J.91 . Undeterminants , &c.: A.59 . Unicursal curves : A.60 : C.78,94,96 : LM.4. cubics, ths. re inflexion : E.28. quartics : LM.16 : N.84 : Z.28. surfaces, transf. of : M.66. Uniform functions : C.92,947,95,963. with an alg. relation : C.912. of an anal. point xy : Ac.1 : C.94. two points : C.96,97. with 66 coupures ": C.96. with discontinuities : C.94. doubly periodic : C.94. from linear substitutions : M.19,20. with a line of singular points , decom position into factors : C.92. monogenous : Ac.4. in the neighbourhood of a singular point : C.89. of two independent variables : C.94,95. *Units of elasticity, electricity, and heat : p.2. *Vanishing fractions : 1582 : M.15 : Q.1 . Vanishing groups : CD.2,3,6,7,8 . ap. to quantics : CD.2,3,6 . *Variation : 76 : C.17. of arbitrary constants : L.38. calculus of : 3028-91 : A.3 , prs42 : An.52 : C.16,50 : CD.3 : CM.2 : J.13,pr15,41,54,65 ,prs74,82 : JP. 17 : L.41 : M.2,152 : Me.prs72 : Mo.57 : N.83 : Q.pr10 : and de, L.38 and C.49. history : N.51. immediate integrability : 3090. and infinitesimal analysis : C.17,40. relative max. and min.: 3069 : L.42. of multiple integrals : J.15,56,f59 : Mém.38. transformations : J.552. two dependent variables : 3051 . two independent variables : 3175. integral, of functions : C.403. of parameters : th2714, 3243 : N.77. of second order : 3087 : A.4 : J.55 : Q.14 : Z.23. of higher order : 3089 : A.27 : Z.26. * Versiera or Witch of Agnesi : 5335. *Volumes :-of solids : 5871-83 : A. 31,32,36 : J.34 : N.f57,80. approximate : C.95. of frustums : A.33 ; of conicoids, J.44 . of right cylinders and cones in abso lute geometry : A.59. of surface loci of connected points : LM.14.

INDEX.

Volumes-(continued) : and surfaces by curvilinear coor dinates C.16. *Wallis's formula : 2456 : A.39,gz39 : JP.28. Waring's identity : N.49,62 : extMe.85. Wave surface : C.13,47,78, ge082,92 : CD.7,8 : CP.6 : L.46 : Me.66,73 , 76,78,79 N.63,82 : Pr.32 : Q.2,33, 4,5,9,15,17. asymptotes : C.97. and cone : Q.23,26. cubature of : An.61. generation and cn : C.90 . lines of curvature : An.59 : C.97. normals and centres of curvature : geoC.64. umbilics, geo : C.882.

935

Wear of gold coins : E.43. Web surfaces : see " Net surfaces ." Weierstrass's

function Enb” cos а”xπ 0 with a 1 and b < 1 : G.18 : J. 63,90. expansion in powers of the modulus : C.822,85,86 : L.792. *Wilson's theorem : 371 : A.48 : CD.9 : J.8,19,20 : Me.83 : N.43. generalisation : J.31 : Me.64 : Mél.2 : N.45. Wronski's methods : C.92 : L.82,83. formula of 1812 : N.74 : Q.th12. Zetafuchsian functions : Ac.5. Zonal conics of tetrazonal quartics : Q.10.

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