Lessons introductory to the modern higher algebra [5 ed.]

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LESSON S

INTRODUCTORY

MODERN

TO

HIGHER

THE

ALGEBRA

«4

BY

GEORGE REGIUS

PROFESSOR

OF

SALMON, DIVINITY

FIFTH

CHELSEA

IN THE

UNIVERSITY

OF DUBLIN

EDITION

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BRONX,

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UNIVERSITY

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NORTHWEST ee FIFTH THE

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THE

FOURTH

FIFTH,

EDITION EDITION

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A. CAYLEY, ESQ., AND J. J. SYLVESTER, ESQ., I BEG

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Digitized by the Internet Archive in 2023 with funding from Kahle/Austin Foundation

https://archive.org/details/lessonsintroductO000geor

LESSONS

INTRODUCTORY

MODERN

HIGHER

TO

ALGEBRA.

THE

PREFACE

TE pressure of other engagements having prevented me from taking any part in the preparation of this new edition of my Higher Algebra, I have to express my obligations to the good offices of my

friend

Mr.

Cathcart

in revising the work

and superintending its progress through the press. A comparison

of the number

of pages

will show

that he has made several additions to the Contents

of the last edition. These will chiefly be found in ‘Applications to binary quantics,” which are now divided into Lessons XvVII., XVII.

GEO. TRINITY

COLLEGE,

May, 1885,

DUBLIN,

SALMON.

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CONTENTS. LESSON I. DETERMINANTS.—PRELIMINARY

ILLUSTRATIONS AND DEFINITIONS. PAGE

Rule of signs

0

3

:

fj

Sylvester’s umbral notation

°

.

3

LESSON I. REDUCTION AND CALCULATION OF DETERMINANTS, Minors [called by Jacobi partial determinants} Examples of reduction . A

.

.

Product of differences of n qaacileies expressed as a determinant Reduction of bordered Hessians . . . Continuants . A . . .

LESSON MULTIPLICATION

III.

OF DETERMINANTS,

The theorem stated as one of linear transformation

Extension of the theorem 2 5 ° Examples of multiplication of aelemmats : ‘ Product of squares of differences of n quantities . Radius of sphere circumscribing a tetrahedron Relation connecting mutual distances of points on a circle orners Of five points in space ° Sylvester’s proof that equation of secular eusqpalities a all roots real

MINOR

LESSON IV. AND RECIPROCAL DETERMINANTS.

Relations connecting products of determinants . Solution of a system of linear equations Reciprocal systems . . . Minors of reciprocal system epee in terms of ae of the original Minors of a determinant which vanishes Forms for expanding a determinant of the fourth eas ° LESSON SYMMETRICAL

AND

SKEW

VY.

SYMMETRICAL

DETERMINANTS,

Differentials of a determinant with respect to its constituents If a symmetric determinant vanishes, the same bordered is a perfect square . Skew symmetric determinants of odd degree vanish 3 . Of even degree are perfect squares Nature of the square root [see Jacobi, Crelle, Il. 354; TORS. 236] Orthogonal substitutions @ 0 5 Number of terms in a symmetrical Gepainant

x

CONTENTS. LESSON DISCRIMINATING

VI.

SYMMETRICAL

DETERMINANTS. PAGE

New proof that equation of secular inequalities has all its roots real Sylvester’s expressions for Sturm’s functions in terms of the roots. Borcharat’s proof 5 0 C . .

LESSON SYMMETRIC

48 49 64

. ’

VII. FUNCTIONS,

Newton’s formulze for sums of powers of roots Improvement of this process 5 . O Determinant expression for sums of powers Rules for weight and order of asymmetric function . Formula for sum of powers of differences of roots Differential equation of functions of differences 2 Symmetric functions of homogeneous equations Differential equation when binomial coefficients are used Serret’s notation ;

57 57 57 58 60 61 62 64 65

° 5 :

é :

.

2 -

. :

5

4 .

. :

5

LESSON VIII. ORDER AND WEIGHT OF ELIMINANTS Eliminants defined 5 : : ; 6 Elimination by symmetric functions > : Order and weight of resultant of two equations S . Symmetric functions of common values for a system of two equations Extension of principles to any number of equations . :

EXPRESSION

66 67 69 72 78

2 . : °

LESSON IX. OF ELIMINANTS AS DETERMINANTS.

Elimination by process for greatest common measure : . Euler’s method . : : Conditions that two equations should have two common factors 0 Sylvester’s dialytic method 9 2 Bezout’s method . Cayley’s statement of it . Jacobians defined . Jacobian and derived equations patistied by common een aes satisfies equations of same degree 3 : Expression by determinants, in particular cases, of reine of 5S Bocas Cayley’s method of expressing resultants as quotients of determinants

LESSON

76 ud 78 79) 81 83 84 84

85 87

X.

DETERMINATION OF COMMON ROOTS. Expression of roots common to a system of equations by the differentials of the resultant : ‘ Equations connecting these differentials re the resultant rane Expressions by the minors of Bezout’s matrix . : General expression for differentials of resultant with respect to any quantities entering into the equations

.

.

General conditions that a system may hae two common roots

.

.

91 91 93

96 97

CONTENTS. LESSON

Xi

XL

DISCRIMINANTS. : ea Order and weight of discriminants 5 Discriminant expressed in terms of the roots

3

9 0

Discriminant of product of two or more functions Discriminant is of form app + a,2

: :

:

5 5

° . Formation of discriminants by the differential equation . Method of finding the equal roots when the discriminant vanishes Extension to any number of variables “ : Discriminant of a quadratic function ; 6 é

LESSON

5 5

0

F

‘ ;

:

S %

PAGE 99 101 101 102 103 104 106 107

XII.

LINEAR TRANSFORMATIONS. Invariance of discriminants Number of independent invariants . Invariants of systems of quantics Covariants =

:

;

‘3

0

4

;

6 0

: 5 : ° Every invariant of a ae is an Seen of the original . Invariants of emanants are covariants ; . 6 Contravariants 5 4 : Differential symbols are Eon eeedient @ variables 5 “ a& + yn + &e, absolutely unaltered by transformation Mixed concomitants : : . 5 . Evectants . Evectant of Aeeata of a arenas “phe SPREE: Parkes

LESSON FORMATION Method by symmetric functions

é 7

d

108 110

:

2

4 0 :

112 114 = J14 116 117, 120 119 7 120 121 122 123

XIII.

OF INVARIANTS

AND COVARIANTS. :

124

Concomitants which vanish when two or more roots are grail . Method of mutual differentiation of covariants and contravariants . . Differential coefficients substituted for the variables in a contravariant give covariants 7

125 126

For binary quantics, as

5

6

and oouinerraniants nob edsentially distinct

c

Invariants and covariants of second order in coefficients . 4 Cubinvariant of a quartic F : Every quantic of odd degree has an jayariank of the 4th orde Q Cubicovariant of cubic 4 : . F F Method of the differential equation : . ; Weight of an invariant of given order é 6 Binary quantics of odd degree cannot have yaa of oda order . Coefficients of covariants determined by the differential equation 5 Skew invariants é Investigation of number of Adependane invariants by fk differential eration Source of product of two covariants is product of their sources , : Cayley’s definition of covariants . A A Extension to any number of variables . : ; :

127 127 129 129

129 130 130 130 130

131 131 132 134 135 136

Xi

CONTENTS. LESSON XIV. SYMBOLICAL

REPRESENTATION

OF

CONCOMITANTS.

Method of formation by derivative symbols . Order of derivative in coefficients and in the variables Table of invariants of the third order é ° Herrite’s law of reciprocity Derivative symbols for ternary quantics Symbols for evectants a .

6

.

: 3

: .

.

:

. .

Method of Aronhold and Clebsch

c

.

E

LESSON XV. CANONICAL FORMS. Generality of a form examined by its number of constants

150 151 151 152 153 153 154, 155 156 . 157 159 160

Reduction of a quadratic function to a sum of squares . Principle that the number of negative squares is unaffected by real eaneueauen Reduction of cubic to its canonical form &

Discriminant of a cubic and of its Hessian differ only in ae General reduction of quanticof odd degree Methods of forming canonizant Condition that a quantic of order 2n be peducible to a sum of n,2n"* powers Canonical forms for quantics of even order . . ° Canonical forms for sextic and octavic ° . . For ternary and quaternary cubics é

LESSON SYSTEMS

OF

XVI. QUANTICS.

Combinants defined, differential equation satisfied by them Number of double points in an involution Geometrical interpretation of Jacobian Factor common to two quantics is square factor ini acobian Order of condition that u + kv may have cubic factor Nature of discriminant of Jacobian Discriminant of discriminant of u + kv Proof that resultant is a combinant c ‘ ; Discriminant with respect to x, y, of a function of u, v Discriminant of discriminant of u+ kv for ternary quantics Tact-invariant of two curves Tact-invariant of complex curves Osculants Covariants of a wey system oeneried with ie of a ternary

LESSON

161 162 162 162 163 164 165 166 167 169 169 170 171 172

XVII.

APPLICATIONS TO BINARY QUANTIOS. Invariants when said to be distinct Number of independent covariants

Cayley’s method of forming a complete system THE QUADRIC

Resultant of two anaes

175 175 177 178 180

CONTENTS. SysTEM

OF

THREH

OR

MORE

QUADRICS

xiii

.

.

.

.

Extension to quantics in general of theorems concerning quadrics THE CUBIC S : ° : : Geometric meaning of covariant edie : Square of this cubic expressed in terms of the other covariants

"asi

4

182 183

5 0

:

184 SOs ELD, 186 2 A . 3 Solution of cubic. 187, 225 : . 6 SYSTEM OF CUBIC AND QuapRie 189 7 A : B : Geometrical illustrations 189 5 . . . ’ . , THE QUARTIC 190 é ‘ ; 3 : 3 = Catalecticants 190 . . : Discriminant of a quartic 191 Relation of covariants of cubic derived from that of nvarianta of a quarks 5 198 ° : “ . Sextic covariant geometrical meaning of Relations connecting quadratic factors of A 194 Reduction of quartic to its canonical form

SL

F

Relation connecting covariants of quartic Symmetrical solution of quartic .

4 .

.

5 °

. .

195 196

°

Criteria for real and imaginary roots.

0

The quartic can be brought to its canonical form by real eulecibesions

A

°

.

°

.

197

0

.

-

SYSTEM OF TWO CUBICS a 5 . . : Resultant of the system . . Condition that u+ rv may be a perfect cape. 5 : ’ Mode of dealing with equations which contain a superfluous variable

197

9

Conditions that a quartic should have two square factors Cayley’s proof that the system of invariants and covariants is porplets ; Application of Burnside’s method 5 Q . . Covariants of system of quartic and its Hessian . : 7 Hessian of Hessian of any quantic SYSTEM OF QUADRIC AND QUARTIC

194

:

;

198 199 ° 200 201 : 202 202, 267

5 A ; .

204 205 205 207

Jacobian and simplest linear covariants : . . Any two cubics may be regarded as differential coefficients of same anante 5 Invariants of invariants of u+ dv are combinants : 2 Process of obtaining concomitant of system from concomitant of pikes quantio

209 210 211 212

Complete list of covariants of system . Plane geometrical illustration of system of two cubics SYSTEM OF FOUR CUBICS : ° ° Illustration of twisted cubics ‘ /

213 214 215 216

SYSTEM OF QUARTIC AND CUBIO

.

.

: °

. .

° :

é

. .

fe

Fy

SYSTEM OF TWO QUARTICS . . . Their resultant ° ° Condition that u + Av oad be pertect eiaaes . Condition that w + Av should have cubic factor . Special form when both quartics are sums of two fourth Rowaral Three quadrics derived functions of a single quartic . .

.

Three quadrics quadric covariants of two cubics

.

C

218, 226

.

. A

9 ’

°

% b

«

219 220 220 221 223 224

226

LESSON XVIII. APPLICATIONS TO HIGHER BINARY QUANTICS, THE QUINTIC

.

Canonical form of quintic

°

6

.

5

°

5

.

Condition that two quartics be first differentials of the same quintic

7

.

227

» (1

227 228

XIV

CONTENTS.

Discriminant of quintic “ j Fundamental invariants of quintic Conditions for two pairs of equal roots Allinvariants of a quantic vanish if more than half its roots be all equal Hermite’s canonical form C 233, Hermite’s skew invariant . Its geometrical meaning Covariants of quintic for canonical fer Cayley’s arrangement of these forms Cayley’s canonical form , Sign of discriminant of any anants aeesratines whether it hada an odd or even number of pairs of imaginary roots Criteria furnished by Sturm’s theorem fora quintic . If roots all real, canonizant has imaginary factors Invariant expression of criteria for real roots Sylvester’s criteria. ‘ Conditions involving variation within certain alintts of a poreeae Cayley’s modification of Sylvester’s method Hermite’s forme-type 5 ° . The Tschirnhausen transformation ° .

Modified by Hermite and Cayley Applied to quartic . . : Applied to quintic . ; . Sextic resolvent of a quintic A Harley’s and Cockle’s resolvent . ° Expression of invariants in terms of roots THE SEXTIC—its invariants and simplest covariants Conditions for cubic factor or for two square factors The discriminant Simplest quartic covariant . . . Quadric covariants The skew invariant expressed in terme of other javarants

Functions likely to afford criteria for real roots

SYSTEM OF TWO QUARTICS

.

5

Jacobian identified with any sextic by means of aeee Functional determinant of three quartics ; Can be similarly identified 9 c ; New canonical forms of sextic by Brill Also by Stephanos

Factors of discriminant of J Beobian Sextic covariant of third order in coefficients Canonical form referred to ternary system Condition for sextic to be sextic-covariant of quartic to be Hessian of quintic

LESSON ON THE

ORDER

OF RESTRICTED

XIX. SYSTEMS

OF EQUATIONS.

Order and weight of systems defined Restricted systems . Determinant systems, / rows, k +1 columns Order and weight of conditions that two equations baie two common aK

284 285 287 291

CONTENTS.

XV

System of conditions that three equations should have acommon root . Systems of conditions that equation have cubic factor or double square factor Intersection of quantics having common curves . 5 : Case of distinct common curves . : Number of quadrics passing through five pointe and tooenine four apy Rank of curve represented by a system of / rows, & + 1 columns 5 System of conditions that two equations should have three common roots System of quantics having a surface common “ 9 : Having common surface and curve ae 4 A Having common two surfaces 4 5 :

. < ; °

System of conditions that three ternary quantics have two common points

Rule when the constants in systems of equations are connected by relations Number of curve triplets having two common points s ; Mr. S, Roberts’ method A s : é 2

LESSON

"993 294 295 297 298 299 300 301 303 305

306, 309

5

307 310 310

XX.

APPLICATIONS OF SYMBOLICAL METHODS, Symbolical expression for invariant or covariant ‘ . . Clebsch’s proof that every covariant can be so expressed 6 cb Formule of transformation “ : ° . 5 Reduction to standard forms : : ° 5 . Transvection . . A ; ' Symbolical expression for eae of lesyaaye . 5 Forms of any order obtained by transvection from forms of lower ome Gordan and Clebsch’s proof that the number of irreducible covariants is finite Every invariant symbol has (ab)? as a factor, where pis at least half n . Symbolical expression for resultant of quadratic and any equation Investigation of equation of inflexional tangents to cubic ‘ . Application of symbolical forms to theory of double tangents to planecurves . Typical exposition of an even binary quantic . : ° of a quantic of order 3p F ° ° 5

314 315 316 318 320 321 323 324 326 326 330 833, 335 337

NOTES. History of determinants . : Commutants “ On rational functional determinate Hessians Z Symmetric functions

Elimination

Combinants

5

.

5

: 4

:

2

Table of transvectants

.

M. Roberts’ table of sums of powersaot differences Table of resultants c Hirsch and Cayley’s tables of meee functions Index

5

:

‘

EB

.

5

.

. .

.

: : é

A

341 342

842 342 343 343 345

345 345

4

347

.

348 3850

346

.

.

c

338 339 340

A

' A

5

,

;

;

:

:

7

.

.

;

°

c

4

: ‘

:

; :

°

5 2

Applications to binary penccs

4

:

.

.

-

; :

:

,

5

6

:

°

:

,

;

;

.

:

:

,

°

°

5

,

.

5 : :

°

:

Discriminants Berzoutiants < Linear transformations 5 Canonical forms

‘ 5

373

ba

' a

i

.

%

wie7

i

"

:

ad

:

-_

;

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:

j

—

2


. We prefix the following lemma: If in any determinant we denote by D the result of making all the leading terms =0, by D; what the minor corresponding to a; becomes when the leading terms are all made =0, by D,, what the second minor corresponding to a,a,, becomes when the leading terms vanish, &c., then the

SKEW

DETERMINANTS.

41

given determinant, expanded as far as the leading constituents are concerned, is A= D+ 3a,,D; + Bay Dy t + GyAyy+ + Fyny where, in the first sum 7 has any value from 1 to 7, in the second sum 7, & are any binary combinations of these numbers, &c. For, the part of the determinant which contains no leading constituent is evidently D; the terms which contain a,; are aiAiiy

where A;; is the corresponding minor, hence the terms which contain @,; and no other leading constituents are got by making the leading constituents =0 in A,,; and so for the other terms. 42. If this lemma be applied to the case of the skew determinant defined in the last article, all the terms D,, D,,, &c. are skew symmetric determinants; of which, those of odd order vanish, while those of even order are perfect squares. The term 4,,4,,..-d,, 13 \", and the determimant is n

BEN Dre

Be

ee,

where D,, D,, &c. denote skew symmetrical determinants of the second, fourth, &c. orders formed from the original in the manner explained in the last article. Ex. 1.

A,

A

hy

Gg,

A,

A,

G31, Azg,

2X

, where a2; = — Ay, ke. Saas 2 2 2 | is = AF +A (yp? + O43? + Ay,?).

Ex, 2. The similar skew determinant of the fourth order expanded is ABA AZ (yg?

2 yg? + yg? + yg” + Gg4? + Ogg”) + (Ayp%gq+ Aa%qq + 4%)”.

43. Prof. Cayley (Crelle, vol. XXXII, p. 119) has applied the theory of skew determinants to that of orthogonal substitutions, of which we shall here give some account. It is known (see Surfaces, p. 10) that when we transform from one set of three rectangular axes to another, if a, b, c, &c. be the direction-cosines of the new axes, and if X=ant+byt+ce, YV=dur+bVyte2, Z=a'xt yt cz;

that we have X*4V724 Pav +y +2, whence a’? +a?+a”"=1, &., ab+a’'b’+a’b" =0, &e.; that also we have

e=aXt+aVt+a'Z,

y=bX4tUV4UZ,

2=cX4+¢Y4+ CZ,

42

SYMMETRICAL AND SKEW SYMMETRICAL

DETERMINANTS.

and that we have the determinant formed by a,b,c; @,0,¢; &.=+1.

It is also useful (in studying the theory of rotation for example) instead of using nine quantities a, 6, c, &c. connected by six relations, to express all in terms of three independent variables. Now all this may be generalized as follows: If we have a function of any number of variables, it can be transformed by a linear substitution by writing

y=a,X+a,¥

a=a,X+a,Y¥+a,2+ ke,

+4,,2+ &e., Ke,

and the substitution is called orthogonal if we have

et+ytet+&=

X°4+Y*4+ 2774 &.,

which implies the equations a,’ +4,7+&.=1,

a,,0,,+4,,4,, 11°12 21°

22

+ &e.=0, &e.

Thus the n® quantities a,,, &c. are connected by 4n(n +1) relations, and there are only $n(n—1) of them independent. We have then conversely

X=4,c+4,y+a,2+&.,

Y=a,x+a,,y + a,2+ &e., &e.,

equations which are immediately verified by substituting on the right-hand side of the equations for xz, y, z, &c. their values.

And hence, the equation X’+Y*+&e.=2'+7'+ &e. gives us the new system of relations a, +a,°+&e.=1,

a,,4,,+4,,0,,+ &e.=0. 11 21 12° 22

Lastly, forming by the ordinary rule for multiplication of determinants, the square of the determinant formed with the n? quantities a,,, &c., each constituent of the square vanishes except the leading constituents, which are each = 1. The value of the square is therefore =1. Thus the theorems which we know to be true in the case of determinants of the third order are generally true, and it only remains to shew how to express the n? quantities in terms of $n(n—1) independent quantities. This we shall effect by a method employed by M. Hermite for the more general problem of the transformation of a quadric function into itself. See his paper ‘Remarques, &c.,’ Camd. and Dub. Math. Jowr., vol. 1X. (1854), p. 63.

ORTHOGONAL

SUBSTITUTIONS.

43

44, Let us suppose that we have a skew determinant of the (n — 1)™ order, b,,,11? b,,,“129 &c. where b,,=— b,,, and b,,=b,,=b,,=13

and let us

suppose

that we

form

with

these

constituents

the

two different sets of linear substitutions,

e=b Et+bn+b,6+ke, y=b,,€4 bnt+b,,€+&e, 2=b,€+b,+ 6,64 &.,

X=b,6+6,74+0,,6+&e., Y=b,F+6,7+6,,6+ &., Z=b,,€ + b,.7+6,,6+ &e.,

from adding which equations we have, in virtue of the given relations between 2,,, 0,,, &c., e+ X=28, y+ Y=2n, &e. If now the first set of equations be solved for &, », &c. in terms of x,y, &c., we find, by Art. 29, AE=8,

2+ 8, y+ B,2+&e.,,

An=8,,0+4+8,,y + &e.

(where @,,, 8,,, &c. are minors of the determinant in question); and putting for 2, x+ X, &c., these equations give

AX= (28,,—4) a+ 28,,y + 28,2 + &e., AY=28,,0 + (28,,—A)y+28,2+&c., &e.,

which express X, Y, &c. in terms of x, y, &c. But if we had solved from the second set of equations for &, 7, &c. in terms of X and Y, &c. we should have found AE=8, X+B8,,Y+B,2t+&e,

An=8,X+8,V+8,2+ &.,

whence, as before,

Aau=(28,,—A)X+28,,Y + 28,,2+ &e., Ay =28,,X+ (28, — 4) ¥+4 28,4+ &e. Thus, then, if we write

28.

—

Bu

8= 41)

28..—A

poe

a Ajy3

28,

rr = Ay)

28,

ie =

Any

we have a, y, &c. connected with X, Y, &c. by the relations a=a,X+a,Y¥+ ke, y=a,X+a,,¥+ ke, &e,, tke, Y=a,0 tayy + &e., &e. X=a,e +4,y We have then a, y, &c., X, Y, &c. connected by an orthogonal substitution, for if we substitute in the value of x the values

44

AND

SYMMETRICAL

SYMMETRICAL

SKEW

DETERMINANTS.

of X, Y, &c. given by the second set of equations, in order that our results may be consistent, we must have

A, +a, 2 +4, + &.=1,

44,4, + 11° 21

12°

24%22 +

2,0), + Ke. =0, Ke.

1323

Thus then we have seen that taking arbitrarily the 4n (n- 1) quantities, 3,,, 0,,, &c., we are able to express in terms of these the coefficients of a general orthogonal transformation of the n” order. Ex. 1. To form an orthogonal transformation of the second order, {i=

i

SARE

then 6;; = Bez =1, Bi2=A, Bo, =—A,

(L+\2) a=

|=14+2

and our transformation is

(1—d2) X+2X¥,

(1+A2) y=—-2AKX4+(1—A2) ¥,

(1+A2) X= (1-22) w— Dy, (+A?) Y=

2w

+ (L—A?)y.

Ex. 2, To form an orthogonal transformation of the third order. A=

i

—-y BM,

Write

Eh

ye

1, —-A,

A 1

Write

|=1+A?4+ p?4+ 0?

Then the constituents of the reciprocal system are 1+A2,

9 v+Apu,

—wt+rAv

—v+Au,

1+pm%,

A+pv

p+rAvy,

—A+py,

14+?

|,

consequently the coefficients of the orthogonal substitution hence derived are

14 —w—v, 24+), 2 (Av—H), 2(\u—v), L+m?—A—v, 2 (uw +2), 2(\v+u), 2ww—-r, 142-2py, where each term is to be divided by 1 + A? + p? + v?,*

45. It is easy to see that for a symmetrical determinant of the orders 1, 2, 3, 4 the number of distinct terms is =], 2, 5, 17 respectively, and the question thus arises what is the number of distinct terms in a symmetrical determinant of the order n. This number has been calculated as follows by Professor Cayley: * The geometric meaning of these coefficients may be stated as follows: Write A=a tan}0, »=6 tan}0, y=c tan }9, then the new axes may be derived from the old by rotating the system through an angle @ round an axis whose direction-cosines are a, 6, c. The theory of orthogonal substitutions was first investigated by Euler (Nov. Comm. Petrop., vol. XV., p. 75, and vol. XX., p. 217), who gave formule for the

transformation as far as the fourth order. The quantities \, pm, v, in the case of the third order, were introduced by Rodrigues, Liouville, vol. v., p. 405. The general

theory, explained above, connecting linear transformations with skew determinant-. was given by Cayley, Crelle, vol. XXXII., p. 119.

NUMBER

OF TERMS.

Consider a partially symmetrical determinant Art. 11) by the notation

Le

45

represented (see

BP)

where in general fy=g/f, but all the letters p, g, ... are distinct from all the letters p’, q’, ... so that these letters give rise to no equalities of conjugate terms; say, if in the bicolumn there are m rows aa, bb, ... and n rows pp’, qq’, ... this is a determinant (m,n); and in the case n=0, a symmetrical determinant. And let (m,n) be the number of distinct terms in a determinant (m, n). Consider jist a determinant for which 2 is not 0, for instance ag |, = ad,sad, apy aq bb ba, bb, bp’, bq

pp q¢

pa, po, pp’, pf q2, 9, 7P', 9

\4

then ga, gb, gp’, q7 are distinct from each other and from every other constituent of the determinant, and the whole determinant is (disregarding signs) the sum of these each multiplied into a minor determinant; the minors which multiply ga, gd are each of the form (1, 2); those which multiply gp’, qq’ are each of the form (2, 1); and we thus obtain (2, 2) =2¢ (1, 2) + 2 (2, 1),

and so in general p (m, n) = md (m—1, n) +np (m,n—1),

whence in particular

# (m, 1) =mg(m—1, 1) +$(m, 0).

Next, if n=0, let us take for instance the symmetrical determinant aa |, =| aa, ab, ac, ad bb ba, bb, be, bd ce ca, cb, ce, ed

dd

da, db, de, dd

46

SYMMETRICAL

AND

SKEW

SYMMETRICAL

DETERMINANTS.

We have here terms multiplied by dd; ad.da, bd.db, cd.dc; and by the pairs of equal terms ad.db + bd.da, ad.dc + cd.da, bd.de + cd.db, the other factors being in the three cases minors of the forms (3, 0), (2, 0), and (1, 1) respectively ;thus we have (4, 0) =

(3, 0) + 34 (2, 0) + 3¢ (1, 1),

and so in general (m, 0) =f (m —1, 0) + (m— 1) (m—2, 0)

+3 (m—1)(m —2) $(m—3, 1), which last equation combined with the foregoing (m, 1) =m

(m—1, 1) + $ (m, 0)

gives the means of calculating ¢ (m, 0), 6(m, 1); and then the general equation ¢ (m, n)=md¢d (m—1, n)+nb(m, n—- 1) gives the remaining quantities ¢ (m, n). It is easy to derive the equation 26(m, 0)—

(m—1, 0)—(m—1)

(m—2, 0)=

f (m—1,0) (m—1) $ (m—2, 0)

+

+(m—1)(m—2) $ (m—3, 0) =f rsisdieleisiesieislecelstaisineieveisieisiere'eiciere

And hence, using the method assuming

+(m—1)...3.2.1 ¢ (0, 0). of generating functions, and m

u= (0,0) + =$(1, 0) +A $ (2, 0). —— $m, 0) tee.5 we find at once

:

that is

ga

a

dx

crip

l-«a

di

gMa(+e+ i) de u Le)

or, integrating and determining the constant, so that for 2=0 u shall become = 1, we have gras u=

v(1— 2)?

whence ¢(m, 0), the number of terms in a symmetrical determinant of the order m, is

pote = [heen coeticlentsoficsa nee V(1— a)°

NUMBER

OF TERMS.

47

The numerical calculation by this formula is, however, somewhat complicated; and it is practically easier to use the equations of differences directly. We thus obtain not only the values of ¢ (m, 0), but the series of values

(0, 9), (1,0), $(0, 1), i) (2, 0),

d (i, 1);

(0, 2),

¢$ (3, 0), &e., which are found to be 1,

1, 2,

(7,

388,

618,

5,

i,

2, 6,

23,

690,

24,

2, 6,

714,

6,

24,

720,

24,

720,

720,

&e., &e.,

as is easily verified. But as regards ¢(m, 0) we may, writing the equation in w under the slightly different form du 2 (1-2) F — (2-2) w= 0,

by

obtain from it the new equation of differences (m, 0) =m

(m—1, 0)—4 (m—1) (m—2) $ (m—3, 0),

and the process of calculation is then very easy. Starting from the values ¢ (1, 0) =1, ¢ (2, 0) =2, which imply ¢ (0, 0) =1, we have m=1,

esc.

dl

2,

Za 2)

3,

§=3.2—-

1.1,

1j7=4.5—

3.1,

, by

35.17 =" 6.2,

=6, 388=6.73 —10.5, &e,,

&e;

LESSON

VI.

DISCRIMINATING SYMMETRICAL DETERMINANTS. 46. Ir we add the quantity % to each of the leading terms of a symmetrical determinant, and equate the result to 0, we have an equation of considerable importance in analysis.* We have already given one proof (Sylvester’s) that the roots of this equation are all real (Ex. 14, p. 28), and we purpose in this Lesson to give another proof by Borchardt (see Liouville, vol. XII. p. 50), chiefly because the principles involved in this proof are worth knowing for their own sake. First, however, we may remark that a simple proof may be obtained by the application of a principle proved in Art. 37. Take the determinant A,+A,

A,

Aer)

Any < r,

a 1?

A355

D5)

&e.

Aya)

&e.

As, a r, &e.

&e. ’ and form from it a minor, as in Art. 37, by erasing the outside line and column; form from this again another minor by the same rule, and so on. We thus have a series of functions of X, whose degrees regularly diminish from the n” to the 1”; and we may take any positive constant to complete the series. Now, if we substitute successively in this series any two values of X, and count in each case the variations of sign, as in Sturm’s theorem, it is easy to see that the difference in the number of variations cannot exceed the number of roots of the equation of the n” degree which lie between the two assumed values of X. This appears at once from what was proved in Art. 37, * It occurs in the determination of the secular inequalities of the planets (see Laplace, Mécanique Céleste, Part 1., Book u1., Art. 56).

STURM’S

FUNCTIONS.

49

that if \ be taken so as to make any of these minors vanish, the two adjacent functions in the series will have opposite signs. It follows, then, precisely as in the proof of Sturm’s theorem, that if we diminish > regularly from +0 to —o, when » passes through a root of any of these minors, the number of variations in the series will not be affected; and that a change in the number of variations can only take place when 2 passes through a root of the first equation, namely, that in which A enters in the n degree. The total number of variations, therefore, cannot exceed the number of real roots of this equation. But, obviously, in all these functions the sign of the highest power of X is positive; hence, when we substitute + oo, we get no variation; when we substitute — oo, the terms become alternately positive and negative, and we get n variations; the equation we are discussing must, therefore, have » real roots. It is easy to see, in like manner, that the roots of each function of the series are all real, and that the roots of each are interposed as limits between the roots of the function next above it in the series.

47. It will be perceived that in the preceding Article we have substituted, for the functions of Sturm’s theorem, another series of functions possessing the same fundamental property, viz. that when one vanishes, the two adjacent to it have opposite signs. Borchardt’s proof, however, which we now proceed to give, depends on a direct application of Sturm’s

theorem. The first principle which it will be necessary to use is a theorem given by Sylvester (Philosophical Magazine, December, 1839), that the several functions in Sturm’s series, expressed in terms of the roots of the given equation, differ only by positive square multipliers from the following. ‘The first two (namely, the function itself and the first derived function) are,

of course, (a — a) (w—8)(x—y) &e., 3 (w@—f) (w—y) &e.; and the remaining ones are

= (a--B)*(@—y)(w-8) &e. ; &(a—B)"(B —y)"(y—4)*(w— 8) &e., &e., where

we

take the product of any & factors of the given

50

DISCRIMINATING

SYMMETRICAL

DETERMINANTS.

equation, and multiplying by the product of the squares of the differences of all the roots not contained in these factors, form the corresponding symmetric function. We commence by proving this theorem.*

48. In the first place, let U be the function, V its first derived function, R,, 2,, &c., the series of Sturm’s remainders; then it is easy to see that any one of them can be expressed in the form AV-—BU. For, from the fundamental equations U=

Q,V-k,,

V=

Q,2, — f,,

f,= Q,f,-f,,

&e.,

we have f,= QV — U, h,=

QR, -

,

V=

(9,2, -

£,= (9,9, - 1) te

1) V-

OnU;

QV =(9,9,9; — Q,- Q;) V

(001)

U,

and so on. We have then in generalt 2, = AV— BU, where, since all the @’s are of the first degree in a, it is easy to see that A is of the degree k—1, and B of the degree k—2, while &, is of the degree n—k. But this property would suffice to determine Ff, R,, &c., directly. Thus, if in the equation k,= @,V—U, we assume Q, = ax +b, where a and 6 are unknown constants, the condition that the coefficients of the highest two powers of « on the righthand side of the equation must vanish (since &, is only of the degree n- 2) is sufficient to determine a and 6. And so in * I suppose

that Sylvester

must

have

originally

divined

the

form

of these

functions from the characteristic property of Sturm’s functions, viz. that if the equation has two equal roots a=, every one of them must become divisible by a—a. Consequently, if we express any one of these functions as the sum of a number of products (a — a) (« — 8) &c., every product which does not include either x—a or «—f must be divisible by (a —@)?; and it is evident in this way that the theorem

ought to be true.

The method

of verification here employed

does not

differ essentially from Sturm’s proof, Liouville, vol. Vil. p. 356, + The theory of continued fractions, which we are virtually applying here, shews that if we

have

constant and=1.

Ry = AzV — BU,

Rez, = Ati V — Bui,U, then

In fact, since Rz,,

Agr = QrAr— Az whence

AxBuy, —

Ag Bry, — Axi, By is

= QzRz — Rey, we have

Bury = Q2Br — Bry,

Axi, Be= Ag+ Bz — AcBr-,

and by taking the values in the first two equations above, namely, where k= 2 and & = 3, we see that the constant value = 1.

SYLVESTER’S

FORMS.

51

general, if in the function 4V— BU we write for A the most general function of the (4-1) degree containing & constants, and for B the most general function of the (4 —2)" degree containing &—1 constants, we appear to have in all 24—1 constants at our disposal, and have in reality one less, since one of the coefficients may by division be made =1.* We have then just constants enough to be able to make the first 24-2 terms of the equation vanish, or to reduce it from the degree n+4—2 to the degree n—%. The problem, then, to form a function of the degree n—k, and expressible in the form 4V—BU, where A and B are of the degrees k- 1, k—2, is perfectly definite, and admits but of one solution. If, then, we have ascertained that any function #, is expressible in the form AV— BU, where A and B are of the right degree, we can infer that R, must be identical with the corresponding Sturm’s remainder, or at least only differ from it by a constant multiplier. It is in this way that we shali identify with Sturm’s remainders the expressions in terms of the roots, Art. 47. 49. Let us now, to fix the ideas, take any one of these functions, suppose

2 (a — 8)’ (By) (y— 4)? (@- 8) (w—e) &e,, and we shall prove that it is of the form 4V—BU, where A is of the second degree, and B of the first in a Now we can immediately see what we are to assume for the form of A, by making «=a on both sides of the equation. The right-hand side of the equation will then become

A (a—B) (4-4) (4-8) (a—¢) &., since U vanishes; and the left-hand side will become

3 (a — 8)’ (8-4) (y— 4)’ (a- 8) (a—«) &e. It follows, then, that the supposition

=a must reduce A to the

form =(8—y)*(a—8) (2—¥), and it is at once suggested that we ought to take for A the symmetric function

2 (B—¥)' («@—B) (@—9). * Just as the six constants in the most general equation of a conic are only equivalent to five independent constants, and only enable us to make the curve satisfy five conditions.

52

DISCRIMINATING

SYMMETRICAL

DETERMINANTS.

And in like manner, in the general case, we are to take for A the symmetric function of the product of £—1 factors of the original equation multiplied by the product of the squares of the differences of all the roots which enter into these factors. It will not be necessary to our purpose actually to determine the coefficients in B, which we shall therefore leave in its most general form. Let us then write down

= (a — 8)? (B —9)*(y —a)? (aw— 8) &e. = 3 (a — 8)’ (x — a) (2 — B) x 3 (a — 8) (2 — y) &e. + (ax + b) (aa) (w— 8) &e., which we are to prove is an identical equation. Now, since an equation of the p” degree can only have p roots, if such an equation is satisfied by more than:p values of a, it must be an identical equation, or one in which the coefficients of the several powers of a separately vanish. But the equation we have written down is satisfied for each of the x values c=a, x=8, &c.,

no matter what the values of a and b may be. And if we substitute any other two values of x, then, by solving for a and } from the equations so obtained, we can determine @ and 8, so that the equation may be satisfied for these two values. It is, therefore, satisfied for n+ 2 values of a, and since it is only an equation of the (n+1)” degree, it must be an identical equation. And the corresponding equation in general, which is of the degree n+ —1, is satisfied immediately for any of the n values x=a, &c.; while B being of the degree —1 we can determine the & constants which occur in its general expression, so that the equation may be satisfied for & other values; the equation is, therefore, an identical equation.

50. We have now proved that the functions written in Art. 48 being of the form AV—BU are either identical with Sturm’s remainders, or only differ from them by constant factors. It remains to find out the value of these factors, which is an essential matter, since it is on the signs of the functions that everything turns. Calling Sturm’s remainders, as_ before, h,, R,, &c., let Sylvester’s forms (Art. 47) be 7,, 7,, &e., then we have proved that the latter are of the form T,=,R,, T,=,,, &c., and we want to determine 2,, A,, &c. We can

STURM’S

THEOREM.

53

at once determine 2X, by comparing the coefficients of the highest powers of « on both sides of the identity 7,=.A,V—B,U; for x" does not occur in 7,, while in V the coefficient of 2"” is n,

and the coefficient of a is also n in A,, which = 3 (2—«a); hence B,=n'*. But the equation 7,=A,V—B,U must be identical with the equation 2,= Q@,V—U multiplied by 2,; we have, therefore, », =n’. To determine in general X, it is to be observed that since any equation 7,=A,V-—B,U is 2, times the corresponding equation for R,, and since in the latter case it was proved (note, Art. 48) that A,B,,,— A,,,B,=1, the corresponding +1 RAL quantity for 7), 7',,, must =A,d,,,. Now from the equations L,= A,V— BU,

T,,+1 =A

V = Baa U,

we have Al,

— Af,

+]

=(A,B£,,, |

k+y

B,) U= Ae, U.

Comparing the coefficients of the highest powers of 2 on both sides of the equation, and observing that the highest power does not occur in A,7,,,, we have the product of the leading coefficients of A,,, and T,=2,A,,,. But if we write we

2(a-8)’=p,, U(a—B) (a-y)' (8-y) =p, &e., have, on inspection of the values in Arts. 47, 49, the

leading coefficient in Z,=p,, in 7T,=p,, &c., and in A,=n, in A,=p,, in A,=p,, &c. Hence p 2

n’

2

Da =NyNgy Dg=Ayhgy Py =p)45) KC, whence A,=* 4, A= at? &e. n

2

The important matter, then, is that these coefficients are all positive squares, and therefore, as in using Sturm’s theorem we are only concerned with the signs of the functions, we may omit them altogether.

51. When we want to know the total number of imaginary roots of an equation, it is well known that we are only concerned with the coefficients of the highest powers of a in Sturm’s functions, there being as many pairs of imaginary roots as there are variations in the signs of these leading terms. And since the signs of the leading terms of 7,, 7, &c. are the same as those of R,, 2,, &c., it follows that an equation has as

SYMMETRICAL

54

DETERMINANTS.

many pairs of imaginary roots as there are variations in the series of signs of 1, , =(a—8), = (a—8)(8—¥)* (y—a)’, &e. This theorem may be stated in a different form by means of Ex. 3, Art. 26, and we learn that an equation has as many pairs of imaginary roots as there are variations in the signs of the series of determinants jae

ee Sepia

Se

a

$1) Ss,

51

545

NIRS 8,

Sy

Cone,

$2)

We,

53)

5,

Pala, CPM pe $59 549 S59 5%

the last in the series being the discriminant; and the condition that the roots of an equation should be all real is simply that every one of these determinants should be positive.

52. We return now, from this digression on Sturm’s theorem, to Borchardt’s proof, of which we commenced to give an account, Art. 47; and it is evident that in order to apply the test just obtained, to prove the reality of the roots of the equation got by expanding the determinant of Art. 46, it will be first necessary to form the sums of the powers of the roots of that equation. For the sake of brevity, we confine our proof to the determinant of the third order, it being understood that precisely the same process applies in general; and, for convenience, we change the sign of A, which will not affect the question as to the reality of its values. Then it appears immediately, on expanding the determinant, that s,=a,, + a,, + Gy, since the determinant is of the form

*— 2? (a,, + a,, + a,,) + &e.

And in the general case s, is equal to the sum of the leading constituents. We can calculate s, as follows: The determinant may be supposed to have been derived by eliminating a, y, z between the equations Ac=4,,e+4,,y+

a3)

AY =a,,0+

Aap YT?)

rz =4,,2+

Ane i A, 2.

Multiply each of these equations by 2X, and substitute on the right-hand side for Aw, Ay, Xz their values, thus we get Vax

(a, P+Gyg

+0,5

)0+(,,4,,+

Doo

+O.)

Y + (a, 173 + Gy pAgyt A445)

ae 2 2 2 » YH (44%, Fy@ 9+ 44,5) a+(a,, +, +4,, )y+ (4,05, +O

oo Nz = (45,44, +050 ,o+ As, )0+ (45,4, +

+ Aya) 2y

2 2 2 9% 9+ Das o5)Y + (Ag, +4,, +4,, ey

BORCHARDT’S

INVESTIGATION.

55

from these eliminating a, y, z, we have a determinant of form exactly similar to that which we are discussing, and which may be written 2 b,, —»r )

Dios

bay

bs ra mM,

b

13

Tet

31?

32?

b

eG

Then, of course in like manner, 8, =, b, = boo +5 bs -

a, Aa Asp + Any. = 2a," cs 2a,, WP2a,,".

The same process applies in general and enables us from s, to compute s,,,. Thus suppose we have got the system of equations Ma=d,,x+

d.y+ d, 2

My=d,,a+d,.y+ d, z 23°)

Ma=d,,xt+d,,y+ A oy

from which we could deduce, as above, s,=d,,+d,,+d,,; then multiplying both sides by 2, and substituting for Ax, &e. their values, we get

We = (d,,4,, + Ttye + Fiq0,,) & + (d,,0,, + Typ + F055) Y 117 11 12°12 13° 13 2i hve 8 12° 22 13 23 a5 (2,4, cigdsp at d, 055) #)

wry = (4,2, a Myo ig d,.4,3) x + CACA a yay BS 01,5) y = (4,05, + CR ES a d,.A55) As

whe = (45,4, zs pA

a d,,05) x + (4,,%, a yyy teAy yg) Y ct (4:45, ft sayy ru 5505) @,

whence

bhew i d, 4, a yg Aay i spy + 2d,,4,, ae 2d),,Arg ie 2d:

53. We shall now shew, by the help of s,, &c., that each of the determinants at the can be expressed as the sum of a number is therefore essentially positive.* Thus write constituents

1) 1, ep OrOpAO Gy

oy

A559

ogy

Seah then it is easy to see that

5

Tay

these values for end of Art. 51 of squares, and down the set of

REO NNO, « , &c. are proportional to 2", z"*x, &e. The coefficient of » is therefore proportional to the result of substituting the singular roots in V, and therefore vanishes. Now, in the case we are considering, the supposition of a=0, b=0, c=0 must make the discriminant vanish, since then all the differentials vanish for the singular roots x=0, y=0. Any other quantic V will vanish for the same values, provided only A=0. The general form of the discriminant then must be such that if we substitute for 6, 64+; for c, c+AC, &e., and then make a, 6, c=0, the result must be divisible by 7; or, in other words, if we put for b, XB; for c, ~C, &c., and then make a=0, the result is divisible by ’, which was the thing to be proved. 118. Concerning discriminants in general, it only remains to notice that the discriminant of a quadratic function in any number of variables is immediately expressed as a symmetrical determinant. And, conversely, from any symmetrical determinant, we may form a quadratic function which shall have

that determinant for its discriminant.

The simplest notation

for the coefficients of a quadratic function is to use a double

suffix, writing the coefficients of x’, y°, &e., G,,, Ay) Ag, KC, and those of xy, xz, &c., a,,,4,,3 @,, and a,, being identical in this notation. The discriminant is then obviously the sym-

metrical determinant By

Uoy

Gs)

&e.

Foy

Ay

Ua

&e.

Asy4

Boos

A559

&e.

&e.

( 108)

LESSON LINEAR

XII.

TRANSFORMATIONS.

119. Invariants. The discriminant of a binary quantic being a function of the differences of the roots is evidently unaltered when all the roots are increased or diminished by the same quantity. Now the substitution of a+ for is a particular case of the general linear transformation, where, in a homogeneous function, we substitute for each variable a linear function of the variables; as for example, in the case of a binary quantic where we substitute for 2, Ax+y, and for y, Nz+p’y. It will illustrate the nature of the enquiries in which we shall presently engage if we examine the effect of this substitution on the discriminant of the binary quadratic, ax®+ 2bxy+cy*.

When

the variables

are transformed, it be-

comes

a (Aw + py) +2b(Ax+ wy) (Moet p’y)te(Not p’y)’5 and if we call the transformed equation a’x7+ 20’xy + c’y*, we have a’ =an'+20rn +007, co Sap? + Abu’ + cy”, BW =aretb (Ap +p) + rp’. It can now be verified without difficulty that a’c’ —b? = (ac — B*) (Ap — N's)’ that is to say, the discriminant of the transformed quadratic is equal to the discriminant of the given quadratic multiplied by the square of the determinant Ay’ — ‘yu, which is called the modulus of transformation.

120. Now, a corresponding theorem is true for the discriminant of any binary quantic. We can see & prior? that this must be the case, for if a given quantic has a square factor, it will have a square factor still when it is transformed; so that whenever the discriminant of the given quantic vanishes,

INVARIANCE.

109

that of the transformed must necessarily vanish too. The one must therefore contain the other as a factor. The theorem however can be formally proved as follows: Let the original quantic be (xy,—yx,) (xy, —yx,) &e., then (Art. 109) the diseriminant is (a,y, — y,x,)” (x,y, — y,%,)° &e. Now the linear factor (xy,—yz,) of the given quantic becomes by transformation y,(AX+pY¥)—a2,(’X+y'Y), and if we write this in the form YX—-—X,Y, we shall have Y,=)y,-Na,, X,=—-py,tw’c, If then the transformed quantic be written as the product of the linear factors (YX - X,Y) (Y,X- X,Y) &e., we have expressions, as above, Long Nay Ais ek re. in terms. of, 75 9. v, &c. We can then, without difficulty, verify that (DGG a x, Y;) = (Ap x rp) (4,2, - L,Y)

It follows immediately that (Y X,-Y_X,)’(Y|X,—Y,X,) &e. is equal to (y,7,—2,y,)° (y,@,—¥,v,) &e. multiplied by a power of Aw’ —d’w equal to the number of factors in the expression

for the discriminant in terms of the roots. A corresponding theorem is true for the discriminant of a quantic in any number of variables. What I have called Modern Algebra may be said to have taken its origin from a paper in the Cambridge Mathematical Journal for Nov. 1841, where Dr. Boole established the principles just stated and made some important applications of them. Subsequently Prof. Cayley proposed to himself the problem to determine & priort what functions of the coefficients of a given equation possess this property of invariance; that when the equation is linearly transformed, the same function of the new coefficients is equal to the given function multiplied by a quantity independent of the coefficients. The result of his investigations was to discover that this property of invariance is not peculiar to discriminants and to bring to light other important functions (some of them involving the variables as well as the coefficients) whose relations to the given equation are unaffected by linear transformation. In explaining this theory, even where, for brevity, we write only three variables, the reader is to understand that the

110

LINEAR

TRANSFORMATIONS.

processes are all applicable in exactly the same number of variables.

way to any

121. We suppose then that the variables in any homogeneous quantic in & variables are transformed by the substitution e=rX+u,Y+v,7+ &e., y=AX+ w,V+v,7+ &e., 2=A,X+4,Y+,7+4+ &e., &e., and we denote by A the modulus of transformation ; namely, the determinant, whose constituents are the coefficients of transformation, 2, (,) Vj, KC.) Ay May Vp» Ke., Ke. Now it is evidently not possible in general so to choose the coefficients X,, #,, &c., that a certain given function ax"+ &c.

shall assume, by transformation, another given form aX" + &c, In fact, if we make the substitution in az"4+&c., and then equate coefficients, we obtain, as in Art. 119, a series of equa tions a’ =an,"+ &c., the number of which will be equal to the number of terms in the general function of the n” degree in % variables. And to satisfy these equations we have only at our disposal the 4? constants 2,, A,, &c., a number which will in general be less than the number of equations to be satisfied.* It follows then that when a function az” +&c. is capable of being transformed into a’ X"+ &c., there will be relations con-

necting the coefficients a, b, &c., a’, b’, Ke. In fact, we have only to eliminate the &’ constants from any &’+1 of the equations a =ad,"+ &c., and we obtain a series of relations connecting a, a’, &c., which will be equivalent to as many independent relations as the excess over A* of the number of equations. Thus, in the case of a binary quantic, the number * The number

of terms in the general equation of the n degree homogeneous

in & variables is ae

aT eae

and it is easy to see that the only

cases where this number is not greater than ? are, first, when n = 2, when it becomes 3k (& + 1), a number necessarily less than #*, & being an integer; and secondly, the case k = 2, n= 3, when both numbers have the same value 4. That is to say, the only cases where a given function can be made by transformation to assume any

assigned form are, first, the case of a quadratic function in any number of yariables ; and secondly, the case of a cubic function homogeneous in two variables,

ABSOLUTE

INVARIANTS.

111

of terms in a homogeneous function of the n degree is x +1. If then, in any quantic az” + &c., we substitute for x, A,X +m, Y, and for y, »,.X+,Y, and if we then equate coefficients with

wX"+ &c., we have n+1 equations connecting a, a’, r,, &e., from which, if we eliminate the four quantities A,, A,» My) May We get a system equivalent to n —3 independent relations between a, b, a, b, &e. It will appear in the sequel that these relations

can be thrown into the form ¢ (a, d, &c.) =¢ (a’, UW, &e.);_ or, in other words, that there are functions of the coefficients a, b, &c. which are equal to the same functions of the transformed coefficients. The process indicated in this article is not that which we shall actually employ in order te find such functions, but it gives an & priort explanation of the existence of such functions, and it shows what number of such functions, independent of each other, we may expect to find. 122. Any an invariant, same function multiplied by that is to say,

function of the coefficients of a quantic is called if, when the quantic is linearly transformed, the of the new coefficients is equal to the old function some power of the modulus of transformation; when we have Did, 0 CMe.) — A

pila, d, ¢, Ke).

Such a function is said to be an absolute invariant when p=0; that is to say, when the function is absolutely unaltered by transformation even though A be not =1. IEf a quantic have two ordinary invariants, it is easy to deduce from them an absolute invariant. For if it have an invariant ¢, which when transformed becomes multiplied by A’, and another yy, which when transformed becomes multiplied by A’, then evidently the q” power of ¢ divided by the p” power of y will be a function which will be absolutely unchanged by transformation. It follows, from what has been just said, that a binary quadratic or cubic can have no invariant but the discriminant, which we saw (Art. 120) is an invariant. For if there were a second, we could from the two deduce a relation ¢ (a, b, &e.) = >(a’, W’, &c.). But we see from Art. 121 that there can be no relation connecting a, b, &c. with a’, b’, &e.,

112

LINEAR

TRANSFORMATIONS.

since, with the help of the four constants 2,, &c. at our disposal, we can transform a given quadratic or cubic, so that the coefficients of the transformed equation may have any values we please. In the same manner we see that a quantic of the second order in any number of variables can have no invariant but the discriminant. On the other hand, suppose we take the binary quartic aax*+ 4ba*y + 6cx°y’ + 4day’ + ey", and that the coefficients become by linear transformation a, UW’, &e, it will be found that we have two invariant functions both distinct from the discriminant; in fact, we have the two equations

a’é —4b'd’ + 3c? = A* (ae — 4bd + 3c”),

a'de+20'c'd’— ad” — ¢b"— c* = AY (ace + 2bcd — ad? — eb? —c’), and from these two we deduce the absolute invariant

(a’ce’ + 20'c'd’ — a’d”— eb"— 0")? _ (ace + 2bed — ad? — eb’ — c*)? (a’e’ — 40'd’ + 307) — (ae — 4bd + 3c’*)* ‘ In this case the invariance of the discriminant may be deduced as a consequence of the preceding equations, for the discriminant is

(ae — 4bd + 3c”)? — 27 (ace + 2bcd — ad? — eb? — c’)’, and consequently the discriminant of the transformed equation is equal to that of the original multiplied by A’, 123. In the same manner as we have invariants of a single quantic we may have invariants of a system of quantics. Let there be any number of simultaneous equations ax” + &c. =0, a’x"+ &c.=0, &e., and if when the variables in all are transformed by the same substitution, these become 4X"+ &c. =0, A’X" + &e.=0, &c., then any function of the coefficients is an invariant if the same function of the new coefficients is equal to the old function multiplied by a power of the modulus of transformation; that is to say, if

¢ (A, B, &e., A’, B’,&e., A” &e.) = A’d(a,b, &e.,a’,0’, &e., a”, &c.), The simplest example of such invariants is the case of a system of linear equations. The determinant of such a system

INVARIANTS

OF A SYSTEM,

113

is an invariant of the system. This is evident at once from the definition of an invariant and from the form in which the fundamental theorem for the multiplication of determinants has been stated in Art. 23. If we are given an invariant of a single quantic, we can derive from it a series of invariants of systems of quantics of the same degree. In order to make the spirit of the method more clear, we illustrate it in the first instance by a simple example. We have seen (Art. 119) that ac—J’ is an invariant of the quadratic ax’+2bxey+cy’, and we shall now thence derive an invariant of a system of two quadratics. Suppose that by a linear transformation aa’+ 2bey+cy’ becomes AX’?+2BXY+CY”", and a’x*+2d’ay+cy’ becomes A’X*+2B’XY+C’Y"; then evidently, by the same transformation (& being any constant),

(a+ ka’) a? will become (4+4A’)

4+2(b+kb’) zy +(c+ke’)y’

X°4+2(B+ kB’) XV+(C+kC’) Y*.

Forming then the invariant (Art. 119)

of the last quadratic, we have

(A+kA’)(C+kC’) —(B+hB’)’= A"{(a4+ ka’) (c+he’)—(b+kb’)*}. But since / is arbitrary, the coefficients of the respective powers of & must be equal on both sides of the equation; and therefore we have not only, as we knew before,

(AC— B’)=A' (ac— 0’), (A’C’~ B”) =A’ (ae - 0”), but also

AC’+CA’—2BB’ = & (ac + ca’ — 20’),

an equation which may also be directly verified by the values of A, B, &c. given Art. 119. We see then that ac’ + ca’ — 2bb’ is an invariant. By exactly the same method, if we have any invariant of a quantic ax" + &c., and if we want to form invariants of the system ax” + &c., a’x" + &e., we have only to substitute in the given invariant for each coefficient a,a+ka’, for b,b+kb’, &e., and the coefficient of each power of & in the result will be an invariant. Writing down, by Taylor’s theorem, the result of

114

LINEAR

TRANSFORMATIONS.

substituting a+ ka’ for a, &c., the theorem to which we have been led may be stated thus: If we have any invariant of a quantic ax"+&c, and if we perform on it the operation eg +0 £4 be. we get an invariant of the system of two Qa

quantics aa” + &c., a’a"+&c. We may repeat the same operation and thus get another invariant of the system, or we may d operate with a” a +05, &e., and thus get an invariant of a system of three quantics, and so on. This latter process gives us the invariants which we should find by substituting for a, a+ka’+ la’, &c., and taking the coefficients of the products of every power of & and /.. In the same manner we get invariants of a system of any number of quantics. 124. Covariants. A covariant is a function involving not only the coefficients of a quantic, but also the variables, and such that when the quantic is linearly transformed, the same function of the new variables and coefficients shall be equal to the old function multiplied by some power of the modulus of transformation ; that is to say, if ax" + &c. when transformed becomes AX” + &c., a function d will be a covariant* if it is such that

¢ (A, B, &e., X, Y, &e.) = A’ (a, b, &e., x, y, &e.). Every invariant of a covariant is an invariant of the original quantic. This follows at once from the definitions. Let the quantic be az”+ &c., and the covariant a’x”+&c. which are supposed to become by transformation AX*+ &c., A’X” + &e. Now an invariant of the covariant is a function of its coefficients

such that f(A’, B’, &c.) = Ad (a’, B’, &e.). * In the geometry of curves and surfaces, all transformations of coordinates are effected by linear substitution. An invariant of a ternary or quaternary quantic is a function of the coefficients, whose vanishing expresses some property of the curve or surface independent of the axes to which it is referred, as, for instance, that the curve or surface should have a double point. A covariant will denote another curve or surface, the locus of a point whose relation to the given curve is independent of

the choice of axes.

and covariants,

Hence the geometrical importance of the theory of invariants

EMANANTS

ARE

COVARIANTS.

115

But A’, B’, &c. by definition can only differ by a power of the modulus from being the same functions of A, B, &c. that a’, b’, &c. are of a, b, &c. Hence when the functions are both expressed in terms of the coefficients of the original quantic and its transformed, we have v (A, B, &c.) = At (a, b, &e.),

or the function is an invariant. Similarly, a covariant of a covariant is a covariant of the original quantic. 125. We shall in this and the next article establish ciples which lead to an important series of covariants.

prin-

If in any quantic u we substitute «+ kx’ for x, y+ ky’ for y, &c., where a’y’z’ are cogredient to ayz, then the coefficients of the several powers of &, which are all of the form ?

(2= +7 = + &e.) u, have been called the first, second, third, &c. emanants* of the quantic. Now each of these emanants ts a covariant of the quantic. We evidently get the same result whether in any quantic we write «+a for a, &c., and then transform a, x’, &c. by linear substitutions, or whether we make the substitutions first and then write X+4X’ for X, &e. For plainly AX+6,V+y,Z74+ kh (AX + w, VY’ +y,Z’)

=), (X+hX')+ my,(V+ kV’) +, (Z+k2Z’). If then wu becomes by transformation U, we have proved that the result of writing x+kx’ for «x, &c. in u, must be the same as the result of writing X+4X’

for X, &c. in U, and since &

is indeterminate, the coefficients of & must be equal on both sides of the equation; or Jaa,

au

wv —t+y Fas

Se

me

ee

ee

ax tk qyt &c.,. ce,

Q.E.D.

126. If we regard any emanant as a function of x’, y’, he. treating x, y, &c. as constants, then any of its invariants will be * In geometry emanants denote the polar curves or surfaces of a point with regard to a curve or surface,

116

LINEAR

TRANSFORMATIONS.

a covariant of the original quantic when x, y, cc. are considered as vartables. ; » au ae We have just seen that a” de t &c. becomes X ix? t &e. when we substitute for a, >,X’+y,¥Y’+&c.,, and for z, rAA+y,Y+&c. It is evidently a matter of indifference whether the substitutions for 2’, &c., and for a, &c., are simultaneous or successive. If then on transforming x’, &c. p

alone, at

8 + be,becomes aX” + &c., then a, &c. will be

such functions of x, &c. as when a, y, &c. are transformed will P

become = , &e.

Now

an invariant of the given emanant

considered as a function of a’, y’, &c. only, is by definition such a function of its coefficients as differs only by a power cf the modulus from the corresponding function of the transformed coefficients a, b, &c. But since, as we have seen, a, &c. become d’U &c. when x, &c. are transformed, it follows that the given ax?’ d’u invariant will be a function of de? — &c., which when x, &c. are

transformed will differ only by a power of the modulus from the corresponding function of —

&e.

It is therefore by

definition a covariant of the quantic. Thus then, for example, since we have proved (Art. 119) that if the binary quantic ax’ +2bay+cy*? becomes by transformation AX’+2BXY+CY?, then

AC- B*=A* (ac- 8"); 2 it follows now, by considering the second emanant (2g +y' 5.)u of a quantic of any degree, that ee d°U Pp aCXPerd Vas

Gre haan du d*u we aay) an 3 haya (ex)

a theorem of which other demonstrations will be given.

127. In general, if we take the second emanant of a quantic in any number of variables, and form its discriminant, this will

CONTRAVARIANTS,

ee

be a covariant which is called the Hesstan of the quantic. It was noticed (Art. 118) that the discriminant of every quadratic function may be written as a determinant. Thus then if, as we have done elsewhere, we use the suffixes 1, 2, &c. to denote differentiation with respect to 2, y, &c., so that, for d*u example, u,, shall denote i? then the quadratic emanant is

u,,a” + 2u,,a'y’ + &e., and its discriminant, which is the Hessian, is the determinant Uy)

Us09

U5)

&e,

Uy

Uo)

Ussy

&e.

Usiy

Usey

Usa

&e.

&e.

128. We have seen (Art. 123) that the determinant of a system of linear equations is an invariant of the system. If then, given a system u, v, w, &c. of as many functions as variables, we take the first emanants

xu, + yu,t+ 2u, + &e., &e.,

their determinant

Uy Uy Uy KC. Vy Wy

Vey Woy

U55 Ws)

&e. &e.

| &e.

is a covariant of the system. This is the determinant already called the Jacobian (Art. 88). The Hessian is the Jacobian of the system of differentials of a single quantic w,, u,, u,, ke. 129. Contravariants. When a set of variables x, y, &c. are linearly transformed, it constantly happens that other variables connected with them are also linearly transformed, but by a substitution different from that which is applied to a, y, &c. If the equations connecting x, y, 2 with the new variables be written as before e=rnAX+u,Yt+y,Z,

y=rXt+u,V+,Z,

2=rAX+y,¥+y,Z,

then variables &, 7, € are said to be transformed by the cnverse

substitution, if the new variables, expressed in terms of the

118

LINEAR

TRANSFORMATIONS.

old, are

HAE+AN+AS,

H=wEte

nto, Z=vF+vnt v,6,

where if in the first substitution the coefficients are the constituents of the determinant (A,u,v,) read horizontally, in the second they are the same constituents read vertically; and where if in the first substitution the old variables are expressed in terms of the new, in the second the new are expressed in terms of the old. Stated thus, it is evident that the relation between the two substitutions is reciprocal. Solving for &, 7, € in terms of 2, H, Z, we get (Art. 29)

AEF=LE+MH+NZ,

An=LE+MUH+

N,Z,

A¢=L,2+ MU,H+ N,Z, where L,, I, &c. are the minors obtained by striking out from the matrix of the determinant (A,y,v,) (the modulus of transformation) the line and column containing X,, u,, &e. Sets of variables x, y, 2; &, 1, ¢ supposed to be transformed according to the different rules here explained, are said to be contragredient to each other. In what follows, variables supposed to be contragredient to x, y, 2 are denoted by Greek letters, the letters a, 8, y being usually employed in subsequent lessons. We proceed to explain two of the most important

cases in which the inverse substitution is employed. 130, When a function of a, y, 2, &c., is transformed by linear substitutions to a function of X, Y, Z, &c., then the differential coefficients, with respect to the new variables, are linear functions of those with respect to the old, but are expressed in terms of them by the cnverse substitution. We have dS td edawed Vapi) de

dX ~ dx dX * dy aX* de dX

+ &e.

But from the expressions for a, y, &c. in terms of X, Y, &e., we have de dy dz

aXe Wy axa

Hence then

eax s

d d d d bice a +A,2 dy +A,* =Te + ee

INVERSE

Similarly

a

TRANSFORMATIONS

119

B, es+ ML, 5 + 4, z + &e., &e.

Thus then, according to the definition given in the last article, the operating symbols Zz = S &c. are contragredient to x, y, 2, &e., that is to say, when the latter are linearly transformed, the former will be linearly transformed also, but according to the different rule explained in the last article. If, as before, u,, u,, &c. denote the differential coefficients of wu, and U,, U,, &c. those of the transformed function U, we have just proved that U,=Ayu, +AU AA,

UV, = ou, + MU, + UsMy, Ke.

Consequently, if u,, w,, vu, all vanish, U,, U,, U, must all vanish likewise. Now we know that w,, u,, u, all vanish together only when the discriminant of the system vanishes; if then the discriminant of the original system vanishes, we see now that the discriminant of the transformed system must vanish likewise, and therefore that the latter contains the former as a factor, as has been already stated (Art. 120). 131. In plain geometry, if x, y, 2 be the trilinear coordinates of any point, and z&+ yn+2f=0 be the equation of any line, &, 7, € may be called the tangential coordinates of that line (see Conics, Art. 70). Now, if the equation be transformed to any new system of axes by the substitution x=A,X + &c., the new equation of the line becomes E(A,X+pm,¥+v,Z)+(A,X+ uw,V¥t+y,Z)+F(A,X+m,V¥+v,7) =0,

so that if the new equation EX+HY+ZZ=0, we have

R=VE+AN +26,

of the right line be written

H=wEtuntys,

Z=vE+v0t v6

In other words, when the coordinates of a point are transformed by a linear substitution,

the tangential coordinates of a line are

transformed by the inverse substitution; that is, they are contragredient to the coordinates of the point. In like manner,

in the geometry of three dimensions, the tangential coordinates of any plane are contragredient to the coordinates of any

120

LINEAR

TRANSFORMATIONS.

point. When we transform to new axes, all coordinates xyz, a'y'z'w', &e. expressing different points, are cogredient: that is to say, all must be transformed by the same substitution w=r,X+ke., x =r,X’ + &e., &e. But the tangential coordinates of every plane will be transformed by the inverse substitution, as we have just explained. Similarly the ray coordinates of different lines for the same system of reference are cogredient, but the axial coordinates are transformed by the inverse substitution, that is, are contragredient to the former. See Surfaces, Art. 57e. The principle just stated will be frequently made use of in the form cE +ynt+26=XE+YH+

ZZ,

where 2, y,.2 being supposed to be changed by the substitution, e=rA,X+p,Y+ &e., &, , § are supposed to be changed by the inverse substitution = =A,E+A,7+A,¢, &c. In other words, in the case supposed, «& + yn + 2 is a function absolutely unaltered by transformation, and analogous statements easily follow in the other cases mentioned. 132. If a function az*+&c. becomes by transformation AX"+&c., then any function involving the coefficients and those variables which are supposed to be transformed by the inverse substitution is said to be a contravariant if it is such that it differs only by a power of the modulus from the corresponding function of the transformed coefficients and variables: that is to say, if f(A, B, &e., Z, H, &e.) = A’¢p (a, b, &e., &, n, &e.). Such functions constantly present themselves in geometry. If we have an equation expressing the condition that a line or plane should have to a given curve or surface a relation independent of the axes to which it is referred (as, for example, the condition that the line or plane should touch the curve or surface), then, when we transform to new axes, it is obviously indifferent whether we transform the given relation by substituting for the old coefficients their values in terms of the new, or whether we derive the condition by the original

rule from the transformed equation.

In this way it is seen

CONCOMITANTS.

121

that the conditions in question are of such a kind that ¢(a, d, &, &c.) differs only by a factor from ¢ (A, B, 3, &c.). 133. Besides covariants and contravariants there are also functions involving both sets of variables, which differ only by a power of the modulus from the corresponding trans formed functions: 7.e. such that (A,B, &e.,X, Y,&c., 2, H, &c.) = Ad (a,b, &e., x,y, &e., E,n, &e.). Dr. Sylvester uses the name concomitant as a general word to include all functions whose relations to the quantic are unaltered by linear transformation, and he calls the functions now under consideration mixed concomitants. I do not choose to introduce a name on my own responsibility ; otherwise I should be inclined to call them divariants. The simplest function of the kind is z& + yn + 2€, which we have seen (Art. 131) is transformed to a similar function, and is therefore a concomitant of every quantic whatever. 134. If we are given any invariant J of the quantic

a,x” + naa"ry+ nbc"2 + 4n (n— 1) a,"*y’+ &e., we can deduce from it a contravariant by the method used in Art. 123. If a,x" + &c. becomes by transformation A,X" + &c., then, since z& + &c. becomes XZ + &c., it follows that

au" + &e. + k(xE + yn + 26)"= A,X" + &e. + k(XE+YH+ ZZ)". Now an invariant of the original quantic fulfils the condition $(A,, 4,, B,, &e.) = A"d (a,) a, b,, &e.).

Forming then the same invariant of the new quantic, it will be seen that

f(A, +k", A, + kB""H, &e.) = A’p (a, + hE", a, + KE"ny, &c.). Since & is arbitrary we may equate the coefficients of like powers of & on both sides of this equation. But, by Taylor’s theorem, these coefficients are all of the form d a + ve n= om, (E ‘Phat n-1 17, n

d

n-2

o a + Se): L ja

122

LINEAR

TRANSFORMATIONS.

We have proved then that they differ only by a power of the modulus from the corresponding function of the transformed equation. They are, therefore, contravariants, since it is assumed all along that &, 7, ¢ are to be transformed by the inverse sub-

stitution. Dr. Sylvester has called contravariants formed by this rule, first, second, &c. evectants of the given invariant. Thus & BE cs is the first evectant. It is to be obda, da, served that in the original quantic the coefficients are supposed to be written with, and in the evectant without, binomial coefficients. Comparing this article with Art. 123 we see that the fay emisé : : : function £& ae &c. may be considered either as a contravariant 0

of the single given quantic, or as an invariant of the system obtained by combining with the given quantic the linear function v&+yn+26. The theory of contravariants, therefore, may be included under that of invariants.

If we perform the operation & fe+&c. upon any covariant 0

we obtain a mixed concomitant, for it is proved in the same way that the result, which will evidently be a function involving variables of both kinds, will be transformed into a function of similar form. Ex. 1. We know that ac—6? is an invariantof aw2bay+cy?; hence cE?—2b£n+an? is a contravariant of the same system. Ex. 2. Similarly, abe + 2fgh — af? — bg? — ch, being the discriminant, and therefore an invariant of ax + by? + cz? + 2fyz + 2gzu + 2hey,

(be — f?) E+ (ca — 9?) n° + (ab — h?) + 2(gh — af) nv + 2 (hf— bg) CE+2 (fg —ch)En is a contravariant of the same quantic. Geometrically, as is well known, the function equated to zero expresses the tangential equation of the conic represented by the given quantic. Ex. 3. Given a system of two ternary quadrics ax? + &c., a’x? + &c., then since a’ (bc —f?) + &, is an invariant of the system (Art. 123), we find on operating with ge £ + &e., that (be’ + b'c — 2Ff") &? + (ca’ + c'a — 299") n? + (ab! + a'b — 2hh’)

+ 2(gh'+ gh — af!— af) nb +2 (hf+ hif— bg’— 09) E+ 2 (fo +f’ —ch’ — eh) En is a contravariant of the system.

é

ain

y CE

We might have equally found this contravariant

-

by operating with a ae &e, on the contravariant of the last example.

Geometrically,

the function equated to zero expresses the condition that a line should be cut harmonically by two conics.

EVECTANTS.

123

135. When the discriminant of a quantic vanishes, it has a set of singular roots 2’y’z’ [geometrically the coordinates of the double point on the curve or surface represented by the quantic]; and in this case the first evectant will be the perfect n” power of (2E+y'n +2). Since we have seen that this evectant is a function unaltered by transformation, it is sufficient to see what it becomes in any particular case. Now if the discriminant vanishes, the quantic can be so transformed that the new coefficients of a", 2**y, a"*z shall vanish; that is to say, so that the singular root shall be y=0, z=0 [geome trically, so that the point yz shall be the double point]. But it was proved (Art. 117) that the form of the discriminant is Ap i ap a3 a,b

5 bX:

Evidently then, not only will this vanish when a,, a,, , vanish, but also its differentials with respect to every coefficient except a, will vanish. This evectant then reduces itself to ae multiplied by the perfect n* power

£", which

is what

i)

(a’E+y'n+2'€)* becomes when 7’ and 2’=0, ande’=1. Thus then, if the discriminant of a ternary quadric vanish, the quadric represents two lines: the contravariant (be — f”) E+ (ca—g’) 1° + &e.

becomes a perfect square; and if we identify it with (a’E+y’n+2’¢)’, we get «y’z the coordinates of the intersection of the pair of lines. If a quantic have two sets of singular roots, all the first differentials of the discriminant vanish, and its second evectant becomes the perfect n™ power of

(WE + yn +25) ("E+ y"n+2°6), where 2’y'2’, a”’yz” are the two sets of singular roots; and 80 on.

(aio aee)

LESSON FORMATION

XIII.

OF INVARIANTS

AND

COVARIANTS.

136. Havine now shewn what is meant by invariants, &c., we go on to explain the methods by which such functions can be formed. ‘Three of these methods will be explained in this Lesson, and a fourth in the next Lesson. Symmetric functions. The following method is only applicable to binary quantics. Any symmetric function of the differences of the roots is an invariant, provided that each root enters into the expression the same number of times.* It is evident that an invariant must be a function of the differences of the roots, since it is to be unaltered when for x we substitute 7+. Now the most general linear transformation is evidently equivalent to an alteration

of each root a into

change the difference between

a ae

Nat pe

By this

any two roots a— 8 becomes

(Apw’ — dp) (a — B) . In order then, that any function of the (Na+ pw’) (B+ pw’) differences may, when transformed, differ only by a factor from its former value, it is necessary that the denominator should be the same for every term; and therefore the function must be a product of differences, in which each root occurs the same number of times. Thus for a biquadratic, = (a—£)?(y—8)* is an invariant, because, when we transform, all the terms of which the sum is made up have the same denominator. But * If in the equation the highest power of a is written with a coefficient a,, we have to divide by that coefficient in order to obtain the expression for the sum, &c.

of the roots; and all symmetric functions of the roots are fractions containing powers of a in the denominator. When we say that a symmetric function of the roots is an invariant, we understand that it has been made integral by multiplying it by such a power of a, as will clear it of fractions; or, what comes to the same thing, if we form the symmetric function on the supposition that the coefficient of 2 is 1, that we make it homogeneous be necessary.

by multiplying each term by whatever power of a, may

FROM

THE

ROOTS

OF BINARY

QUANTICS.

125

=(a— 8)’ is not an invariant, the denominator for the term (a—8)* being (Na+y’)? (8+ yu’), and for the term (y—68)*

being (Ny + mw’)? (N84 pw’). 137. Or perhaps the same thing may be more simply stated by writing the equation in the homogeneous form. We saw (Art. 120) that if we change x into Ax+ py, y into Va+p'y, the quantity a,y,-—2,y, becomes (du - du) (x,y,—2,y,), and consequently any function of the determinants x,y,-2,y, &e. is an invariant. Now (Art. 61) any function of the roots expressed in the ordinary way is changed to the homogeneous form by writing for a, 8, &c. - ; me &c., and then multiplying 1

2

by such a power of the product of all the y’s as will clear it of fractions. If any function of the differences in which all the roots do not equally occur be treated in this way, powers of the y’s will remain after multiplication, and the function will not be an invariant. Thus, for a biquadratic, =(a—()? becomes Ly,’y,"(x,y,—x,y,); but the function = (a—£)*(y—8)’, in which all the roots occur, becomes 3 (x,y, — X,y,)” (%Y,— ©4Ys)°9 and this being a function of the determinants only, ¢s an invariant. It is proved in like manner, that any symmetric function formed of differences of roots and differences between a2 and one or more of the roots is a covariant, provided that each root enters the same number of times into the expression. Thus for a cubic 3 (a — 8)’ (a— +)’ is a covariant. 138. We can, by the method just explained, form invariants or covariants which shall vanish on the hypothesis of any system of equalities between the roots. Thus, let it be required to form an invariant which shall vanish when any three roots are all equal, it is evident that every term must contain some one of the three differences a— 8, 8—y, y—a@} and in like manner for every other set of three that can be formed out of the roots. Thus, in a biquadratic, there are four sets of three roots: the difference a— 8 belongs to two of these sets, and y—6 to the other two; therefore = (a—()*(y—-)* is an invariant which * L (a — B) (y — 6) would vanish identically.

126

FORMATION

OF INVARIANTS

AND

COVARIANTS.

will vanish if any set of three roots are all equal. In like manner, for a quintic there are ten sets of three: a—f belongs to three sets, y—6 to three other sets; the remaining sets are aye, ade, Bye, Bde, two of which contain y—e and the other

two

5—e.

The function then = (a— 8)‘ (y— 8)? (8-8)? (y—e)? is

an invariant which will vanish if any set of three roots are all equal. This invariant (Arts. 57, 58) is of the fourth order and its weight is 10. So, again, if we wish to form a covariant of a biquadratic which shall vanish when two distinct pairs of roots are equal, the expression must contain a difference from each of the pairs a-B, y-8; a-—y, B-6; a-—5, 8—y. Such an expression

would be

or

= (a— 8) (B-y)* (y—4)* (@— 8)%, = (a—8) (2—y) (a— 8) (@- B)* (w@— 9)’(@- 8)",

which are covariants of the fourth and sixth degrees respectively in the variables; and of the fourth and third in the coefficients, and every term of each vanishes when two distinct pairs of roots are equal. 139. Mutual differentiation of covariants and contravariants. When we say that ¢ (a, 0, &, n, &e.) is a contravariant, &, 7, &e. may be any quantities which are supposed to be transformed by the reciprocal substitution. Now we have shewn (Art. 130) Ch KES that the differential symbols , &e. are so transformed,

EDF

We may, therefore, in any contravariant substitute these differential symbols for &, 7, &c., and we shall obtain an operating

symbol unaltered by transformation, and which, therefore, if applied either to the quantic itself or to any of its covariants, will give a covariant if any of the variables remain after differentiation; and if not, an invariant. Similarly, if applied to a mixed concomitant, it will give either a contravariant or a new mixed concomitant, according as the variables are or are not removed by differentiation. Or, again, in any contravariant instead of obtaining an operating symbol by substituting for ad a dU aU

E,n, &c., ae

&e., we may substitute aay

&c. where

IN ANY

NUMBER

OF VARIABLES.

127

U is either the quantic itself or any of its covariants, and so obtain a new covariant. The relation between the sets of variables a, y, 2, &e., &, n, & &e. being reciprocal, we may, in like manner, substitute in any covariant, for x,y,z, &c., a 5,: oe&e., when we get an operative symbol which when applied to any contravariant will give either a new contravariant or an invariant. Thus then, if we are given any covariant and contravariant, by substituting in one of them differential symbols and operating on the other, we obtain a new contravariant or covariant ;which again may be combined with one of the two given at first, so as to generate another; and so on. 140. In the case of a binary quantic, this method may be stated more simply. The formule for direct transformation being

w=AX+H,Y,

y=rX+4,Y,

those for the reciprocal transformation are (Art. 129)

EF=AF+A, whence A€=y,5—-d,H, which may be written

H=mE+,% An=- 4,2 +,H,

An=,H+ #,(— 8); A(—£)=),H+ , (—4). Thus we see that, with the exception of the constant factor A, » and — € are transformed by exactly the same rules as x and y; and it may be said that y and — a are contragredient to x and y. Thus then, in binary quantics, covariants and contravariants are not essentially distinct, and we have only in any covariant to write 7 and —€& for # and y, when we have a contravariant, or vice versd. In fact, suppose that by transformation any homogeneous function whatever ¢ (a, y) becomes @©(X, Y), the formule just given shew that ¢(y, — &) will become A’ (H, — 3), where p is the degree of the function in z and y. If then $(a, y) is a covariant, that is to say, a function which becomes by transformation one differing only by a power of A from a function of like form in X and Y, evidently

¢ (7, —&) will by transformation become one differing only by

128

FORMATION

OF

INVARIANTS

AND

COVARIANTS.

a power of A from one of like form in Z and H; that is to say, For example, the! contravariant, it will be a contravariant. noticed (Art. 134, Ex. 1), c&*—2b&n +n’, by the substitution just mentioned, becomes the original quantic. Instead then of saying that the differential symbols are contragredient to and y, we may say that they are cogredient to y and —a; and if either in the quantic itself or any of its

covariants we write 5 - 2 for x and y, we get a differential symbol which may be used to generate new covariants in the manner explained in the last article. Or we may substitute du He

du oe for a and y, and so get a new

j covariant.

The

following examples will sufficiently illustrate this method: Ex, 1. To find an invariant of a quadratic,

or of a system of two quadratics,

Suppose that by transformation az? + 2bzy + cy* becomes AX? + 2BXY+CY¥?,

then

since we have seen that A 5 —A a are transformed by the same rules as x and y, it follows that the operative symbol d? ad d? a? A? (aaye Si dady +ce—, Jz2) becomes by transformation (4ee ay?

a? ad pe 2B aray taxi) .

If then we operate on é. given quadratic itself, we get

442 (ac — 6?) = 4 (AC— B?), which shows that ac — 3? is an invariant; or if we operate on a’x? + 26’xy + c’y? and the transformed function, we get 242 (ac’ + ca’ — 26d’) = 2 (AC’ + CA’ — 2BB’), which shews that ae’ + ca’ — 260’ is an invariant. We might also infer that a (bx + cy)? — 2b (bx + cy) (ax + by) + c (ax + by)? is a covariant; but this is only the quantic itself multiplied by ae — 82, Ex. 2. Every binary quantic of even degree has an invariant of the second order in the coefficients.

We have only to substitute, as just explained, a ,- 2 for x and y, and

operate on the quantic itself.

ae — 4bd + 3c? is an invariant;

Thus for the quartic (a, d, c, d, eXa, y)* we find that

or for the general quantic (a), a,...dn—, an La, y)",

we find that an — ”,4n-, + 42 (m — 1) a54n — &e. is an invariant ;where the coefficients are those of the binomial, but the middle term is divided by two.

If we apply this method to a quantic of odd degree ; as, for example, if we operate ; ; & a 3 on the cubic ax? + 3ba?y + 3cay? + dy?, with d ae 3c dtdy + 3b ea —a a pelt will be found that the result vanishes identically. We thus find, however, that a system of two cubics has the invariant (ad! — a’d) — 3 (bc’ — b’c). Or, in general, that a system of two quantics of odd degree, ayx + de , b,x" + &c., has the invariant (aobn — Andy) — 0 (ay0n-1 — An-yb,) + bn (n — 1) (aybn-2 — Gn-2b2) &ec.,

which vanishes when the two quantics are identical.

BY

MUTUAL

DIFFERENTIATION.

129

141. When, by the method just explained, we have found an invariant of a quantic of any degree, we have immediately, by the method of Art. 126, a covariant of any quantic of higher degree. Thus, knowing that ac— 2? is an invariant of a quadratic, by forming that invariant of the quadratic emanant of du d*u d*u 2 any quantic, we learn that ) is a covariant

at ay 7 dey

of any quantic above the second degree. In like manner, from the invariant of a quartic ae— 46d + 3c’, we infer that for every quantic above the fourth degree

d‘u d*u

du

de dy

d*u

datdy dudy

d‘u

* (acta)

\?

is a covariant, &c. In this way we see that a quantic in general has a series of covariants, of the second order in the coefficients, and of the orders 2 (n—2), 2(n—4), 2(n—6), &e. in the variables. These covariants may be combined with the original quantic and with each other, so as to lead to new covariants or invariants. Ex. 1, A quartic has an invariant of the third order in the coefficients. that its Hessian

We know

(ax? + 2baxy + cy?) (cx? + 2Wday + ey?) — (bu? + 2cxy + dy?)*,

or (ac — b?) x4 + 2 (ad — be) xy + (ae + 2bd — 3c?) xy?+2 (be — cd) zy? + (ce — d?) x, is a covariant. two times

Operate on this with (a, b, ¢, d, es p= £ is and we get seventy-

zs ace + 2bed — ad? — eb? — c,

which is therefore an invariant, Ex. 2. Every quantic of odd degree has an invariant of the fourth order in the coefficients.

n-1

The quantic has a quadratic covariant —

n-)

aet — &c, of the second

order in the coefficients ;and the discriminant of this quadratic will be an invariant of the original quantic (Art. 124), and will be of the fourth order in its coefficients. In fact, it is proved in this way that every quantic has an invariant of the fourth order; for if we take any of the covariants of this article, which are all of even degree, its invariant of the second order will be of the fourth order in the coefficients of the original quantic. But when the quantic is of even degree, it may happen that the invariant so found is only the square of its invariant of the second order. Ex. 3, To form the invariant of the fourth order for a cubic.

Its Hessian is

or

(ax + by) (cw + dy) — (ba + cy)?;

(ac — 7) a + (ad — bc) xy + (bd — 0?) y?,

130

FORMATION

OF INVARIANTS

AND COVARIANTS,

Hence (ad — be)? — 4 (ac — b?) (bd — &) is an invariant of the cubic. In fact, it is its discriminant a’d? — Gabed + 4ac® + 4b%d — 3b%c?,

142, From any invariant of a binary quantic we can generate a covariant. For from it we can form (Art. 134) the dI evectant contravariant £” —— + &c.; and then in this substi-

as

tuting y, —x for — and y, we have a covariant. For example, from the discriminant of a cubic which has been just written we form the evectant E* (ad* — 3bed + 2c*) + 3E*n (— acd + 20°d — bc’) + 3&n? (— abd + 2ac? — b’c) + (a°d— 3abc + 20°), whence we infer that the cubic has the cubic covariant

(a°d—3abe+20*, abd-2ac’+b'c, —acd+2b°d-be*, 3bed—ad*—2c"¥ x,y)”.

143. The differential equation—We saw (Art. 62) that invariants satisfy certain partial differential equations, and these furnish a third method of forming these functions based on the following principle. Jf n be the order of a binary quantic, 0 the order in the coefficients of any of its invariants, then the weight (see Art. 56) of every term in the invariant is constant and = $n. For if we alter x into Az, leaving y unchanged, since this is a linear transformation, the invariant must, by definition, remain unaltered, except that it may be multiplied by a power of A, which is in this case the modulus of transformation. It is proved then, precisely as in Art. 57, that the wezght, or sum of the suffixes, in every term is constant. Again, the invariant must remain unaltered, if we change x into y, and y into a, a linear transformation, the modulus of which is —1. The effect of this substitution is the same as if for each coefficient a, we substitute a,. Hence the sum of a number of suffixes a+B+y+&, =(n—a)+(n—8)+ (n—y) + &e., whence 2(a+B+y+&c.)=n6.

Q.E.D.

Cor. n and @ cannot both be odd, since their product is an even number; or, a binary quantic of odd degree cannot have an invariant of odd order.

BY

THE

PARTIAL

DIFFERENTIAL

EQUATION.

131

144, The principle just established enables us to write down immediately the literal part of any invariant whose order is given. For the order being given, the weight is given also. Thus, if it were required to form for a quartic an invariant of the third order in the coefficients, the weight must be 6, and the terms of the invariant must be Aa,a,a, + Ba,g,4, + Ca,a,a, + Da,a,a, + Ea,a,0,,

where the coefficients A, B, &c. remain to be determined. The reader will observe that there are as many terms in this invariant as the ways in which the number 6 can be expressed as the sum of three numbers from 0 to 4 inclusive; and generally that there may be as many terms in any invariant as the ways in which its weight 4n@ can be expressed as the sum of @ numbers from 0 to n inclusive. We determine the coefficients from the consideration that since an invariant is to be unaltered by the substitution either of x+ for x, or of y+X for y, evidently, as in Art. 62, every invariant must satisfy the two differential equations

0, 5,+24, tat3a ep+ &.=0, na Fut

1)a, e + &.=0,

it being supposed that the original equation has been written with binomial coefficients, In practice only one of these equations need be used; for the second is derived from the first by changing each coefficient a, into a,_,.

It is sufficient then to

use one of the equations, provided we take care that the function we form is symmetrical with regard to 2 and y; that is to say, does not change (or at most changes sign)* when we change a, into a,,. And this condition will always be fulfilled if we take care that the weight of the invariant is that which has been just assigned. Thus then, in the example chosen for an illustration, if we operate on Aa,a,a,+ &c. with

* When we change z into y and y into g, this is a transformation whose modulus is 0, 1 |or—1. Any invariant, therefore, which when transformed becomes multiplied |1, 0 | by an odd power of the modulus of transformation will change sign whey we interchange # and y, Such invariants are called skew invariants.

32

OF INVARIANTS

FORMATION

AND

COVARIANTS.

a, é + &., we get di 1 (2B+424) a,a,a,+ (D+6C+ 44) a,a,, + (2D + 4B) a,a,a, + (6£+ 3D) a,a,a, =, whence if we take A=1, the other coefficients are found to be B=-1, D=2, C=—1, H=—1, and the invariant is a,4,0, + 2a,a,0, — 4,4,4, — 4,4,%, — 4,4,4,. 145. In seeking to determine an invariant of given order by the method just explained, we have a certain number of unknown coefficients A, B, C, &c. to determine, and we do so by the help of a certain number.of conditions formed by means of the differential equation. Now, evidently, if the number of these conditions were greater than the number of unknown coefficients, the formation of the invariant would in general

be impossible; if they were equal we could form one invariant; if the number

of conditions were

less, we

could form more

than one invariant of the given order. We have just seen that the number of terms in the invariant, which is one more than the number of unknown coefficients, is equal to the number of ways in which its weight 4n@ can be written, as the sum of 6 numbers, none being greater than n. But the effect of

the operation a

+ &e. is evidently to diminish the weight |

by one, the number of conditions to be fulfilled is, therefore, equal to the number of ways in which 4n6—1 can be expressed as the sum of 9 numbers, none exceeding m. Thus, in the example of Art. 144, the number of conditions used to determine A, B, &. was equal to the number of ways in which 5 can be expressed as the sum of three numbers from 0 to 4 inclusive. To find then generally whether an invariant of a

binary quantic of the order 6 can be formed, and whether there can be more than one, we must compare the number of ways in which the numbers $n, 4n@-1 can be expressed as the sum of @ numbers from 0 to n inclusive.* * It was in this way Prof. Cayley first attempted to investigate the number of invariants and covariants of a binary quantic.

BY

THE

PARTIAL

DIFFERENTIAL

EQUATION.

133

146. Similar reasoning applies to covariants. A covariant, like the original quantic, must remain unaltered, when we change x into px, and at the same time every coefficient a, into pa, If then the coefficient of any power of x, x in the covariant be a dgcy, &e., it is obvious, as before, that #+a+8+4 &c. must be constant for every term; and we may call this number the weight of the covariant. Again, in order that the covariant may not change when we alter x into y and y into «, we must have

e+tatB+y+ &. =(p—p) + (n—a) + (n—f)+&e., where p is the degree of the covariant in « and y; whence if @ be the order of the covariant in the coefficients, we have immediately its weight =4(n@+~p). Thus if it were required to form a quadratic covariant to a cubic, of the second order in the coefficients, n=3, 6=p=2, and the weight is 4. We have then for the terms multiplying 2, a+8=2, and these terms must be a,a, and a,a,. In like manner the terms multiplying zy must be a,a,, a,a,, and those multiplying y* must be a,a,, a,¢,. In this manner we can determine the literal part of a covariant of any order. The coefficients are determined as follows: 147. From the definition of a covariant it follows that we must get the same result whether in it we change « into x+ Ay, or whether we make the same change in the original quantic and then form the covariant. But this change in the original quantic is equivalent (Art. 62) to changing a, into a,+a,A, a, into a,+2a,.+a,,

&e.

Hence, in the covariant also the

change of x to «+ Ay must be equivalent to changing a, into a,+a,r, &c. Let the covariant then be Ax? + pA aly +4p (p—-1) Aa” *y’ + &e. Let us express that these two alterations are equivalent, and let us confine our attention to the terms multiplying A. Then if, as in Art. 64, we use as an abbreviation to denote the

operation a,dai Laie £ + &c., the symbol es we get

dA, _, dA, dA, dA, aA, ade =0, dé =A, dt 2=2A-Ke., de =(p-1)A,_,, dé =pA,_,3

134

FORMATION

In like manner,

OF INVARIANTS

writing

AND

COVARTANTS.

d 5 for na, it

d @— 1)a, Ta +

0

we have

1

dA, _ 4 @A,

: ‘A Thus we see that when A, is determined so as to satisfy = =03 in other words, when A, is a function of the differences of the roots of the quantic (Art. 58), all the other terms of the covariant are known. The covariant is in fact

dd, ty 4 Ay Py? soe

a8 hE

PA, hy! + &e.

12 + ay 1.2.3.

It will be observed that the weight of the covariant being 4 (n@+p) the weight of the term A, is 4(n@->p), since the weight of A, together with p makes up the weight of the covariant. This term A,, whence all the other terms are derived, was named by Prof. M. Roberts the source of the covariant. He observed also that the source of the product of two covariants is the product of their sources. For if we multiply the covariant last written by Bx

dB

2

dt at ay

7-2,

ae d B, x

dy*®

2

J

1.2

+ &e.,

we get, as may be easily seen, Batt?

4

ABs

d(A.B) gpiety 4LIAB) oy)

gt

ay

TEL,

+ &e.

Hence, if we know any relation connecting any functions of the differences A,, B,, C,, &c., the same relation will connect the covariants derived from these functions. Ex, 1. To find the quadratic covariant of a cubic. A, is of the form

a,a,+ Ba,a,.

Operate

We have seen (Art. 146) that

on this with he + 2a, Be and we ay

ay

get (2+ 2B) aa,=0, whence B=—1 and A,=a,a,—a,a,. Operate then with d 2 d d 3a, daa ada. + a; aa = and we have 2A, = a,a)—a,a,. Operate with the same on A,, and we have A; =4,a,—a,a,, (4% — 4)

The covariant, therefore, is

%? + (agas — 4,0,) ry + (aa; — 2,02) 9,

LAW

OF COEFFICIENTS

OF BINARY

COVARIANTS.

135

Ex, 2. To find a cubic covariant of a cubic of the third order in the coefficients. Here n = 3, 6=3, } (n6+p)=6, The sum then of the suffixes of the coefficient of 2? will be 3; and this coefficient must be of the form Aa,aya) + Ba,a,a) + Ca,a,a,.

Operate with a, & + 2a, & + 8a,

, and we get 3

(34 + B) aa a) + (2B + 8C) a,a,a, whence if we take

A =1, we have B = — 3, C= 2, or Ag = agpdy — 30,0,% + 20,0,0.

Operate on this three times successively with 3a, = + 2a, ie+ a3 dae =— and we have the remaining coefficients, and the covariant is (see Art. 142)

(309% — 30,0,0 + 2a,4,4,) 23+ 3 (a30,@ — 2a,G44) + a,0,4,) xy

+3 (205044, — q49% — Ugg) xy” + (3A g%y0, — 242,02 — 405%) °.

148. We have seen that a quantic has as many covariants of the degree p in the variables and of the order 6 in the coefficients as functions A,, ie weight is 4(n@—p) can be

found to satisfy the equation oe =0.

And, as in Art. 145, we

see that this number is equal to the difference of the ways in which the numbers 3 (n@— p) and 4 (n@ —p) —1 can be expressed as the sum of @ numbers from 0 to inclusive. It may be remarked that p cannot be odd unless both m and @ are odd. Hence only quantics of odd degree can have covariants of odd degree in the variables, and these must also be of odd order in the coefficients. 149.

The results arrived at (Art. 147) may be stated a little

differently.

The operation y o performed on any quantic is

equivalent to a certain operation performed by differentiating with respect to the coefficients. Thus, for the quantic (@,) @,) %..- Kx, y)", we get the same result whether we operate

on it with y = or with a ‘z + 2a i + &c.

This latter opera-

tion then may be written Ez| ; and the property already proved for a covariant may be written that we have for it y= - ly=, |=(.

In other words, that we

get the same

136

FORMATION

OF INVARIANTS

AND

COVARIANTS.

: : d : result whether we operate on the covariant with Y Fm with d : : : a, de + 2a, ie+&c. In his Memoirs on Quantics, Prof. Cayley has started with this property as his definition of a covariant; a definition which includes invariants also, since for them we have

2 = 0, and therefore also ly=| = 0.

150. It can be proved, in like manner, that covariants of quantics in any number of variables satisfy differential equations , : d d d d which may be written y ee =|y =| eo ae E=| aUKG. Thus, for the quantic (a, 5, c, f, 9, Aa, y, 2), we have

cis st be egies ait leaned Mi Y de 2 hd apagh Neldaca dgadaiee and every covariant must satisfy these two equations. every invariant must satisfy the two equations

a vf CRI Sa

While

OG ee h a toe pi =0;

dg'

df

7 de

as may easily be proved from the consideration that the invariant remains unaltered if we substitute for 2, x+Ay or x+ pz.

(ez ay

LESSON SYMBOLICAL

XIV.

REPRESENTATION

OF

CONCOMITANTS,

151. Ir remains to explain a fourth method of finding invariants and covariants, given by Prof. Cayley in 1846 (Cambridge and Dublin Mathematical Journal, vol. 1. p. 104, and Crelle, vol. xxx.); which not only enables us to arrive at such functions, but also affords the basis of a regular calculus by means of which they may be compared and identified. Let x, ¥,3 %,, y, be any two cogredient sets of variables; : : : Che Mai None) then, if we write briefly for FN ES ae E, my &y &e., 1

1

2

it has been proved (Arts. 130, 120, 139) that &, ,, &, 7, are transformed by the reciprocal substitution; that &y,—£&, is an invariant symbol of operation; and that if we operate with any power of this symbol on any function of x,, ¥,, x,, Y,, We obtain a covariant of that function. We shall use for &.,- &.7, the abbreviation 12. Suppose now that we are given any two binary quantics U, V, we can at once form covariants of this system of two quantics. For we have only to write the variables in U with the suffix (1), those in V with the suffix (2), and then operate on the product UV with any power of the symbol 12; the result must be an invariant or covariant. ‘Thus if we operate wy oo : : UdV dud : simply with 12 we obtain the Jacobian Leica fii ae which

we saw (Art. 128) was a covariant of the system of quantics. Again, let

U= ax) + 2bx,y,+cy,; Vaan,’ + 0'a,y, + cy,’, then if we operate on UV with length, is oe ele Bhi

a a 1,

12°, which, written at full

oa Deny

Me,

—260’, which is thus proved to be an the result is ac’+ ca’ invariant of this system of quantics. In general, it is obvious

138

SYMBOLICAL

REPRESENTATION

OF CONCOMITANTS

that the differentials marked with the suffix (1) only apply to U, and those with the suffix (2) only to V; and it is unnecessary to retain the suffixes after differentiation ;* so that 12° applied to two quantics of any degree gives the covariant

BUC ViedeU edWaren day aaa dx* dy? ay” ax dxdy dady’ Similarly the symbol 12° applied to two cubics gives the invariant (ad’ — da’) — 3 (be — eb’),

or to any two quantics gives the covariant

a Ura V4 au, ay. UTTAR ‘dx dy ~ datdy dudy? te dady dady

RCH SICIAGe dy> da*’

and so in like manner for the other powers of 12.

152. We can by this method obtain also invariants or covariants of a single function U. It is, in fact, only necessary to suppose in the last Article the quantics U and V to be identical. Thus, for instance, in the example of the two quadratics given in the last Article, if we make a=a’, b=)’, c=¢, the invariant 12? becomes 2 (ac —2"). And, in like manner, the expression there given for the covariant 12° of a system U, V, by making U= J, gives the covariant of a single quantic

a0 aU

datas

7d

UN.

(aoa)

In general, whenever we want by this method to form the covariants of a single function, we resort to this process :—We first form a covariant of a system of distinct quantics, and then suppose the quantics to be made identical. And in what follows, when we use such symbols as 12” &c. without adding any subject of operation, we mean to express derivatives of a * If W be any function containing a, y;; x2, ya; we get the same result whether

we linearly transform

these variables, and afterwards omit all the suffixes in the

transformed equation ; or whether we omit the suffixes first, and afterwards transform zx and y. This results immediately from the fact that 2, y,; a, yo; x, y are

cogredient. It follows then at once that if W, written as a function of a, y,; Hos pe be a covariant of U, V; that is to say, if the expression of the coefficients of W in

terms of the coefficients of U and V be unaffected by transformation, then W is also a covariant when the suffixes are all omitted,

FOR

ANY

NUMBER

OF FUNCTIONS.

139

single function U. We take for the subject operated on the product of two or more quantics U,, U,, &c., where the variables Ly Y,; Ly y,3 &c. are written in each respectively, instead of x and y; and we suppose that after differentiation all the suffixes are omitted, and that the variables, if any remain, are all made equal to x and y, 153. From the omission of the suffixes after differentiation, it follows at once that it cannot make any difference what figures had been originally used, and that 12” and 34" are expressions for the same thing. rf the use of this method we have constantly to employ transformations depending on this obvious principle. ‘Thus, we can show that when n is odd, 12" applied to a single function vanishes identically. For, from what has been said, 12"=21"; but 12 and 21 have opposite signs, aS appears immediately on writing at full length the symbol for which 12 is an abbreviation, It follows then that 12” must vanish when x is odd. Thus, in the expansion of 12°, given at the end of Art. 151, if we make U= J, it will obviously vanish identically. The series 12°, 12‘, 12°, &c. gives the series of invariants and covariants which we have already found (Art. 141). It is easy to see that, when n is even, 12” applied tO (G,, @,, A,» (2, ¥) gives Gd,

— NAG, i+ on (n—I)a 2 A, -. — Ks,

where the last coefficient must be divided by two, as is evident from the manner of formation. In particular, we thus have the invariants, for the quadratic, ac—6°; for the quartic, ae— 4bd+ 8c’; for the sextic, ag — 64/+15ce—10d*; and so on. 154. The results of the preceding Articles naturally extend to any number of functions. We may take any number of quantics U, V, W, &c., and, writing the variables in the first with the suffix (1), those in the second with the suffix (2), in the third with the suffix (3), and so on, we may operate on their product with the product of any number of symbols 12%, 238, 31’, 14, &c., where, as before, 23 is an abbreviation for £,n,—&n,, &c. After the differentiation we suppress the suffixes, and we thus get a covariant of the given system of

140

SYMBOLICAL

REPRESENTATION

OF CONCOMITANTS

quantics, which will be an invariant if it happens that no power Any number of the of x and y appear after differentiation. quantics U, V, W, &c., may be identical; and in the case with which we shall be most frequently concerned, viz., where we wish to form derivatives of a single quantic, the subject operated on is U,U,U, &c., where U, and U, only differ by having the variables written with different suffixes. It is evident that in this method the order of the derivative in the coefficients will be always equal to the number of different figures in the symbol for the derivative. For if all the functions were distinct, the derivative would evidently contain a coefficient from every one of the quantics U, V, W, &c.; and it will be still true, when U, V, W are supposed identical, that the degree in the coefficients is equal to the number of factors in the product U,U,U, &c., which we operate on. Thus the derivatives considered in the last Article being all of the form 12? are of the second order only in the coefficients. Again, if it were required to find the degree of the derivative in x and y. Suppose, in the first place, that the quantics were distinct, U being of the degree n, V of the degree n’, W of the degree n”, and so on; and suppose that in the operating symbol the figure 1 occurs a times; 2, 8 times; and so on; then, since U is differentiated a times, V, 8 times, &c., the result is of the degree (n—a) + (n’ — 8) + (n” —- y) + &c.

When

the quantics

are identical, if there are p factors in the product U_U....U,, which we operate on, the degree of the result in a and y will be mp—(a+8+y+&c.). While again, if there be r factors such as 12 in the operating symbol, it is obvious that (a+B+y+&e.)=2r. It is clear that if we wish to obtain an ¢nvariant, we must have a=B=y=n. 155. To illustrate the above principles, we make an examination of all possible invariants of the third order in the coefficients. Since the symbol for these can only contain three figures, its most general form is 12*.23°.317; while, in order that it should yield an cnvariant, we must have

Cook ee SiC iy; whence a=8=y.

The general form, then, that we have to

INVARIANTS

OF THE

THIRD

ORDER.

141

examine is (12.23.31)*%. Again, if a be odd, this derivative | vanishes identically; for, as in Art. 153, by interchanging the figures 1 and 2, we have (12.23.31)*= (21.13.32)%; but these have opposite signs. It follows, then, that all invariants of the third order are included in the formula (12.23.31)*, where @ is even. Thus, 12°.23%.31? is an invariant of a quartic, since the differentials rise to the fourth degree; 12%.23°.31* is an invariant of an octavic; 12°.23°.31° of a quantic of the twelfth degree, and so on; only quantics whose degree is of the form 4m having invariants of the third order in the coefficients. If we wish actually to calculate one of these, suppose 12?.237.31°, write, for brevity, &, 7,, &c., instead of Ee » a , &c., and we ee dx,’ dy have actually to multiply out : é

(Ctamicei) alee ies

ics

ess)« 4

In the result omit all the suffixes, and replace & by a Ke. or, when we operate on a quartic, by a, the coefficient of 2x‘, &e, There are many ways which a little practice suggests for abridging the work of this expansion, but we do not think it worth while to give up the space necessary to explain them; and we merely give the results of the expansion of the three invariants just referred to. 12”.23’.31? yields the invariant of a quartic already obtained (Art. 141, Ex. 1, and Art. 144), viz. :— 4,4, + 24,0,0,- 4,4, — 4,4, —a,. 12*.23'.31* gives

a, (aa, — 40,4, + 34,0,) + a, (— 4a,a, + 120,0, — 8a,0,) + a, (30,4, — 84,4, — 224,07,+24a,0,) + a,(24a,a,—36a,a,) + 15a,a,q,. And 12°.23°.31° gives

a,,(4,4,— 64,4,+ 15a,4,— 10a,0,)+ a,,(—6a,0,+ 304,,—54a,0,+ 304,a,) +a,, (15a,a, — 54a,a, + 24a,4, + 150a,a, — 135a,a,) +a, (— 10a,a, + 30a,a, + 150a,a, — 430a,a, + 270a,a,) +a, (— 135a,a, + 270a,a, + 495a,a, — 540a,a,)

+a, (— 540a,a, + 720a,a,) — 280a,0,4,.

156. Though the above-mentioned is the only type of invariants of the third order, there is an unlimited number of

142

SYMBOLICAL

REPRESENTATION

OF CONCOMITANTS.

covariants, the simplest being 12°.13, which, when expanded, is

a Ut dx

OC

Uae... ( ad’ U Paes

dy? dy dxdy

\" dady dy

ad’U a a dady* \dx dy 3

dy’ da

d*U a) dady dx)

d*U d*U dU dy> dx dx*

When this is applied to a cubic, it gives the evectant obtained already (Art. 142). The general type of invariants of the fourth order in the coefficients is (12.34)* (13.24) (14.23). Thus the discriminant of a cubic is expressed in this notation as (12-34) (13.24); but we cannot afford space to enter into greater details on this subject.

157. The principles just laid down a remarkable theorem first demonstrated which we shall refer as “ Hermite’s Law number of invariants of the n” order in

afford an easy proof of by M. Hermite, and to of Reciprocity.” The

the coefficients possessed by a binary quantic of the p” degree vs equal to the number of invariants of the order p in the coeffictents possessed by a quantic

of the n™ degree. We have already proved that if any symbol 12°.23”.34° &c. denotes an invariant of the order p of a quantic of the degree , then the number of different figures 1, 2, 3, &c., is p, and each figure occurs 2 times. But we might calculate by the method of Art. 136 an invariant 5 (a —8)* (8—y)’(y—8)° &e., where we replace each symbol 34 by the difference of two roots (y—96). This latter is an invariant of a quantic of the p” degree, since there are by hypothesis p roots; and it is of the order n in the coefficients of the equation (Art. 58). Thus, for example, a quadratic has but the single independent invariant («—)’, though of course every power of this is also an invariant; and the general type of such invariants is

(a—)*". Hence, only quantics of even degree have invariants of the second order in the coefficients, and the general symbol for such invariants is 12”, So again, cubics have no invariant except the discriminant (a — 8)*(8—+¥)? (y—a)’ and its powers; and the discriminant is of the fourth order in the coefficients. Hence, only quantics of

HERMITE’S

the degree 4m

12”".23"".31"".

LAW

OF RECIPROCITY.

143

have cubic invariants whose general type is

It will be proved that quartics have two inde-

pendent invariants, one of the second and one of the third order, in the coefficients; and, of course, any power of one multiplied by any power of the other is an invariant. Hence, quartics have as many invariants of the p” order as the equation 2x+3y=p admits of integer solutions; this is, therefore, the number of invariants of the fourth order which a quantic of

the p” degree can possess. 158. Hermite has proved that his theorem applies also to covariants of any given degree in x and 7; that is to say, that an n* possesses as many such covariants of the p” order in the coefficients as a p” has of the n order in the coefficients. For, consider any symbol, 12.23.34” &c., where there are p figures, and the figure 1 occurs a times, 2 occurs 6 times, and so on; then we have proved that the degree of this covariant in x and y is (n -a)+(n—6)+&c. But we may form the symmetric function

5 (a— 8) (8—y)" (y- 8) (wa) (w- )"" &e.,

which has been proved (Art. 137) to be a covariant of the quantic of the p” degree, whose roots are a, 8, &e. Every root enters into its expression in the degree n, which is therefore the order of the covariant in the coefficients, and it obviously contains x and y in the same degree as before, viz. (n—a)+(n—b)+&c. Thus, for example, the only covariants which a quadratic has are some power of the quantic itself multiplied by some power of its discriminant, the general type of which is

(a — B)” (w@—a)' (e@— BY,

the order of this in the coefficients is 2p + q, and in x and y is 2q. Hence we infer that every quantic of the degree 2p+q has a covariant of the second order in the coefficients, and of the degree 2g in x and y, the general symbol for such covariants being 12”. When g =1, we obtain the theorem given (Art. 141), that every quantic of odd degree has a quadratic covariant. 159.

Concomitants of quantics in three or more variables are

expressed in a manner similar to that already explained.

If

SYMBOLICAL

144

OF CONCOMITANTS.

REPRESENTATION

LY 2) CY Koy LyYa%q) be cogredient sets of variables, then, by the rule for multiplication of determinants, the determinant oa (¥, Te

eee) + 2, (Y,2, — 9,25) + &, (ye

Y221)

is an invariant, which, by transformation, becomes a similar function multiplied by the modulus of transformation. And if in

: d d p the above we write for x,, ees for ¥,, TAR and so on, we obtain 1

2

an invariantive symbol of operation, which we shall write 123. When, then, we wish to obtain invariants or covariants of any function U, we have only to operate on the product U,U,U,...U, with the product of any number of symbols 123 124¢ 235" &c, and after differentiation suppress all the suffixes. Thus, for example, let U,, U,, U, be ternary quadrics, and let the coefficients in U, be a, d, c, 2f, 2g, 2h, as at p. 99, then 123’ expanded is a

(b'c"+

6’

ms of

Ff") ae b (c'a”’+

ay fi (g/h” th gh

ee oh

(Sige

+

fq

_s af”

a3 ch’

—

(=

gts)

2

eh’)

29'9"’) feCc(a’b"+

de 2g (Wf

ab’

+ h’f’—

—2h’h’’)

b’g”—-b’g’)

:

and this when we suppose the three quantics U,, U,, U,, to be identical, or a=a’=a” &c. reduces to six times

abe + 2fgh — af” -— bg’ — ch’. ad’ U If in the above we replace veia, the coefficient of 2”, by Te &e. we get the expansion of 123° as applied to any ternary quantic. This covariant is called the Hessian of the quantic. It is seen, as at Art. 153, that odd powers of the symbol 123 vanish when it is applied to a single quantic. We give as a further example the expansion of 123* applied to the quartic,

act + by* + ca! + 4 (a,x*y + axe + b,y°2 + bya + 0,2°x + ¢,2°y) + 6 (dy?2* + ex*x® + fa*y*) + 12axyz (le + my + nz). Then 123* is abe — 4 (ab,c, + be,a, + ca,b,) + 3 (ad* + be* + of?) +4 (a,b,c, +a,,¢,)

—12 (a,nd + amd + b,ne + b,le + c,mf+ ¢,/f)

+ 12 (1d,c, + me,a, + na,b,) +12 (dl + em’ + fn?) + 6def—12lmn.

IN ANY

NUMBER

OF VARIABLES.

145

160. We can express in the same manner functions containing contragredient variables; for if a, 8, y be any variables contragredient to a, y, 2, and therefore cogredient with i 5 S 5 it follows, as before, that the determinant

(oo

-

dy,dz,

fe \0(f dy,dz,

2-2

dz,dx,

2),,(24

dz,dax,

dd

Y dx, dy, a rei

(which we shall denote by the abbreviation a12) is an invariantive symbol of operation. Thus, if U,, U, be two different quadrics, aJ2° is the contravariant called ® (Conics, Art. 877), which expanded is “a (O'c"+ bc=

oo 2By

(fh

tL 2a8

Wi

254)

+ J

+

h’-

he +f"

B?(ca"+

af’

=

—

Clie

ce’

af’)

=

— 297’)

fe 2ya

+r? (a’b"+ “b=

(Af

+

h’f'-

2h’h”)

0g" —

b’g’)

CM;

and which, when the two quadrics are identical, becomes the equation of the polar reciprocal of the quadric. In like manner, the quantic contravariant to a quartic, which I have called

S (Higher Plane Curves, p. 78), may be written

symbolically a12*, and the quantic 7’ in the same place may be written 412° «23° a431*. In any of these we have only to replace n

the coefficient of any power of a, x” by S &c. to obtain a symbol which will yield a mixed concomitant when applied to a quantic of higher dimensions. Thus 12° is

eijeeve.U {e dz

fa

Us

(ea)

:

which, when applied to a quadric, is a contravariant, but, when applied to a quantic of higher order, contains both a, y, 2, as well as the contragredient a, 8, y, and, therefore, is a mixed concomitant. In general, if we have the symbolical expression for any invariant of a binary quantic, we have only to prefix a contravariant symbol a to every term, when we shall have a contravariant of a ternary quantic of the same order. And in particular it can be proved that if we take the symbolical expression for the discriminant of a binary quantic, and prefix in

146

SYMBOLICAL

REPRESENTATION

OF

CONCOMITANTS.

this manner a contravariant symbol to each term, we shall have the expression for the polar reciprocal of a ternary quantic. Thus, the symbol for the discriminant of a binary cubic is 12°.34°.13.24, and the polar reciprocal of a ternary cubic is 0.127.a34°.a13-024, which is obviously of the sixth order in the

variables a, 8, 7, and of the fourth in the coefficients. 161. If in any contravariant we substitute aa Zs = for a, 8, y, and operate on U, we get a covariant (Art. 139); and the symbol for this covariant is got from that for the contravariant by writinga new figure instead of «& Thus from «23? is got 123”, from «23. a24 is got 123.124, &e. Conversely, if in the ere for any invariant we replat any figure by a contravariant symbol a, we get the evectant of that invariant.

Thus,

123.124.234.314

is an invariant of a cubic, and the evectant of that invariant is 123.012.023.431.

In the case of a binary quantic, this rule assumes a simpler form; for if we substitute a contravariant symbol for 1 in 12, it becomes, when written at full length, & 5 - n =, but since

& and 7 are cogredient with —y and a, this may be written d re +95, and may be suppressed altogether, since it only diz affects the result with a numerical multiplier. Hence, given the symbol for any invariant of a binary quantic, its evectant is got by omitting all the factors which contain any one figure. Thus, being the discriminant of a cubic,its evectant, got by omitting the factors which contain 4, is 12”.13. dU dU dU Tf in a contravariant of any quantic we substitute—— dx? dy? dz for a, 8, y, we also get a covariant, and the symbol for it is obtained from that for the contravariant by writing a different

MODIFIED

NOTATION.

new figure in place of every a. 134.234, and so on.

147

Thus, from 34” we get

162. In the explanation of symbolical methods which has been hitherto given, I have followed the notation and course of proceeding originally made use of by Prof. Cayley. I wish now to explain some modifications of notation introduced by Aronhold and Clebsch, who have employed these symbolical methods with great success, but who perhaps at first scarcely sufficiently recognized the substantial identity of their methods with those previously given by Prof. Cayley. The variables are denoted ,, x,, x,, &c., while the coefficients are denoted by suffixes corresponding to the variables which they multiply. Thus the ternary cubic, the ternary quartic, &c., may be briefly denoted 24,,,2;0;,2)) TVAjpyVjL,0;0,, “Ke., where the numbers ¢, k, l, m are to receive in succession all the values 1, 2, 3, Kc. It will be observed that in this notation a; =@,;=4,,;;, 80 that when we form the sums indicated we obtain a quantic written with the numerical coefficients of the binomial theorem. Thus when we form the sum 3a,,,7,x,7, the three terms a,,,2,2,2,, yoL,%_C,y Up,,%,0,0, are identical, as in like manner are the six Lo heat tet Seb y Met Shhee tae terms D930

LV 5

Di39% UX oy

Dy igVyX Lay Ay3,C, LL,

Ds pL 0 Ly

Bo, L,L,X yy

so that the sum written at length would be A, 1,0, 2,2, or Dyn

VLU,

” Fy ggU VX, i 3A, 1.

0,2, Feet

66,950, 0,U ye

And so, in like manner, in general. Now Aronhold uses, as an abbreviated expression for the quantic in general, (a,x, “2 A,X, ae av,

. es )"

where, after expansion, we are to substitute for the products a,a,a,, &c., the coefficients a,,,, Thus the ternary cubic given above may be written in the abbreviated form (4,2, + 4%, os a,%,)° )

the terms

A,0,0,0,0,0, + 3a,a,a,0,0,0, + &e.

in the expansion of the cube being replaced by a,,,%,7,2,, 34,,,0,0,0,, &e. The quantity a,x,+0,7,+0,2, is written a,

148

SYMBOLICAL

REPRESENTATION

OF

CONCOMITANTS.

or sometimes simply a, and the quantic is symbolically expressed as a,". The cubic might equally have been written (6,0, + 0,0, + b,x,)°, (cv, + 6,2, +¢,0,)°, &e., it being understood that we are in like manner to substitute for ,0,0,, c,c,c,, &c., the coefficients @,,,, @,,,., &c. Now the rule given by Aronhold for the formation of invariants is to take a number of determinants, whose constituents are the symbols a,,a,,a,; 0,,0,, &c., to multiply all together, and after multiplication to substitute for the symbols a,a,a, 0,,b,b,, the coefficients aj, Gmnp, Sc. Thus Aronhold first discovered a fundamental invariant of a ternary cubic by forming the four determinants 3% + a,),c,, Ztbed,, Died,a, 2+d,a,b,; multiplying all together and then performing the substitutions already indicated. This is the same invariant which, in Prof. Cayley’s notation, would be designated as 123.234.341.412. In order to obtain an invariant by this method, it 1s obviously necessary (as in Art. 154) that the a@ symbols, 4 symbols, &c. respectively should each occur m times. A product of determinants not fulfilling this condition is made to express a covariant by joining to it such powers of a,, b,, &c. as will make up the total number of a’s, b’s, &e. to n. Thus the Hessian of a binary quadratic, which in Cayley’s notation is 12? is in Aronhold’s (ab)’; but the Hessian of any other binary quantic, which in Cayley’s notation is still 12’, is in Aronhold’s (a0)? a,"b,".

163. In order to see the substantial identity of the two methods, it is sufficient to observe that by the theorem of homogeneous functions any quantic w of the »” order differs only

b

: ae, d d ai\ "am y a numerical multiplier from { 2, da + 8 Jp tM ) Uy 80 1

2

3

that if we write it (4,7,+4,7,+,#,)", the symbols a,, a,, a, differ only by a numerical constant from the differential sym-

d bols de , &c.

: And we evidently get the same results whether

1

with Prof. Cayley we form determinants whose constituents are d d d Sperone | or 3

’ with

Aronhold,

whose

constituents

are

MODIFIED

NOTATION.

149

And the artifice made use of by both is the same. If we multiply together a number of differential symbols (5 + » 5)(g + pb 5) , &c., and operate on JU, it is evident

the result will be a linear function of differentials of U of an order equal to the number of factors multiplied together; and that in this way we can never get any power higher than the first of any differential coefficient. When, then, it is required to express symbolically a function involving powers of the differential coefficients, the artifice used by Prof. Cayley was to write the function first with different sets of variables, and form such a function as (at day) (Gta)

U_U,, and

after differentiation to make the variables identical. So in like manner Aronhold in his symbolic multiplication uses different symbols which have the same meaning after the multiplication has been performed. By multiplying together symbols Gjy A 4,, &e., we can only get a term such as q,,, of no higher than the first order in the coefficients. When, then, we want to express symbolically functions of the coefficients of higher order than the first, the artifice is used of multiplying together different sets of symbols a,, a,, a,; 0,, 6, 0, &c., the products A,,8y 0,215 6,C,Cy KC., all equally denoting the coefficient a,;,,. The notations explained in this Lesson afford a complete calculus, by means of which invariants and covariants can be transformed and the identity of different expressions ascertained. We shall in a subsequent Lesson give illustrations of the applications of this method, referring those desirous of further information to Clebsch’s valuable Z'heorte der bindren algebratschen Formen, in which work this symbolical method is the basis of the whole treatment of the subject.

( 150)

LESSON CANONICAL

XY. FORMS.

164. SINCE invariants and covariants retain their relations to each other, no matter how the quantic is linearly transformed, it is plain that when we wish to study these relations it is sufficient to do so by discussing the quantic in the simplest form to which it is possible to reduce it. This is only extending to quantics in general what the reader is familiar with in the case of ternary and quaternary quantics; since, when we wish to study the properties of a curve or surface, every geometer is familiar with the advantage of choosing such axes as shall reduce the equation of this curve or surface to its simplest form.* The simplest form then, to which a quantic can without loss of generality be reduced, is called the canonical form of the quantic. We can, by merely counting the constants, ascertain whether any proposed simple form is sufficiently general to be taken as the canonical form of a quantic, for if the proposed form does not, either explicitly or implicitly, contain as many constants as the given quantic in its most general form, it will not be possible always to reduce the general to the proposed form.t * It must be owned, however, that as in the progress of analysis greater facility is gained in dealing with quantics in their most general form, the advantage diminishes of reducing them to simpler forms, + It is not true, however, conversely, that a form which contains the proper number of constants is necessarily one to which the general equation may be reduced. For when we endeavour by comparison of coefficients to identify such a form with the general equation, although the number of equations is equal to the number of quantities to be determined, it may happen that the constants enter into the equations in such a way that all the equations cannot be satisfied. Thus

(ce—a)?+(y-—BP=k+myt+n is a form containing five constants, and yet is not one to which the general equation of a ternary quadric can be reduced; since the constants enter the equation in such a way that though we have more than enough to make the coefficients of « and y and the absolute term identical with those in any proposed equation, we have no means of

identifying the coefficients of x’, zy and y*,

A more important example is

eit yf + 2+ ul + vf,

OF QUADRICS.

151

Thus, a binary cubic may be reduced to the form X*4Y°; for the latter form, being equivalent to (Jz + my)* + (a + m’y)*, contains implicitly four constants, and therefore is as general as (a, b,c, dx, y)*°. So, in like manner, a ternary cubic in its most general form contains ten constants; but the form X*+¥*%+Z°+6mXYZ contains also ten constants, since, in addition to the m which appears explicitly, X, Y, Z implicitly involve three constants each. This latter, then, may be taken as the canonical form of a ternary cubic, and, in fact, some of the most important advances that have been made in the theory of curves of the third degree are owing to the use of the equation in this simple and manageable form. 165. The quadratic function (a, 5, cf, y)? can be reduced in an infinity of ways to the form a’+y’, since the latter form implicitly contains four constants, and the former only three. In like manner the ternary quadric which contains six constants can be reduced in an infinity of ways to the form x’+y’+2*, since this last contains implicitly nine constants; and, in general, a quadratic form in any number of variables can be reduced in an infinity of ways to a sum of squares. It is worth observing, however, that though a quadratic form can be reduced in an infinity of ways to a sum of squares, yet the number of positive and negative squares in this sum is fixed. Thus, if a binary quadric can be reduced to the form x’ + y’, it cannot also be reduced to the form u’ —v”, since we cannot have 2*+y” identical with u’—v", the factors on the one side of the identity being imaginary, and those on the other being real. In like manner, for ternary quadrics we cannot have 2+ y’— 2’ =u’ + "+ w’, since we should thus have o+y=2?+u?+v'+w’, or, in other words, vty a2 + (le+ my 4+nz)?+(UVet+m’y+ n'2)"+ (a+ my +n"2)’,

and if we make aw and y=0, one side of the identity would where z, u, v are linear functions. In the case of a ternary quantic this form contains implicitly fourteen independent constants, and therefore seems to be one to which the quartic in general can be reduced. But Clebsch has shewn that a condition must be fulfilled in order that a quartic should be reducible to this form, namely, the

vanishing of a certain invariant,

See also Surfaces, Note to Art. 235,

152

CANONICAL

FORMS.

vanish, and the other would reduce itself to the sum of four positive squares which could not be =0. And the same argument applies in general.

166. We

commence

by shewing that, as has been just

stated, a cubic may always be reduced to the sum of two cubes. To do this is, in fact, to solve the equation, since when the quantic is brought to the form X°+Y’%, it can immediately be resolved into its linear factors. Now, if the cubic (a, 6, c, d{a, y)? become by transformation (A, B, C, DX, Y)’, then, since (Art. 126) the Hessian (ax + by) (cw + dy) — (bx+cy)* is a covariant, it will, by the definition of a covariant, be transformed into a similar function of A, B, C,D, X, Y. That is to say, we must have

(ac— 0’) a + (ad— bc) xy + (bd —c*) y?

= (AC— B’) X?+(ADBC) XY+(BD-C’)Y?. Now, if in the transformed cubic, B and C vanish, the Hessian takes the form ADXY; and we see at once that we are to take for X and Y the two factors into which the Hessian may be broken up. When we have found X and Y, we compare the given cubic with AX*+ DY’%, and determine A and D by comparison of coefficients. Ex. To reduce 4x3 + 9x? + 18x + 17 to the form AX*+4+ DY,

The Hessian is

(4x + 3) (62 + 17) — (8a + 6)2, or

15a? + 50x” + 15,

whose linear factors are x + 3, 3c +1.

Comparing then the given cubic with

A («+ 3)3 + D (8x + 1), we have A+ 27D =4, 274 + D=17,

in the ratio of 5 to 1.

whence 728D = 91, 728A = 455, or A isto D

The given cubic then only differs by a factor (viz. 8) from 5 (2 + 3)? + (8a + 1)3,

and it is obvious that the roots of the cubic are given by the equation 8a + 1+ (@ + 38) 3J(5) = 0.

167. It is evident that every cubic cannot be brought by real transformation to the form 4AX*°+DY’°%, for this last form has one real factor and two imaginary; and therefore cannot

OF QUANTICS

OF ODD

DEGREE.

153

be identical with a cubic whose three factors are real. discriminant of the Hessian

The

4 (ac — B°) (bd - ¢*) — (ad— bc)? is, with sign changed, the same as that of the cubic. When the discriminant of the cubic is positive, the Hessian has two real factors, and the cubic one real factor and two imaginary. When it is negative, the Hessian has two imaginary factors, and the cubic three real. When it vanishes, both Hessian and cubic have two equal factors, and it can be directly verified that

the Hessian of X?Y is X?.* It is to be observed, that a quantic of the same degree cannot always be reduced to the same canonical form. The impossibility of the reduction indicates some singularity in the form of the quantic. Thus a cubic having a square factor cannot be brought to the form Az’+ Dy’: a different canonical form must be adopted, and the most simple one is the form a*y, to which the cubic in question is obviously at once reducible.

168. In the same manner as a cubic can be brought to the sum of two cubes, so in general any binary quantic of odd degree (2n —1) can be reduced to the sum of n powers of the (2n—1)” degree, a theorem due to Dr. Sylvester. For the number of constants in any binary quantic is always one more than its degree, or, in the present case, 2n; and we have the same number of constants if we take n terms of the form (la+ my)". The actual transformation is performed by a method which is the generalization of that employed (Art. 166). For simplicity, we only apply it to the fifth degree, but the method is general. The problem then is to determine 4, », w, so that (a, 0, c, d, e, fXa, y)° may =u°+v°+w. Now we say that if we form the determinant ax+ by, be+cy, cx + dy but cy, ce+dy, dx+ ey ca + dy, dut+ey, eat+fy

te ee Se

SS

SS

Be es

|, ee

ae

* In general, when a binary quantic has a square factor, this will also be a square factor in its Hessian, as may be verified at once by forming the Hessian of 2p,

154

CANONICAL

FORMS.

For let

the three factors of this eubic will be w, v, w.

u=le+my,

v=Ua+m'y,

w=at+m’'y;

then, differentiating the identity (a, b,c, d, e, fix, yr au tv +0

four times successively with regard to x, and dividing by 120, we get

az + by =ltu + Uy 4 Uw. Similarly differentiating three times with regard to x, and once with regard to y,

bx + cy=l?mut U%m’o4 mw; and so on. The J determinant, ’ then, ’ written above, , may May be p put into the form Yu + tv +

Uw

, ysl mw ) Pmutl?m'os Um", ) Pmutl?m

Pmut l°m'y+ U'm''w, Pi?utl?mo4 Um’?w, lndut mv 4l'm'w Pmutl?m?vt4l?m'?w, lm®utl'm’y + U'm'?w, m*u+ m'*v + m'*w

But (Art. 22) this is the prodact Pu

;

Uy

;

U"?w%

P 4

iPe ;

fe

Imu, Um'v, W’m'’w | | ln, Um’, Um" Mtb,

or is

my

;

mw

m’,

m?

:

m”

-

uvw (lm’ —Um)? Um" — Um’)? (Um — lm)’.

When, then, the determinant written in the beginning of this Article has been found, by solving a cubic equation, to be the product of the factors (a +Ay) (a+ py) (x+vy), we know that uw, v, w can only differ from these by numerical coefficients, and we may put

(a, b,c, dye, fa, y)! =A (w+ ry) + Bla + py) +C (a + vy)’; and then A, B, C are found by solving any of the systems of simple equations got by equating three coefficients on both sides of the above identity. The determinant used in this Article is a covariant, which is

called the canonizant of the given quintic.

CANONIZANT

OF QUINTIC.

15%5)

169. The canonizant may be written in another, and perhaps simpler form, namely, 3

2

Y, ~YX,

a

2

YX,

-— 2

3

Shae Ra 5

b,

¢,

de

e,

d,

OT

aa

This last is the form in which we should have been led to it if we had followed the course that naturally presented itself, and sought directly to determine the six quantities A, B, C,A, m, v, by solving the six equations got on comparison of coefficients of the identity last written in Art. 168. For the development of the solution

in this form, to which

we

cannot

afford the

necessary space here, we refer to Sylvester’s Paper (Philosophical Magazine, November, 1851). Meanwhile, the identity of the determinant in this Article with that in the last has been shown by Prof. Cayley as follows. We have, by multiplication of determinants (Art. 22), ¥’, — ya, yo, — 2°

Wy ee ,

ad

1, 0, 0, 0

hee ee:

OM Sea?

Capek 0, 0 0,2, 04/0

a«.

Ff

0, 0, #, y

Phi?

Be

m0

he

0, aw+ by, bu+cy, ca + dy 0, ba+cy, ce+dy, dx+ ey 0, cot+tdy, dx+ey, ex+fy

|,

which, dividing both sides of the equation by y’, gives the identity required.

170. We have still to mention another way of forming the canonizant. Let this sought covariant be (A, B, C, Da, y)’, where we want to determine A, B, C, D; then (Art. 140)

(4 B,C, Dye - ae will also yield a covariant.

But if this

operation is applied to (w+Ay)" where «+Ay is a factor in (A, B, C, Dx, y)*, the result must vanish, since one of the

156

CANONICAL

FORMS.

factors in the operating symbol is 5 —X 2 .

Since, then, the

given quantic is by hypothesis the sum of three terms of the form (a+Ay)*, the result of applying to the given quantic the operating symbol just written must vanish. Thus, then, we have

A (d, e, fXx, y)’— B(o, d, ex, y)' +O (b, c, dXa, yy’ — D(a, b, ca, y)’=0,

or, equating separately to 0 the coefficients of x’, xy, y’, we have Ad— Bc +Cb- Da=0, Ae — Bd + Oc — Dd =0, Af - Be +Cd— De =0,

whence (Art. 28) A is proportional to the determinant got by G500,.¢ suppressing the column A or | 6, c, d | and so for B, C, D, C) Ge which valueg give for the canonizant the form stated in the last Article.

171. We proceed now to quantics of even degree (2n). Since this quantic contains 2n+1 terms, if we equate it to a sum of n powers of the degree 2n, we have one equation more to satisfy than we have constants at our disposal, On the other hand, if we add another 2n” power, we have one constant too many, and the quantic can be reduced to this form in an infinity of ways. It is easy, however, to determine the condition that the given quantic should be reducible to the sum of n, 2n” powers. Thus, for example, the conditions that a quartic should be reducible to the sum of two fourth powers, and that a sextic should be reducible to the sum of three sixth powers, are respectively the determinants O20 ac

Ceol

b, c, d |=0, d ieee

ao, Bing = 0, ee Yes) def, 9

OF QUARTIC.

157

and so on. For, in the case of the quartic, the constituents of the determinant are the several fourth differentials of the quantic, and expressing these in terms of u and v precisely as in Art. 168, it is easy to see, Art. 26, Ex. 5, that the determinant must vanish, when the quartic can be reduced to the form u‘+v*. Similarly for the rest. This determinant expanded in the case of the quartic is the invariant already noticed (see mitt 141,eiix, 1); ace + 2bcd —ad’ - eb? —c*.

172. When this condition is not fulfilled, the quantic is reduced to the sum of powers, together with an additional term. Thus, the canonical form for a quartic is naturally taken to be u*+v'+6ru’v. We shall commence with the reduction of the general quartic to this canonical form; the method which we shall use is not the easiest for this case, but is that which shows most readily how the reduction is to be effected in general. Let the product, then, of wu, v, which we seek to determine, be (A, B, CX, y)’, and let us operate with (A, B, Oxy.»- = on both sides of the identity (a, }, c, d, ea, y)*=u' + v* + 6rAU’0", Now, as before, this operation performed on u* and on v* will vanish, and when performed on 6Au’v’ it will be found to give 12\’uwv, where ’=2(44C—5’)r. Equating then the coefficients of x’, xy, and y’ on both sides, after performing the operation, we get the three equations

Ac—Bb+Ca=

WA,

Ad—Bc+Cb=43NnB, Ae—Bd+Cc=

WO,

whence eliminating A, B, C, we have to determine 2’, the determinant a,

b,

b,

c—N c+43r, d

c-N,

d,

é

= 0,

158

CANONICAL

FORMS.

which, expanded, is the cubic nN? — NX (ae — 4d + 8c”) — 2 (ace + 2bed — ad’ - eb’ —c*) = 0,* the coefficients of which are invariants. Thus, then, we have a striking difference in the reduction of binary quantics to their canonical form, between the cases where the degree is odd and where it is even. In the former case, the reduction is unique, and the system w, v, w, &c. can be determined in but one way. When u is of even degree, however, more systems than one can be found to solve the problem. Thus, in the present instance, a quartic can be reduced in three ways to the canonical form, and if we take for A’ any of the roots of the above cubic, its value substituted in the preceding system of equations enables us to determine A, B, C. 173. If now we proceed to the investigation of the reduction of the quantic (a,, a,, a,...{a, y)”, the most natural canonical form to assume would be u”™+v"+w™”+ &e. + Au’'v'w’ K&e., there being n quantities wu, v, w, &c. But the actual reduction to this form is attended with difficulties which have not been overcome, except for the cases n=2 and n=4. But the method used in the last Article can be applied if we take for the canonical form wu”+ v0" + &c. + ~AVuew &e., where, if uvw &t.= (Aa, Anh, Y) V is a covariant of this latter function such that when

is operated on by (A,, 4, ee

Vuvw &e.

=e) the result is propor-

tional to the product www &e. Suppose, for the moment, that we had found a function V to fulfil this condition, then, proceeding exactly as in the last Article, and operating with the differential symbol last written on the identity got by equating the quantic to its canonical form, we get the system of equations Aja, —A,a,,+ A,o,,-&.=NvA, J n-1 2

Ad

—-Aa,

A iyg—

+A,a,_,— &eo.= 7A An

A,Q,,,+ 4,0, nt1

Lo

—&e.=

2

,

n(n—1) Xr“a &e.,

* N.B.—The discriminant of this cubic is the same as that of the quartic.

OF QUANTICS

OF EVEN

DEGREE.

159

whence, eliminating A, 4,, A,, &c., we get the determinant /,

Par),Sa Oa

Zo

coacsanoe90N" vere Q,

Qs

Cah;

a nt2 ;

a nH)

NY m n(n —

ayn)

Ces

CsPR EIOC ceotes ees ve Oo t)/F\EN |

,

Cg syle ote sisleieisislovisiciele a, te

t)

?

306 a C

and having found 4’ by equating to 0 this determinant expanded (a remarkable equation, all the coefficients of which will be invariants), the equations last written enable us to determine the values of A,, A,, &c., corresponding to any of the n+1 values of

174. To apply this to the case of the sextic, the canonical form here is u° + v°+ w° + Vuvw, where, if uvw be (A,, A, A,, A,Xa, y)’s

V is the its value afford an show that

evectant of the discriminant of this last quantic, and is written at full length (Art. 142). Now it will excellent example of the use of canonical forms if we in any cubic the result of the operation (2) G5 Fas aS 9 ets ay

performed on the product of the cubic and the evectant just mentioned, will be proportional to the cubic itself. For it is sufficient to prove this, for the case when the cubic is reduced to the canonical form x*+¥y*, in which case the evectant will be x’— y’, as appears at once by putting b=c=0, and a=d=1 in the value given, Art. 142. The product, then, of cubic and * The determinant above written may be otherwise obtained as follows. a’, y’ be cogredient to x, y, and let us form the function

Let

d , a Ke \n («, Gackg) a U +X (ay!, — yx’),

which (Arts. 125, 131) we have proved to be linearly transformed into a function of similar form. Equate to zero the n+ 1 coefficients of the several powers x”, a”-“!y, &c., and from these eliminate linearly the n + 1 quantities x’, «’™ly’, &c., and we obtain the determinant in question.

160

CANONICAL

FORMS.

ste. d° evectant will be a°— y°, which, if operated on by dys

“Po: da? gives

And the theorem a result manifestly proportional to «*+y*. now proved being independent of linear transformation, if true for arty form of the cubic, is true in general. The canonical form, then, being assumed as above, we proceed exactly as in the last Article, and we solve for \ from the equation as

a5

a;

;

ey)

a,

Ose xX

ok

a,+ gr, a,

a 2?

Qa— thea 3 Bec?

Ad

a5

a, ts r,

ayy

as,

a,

=O 4

which, when expanded, will be found to contain only even powers of A. If we suppose ww reduced as above to its

canonical form a°+y’*, the three factors of which are L+Y,

L+wy,

x+ wy,

where @ is a cube root of unity, then it is evident from the above that the corresponding canonical form for the sextic is

A(x+y)+ Bat ay) +C(x + wty)' + D (a*—y’). It can be proved that if u, v, w be the factors of the cubic, then the factors of the evectant used above are v—w, w—u, u—v, so that the canonical form of the sextic may also be written

ue + v0 +w* + duvw (uv) (v—w) (w— Uw). 175. In the case of the octavic the canonical form is ur +o + w+ 2° + rw ewe’,

for if we operate on w’v*w*s* with a symbol formed according to the same method as in the preceding Articles, the result will be proportional to uvwz. As for higher canonical forms we content ourselves with again mentioning that for a ternary cubic, viz. a +y° + 2°+ 6mayz, and that given by Sylvester for a quaternary cubic, et+yt+etuetr’

( 161)

LESSON SYSTEMS

XVI.

OF QUANTICS.

176. Ir still remains to explain a few properties of systems of quantics, to which we devote this Lesson. An invariant of a system of quantics of the same degree is called a combinant if it is unaltered (except by a constant multiplier) not only when the variables are linearly transformed, but also when for any of the quantics is substituted a linear function of the quantics. Thus the eliminant of a system of quantics u, v, w is a combinant. For, evidently the result of substituting the common roots of vw in u+Av+pw

is the same as that of substituting

them in w; and the eliminant of u+2Av+ uw, v, w is the same as the eliminant of uew. In addition to the differential equations satisfied by ordinary invariants, combinants must evidently also satisfy the equation

Gol

Modal.

eal

da * db * de

+ &c. = 0.

It follows from this that in the case of two quantics a combinant is a function of the determinants (ab’), (ac’), (bd’), &c.; in the case of three, of the determinants (ab’c’), &c.; and will accordingly vanish identically, if any two of the quantics become identical. If we substitute for u,v; Aut pv, Nu+p’v, every one of the determinants (ad’) will be multiplicd by (Ap’— A’p); and therefore the combinant will be multiplied by a power of (Aw’— Au) equal to the order of the combinant in the coefficients of any of the quantics. Similarly for any number of quantics. ‘There may be in like manner combinantive covariants, which are equally covariants when for any of the quantics is substituted a linear function of them. For instance, the Jacobian (Art. 88) U4)

Uy)

U,

162

SYSTEMS

OF QUANTICS.

if we substitute for u, lu+mv+nw, for v,lutmv+n'w, &e. by the property of determinants, becomes the product of the determinants (/m’n’), (u,v,w,). The coefficients of a combinantive covariant are also functions of the determinants (a0’), (ac’);

(ab’e’’), &e.

177. If w=(a,b, c...Ka, y)", v=(a’, vc... fa, y)” be any two binary quantics of the same depos then u+kv or (a+ ka’, b+ kb’...Xa, y)", where we give different values to /, denotes a system of quantics which are said to form with u, v an involution. Now there will be in general 2 (n— 1) quantics of the system, each of which will have a square factor. For the discriminant of a quantic of the n” degree is of the order 2(n—1)

in the coefficients (Art. 105).

If then we sub-

stitute a+ka’ for a, b+kb’ for b, &e., there will evidently be 2 (n—1) values of &, for which the discriminant will vanish. If we make y=1 in any of the quantics, it denotes 2 points on the axis of 2 We have just proved that in 2 (n—1) cases, two of the m points denoted by u+kv will coincide; or, in other words we may say, that there are 2(n—1) double points in the involution. When u+kv has a square factor x—a, we know that a satisfies the two equations got by differentiation, viz. u, + kv, =0, u,+kv,=0, and therefore will satisfy the equation got by eliminating k between them, viz. uv,—u,v,=0. Now u,v,—uU,v,, Which is of the degree 2 (n— 1), is the Jacobian of u, v; and we see that by equating the Jacobian to 0, we obtain the 2(n-—1) double points of the involution determined by UO 178.

If wand v have a common factor, this will appear as a

square factor in their Jacobian. First, let it be observed, that since nu=au,+ yu, nv=a2v,+yv, then if we write J for u,v, — U,v,, we shall have n Gas — vu,) =a, n (uv, —vu,)=— yd. * In like manner, for a ternary quantic, the Jacobian of w, v, w is the locus of the double points of all curves of the system «+ kv + lw which have double points, And similarly for quantics with any number of variables,

INVOLUTION.

163

Differentiating the first of these equations with regard to y, and the second with regard to a, we get a (wv,, a VU,») = ad,

be (ur,, a vu,,) ae

yJ,

It follows from the equations we have written, that any value a of x which makes both uw and v vanish, will make not only J vanish but also its differentials J,, J,, and therefore «—a must be a square factor in J. Or more directly thus: let u=8¢, v=Aw, where B=la+my; then u,=1¢+ Bd,, u,.=md + B$,, v1, =lpt By, and

UjVy- UU,= B(¢,¥,-

whence

’,¥,)+ BU py,—,v)

v,= mb + BY, 3

+Bmn(o,"—o¥,),

(n — 1) (u,v, — U,v,)

= (n— 1) B’($,%,— O:,) + B (la + my) ($0, — $,4,) = np” (dv, = $,,)-

It follows from what has been said, that the discriminant of the Jacobian

of uw, v must

contain

£& their resultant as a factor;

since whenever & vanishes, the Jacobian has two equal roots. Thus in the case of two quadratics. (a, b, chaz, y),

(a’, 0, cha, y)’,

the Jacobian is (ab’) x” + (ac’) xy + (bc) y’, whose discriminant is 4 (ab’) (bc’) — (ac’)’, which is the eliminant of the two quadratics. In the case of quantics of higher order, the discriminant of the Jacobian will, in addition to the resultant, contain another factor, the nature of which will appear from the following articles. 179. It has been said that we can always determine 4, so that «+ kv shall have a square factor. But since two conditions must be fulfilled, in order that w+ kv may have a cube factor, k cannot be determined so that this shall be the case unless a certain relation connect the coefficients of u and v. This condition will be of the order 3(n— 2) in the coefficients both of u and v. If (x—a)® be a factor in w+hkv + lw, 2—a will be a factor in the three second differential coefficients, or «=a will satisfy the equations u,, thu, + lw,,=0, u,,+kv,, + lw, =0, u,, + kv, + lw, =9,

164

SYSTEMS

OF QUANTICS.

whence eliminating & and J, «=a

will satisfy the equation

My

Uy

rs

Yor

Yin

Vie

Uso Vo) Woo

|= 0-

If then we use the word treble-point in a sense analogous to that in which we used the word double-point (Art. 177), we see that the equation which has been just written gives the treble points of the system u+hv+lw; and since the equation is of the degree 3(n—2), there may be 3 (n — 2) such treble points. But we could find the number of treble points otherwise. Suppose we have formed the condition that w+ kv should admit of a treble point, and that this condition is of the

order p in the coefficients of w. If in this condition we substitute for each coefficient (a) of wu, a+la”, we get an equation of the degree p in 7; and therefore p values of 7 will be found to satisfy it. In other words, p quantics of the system u+kv+lw will have atreble point. It follows then from what has just been proved that p=3(n—2). And the same argument proves that the condition in question is of the order . 3 (n — 2) in the coefficients of v. This condition is evidently a combinant; for if it is possible to give such a value to 4, as that u+sv shall have a cube factor, it must be possible to determine f, so that (w+ mv) + kv shall have a cube factor.

180. If u+kv have a cube factor (x- «)°, then the Jacobian of u and v will contain the square factor (w—«a)*. For the two differentials w,+kv,, u,+ kv, will evidently contain this square factor, and therefore it will appear also in the Jacobian, which may be written (u,+kv,)v,—(u,+kv,)v, If then S=0 be the condition that «+ %v may have a cube factor, S will be a factor in the discriminant of the Jacobian, since if S=0 the Jacobian has two equal roots, and therefore its discriminant vanishes. If & be the resultant, the discriminant of the Jacobian can only differ by a numerical factor from RS. For since the Jacobian is of the degree 2(n—1), its discriminant is of the degree 2 {2 (n—1)—1} in its coefficients, which are of the first

order in the coefficients of both wu and v

Now JB is of the order

INVOLUTION.

165

m in each set of coefficients, S of the order 3(n—2). Both these are factors in the discriminant; and it can have no other, since n+3 (n—2)=2 {2(n-1)-1l}. 181. The discriminant of w+v, considered as a function of k, will have a square factor whenever uw and v have a common factor. In fact (Art. 111) the discriminant of u+kv will be of the form (a+ha')@+(b+k0')’y. But if uw and v have a common factor, we can linearly transform u and v so that this factor shall be y, that is to say, so that both a and a’ shall vanish. The discriminant will therefore have the square factor (b+ %0’)?; and since the form of the discriminant is not affected by a linear transformation of the variables, it always has a square factor in the case supposed. It follews that if we form the discriminant of w+ hv, and then the discriminant of this again considered as a function of k, the latter will contain as a factor & the resultant of w and v. For it has been proved that when R=0, the function of k has two equal roots, and therefore its discriminant vanishes. For example, the discriminant of a quadratic ac— 0’ becomes, by the substitution of a+ ka’ for a, &c., (ac —b’) +k (ac’ + ca’ — 200’) + k (a’c’ — b”), whose discriminant is 4 (ac — b”) (a’c’ — b”) — (ac’ + ca’ — 2b0')?. But this is a form in which, as was ae by Boole, the resultant of the two quadratics (a,b, c{a,y)*, (a’, UY, oa, y)’ can be written (cf. Ex. 6, p. 24). This form, allthe component parts of which are invariants, is sometimes more convenient than that given (Art. 178). In the case of quantics of higher order, the discriminant of the discriminant will have & as a factor, but will have other factors besides. 182.

If w have either a cube factor or two distinct square

factors, the discriminant of w+ 4v will be divisible by #*. if the discriminant of u be A, that of u+ kv is

A+k(a Bip

da

db

+ &e.) + &e.

For

166

SYSTEMS

OF QUANTICS.

Now when wu has a square factor A vanishes; and it appears from the expressions in Art. 114, that if either three roots of u are equal a = 8 =¥, or two distinct pairs be equal

then all the differentials of A, SF &c., vanish;

a= 8,

y= ds

and therefore

the coefficient of 4 in the expression just given vanishes.

The

discriminant therefore contains %’ as a factor. It is evident hence that if «w+av have a cube or two square factors, the discriminant of w+ kv will be divisible by (k—a)’; since w+ kv may be written wt+av+(k—a)v. If then, as before, S=O express the condition that the series w+ kv may include one quantic having a cube factor; and if 7=0 be the condition that it should include one having two square factors, both S and 7 will be factors in the discriminant with respect to k of the discriminant of u+kv. For we have just seen that the discriminant has a square factor if either S=0 or 7Z=0. We proved in the last Article that the discriminant has & as a factor; and, in fact, the discriminant will be, as Prof. Cayley has observed, RS* 7”. I do not know whether there is any more rigid proof of this than that we see that there is no other case in which the discriminant of w+kv has a square factor; that we find in the case of the third and fourth degrees that Sand 7’enter in the form S*, 7”; and that we can thus account for the order in general. For the discriminant of «+ kv is of the order 2(n—1) in &, and the coefficients are of the order 0, 1, 2 (n—1) inthe aoeicents of either quantic. The degen! flion with respect to k& will be of the order 2(n—1) (2n—3) in the coefficients of either quantic. But R is of the order n, S of the order 3 (n—2), and it will be proved in a subsequent lesson that 7’ is of the order 2 (n—2) (n- 8) ’ and

2 (n — 1) (2n—3) =n+9(n- 2) +4 (n—2)(n—3). 183. It was stated (Art. 176) that every combinant of u, v becomes multiplied by a power of (Aw’ —XA'u) when we abe stitute Au + wv, Nu+t+yp'v for u, v. It will be useful to prove otherwise that the eliminant of u, v has this property. First, let it be observed that if we have any number of quantics,

ELIMINANTS.

167

one of which is the product of several others, u, v, ww'w", their resultant is the product of the resultants (wow), (wow'), (wow"). For when we substitute the common roots of u, v in the last and multiply the results, we evidently get the product of the results of making the same substitution in w, w’, w”. Again, the resultant of u, v, kw is the resultant of u, v, w multiplied by k™ since the coefficients of w enter into the resultant in the degree mn. If now F (wu, v) denote the resultant of u, v, which are supposed to be both of the same degree n, we have wR (Au + po, Nut pv) = RB (Apu + wp’v, Nu + p’v)* = Rh {(Aw —d’p) u, Nut w’r} = (Ap — Vp)" BR (wu, Nut p’r)

= (Au —d’p)" ew" B (u,v); whence

&(Au+ pv, Vu+ pv) = (Ap — Np)” # (u, v).

By the same method it can be proved that the eliminant of Aut putvyw, Nut pvtvw, Wut pv tv'w is (Ap’v’)” times that of u, v, w, and so on. 184. If U, V be functions of the orders m and n respectively in w, v, which are themselves functions of x, y of the order p, and if D be the result of eliminating u, v, between U, V; then the result of eliminating x, y between U, V will be D” times the mn” power of the resultant of wu, v. For U may be resolved into the factors u—av, u— Pv, &e., and V into u—a’v, u—’v, &e. And, Art. 183, the resultant of U, V will be the product of all the separate resultants u— av, w—a’v. But one of these is (a—a’)” R(u, v). There are mn such resultants. When therefore we multiply all together, we get the mn” power of R (u,v) multiplied by the p” power of (a—a’) (a—a@”), &e. But this last is the eliminant of U, V with respect to u, v.

185. Similarly, let it be required to find the discriminant, with respect to x, y, of U, where U is a function of u, v. First, let it be remarked

(see Art. 110) that the discriminant

of the product of two binary quantics u, v is the product of the * The resultant of u+ kv, v, being the same as the resultant of wu, v, Art. 176, we next subtract m times the second quantic from the first.

168

SYSTEMS

OF QUANTICS.

discriminants of « and v multiplied by the square of their resultant. If then U=(u—av) (w —8v) &e., the discriminant of U will be the product of the discriminants of w— av, w—fv, &e. by the square of the product of all the separate resultants u— av, Bat, as before, any of these will be (2—£)’ £ (u, v). u—Bv. If then m be the degree of U considered as a function of w, v; there will be 4m (m— 1) separate resultants, and the square of the product of all will be (a- 8)” (a—y)”, &. x BR" (u, v). But (a — 8)? (a—¥)’, &e. is the discriminant of U considered as

a function of w, v. If then we call this A, we have proved that the product of the squares of the separate resultants is A’R”™”™, Let us now consider the product of the discriminants of u—av, w— Bv, &c ; this is the result of eliminating @ between the discriminant of w— Ov, which is a quantic of the order 2(p—1) in @ and the quantic of the m” order got by substituting w=6v in U. Or this product has been otherwise represented by Dr. Sylvester. If a,, 6, be the coefficients of x in u, v, then (Art. 108) the resultant of w—av, u,—av, will be a,— ab, times the discriminant of w—av. But R (u,v) & (u-av, u,—av,) =F (u—av,u,v—avv,) = B (u—av, uv—uv,).

Now (Art. 178) p (uv—uv,)=yJ where J is uv,—u,v,, and R(u-—av,y) isa,—ab, It appears thus that the discriminant of w— av differs only by a numerical factor from the resultant

of (u—av, J) divided by R(u, v). The product then of all the discriminants will be the—resultant of J and the product u— av, u—Bv, &c., in other words, the resultant of U, J—divided by the m™ power of R(u,v). Thus we have Dr. Sylvester’s result (Comptes Rendus, LYIII., 1078) that the discriminant of U with respect to x: y is A’R(u, v)"" R(U, J). But it will be observed that the result expressed thus is not in its most reduced form since & (U, J) contains the factor R (u, v)”.

186. We have next to see what corresponds in the case of ternary and quaternary quantics to the theory just explained for systems of binary quantics. Let then uw and v be two ternary quantics, and let us suppose that we have formed the

ELIMINANTS.

discriminant quantics w, will have a wili have a v touch each equation of

of u+kv. Then (for certain relations between the v) this discriminant considered as a function of & square factor. In the first place the discriminant square factor, if the curves represented by w and other. For we have seen (Art. 117) that if the a curve be az" + bz""x+ cz” "y+ &.=0, its dis+

criminant is of the form a0 + 0°6 + be +c’y.

then of

169

The discriminant

u+kv will be of the form (a + ka’) 0+ (b+ kb’)? $+ &e.

But if we take for the point xy, a point common to w and », both a and a’ will vanish; and if we take the line y for the common tangent, both 4 and 0’ vanish; and the discriminant will be of the form (c+ kce’)’ x; and therefore will always have a square factor in the case supposed.

187. Again, the discriminant will have a square factor if u have either a cusp or two double points. The vanishing of the discriminant A of a ternary quantic gives the condition that U,, U,, vu, Shall have a common system of values. If, however, u have either two double points, or a cusp, w,, u,, u, will have two systems of common values, distinct or coincident, and therefore (Art. 103) not only will A vanish, but also its differentials with respect to all the coefficients of vu. The discriminant then of u+kv

being in general

A+% (

a ew &e.) + &e. will in this

case be divisible by #7. And as in Art. 182, it will be divisible by (k—a)’ if the curve u+ av have either a cusp or two double points. Let then R=O0 be the tact-invariant of w and », that is to say, the condition that the two curves should touch; S=0 the condition that in the system of curves u+kv shall be included one having a cusp; and 7’=0 the condition that there shall be included one having two double points. It has been proved that R, S, 7 are all factors in the discriminant of the discriminant of w+v, considered as a function of k. In fact this discriminant will be RS°’7”. For an investigation of the orders of , S, 7, when both curves are of the same degree, see Higher Plane Curves, Art. 399. The tact-invariant & is of the order 3n (n— 1) in the coeffi-

170

SYSTEMS

OF QUANTICS.

cients, 8 of the order 12(n—1)(n—2), and TJ of the order 2 (n—1) (n—2)(3n?—3n—11): the discriminant of w+kv in regard to #:y is of the order 3(n—1)* and the discriminant of this with regard to k of the order 3 (n — 1)? (3n°—6n +2), and we have identically

3 (n — 1)’ (3n? — 6n + 2) = 8n (n —1) + 86 (n— 1) (n—2) +3 (n—1) (n —2) (3n*® — 8n — 11), showing that the order of the discriminant of RST”.

is equal to that

188. The theorem given, Art. 110, for the discriminant of the product of two binary quantics cannot be extended to ternary quantics; for the discriminant of the product of two will, in this case, vanish identically. In fact, the discriminant is the condition that a curve shall have a double point; and a curve made up of two others has double points; namely, the intersections of the component curves. Or, without any geometrical considerations, the discriminant of wv is the condition that values of the variables can be found to satisfy simultaneously the differentials wv,+vu,, uv,+vu,, Ke, But these will all be satisfied by any values which satisfy simultaneously wand v; and such values can always be found when there are more than two variables. But the theorem of Art. 110 may directly be extended to tact-invariants. The condition that wu shall touch a compound curve vw will evidently be fulfilled if u touch either v or w, or go through an intersection of them. For an intersection counts, as has been said, as a double point on the complex curve; and a line going through a double point of a curve is to be considered doubly as a tangent. Hence if Z'(u, v) denote the tact-invariant of u, v, we have PL(u, vw) = T(u, v) T(u, w) {BR (u, 2, w)}’,

when F (u, v, w) is the resultant of u, v, w. And the result may be verified by comparing the order in which the coefficients of uw, v, or w occur in these invariants. Thus, for the coefficients of wu, we have

(n+p) (n+p +2m —3)=n(n+2m—3)+p(p+2m-—8) +2np.

TACT-INVARIANTS,

ivAl

189. The theory of the tact-invariants of quaternary quantics is given in Geometry of Three Dimensions, p. 544; and there is not the least difficulty in forming the general theory of the class of invariants we have been considering, to which Dr. Sylvester proposes to give the name of Osculants. Let there be ¢ quantics, U, V, W, &c. in & variables; then the osculant is the condition that for the same system of values which satisfy U, V, &c. the +, &c., &c. shall be connected by tangential quantics eU/+yU an identical relation

(aU

+ &e.) + w (a Vi + &e.) + v (w@W/ + &e.) + &e. = 0.

In other words, the osculant is the condition that the equations U=0, V=0, &c., and also the system U, Viet

U, Ven

U, &e. ae Oe,

||=0

W,, W,, W,, &e.

can

be simultaneously satisfied.

This latter system having &

columns and 7 rows is equivalent to 4-7+1 equations; therefore this system combined with the given ¢ equations is apparently equivalent to +1 equations in & variables. It is really, however, only equivalent to & equations; for writing U=0 in the form zU,+ yU,+ &c.=0, and similarly for V, &c, we see that when the system of determinants is satisfied, and all but one of the equations U=0, V=0, W=0, &c., the remaining one must be satisfied also. The system then being equivalent to & equations in & variables cannot be simultaneously satisfied unless a certain condition be fulfilled. The order of this condition, in the coefficients of U, is found by the same method as in We write for U, U+ du, and Geometry of Three Dimensions. we examine how many values of the variables can simultaneously satisfy the 7+ 1 equations V, W, &c., and the system equivalent

to k—1 equations Us. Uy &e. || = 0. a

Vi, W,

0

Sea

SLC.

Vy Vy &e. Wi, Wy &e.

172

SYSTEMS

OF QUANTICS.

The order of the 7-1 equations V, W, &c, is the product of their degrees n, p, &c.; and the order of the osculant in the coefficients of U is the product of this number by the order of the system of determinants, which is found by the rule given in a subsequent Lesson on the order of systems of equations. When we are given but one quantic, the osculant is the discriminant ;when we are given & quantics in & variables, the osculant is the resultant. The theorem of Art. 110 may be extended to osculants in general; viz. that if we form the osculant of k—1 quantics in & variables, and if the last be the product of two quantics U, V, then the-osculant of the entire system will be the product of the osculant of the system of the other k —2 with U, that of the system of s- 2 with V, and the square of the resultant of all the quantics.

190. We have already seen (Art. 151) how the invariants and covariants of a single quantic are derived from those of a system of quantics in the same number of variables; and we wish now to point out how the invariants and covariants of a single quantic are connected with those of a system of quantics in a greater number of variables. Suppose, in fact, we had twa ternary quantics, geometrically denoting two curves, we can, by eliminating one variable, obtain a binary quantic satisfied by the points of intersection of these curves; and it is evident, geometrically, that the invariants of the binary quantic (ex: pressing the condition, for instance, that two of these points should coincide, or should have to each other some permanent relation) must also be invariants of the system of two ternary quantics. Conversely, we may consider any binary quantic as derived from a system of two ternary quantics; for we have

only to assume X=$(c,y), Y=¥(x,y), Z=x(0%,y), equa

tions which in themselves imply, by elimination of 2 and y, one fixed relation between X, Y, Z, and from which, combined with the given binary quantic equated to zero, we can obtain a second such relation. The simplest example of such a transformation is that investigated by Mr. Burnside Quarterly Journal x. (1870) p. 211; (compare Conics, note pp. 386-7), where $, , x

BINARY

CONNECTED

WITH

TERNARY

SYSTEMS.

ss

are quadratic* functions of x and y. The substitution is then reducible by linear transformation to X=2*, Y=2zry, Z=y’, giving the fixed relation 4XZ7- Y’=0. By making these substitutions for x? &c. in a binary quantic of even degree, we have at once a second relation between X, Y, Z; if the quantic be of odd degree, it can be brought to an even degree by squaring. The resulting relation is obviously not unique, but is of the form ¢$,,+¢,,,.(4XZ- Y*), where ¢,,, is any one form of the relation, and the coefficients in @,, are arbitrary. Geometrically, the binary quantic of the m” degree is thus made to represent m points on a conic, determined when m is even by the intersection of the conic with a curve of the order 4m, and when m is odd with a curve of order m touching the conic in m points. Among these forms there is always one whose invariants and covariants are also invariants and covariants of the given binary quantic.T Thus the binary quadratic ax’ + 2bxy + cy’ is replaced by the system aX + bY+cZ,4XZ— Y"’, and geometrically denotes the two points of intersection of a line with a conic. The discriminant of the quadratic is also an invariant of the system; that whose vanishing expresses the condition that the line shall touch the conic. So, in like manner, the system of two binary quadratics ax’+ 2bay + cy’, a’x*+2b’xy+cy’, gives rise to the system of a conic and two lines. The invariant of the binary system ac’ +ca’— 260’ (Art. 151) is also an invariant of the ternary system; viz., its vanishing expresses that the lines are harmonic polars with respect to the conic. If three lines Z, M, N be mutually harmonic polars with respect to a conic, we know (Conzcs, Art. 271) that the equation of the conic may be written in the form V= /L’ + mM’ + nN*=0, whence we infer immediately that if three binary quadratics be * If linear functions had been taken, the transformation could be reduced to X=a2, Y=y, Z=0, and the binary quantic of the n+ degree would represent nm points on the line Z (see Art. 177). + This form can be found by operating on ym+ pom-2 (4K4Z— ¥ 2) with the form qd?

reciprocal to 4XZ— Y?, viz. ixaz oy and equating to zero the coefficients of every term in the result.

174

SYSTEMS

OF QUANTICS.

connected in pairs by the relation ac’ + ca’ — 2bb’ = 0, their squares are connected by an identical relation 1L*+mM*+nN*=0, for V vanishes identically when we return to binary quantics. To the Jacobian of two binary quadratics answers, for the system of two lines and a conic, the line 2(ab’) X+(ac’) ¥Y+2(dc/) Z, which is also a covariant of that system. In fact, it is the polar with respect to the conic of the intersection of the two lines. More generally, the Jacobian of any system wu, v will be transformed into the Jacobian of the system formed by JU, V, and the fixed conic. For let w,, w,, «,, denote the differentials of u with respect to X, Y, Z, which, it will be remembered, denote x”, 2xy, y’ respectively, then the Jacobian is Y's — LY; an LU, + YU,

LU, + YU,

Xv, + Y%,) Lv, + XY,

=|

u 1? Ee

U,

U,

23s

RCAC

4,-4% Y, Xx a

U

rP)

Uy

U,

U, phoney a let)

Us

99 Us

PY

but the terms in the first line are proportional to the differentials of 4XZ— Y’. The same method being applied to the discussion of the biquadratic, it is found to be equivalent to the system of two conics, viz. the fixed conic 4XZ— Y’, and the conic aX? +cY¥*?+4+eZ" + 2dYZ+

2cZX + 2bXY=0,

the discriminant of the latter conic being also an invariant of the quartic (Art. 171). So again the system of two binary quartics is equivalent to a system of three conics. We shall have occasion in the next Lesson to give further illustrations of this method: it has been applied by Mr. W. R. W. Roberts to the system of two cubics which involve properties of a twisted cubic, Proc. Lond. Math, Soc. vol. X111. (1881), and the relation between binary and quaternary forms is developed with the general symbolic formule in a paper by Dr. Lindemann, Math. Ann. (1884) XXUI. p. 111, &e.

Gaizoe |

LESSON APPLICATIONS

XVII.

TO BINARY

QUANTICS,

191. HaAvine now explained the most essential parts of the general theory, we wish to illustrate its application by enumerating the different invariants and covariants of binary quantics for the lower degrees. If S and 7 be invariants of the same degree, or covariants of the same degree and order, and & any numerical factor, then S+%7, which will of course be also an invariant or covariant, will not be reckoned in our enumeration as distinct from the invariants S and 7. And, generally, any invariant or covariant which can be expressed as a rational and integer function of other invariants and covariants of the same or lower degrees is said to be reducible, and will not be considered as distinct from these latter functions. It is otherwise if the expression be not rational and integer. Thus, if S be an invariant of the second and 7’ of the third degree, then though S°+kT? would not be regarded as a new invariant, yet if it be a perfect square, and we have h’= 8° +7", we count & as a new invariant distinct from S and 7’; and call it irreducible. It was proved in Art. 121 that a binary quantic has n—3 absolute invariants, and in Art. 122 that from any two ordinary invariants an absolute invariant can be deduced. We should infer, therefore, that the number of independent ordinary invariants is one more than the number of absolute invariants; or, in other words, that a binary quantic of the n” order has nm —2 invariants, in terms of which every other invariant can be expressed. But as it does not follow that the expression is necessarily rational, we do not in this way obtain any limit to the number of irreducible invariants. And so as regards the covariants (including in this expression the invariants) we shall presently see that for a quantic of the n” order there are, inclusive of the quantic itself, » covariants, such that

176

APPLICATIONS

TO BINARY

QUANTICS.

every other covariant multiplied by a power of the quantic is equal to a rational and integer function of the » covariants; thus, each such other covariant is a rational, but not an integer function of the n covariants; and we do not hereby obtain any limit to the number of the irreducible covariants. We have stated (Art. 145) the method by which Prof. Cayley originally attempted to determine the number of distinct covariants and invariants. He did not at the time succeed in obtaining any limit to their number for quantics above the fourth order. Subsequently Gordan proved (see Crelle, vol. LXIx., or Clebsch, Theorie der bindren algebraischen Formen, p. 255, also his Programm for the University of Erlangen, Ueber das Formensystem bindrer Formen, Leipzic, 1875), that for a binary quantic or system of binary quantics, the number of distinct invariants and covariants is always finite; and he has given a process by which when we have the complete system of invariants and covariants for a quantic of any degree, we can find the system corresponding to the next higher degree. His proof, which is founded on an analysis of the different possible expressions by the symbolical method explained Lesson xIv., will be found in a subsequent Lesson on that method. Later still, Prof. Sylvester has investigated the whole subject by Prof. Cayley’s method, founded on the theory of the partition of numbers, in various memoirs in the Comptes Rendus, vol. LXXXIV. pp. 974-5, 1113-6, &c., and subsequent volumes, as also in the American Journal of Mathematics.

192. It will be convenient to bear in memory what was proved, Art. 147, that a covariant is completely known when its leading coefficient, or, as we have there called it, its source, is known; this coefficient being any function of the differences of the roots of the quantic.* Thus take the quantic (a, b,c... Ka, y)", we know that, in the case of the quadratic, (ac — 6”) is an invariant; and if we desire to form the covariant * Such functions have been called semi-invariants or seminvariants, as they remain unaltered (see Art. 62) when we substitute x +A for x, but not necessarily when we substitute y + for y; and as they satisfy one of the differential equations given in that article, but not necessarily the other.

INTEGRALS

OF PARTIAL

DIFFERENTIAL

EQUATION.

177

(ac — b*) a? + &e., having this leading term, we observe that the weight of the given source and its degree in the coefficients are each =2, so that writing @=2 in the formula (Art. 147) 2=}(n@—p) we have p=2(n—2). The other coefficients are found by the method explained in that article; thus the covariant is found to be (ac — 0) a?" + (n — 2) (ad — bc) a *y

+ (n— 2) {(bd —c*) + 4 (n— 8) (ae—c’*)} 2” *y? + &e. It follows also from what has been stated in the article referred to, that any algebraic relation between the sources of different covariants implies a corresponding relation between the covariants themselves. Prof. Cayley has used this principle in attempting to form the complete system of the covariants of a binary quantic; and though it does not lead to any general theory it furnishes the most elementary and satisfactory proof of the numbers of concomitants for functions of the first four degrees. The leading coefficient of any covariant being a function of the differences must (Art. 62) satisfy the differential equation (ad, + 2bd, + 3cd,+ &c) U=0: and we assume that JU is a rational and integer function of a, b,c, &e. Now, if we solve the partial differential equation, we find that U must be a function of

a, ac— 0b’, a’d— 3abce + 20°, a’e — 4a°bd+ 6ab’c — 3b", &e , where the law of formation of the successive terms is obvious; and, in fact, the covariants of which these terms are the leaders are each the Jacobian of the preceding covariant in the series, combined with the original quantic, We shall refer to these quantities as J,, L,, L,, &c. and we see that the leading coefficient of any covariant must be a function of these quantities: and it must of course be a rational function of them. The question is whether there are any rational, but not integer functions of Z,, L,, £,, &c., which are rational and integer functions of a,6,¢; and a little consideration shows that the only admissible form is that of a rational and integer function divided by a power of Z,, that is a. For, the leading

178

APPLICATIONS

TO BINARY

QUANTICS.

coefficient in question is a rational function (a, b, c,...)3 and if we make in it

of the coefficients

}=0, it becomes a rational

function of a, c, d, &c., and by multiplying by a suitable power of a it can be made an integer function of a, ae, a’d, a’e, &c. But these are the values of L,, L,, &c. on the supposition of b=0. Thus we see that the leader of any covariant can only be the quotient by a power of a of an integer function of these m quantities. Conversely, the problem of finding all possible covariants is the same as that of finding the new functions which arise when rational and integer functions of L,, L,, &e.

are formed which are divisible by a. To find these functions we make a=0 in L,, L,, &c..and eliminate 6 between any pair; we thus get a function of L,, &c. which vanishes on the supposition of a=0, and therefore is divisible by a power of a. By performing the division we obtain the leader of a new covariant. This again may be treated in like manner, by putting a@=0 and examining whether it be possible to eliminate the remaining coefficients. This method will be better understood from the applications which will be made presently. It is obvious that the same considerations apply to the still simpler forms—of lowest degrees—of particular integrals of the partial differential equation a, ac—b*,a°d—3abe+20", ae—4bd+ 8c’, a'f— 5abe + 2acd — 6be’ + 8b°d, &c. of which the second, fourth, &c. are the successive quadrinvariants of even quantics as they arise, and the third, fifth, &c. are the sources of the evectants of the successive quartinvariants of the corresponding odd quantics as they arise. See Art. 142.

193. We have already stated the principal points in the theory of the quadric form (a, b, c{z,y)*. Since there are but two roots and only one difference, there can be no function of the differences of the roots but a power of this difference; and the odd powers, not being symmetrical functions of the roots of the given quadratic, cannot be expressed rationally in terms of its coefficients. It thus immediately follows that the quadric has no covariants other than the quantic itself, and no

invariant other than the powers of the discriminant, ac—*,

SYSTEM

which

OF TWO

QUADRICS.

179

is proportional to («— 8)’. We have already shewed

(Art. 157)

that it follows, by Hermite’s

law

of reciprocity,

that only quantics of even degree can have invariants of the second order in the coefficients. These are the system whose symbolical form is 12", explained Art. 153, ac— b*, ae— 4bd+ 3c’, ag — 6bf+ 15ce — 10d’, &e. If we make y=1 in the quadratic it denotes geometrically a system of two points on the axis of x, and the vanishing of the discriminant expresses the condition that these points should coincide, Art. 177. System of two quadrics. This system (a, 0, cha, y)’, (a’, b, cha, y),

has the invariant 12° or ac’ + ca’— 2bb'. When each quantic is taken to represent a pair of points in the manner just stated, the vanishing of this invariant expresses the condition (see Conics, Art. 332) that the four points shall form a harmonic system, the two points represented by each quantic being conjugate to each other. We have also proved (Art. 177) that the covariant 12 (or the Jacobian of the system) represents the foci of the system in involution determined by the four points. It is easy to see, as in Art. 169, or by Conics, Art. 333, that the Jacobian may be written in the form y’, — LY, x J(u, vj)=|a, 6, €

Gan

Dn OC:

Now by the ordinary rule for multiplication of determinants we have Y') — XY; ac”

a a’,

or

| ae",

2xy, ¥

0, ¢|X|¢,

—20,

b,

a

c

c;

ob!

;

a’

uw,

v

@|=|4, 2D,

0,

A

v,

A

;

9 D'

-

2J? = —2u’D' + 2uvd — 20D,

where J denotes the Jacobian, D and D’ the discriminants of

the quantics, and A the intermediate invariant ac’+ ca'— 200’.

180

APPLICATIONS

TO BINARY

QUANTICS.

This equation includes the theorem stated Art. 190, for the case A=0. The equation just given may also be easily verified by means of the canonical form. We have seen (Art. 177) that there are two values of k, for which u+ kv is a perfect square, and if these squares be 2 and y’ the system may be written aa?+cy’, a’x’+c'y’, or more simply x+y’, ax*+ cy’. We have then A=a+c, J=(c—a) ay, by means D=1, D’=ac, of which values the preceding equation is at once verified. So again the Jacobian of u, J is for the canonical form 3,(c— a) (w*—y’), and therefore is in general 4Au—Dv. The invariant A taken between uw and J vanishes identically, as is geometrically evident. All other invariants or covariants of a system of two quadrics may be expressed in terms of wu, v, J, D, D’, A. Thus the eliminant (ac’ — ca’)? + 4 (ba’ — ab’) (be’ — cb’), may also be written in the form

(ac’ + ca’ — 2bb’)? — 4 (ac — 8°) (a’c’ — 8”). In other words, the eliminant is the discriminant either of the Jacobian

(ab’) a* + (ac’) ay + (be) y’, orof

(ac — 6°) A? + (ac’ + ca’ — 200’) Apt (a’c’ — b”) w*,

The former expression is linearly transformed into the latter by the substitution X4 + wd’, — (Aa + wa’) for x and y. System of three quadrics (a, 6, cha, y)*, (a’, bY, cha, y)*, (a, 0”, ca, y)* This system has, in addition to the invariants and covariants corresponding to the respective pairs of quadrics, the determinant 23.31.12,

whose vanishing expresses the condition that the three pairs of points represented by the quadratics shall form a system in involution (Ex. 7, p. 25, also Contcs, p. 310). This invariant

THREE

OR MORE

QUADRICS.

181

formed for uw, v, J is another expression for the eliminant of u and v. The expression found for J? of two quadrics may be generalized, if in the second determinant we write in the second and third rows 0”, —2b”, a”; ¢”, —20’’, a”. We thus find that if there be four binary quadrics u,, w,, 24, %) 2S) J 34 ia

0 ?

u, ?

u,

Us,

Ds

De

Ug)

Ds

Ine,

We get, similarly, an expression invariants 1237-4562 RR a,

2B ols

=

b, C,

Cy

a, by C

— 2b,, a

Can

255 bs C,

for the product of two Dy

2b,, Bhs ||

| Ces — 26, a,

D5) Da

Days Ve

Dy

26

Ds.) De

’

To these formule may be added the following, the truth of which is easily seen, Uy, + Udy, + UT = 9; 1 23 2° 31 3” 12 UDeoa ae u,fi.,, J Uli. a UP 9g = 0,

Biogty a Deis a Did, + Dios 7 ie

= (OED. a DS) ut

(D,D,, ak DED)

&e, u,

a (Dep

~ LUNI SSS) Usy &e.

From the linear relations connecting the w’s and J’s follow the quadric identities St

11° pore 23

ID22°gh31 25 Ded 83° 12 OS WD 23°AL81 Ah12 SID 0=

Us)

0

dD, 1b

Uy

Us,

Ds

4,

Di;

Day)

Dy

u,

Dy,

Dy

D,s) u,

31 dh12° dh28

Ao RID12° dh23° dl819

Thus a system of three binary quadrics at once gives rise to a conic and their three Jacobians to its reciprocal form. The equation of the conic is referred to any line and two others through its pole when the three binary quadrics are any two arbitrarily taken and their Jacobian. The conic breaks up into right lines if the three binary quadrics form an involution.

182

APPLICATIONS

TO

BINARY

QUANTICS.

194. It is to be remarked that, by means of Kuler’s theorem for homogeneous functions, the theory of those covariants of any quantic, the expression of which contains differential coefficients in not higher than the second degree, reduces itself to the theory of the quadric; and so every relation between the covariants of a quadric has answering to it a relation between such covariants of a quantic in general. Similarly, a relation between covariants of a cubic gives a relation between general covariants not involving differential coefficients in more than the third degree, and so on. Thus, the expression obtained for the square of the Jacobian of two quadrics gives the identical relation

{(aze + by) (in-+ oy) — (a’e + By) (Bar + cy)}

=— (ac — 0°) (a’x? + 2V’xy + cy’) + &e. But if a, b,c; a’, 0’, c’ denote the second differential coefficients of any two quantics, we have ax+ by=(n—1)u,, Wa?+ 2’ xy+cy?=n' (n’—1) v, &e., whence we have an expression for the square of the Jacobian of any two quantics (2 —1)* (n’ —1)? J? =—n” (v’—1)

+ nn’ (n— 1) (n’— 1) Aww— 2’? (n—-1) B’'v’, where H denotes the Hessian ac—b* and A, as before, the covariant ac’ + ca’ — 2b)’. So again, since the Jacobian involves only differential coefficients in the first degree, the Jacobian of J, u, involves them only in the second, and therefore can be expressed by means of the theory of the quadric. Writing Z, M for the first differential coefficients, we have i J=LM’-IL'M; T(J, u) = (aMM’—b(LM’'+ I’M) +cLL’} —{a’M?—20 LM +L"). But the values of the two members of the right-hand side of the equation are immediately found by the canonical form of the quadric, and are respectively nt ani

n eo

and

a

ee

n(n’ —1) (n—1)

Av,

THE

whence

CUBIC.

JI (J, uy) = aa

Hv- —

—

195. The cubic. the cubic

We

come

183

n

Aen.

_

next to the concomitants

of

U=(a, b,c, dha, y)*.

It has but one invariant (Art. 167), viz. the discriminant D =a’d* + 4ac’ — 6abed + 4db® — 307°. If the cubic were written without binomial coefficients, the discriminant would be 27a’d* + 4ac’- 18abcd + 4db°—0’c*, It is to be noted that the function here written is, with sign changed, the product by a* of the squares of the differences of the roots of the cubic. A useful expression may be derived from the last remark. Consider the three quantities B—y, y —a, a—8, they are the roots of a cubic for which a=1, 6=0,

c=— % (8-4) + (y— 4)?+ (a-8)'}, d=(B-4) (y—4) (2-8). Hence = (2a B—y)* (28 -y- a)’ (27-4 —B)? =4{(8- 9)+ (y- 4)?+ (a- B)P -27 (8-4) (y- 4)’(a- BY The Hessian 12? or JZ, is Gyan ae bora

(ac — 0°) a’ + (ad — be) ey + (bd—c’) y=

Y') — BY, we This has the same discriminant as the cubic itself (Art. 167). The cubic covariant 127.13, or the evectant of the discriminant, which we call J, is (see Art. 142)

(a’d—8abc+20°, abd+b°c—2ac’, 2b°d—acd—be*, 3bed—ad’—20° Ka, y)’; which may also be written in the determinant form

0,

d,

= 20, b

fe

This cubic may be geometrically represented as follows :—If we take the three points represented by the cubic itself, and take the fourth harmonic of each with respect to the other two, we

184

APPLICATIONS

TO BINARY

QUANTICS.

get three new points which will be the geometrical representation of the covariant in question. This theorem is suggested by its being evident on inspection, that if the given cubic take the form xy (x+y), then «—y will be a factor in the covariant, as appears by making a=d=0, b=c=1 in its equation. But x+y, «—y are harmonic conjugates with respect to a and y. Now, if a, 8, y, 5 denote the distances from the origin of four points on the axis of x, any harmonic or anharmonic relation* between them is expressed by the ratio of the products (a—8) (y—8) and (a—y) (@—8): and this ratio (see Art 136) is unaltered by a linear transformation; that is, when for each :

.

x

;

distance a we substitute woe P’.

,

Such relations, then, being

BB

unaltered by linear transformation, if proved to exist in one case, exist in general. We find that the other factors in the evectant of xy(x+y) are x+ 2y, 2x+y, so that our result may be written symmetrically, that the evectant of xyz (where x, y, 2 are connected by the linear relation a+y+z2=0) is (y—z)(¢-«x) (w-y). These considerations lead us to the expression for the factors of the covariant in terms of the roots of the given cubic: for if 8 be the distance from the origin of the point conjugate to a with respect to 8 and v; solving for 6 from a8 + ay—2By the equation 5 we get d=

“a= Bay”

Ray wey) H a-y¥ whence the covariant must be

a’ {(2a— B —y) a+ (2By — a8 — ay) y} {(28~a—y)x

+ (2ya— By — Ba) y} {2y¥-a— 8) a+ (248 —ya— 8) y} =— 27d, as may be verified by actual multiplication and substitution in terms of the coefficients of the equation. * The anharmonic ratio of four quantities has any of the six values, according to the order

assumed,

1

i»ae

1

A-1

1-), Pe

r

ae

A

4

which are in general

all different. They may come to have equal values either if X =1 when two values of the quantities are equal and the other values of the anharmonic ratio

are 0 and o; or if other values of the quantities form an are one imaginary

A=—1, when the quantities form a harmonic series, and the anharmonic ratio are 2 and 3; or if A7—A+1=0, when the equi-anharmonic series, three values of the anharmonic ratio cube root of —1, and three its other imaginary cube root.

THE

CUBIC.

185

Similarly for the quadric covariant, if w be an imaginary cube root of unity and we solve for 6 from

(2—¥) (6-8) + (8-7) (5- a) =0, we get

(2+ w8+ wy) 8+ By + wyat+ o'a8=0

to determine a distance equi-anharmonic to a, 8, y. have the covariant

{(a+ @B + wy)

Hence, we

2+ (By + wya+ wa8) y}

x [(a+ w°8 + wy) x + (By + w’ya+ wa8) y}, double this is found to be = ©(a—a)*(8—y)’, which, multiplied by a’ and expressed in terms of the coefficients, = — 18H, 196. We can now see that our list of covariants is complete. The leading coefficient of any covariant is a function of the differences B—y, y-—a, a—-f. Since the sum of these quantities is zero, any symmetric function of them can be expressed in terms of the sum of their squares, and their continued product. But since this product is only half symmetrical with respect to the roots of the given cubic, that is to say, is liable to change sign by an interchange of the roots, it can enter only by its square into a function expressible in terms of the coefficients. We thus see, that if the leader be a symmetric function of the differences, the covariant can be expressed as a rational function of U, H, D. But there is another function, viz. the product of the differences (22-—8-—+y) (28—y-4)(2y-—a— 8), which though only half symmetrical with respect to the differences, is symmetrical with respect to the roots of the given quantic. ‘This is the leading term of the covariant J. But obviously the square of this function can be expressed in terms of the sum of squares, and product of differences. ‘The expression has, in fact, been given in the last article. It is easy to prove, that in the case of the cubic written with binomial coefficients, we have a? (a— 8)?=18 (b’—ac), a*(B—y)’ (y- a)’ (a— 8)’ =—27D,

a (2a B- 9)(28-74) 2y—a— 8)=—27(a'd— Babe + 25")

186

APPLICATIONS

TO

BINARY

QUANTICS.

by the help of which values, the expression obtained in the last article gives the relation between the covariants, due to Prof. Cayley, J’? -DU’=—4H”°.

This relation may also be easily verified by using the canonical form U=az* + dy’, in which case we have D=a'd*, H=adzy, J=ad (ax*— dy’). Any other relation between covariants may be similarly investigated. Thus we can prove that the discriminant of J is the cube of the discriminant of U, the former discriminant being for the canonical form a*d*. So again we see that the Hessian of J differs only by the factor D from the Hessian of U. Prof. Cayley has used the relation just found between J, D, U, and H, to solve the cubic U, or, in other words, to resolve it into its linear factors. For, since J’—DU”® is a perfect cube, we are led to infer that the factors Jt+U/D will also be perfect cubes, and, in fact, the canonical form shows that they are 2a°da* and 2ad*y*. Now, since xa}+ yd3 is one of the factors of the canonical form, it immediately follows that the factor in general is proportional to

(UVD+J)§+(UVD-ZJ)A, a linear function which evidently vanishes on the supposition Ue (, Ex. Let us take the same example as in Art. 166, U = 423 + 9x?y + 18xy? + 17y3, Here we have D = 1600, J = 110x3 — 90x*y — 630xy? — 670y3, whence

U{D+J=10(8e+y)?;

U{D—J=50 (x + 3y)?;

and the factors are 32+ 4+ (a + 3y) 3J5.

197. The entire system of covariants for a cubic is also immediately found by Prof. Cayley’s method explained Art. 192. We start with the three covariants U, H, J, whose leading coefficients are L,=a, L,=ac—0°, L,=a'd—3abe+20%. If we make a= 0, the last two become — ’, 26°, whence by eliminating b we have 40,°+ L,=0. Thus we see that 4H°+ J? is divisible by a, and actually it is found to be divisible by a’, the quotient being D or a’d’ + 4ac’ + 4db°— 38’c? — Gabed. We have thus obtained the new invariant D, together with the equation of

SYSTEM

OF CUBIC

AND

QUADRIC.

187

connection 4H°+J?’=DU*. If in D we make a=0 it becomes Ad’ — 3b*c*, and since this combined with the preceding gives rise to no new relation between Z,, L,, D, we learn that the system of covariants is complete. 198.

System of cubic and quadric.

Let these be

U=(a, b, 0, d¥a, y); V=(A, B, Cha, 9)’; then the following is a list of the different independent covariants of the system. The figures added to each denote its order in the coefficients of the cubic and in those of the quadric. Three cubic covartants, viz. the original cubic UV, (1, 0); its cubicovariant (3, 0) which we call J, printed in full Arts. 195, 142, and the Jacobian of U, V, (1, 1) which is

(Ab—Ba) x*°+(2 Ac— Bb— Ca) x*y+(Ad+ Bc—2 Cb) xy’+(Bd —Ce) 7’. Three quadric covariants, viz. the original quadric V, (0, 1); the Hessian of the cubic (2, 0) and the Jacobian of these two (2, 1) which is

(A (ad— bc) -2B(ac—0’),

A (bd- c’)-C(ac- 0’), 2B (bd — c’) —C(ad — be) fx, y)*.

Four linear covariants, viz. L,(1,1) which is obtained by substituting differential symbols in the quadric and operating on the cubic,

L, = (aC —-26B+ cA)a+(bC—2cB+dA)y; L,(1,2) which is obtained L, on the quadric,

by operating in like manner with

L, = {aBC— 6 (2B’+ AC) +3cAB—dA’} x + {a0®-—3bBC4+¢(AC+ 2B’) -dAB} y, and L,(3, 1), and L,(3, 2) which are obtained in like manner from the quadric and the cubicovariant J, and which may be written at length by substituting for a, 6, &c. in the values of Z,, L, just given, the corresponding coefficients of J. Five invariants, viz. 4 (0, 2) the discriminant of the quadric, D (4,0) that of the cubic, Z(2, 1) which is the intermediate invariant between the system of two quadrics V, 1

[=A (bd- c*) — B (ad bc) +C (ac— 2),

188

APPLICATIONS

TO BINARY

QUANTICS.

R (2, 3), the resultant of the cubic and quadric, which formed by the methods of either Art. 67 or Art. 86, is

R=a°C° - 6abBC’ + bacC (2B?— AC) + ad (6ABC- 8B?) + 90°A C? —18bcABC+ 6bdA (2B*- AC)

+ 9c°A’C—6cdBA’ + d’A’, and, finally, M(4, 3), the resultant ae

either of L,, L,, or of

18

M=

a’dC*—8a°bcC*— 6a°bdBC"+ 60°? BO" + 2ab’ C*+ 6ab’c BC?

+ 3ab°dA C*+ 12ab"dB’ C—6abe?AC?—24abc2B* C+12ac° ABC + 8ac’ B’—3ac’dA? C—12ac’d A B*+ 6acd’ A’ B-ad’ A°—60

BC?

+ 36°cA C? + 120°cB’ C— 120°d ABC — 8b°dB® + 60’cd A? C + 246°cd AB? — 60°d’?A’B— 3bc° A’ C — 1260 AB’ — 6b¢°d A?B + 8bced* A’ + 6° A?B—- 2°d A’. This last invariant J is a skew invariant changes sign if we interchange x and y; are also skew functions. In comparing may conveniently make A and C=0, taking for « and y the two factors of case the fundamental invariants are

A=-B*,

(see Note, p. 131) and the functions J, L,, L, different invariants we which is equivalent to the quadric. In this

D=a’d’ + 4ac’ + 4db° — 3b°c? — 6abed,

I=-B(ad-be),

R=-8B'ad,

M=8B* (ac — db"),

Thus we have in the same case

DL, =- 2B (ba + cy), L,=— 2B? (bx —cy), and LZ the resultant of these two is —8B*bc, whence we see immediately that Z can be expressed in terms of the funda mental invariants; in fact, L=R+8AIL. So, again, we see that the square of M can be expressed in terms of the other invariants, giving a relation between them. For we have

8 (ac’ + db’) = 2 (D—a’d’ + 3b’c’ + babcd), whence M*=4B° (D—a’d’ + 30°c’ + abcd)? — 256 Badb'c’,

and if in this equation we substitute for ad, ae for bc, — R+8AaTr sBe

?

THE QUARTIC.

189

and for B’, — A, we have the required relation

M* =- 40°D’ + D(R’4+12RAT+ 24472") — 4RI*- 36a1'. Geometrically, see Art. 190, if the cubic U be represented by three points on a conic its J covariant determines on the same conic the harmonic conjugates of each of the three with respect to the other two; the f covariant determines the double points of the involution of these six points. Or we may state it thus (see Contcs, p. 387), the triangle touching the conic at the vertices of U is in perspective with U, the Jines connecting corresponding vertices mark off J on the conic and intersect in the centre of perspective Z,, the axis of the perspective meets the conic in the points H: #, is also the pole of H with respect to the conic. Any quadratic V gives a right line meeting the conic in two points, and the line joining its pole to the centre of perspective H, is the Jacobian of Hand V and is the axis of a new perspective whose centre is on the conic and given by the linear covariant L,. The line joining Z, and the pole of H meets the conic again in L,. JL, is the harmonic of ZL, with respect to V, and L, is the point where HL, meets the conic again. The invariant Z vanishes for any right line which passes through H,.

199. The quartic. We come next to the quartic, which, as we have seen, pp. 128-9, has the two invariants

S=ae—4bd+3c? and T'=ace+ 2bed -ad’—-eb’-c’. We have shown (Art. 172) that the quartic may be reduced to the canonical form x*+ 6ma’y’+y*, and for this form these invariants are S=1+ 3m’, T=m—~m’. These invariants, expressed as symmetric functions of the roots, are 24S =a’ (a— 8)’ (y— 5)’, or 128 =a? [By+ad+ w (ya + Bo) + w (a8 +y6)} (By+ad+w? (ya+Bd) + w (a8 +y9)}, also

PS = ((a-) (y - 8) ~ (a7) (8— A)? + (a—7) 0-f)—(a-3) BF + {(a— 8) B—-¥) - (2-8) (y- 9},

and 4327’=a°d (a— 8)’ (y—8)’ (a—y)(8—5), veniently,

or, more

con

432T=a%{(¢-B)(y—0)—(a—y)(3-P)} (a~y)(8-B)—(a-8)(B—7)}{(a—8)(B-y)—(a-B)(y-8) Jo

190

APPLICATIONS

TO

BINARY

QUANTICS.

In the latter form it is easy to see that Z'=0 is the condition that the four points represented by the quartic should form a harmonic system, thus Z’ may be called the harmonic invariant of the quartic, and in like manner § its equi-anharmonic invariant, see Note, p. 184. It was stated (Art. 171) that 7’=0 is the condition that the quartic can be reduced to the form

z+ y‘,* and that 7’can be expressed as a determinant ay db,¢ 6, fed Cyd, ~e If A be the modulus of transformation, then (Art, 122) S and T become by transformation A*S, A°7, respectively; and the ratio S°; 7” is absolutely unaltered by transformation. 200. To express the discriminant in terms of S and T. It has been already remarked (Art. 111) that the discriminant of a quantic must vanish, if the first two coefficients a and } vanish; for, in that case, the quantic, being divisible by y’, has a square factor. On the other hand it is also true, that any invariant which vanishes when a and 6 are made =0, must contain the discriminant as a factor. Such an invariant, in fact, would vanish whenever the quantic had any square factor (x—- ay)’; for, by linear transformation, the quantic could be brought to a form in which this factor was taken for y, and in which therefore the coefficients a and 6=0. But an invariant which

vanishes whenever any two roots of the quantic are equal, must, when expressed in terms of the roots, contain as a factor the difference between every two roots; that is to say, must contain the discriminant as a factor. It is easy now, by means of S and JZ, to construct an invariant which shall vanish when we make a andd=0. For on this supposition S becomes 3c’, and Z’ becomes —c*; therefore S*—277" vanishes. Now this invariant of the sixth order in the coefficients is of the same order as that which we know * Dr. Sylvester gives the name catalecticant to the invariant, which expresses that @ quantic of order 2n can be reduced to the sum of n powers of the degree 2n.

DISCRIMINANT

OF A QUARTIC.

191

(Art. 105) the discriminant to be. It must therefore be the discriminant itself, and not the product of the discriminant by any other invariant. The discriminant is therefore (ae — 4hd + 8c*)* — 27 (ace + 2bcd — ad? — eb? — &*)’, We can in various ways verify this result. For instance, it appears from Art. 185,* that the discriminant of the canonical form a*+ 6mz*y’+y* is the square of the discriminant of the quadratic 2° + 6may+y’; that is to say, is (1—9m’)’, But (1 — 9m”)? = (1 + 3m?)* — 27 (m — mi)? We should also be led to the same form for the discriminant, by writing the quartic under a form more general than the canonical form, viz. Az*+ By*+Cz‘, where «+y+2=0. In this case we have a=A+C, e=B+C, b=c=d=C, and we easily calculate S=BC+CA+AB, T=ABC. But if we equate to nothing the two differentials, viz. Az*-Cz*, By’ —Cz, we get x, y’, 2 respectively proportional to BC, CA, AB; and, substituting in a+y+2=0, we get the discriminant in the form (BC)}+(CA)3+(AB)s=0, which is (BC+CA+ AB) —27A4’°B’C’?=0 or S°-27T?=0. 201. From the expression just given for the discriminant of a quartic in terms of S and 7 can be derived the relation (Art. 196) which connects the covariants of a cubic. If we multiply two quantics together, the invariants of the compound quantic will be invariants of the system formed by the two components. If then we multiply a quantic by x& + yn, the invariants of the compound will (Art. 134) be contravariants of the original quantic;

and when

we change & and

into y

and — a, will be covariants of it. If we apply this process to a cubic, the coefficients of the quartic so formed will be

ay, 4(3by—ax)

$(cy—bx),

$(dy—3cx),

—dz;

* We may also see this directly, thus: The resultant of ax* + by*, a’xk+ b'y# is the k*® power of ad’ — ba’, since the substitution of each root of the first equation in the second gives ab’— ba’. Now the discriminant of ax + 6cx*y? + ey’ is the resultant of ax? + Sexy’, 3cx?y + ey’. If we substitute 2 = 0 in the second, and y= 0 in the first, we get results e, a, respectively, and the resultant of ax? + 3cy?, 3ex? + ey? is (ae — 9c?)?, ‘Lhe discriminant is therefore ae (ae — 9c?)?.

192

APPLICATIONS

TO

BINARY

QUANTICS.

and the invariants S and 7’ of this quartic are found to be the covariants — 3H, #,J of the cubic. But the discriminant of the product of any quantic by e& + yn, by Art. 110, becomes, when treated thus, the discriminant of U, multiplied by U*. Expressing then the discriminant of the compound quartic in terms of its S and Z, we get the relation connecting the H, J, and discriminant of the cubic.

202. A quartic has two covariants, viz. the Hessian H, whose leading coefficient is ac—6’, and J the Jacobian of the quartic and its Hessian, whose leader is a’d— 3abe + 20°. The Hessian is the evectant of 7, its value is H= (ac— b*) x + 2 (ad — bc) x*y + (ae + 2bd — 30’) xy?

+ 2 (be —cd) xy’+ (ce—d”) y* =>

0 1¥°,

yy ,a,

orisis

"a, 6, AP a ia

~%Y;

ae”

by,e€

—avy,b, @,c,

dy,el,

c,d

ed a7 La a

laid, e,

3y", —2xy, x , 0 0, y° , —2xy, 32°

oc, —3d,

sole 3c, —b

x’, 2xry, y", 0 0, a, 2ay, y*

|,

and becomes, for the canonical form, m (a* + y*) + (1 — 8m’) xy’. Expressed in terms of the roots, it is —48H=a'3 (a— 8)’ (w@—- yy)’ (x 8)’. The covariant J, which symbolically is 12? 13, written at length, is J = (a°d— 3abc + 20°, a’e + 2abd—9ac*+6b°c, 5abe— 15acd+ 107d, —10ad’+10b%e, — 5ade + 15bce — 10bd’, — ae’ — 2bde + 9c’e — 6ed®, — be’ — 2d’ + 3cdeKa, y)* =$|

w, da'y, 3xy’,

¥° , 0

—d,

a,

OF,

2&5 oh y, 8274, 3c,—3b,

0

—é, 2d, 0, -2b, 4 OF err ad (eo b and for the canonical form (1 — 92m”) ay (a* — y*),

COVARIANTS

OF A QUARTIC.

193

We have just seen that the x and y of the canonical form which we use are factors of J, but it will be remembered (Art. 172) that the problem of reducing a quartic to its canonical form depends on the solution of a cubic equation; hence, the factors of J are the x and y of the three canonical forms, This may be connected with the theory explained in Art. 177. If U and V are any two quartics, six values of A can be found, such that U+ AV shall have a square factor, and those six factors are the factors of the Jacobian of U and V. But when V is the Hessian of U, the sextic in question becomes a perfect square, and there are three values of A, for each of which U+ AV contains two square factors, but these factors are still the factors of the Jacobian of U and V. The geometrical meaning of J may be stated as follows: let the quartic represent four points on a line A, B, C, D, then these determine three different systems in involution (according as B, C or D is taken as the conjugate of A), and the foci of these three systems are given by the covariant J. From the last reniark we can express the factors of J in terms of the roots. In fact, by Ex. 7, p. 25, the double points of the involution formed by 8, y; a, 5 are determined by the wee, L quadratic |By, B+y, 1}=0. Similar equations determine ad, a+6, 1 the foci for the other two systems. Ex. 1. To break up the quartic into two quadratic factors. Let (px? + 2gry + ry) (p'a* + 2q'zy + ry”) be identified with the quartic, and substitute in their places for pp’=a, pq'+qp'=2b, qr'+rq'=2d, rr’'=e, pr’ +rp’ =2(c+2p), 9q’=c—p in the identity |p, p’ | |p’, p |= 0, expanded

|| ay Tet as in Art. 25. The réducing cubic is found to be a, 4p? — So+ T=0, b, c+2p, Now, when we write

9,4 Geel 6, ec-p, a,

¢+20 @ e

(y — «) (8 — 8) — (a — B) (y — 8) = 12p,,

(a — B) (y — 8) —(B—y) (a ~8) =12x (B—y)(a — 8) — (y — a) (8 — 8) = 12,

in the expressions for S and 7’ in terms of the roots, Art. 199, they become

Hence,

S = — 4a? (pop3 + papi + P1P2), T = — 40°*p pope: since p, + pg +p3= 0}; [1, 22, A, are the roots of the cubic

40? — So + T=0.

|=0 or

194

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QUANTICS.

Ex. 2. To discuss the relations between the quadratic factors of J. Writing out the determinant forms, let us call

u=(B+y—a—6)2?—2 (By— ad) xy+[By (a+8)— ad B+y)] Yr=4,27+ 2, 2y + cy”, v=(y+e—fB—8) 2-2 (ya —f0) ay +[ya (8B+6) —Bo (y+ @)] y? = a,0"+ 2boxy +Coy’, w=(a+B—y—6)a*—2(aB—y8d) y+ [aB (y+) —yd(a+ f)] y= a327+ 2beay + cay’. We have thus

v—w=—2(B—y) (w—ay)(x#—dy), w—u=—2(y—a)(x—By)(x—dy), u-v=—2(a—B)(z—yy)(a—-dy), vtw= 2(a—d)(x—By)(x—yy), wtu= 2(B—d)(x—-yy)(x—-ay), wtv= 2(y—d)(z@-ay)(z-By)Hence

Ce

eee

ae

ep

Po— Ps Ps— Pi Pi Po @ Thus it appears at once that the identical relation (compare p. 181) between u, v, w is pu? + pv? + pzw?=0. Hence, as this relation involves only the squares, the quadratics are harmonic in pairs. The same thing is found by actual calculation :

AyCo + Ca — 26,0, =0, also

&e.,

a,c, — b,? = (y — a) (8B — 6) (a —B) (y x8) = 16 (93 — py) (01 — pa), &e.,

or, writing 4 =(e2—ps)(e3— 1) (P1— P2)s (4101 — 21”) (02—ps3) =e. = 164. The value already given for H in terms of the roots may be written: a? (u? + v? + w?) = — 48H.

Combining this with the values of U given above we get 16aU=

au+16H

ae+16H

Pi

a

Pe

a*w?+ 16H _

P3

:

Ex. 3, We have seen that J can differ only by a numerical

factor from the

product of the three quadratics u, v, w. To determine it we may compare the Ieading terms of the two forms, or, expressing the symmetric function in terms of the coefficients, find that

a (B+ y—a-—6)(y+a—B-6) whence

(a+ B—v-— 6) = 82 (a%d — Babe + 263),

@uvw = 32J,

203. Solution of the quartic. This is the same problem as that of the reduction of the quartic to its canonical form ax’ + 6cx*y*+ ey‘, for in this form it can be solved like a quadratic. One method of reduction has been explained (Art. 172); the reduction may also be effected by means of the values given for § and 7. Imagine the variables transformed by a linear transformation whose modulus is unity, and so that the new 6 and d shall vanish; then we have S=aet+3c’, T=ace—c’; and the new c is given by the equation 4c°- Sec+ T=0. We get the x and y which occur in the canonical form from the equations

U=aa' + 6ca*y’ + ey*, H=acax* + (ae— 3c’) xy’ + cey*, whence

cU— H= (90? - ae) x*y’.

SOLUTION

OF THE

QUARTIC.

195

Our process then is to solve for c from the cubic just given, and with one of the values of c to form cU—H which will be found to be a perfect square. Taking the square root and breaking it up into its factors we find the new x and y, and consequently know the transformation, by means of which the given quartic can be brought to the canonical form. Having got it to the form ax*+ 6cau*y’+ ey*, we can of course, if we please, make the coefficients of 2* and y* unity, by writing x and y’ for a 4/(a), and y’ /(e). Ex.

Solve the equation xt + 8x3y— 12x7y? + 104xy3 — 2074 = 0,

We c=8

have here is a root.

380 — H=9

S=— 216, 7 =— 756, and our cubic is 4c? + 216¢ = 756, of which The Hessian is

H = — Gz + 60a3y + 72x74? + 24ay3 — 636y4, (at — 4a%y — 12x77? + 32xy3 + 64y4) = 9 (x? — Qry — 8y?)?,

The variables then of the canonical form are X= a+ 2y, Y= a —4y, which give 62=4X+2Y, 6y=X—Y; whence, substituting in the given quartic, the canonical form is found to be 3X44+2X*Y?- Y4, The roots then are given by the equations (x + 2y) J(8) =a2—4y,

(«+ 2y) (-—1l)=a—4y.

204. Since J is proportional to the continued product of the xz and y of the three canonical forms, and since we have just seen that the square of the product of one set of x and y is cU—H, where ¢ is one of the roots of the cubic 4c°— Sc+ 7=0, we have J’ proportional to 4H°— SHU*+ TU*. By calculating with the canonical form, we find the actual value to be —J’.

Or, again, we saw in Ex. 2, Art. 202, that 16 (ap, U— H) =a'w’, 16 (ap, U- H) =a’’, 16 (ap,U— H) =a'w’,

and in the following example that a’uvw = 32J, hence by the values of p,, p,, p, of Ex. 1 of same article

J*=—4H°+U0°HS -U°T, 205. Prof. Cayley has given the root of the quartic in a more symmetrical form. It has been shown that ap,U— UH, ap,U—H, ap,U- H are perfect squares severally of u, v, w.

196

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TO BINARY

QUANTICS.

If, further, we enquire under what is a perfect square, we find that

conditions

Xu + pv + vw

r (a,c, — 8,”) + pw? (a,c, — b,) + Vv (a,c, — b,?) =0

must be satisfied (compare Ex. 2, Art. 202), or, as it may be written, ‘ F pe v

a

Ps =P

= 0.

iePesPx peRicaPs

If we further wish to make

AV (ap,U- H)+ pwV/(ap,U-H)+v/(ap,UHH) vanish with U, we must have 4+4+v=0, we find

whence, solving,

thus the squares of the factors of U are the values of (P.— Ps) V (ap, U- ff) aL (We p,) V (ap, U—H) (a + 6 +c), for example, is divisible by atb4c+, we have no right to infer that in general H is divisible by L, unless in cases where the quotient abc (a + 6 + ¢) has been also proved to be an invariant,

FUNDAMENTAL

INVARIANTS

OF A QUINTIC.

238

given any quintic, and transform it to the canonical form by a substitution whose modulus is unity, the numerical values of the new a, d, c are given by the cubic

ot

a Lt a0,

Now the order of any symmetrical function of a, 3, c will be equal to its weight in the coefficients of this cubic, and when this weight is a multiple of 4, it is easy to see that the symmetric function is a rational function of J, K, L. Being given, therefore, any invariant whose order in the coefficients is a multiple of 4, it has been proved that we can write down a rational function of J, K, L, which, for the canonical form, shall have the same value as this invariant, and therefore be always identical with it. And since it would be manifestly absurd to suppose an integral function of the coefficients to be equal to an irreducible fraction, it follows that every non-skew invariant is an integral function of J, K, L. If we make the first three coefficients a, 6, c each equal 0, J, K, LZ all vanish. Hence when three roots of a quintic are all equal, these three invariants vanish.* If we make a, }, e, f all equal 0, J becomes — 32c’d’, and L, —16c°d*, and therefore J*—2048L vanishes.

Quintics therefore which have two pairs

of equal roots must not only have the discriminant =0, but also J’ = 2048L. 229. The simplest skew invariant is got by forming the resultant of the quintic ax°+ by’ + cz’, and its canonizant abcxyz. Substituting successively the three roots of the canonizant in the quintic, and multiplying together, we get for the resultant a’b’c’ (b—c)(c-— a) (a—6). This invariant, therefore, is of the eighteenth order. Previous to its discovery by M. Hermite,t * Tn general all the invariants of a quantic vanish, if more than $n of its roots be all equal. For when half the coefficients, counting from one end, simultaneously vanish, no term of the proper weight (Art. 143) can be made with the remaining coeflicients. + See Cambridge and Dublin Mathematical Journal, vol. 1X. p.172, M, Hermite works with a new canonical form, the « and y of which are the two factors of the quadratic covariant. The quintic then is supposed to be such that ae — 4bd + 3c, bf — 4ce + 3d? both vanish, and the quadratic covariant reduces to zy, d?

The advantage of this is that ;

the operating symbol thence derived is simply TEL and some of the covariants

234

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QUANTICS.

the possibility of the existence of skew invariants had not been recognised. I took the trouble to calculate this invariant, and the result is printed (Philosophical Transactions, 1858, p. 455*), but as it consists of nearly nine hundred terms I cannot afford room for it here. The leading terms are a’d*f* — a®c°f"; in this, as in every skew invariant, the complementary terms having opposite signs, and the symmetrical terms vanishing. Moreover if the alternate terms in any equation be wanting every skew invariant vanishes. For in this case the weight of each coefficient is even, but the weight of any skew invariant is an odd number. Thus J vanishes if b, d, f vanish; that is to say, if the quintic can be reduced to the form a (a — a’) (x*— 8’), in other words, if we consider the quintic as denoting five points on a right line, the vanishing of J is the condition that one of these points should be a self-conjugate point of the involution determined by the other four. By the argument used, it is proved that every skew invariant of a quintic must be the product of this invariant I by a rational function of J, K, L. 230. The square of J being of the thirty-sixth degree can be expressed rationally in terms of J, K, L (Art. 228). The actual expression is easily found. By forming the discriminant of the cubic (Art. 228)

oF d+

anL

we obtain the product of the squares of the differences of a, b, c in terms of J, K, Z, and thus have

PL = H’K* + 18HKL* - 27L*—4K°L? - 4H’; or putting for H its value {(K°-JLZ), and dividing by Z, we have

162° =JK* + 8LK*-2)°LK? -72JKL? — 4321 + J*L?, In the last equation of Art. 222 when we make B=0 for two quartics derived from a quintic we find by the same article obtained by thus differentiating assume a very simple form. Notwithstanding, I have preferred using Sylvester’s canonical form, which I find much more conyenient. * Where the coefficient of 0’d7’e?f should have been printed 12500.

COVARIANTS

OF A QUINTIC.

H=D and by Art. 220 I+ C=0; function written here.

235

whence M? becomes the

231. We come now to the covariants. We have already (Art. 224) mentioned the quadratic covariant S and the cubic covariant 7. Considering this system of a cubic and quadratic, we have (Art. 198) a series of covariants which give completely all the covariants of the quintic which are not higher than the third order in the variables. The five invariants of Art. 198 reduce to four J, K, L, I already mentioned, the discriminant of the cubic, and the resultant of cubic and quadratic, both reducing to L. The four linear covariants of the system of cubic and quadratic give four linear covariants of the quintic, of the orders 5, 7, 11, 13, which for the canonical form are respectively abe (bex + cay + abz), abe {(b’c*+ a°be) (y — 2) + (c’a’+ B’ac) (2 — x) + (a°b"+ cab) (a — y)}, a’b’c! {be (y — 2) +ca(z-—x) + ab (x—-y)}, a’b'c’ fax + by + cz}.

These are the only distinct linear covariants of the quintic. If we eliminate either between the first and last of these, or between the second and third, or between the first of them and the canonizant, we get Hermite’s J; and if between the first linear covariant and the quintic itself we get J (J*’-3K). Thus, then, if Z vanish, or if J’=3, the quintic is immediately soluble, one of the roots being given by that linear covariant. Hermite has studied the quintic by transforming the equation, so as to take the first two linear covariants for x and y, when all the coefficients in the transformed equation are found to be invariants. The transformation becomes impossible when the two linear covariants are identical, which will be when their resultant / + 9L vanishes. The system of cubic and quadratic have (Art. 198) three quadratic covariants, viz. in addition to S itself, the Hessian

of T or a’l’c’ (x? + y’ +2’), and the Jacobian of this and S, or ab'c? {bea (y — 2) + cay (2 -— &) + abz (a— y)}. These are the only distinct quadratic covariants of the quintic.

236

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QUANTICS.

Lastly, there are three cubic covariants, viz. in addition to T itself, its cubic covariant a°b°c° (y—2) (2 —x) («—y); and the Jacobian of S and 7, abe {beyz (y —2) + cau (2-— x) + abary (x — y)}.

These are the only cubic covariants of the quintic. We have now enumerated fourteen forms, whose order in the variables is not higher than the third; adding to these the quintic and its Hessian, there are still seven forms to be mentioned. If we operate with S upon H, we get a quartic of the fourth order in the coefficients, which only differs by a multiple of the square of § from adc (ax* + by* + c2*). A second quartic covariant is the Jacobian of this and S, or

abe {a* (b- c) a* +B? (c— a) y*+c* (a—5) 2}. These are the only two quartic covariants. We have a quintic covariant by taking the Jacobian of S and U, viz.

a’ (b—c)2° +0? (c—a) y° +c (a—d)2° — abe (y —2) (2-2) (x—y) (ya + 2x + wy).

A second quintic covariant is found by taking the Jacobian of U and the quadratic covariant a0’c*(a2*+y*+2"). This gives a'b*c® {ax* (y — 2) + by* (2 — x) + cz" (a —y)}.

Of sextic forms there only is, in addition to the Hessian, the Jacobian of S and H, or of 7 and VU.

abe {az (y — 2) + by (2 — 2) + c2° (7 y)}. There is one septic form, viz. the Jacobian of U and the simplest quartic covariant, or abe {bey*e* (y — 2) + caz*x? (2— x) + aba’y’ (x—y)}.

And lastly, one nonic, namely, the Jacobian of U and H, or atba'y® — a’cax'e® + b’cy'a* — Vaya" + caz'x* — ebay? + abcac*y*s* (y — 2) (ex) (w#—y). 232. The forms might also have been arranged, as Prof. Cayley has done, according to their order in the coefti-

SOURCES

OF COVARIANTS

OF A QUINTIC.

237

cients. We give here, in his order, the leading terms the less complicated.

of

(1) w, Quintic, a.

(2) (3) (4) (5) (6)

S, Quadratic, ae — 4bd + 3c*. 4, Sextic, ac—0*. Z, Cubic, ace - ad’ — b’e + 2bcd — c*. Quintic, a?f- 5abe + 2acd + 867d - 6c’. Nonic, ad — 3abc + 26°.

(7) Invariant J already given; fourth degree in coefficients.

(8) Quartic, a* (e*—d/) + a (3bef— 3bde — 4c*e + 4cd’) + 5b°ce + 267d? — 2b°f— 8be"'d+ 8c". This differs by the square of S from the corresponding quartic covariant, Art. 231; and is 12? of Z’and wu. (9) Sextic, a* (¢f—de) — ab?f— 2abce + 4abd* — ac’d + 3b%e

— 6b'cd + 3bc°. (10) Linear, a’ (cf? —2def +e) + a(— Bf? — 4bcef'+ 8bd*/) + a(—2bde*—2c'df +1407e") + a(—22cd’e + 9d*)\+ 6b°ef — 12b7cdf — 15b’ce’ + 10b*d’e + 6b0°f+ 30bc*de — 20bcd* — 15c*e + 10c*d?.

(11) Cubic, a? (cef— 3d?f+ 2de*) + a(— bef+ 14bcdf— 11bce’) +a(—bd’e—90°f+ 14c*de— bcd’) — 8b°df+ 9b%e’+ 60°? f— 16b°cde + 8b°d° + 3bc’e — 2bc*d’. This is the Jacobian of S and 7.

(12) Septic, a? (2c?f— 5ede + 3d*) + a (—4b%cf+ 5b°de + 5dc*e) + a(— Thed’ + cd) + 2b°f— 5b°ce — 2b°d’+ 8b7c'd—3bc*. (13) Quadratic, a* (—?f? + icdef— 3ce’ — 3d*f+ 2d’e’) +a (2b%cf? — 5b°def+ 30° — 5be’ef+ Thed?f) +a (—bede’ — bd*e — c'df+ 6c'e* — 8c"d"e + 3cd") — bf? + 5b°cef + 2b°d?f — 3b°de* — 8b°c'df — Abc’e* + 7b'cd’e — b'd* + 8bc'f+ dbe'de — 4be*d® — 3c%e + 2c*d’.

(14) Quartic, a’ (— df?+ &f) + a? (3bcf?+ 2bdef— 5be’— 80'ef) + a’ (2cd*f+ 12cde’—6d*e)+a(—2b°f?—20"cef—60"d" f+130" de*) +a (20bc*df+ 4be’e’ — 52bcd’e + 24bd* — 9c?f+ 20c*de — 10c*d’)

+ 6bef — 120°cdf — 15b°ce? + 100°d’e + 60'c°f+ 3067c'de — 20b°’cd* — 15dc*e + 10bc°d’.

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(15) Linear, a° (cf* — 4def? + 3e°f) + a? (-0°f° — 8bcef”) + a? (160°? + 4bde?f— 15be*—6c°d f? + Ac°e?f—22cd’ef) + a® (26cde* + 9d*f— 12d °c”) + a (TD°ef? — 300°cdf? + b’ce?f) +a (—T40°def + 8407de* + 18b°f? + 160bc"def — 98bc"e") + a (—20bed*f— 94bed’e? + 51bd*e — 81ctef+ 18c°d?f+ 140c*de*) +a(—100c'd*e + 18cd°) + 8b*df? — 18b*e?f— 6b°e2f? + 32b°cdef + 45b%ce® + 1120°d°f— 1500°d*e — 6b°c°ef— 28407c'd?f+ 5007c'de* + 320B*cd*e — 1200°d° + 216bc'df— 1dbc*e”—310bc%d’e + 130bc"d* — 54c°f+90e°de — 40c'd®. (16) Quintic, a° (cdf?—2ce?f+ 2d*ef—de*) + a?(—Bdf?+207ef) + a? (— 8be?f? — bbedef+ 13bce® — 8bd*f+ 2bd*e”) +a (16cef—20'd*f— 38c'de’ + 34cd°e — 9d*) +a (5b°of? + 2b'def — 120°? — 240°’ef+ 52b°cd*f + 7b’cde’) +a (—220'd%e — 52bc°'df+ 34bc°e? + 8he'd’e — bed* + 18¢°f) +a (—25ce%de + 10c°d*) — 20°F" + 10b*cef— 28b*d?f+ 300%de? + 820°c'df— 35b°c’e’ — 500°cd’e + 300°d* — 1207c*f+ 700°ce°de — 400°c*d*® — 15be'e + 10be*d’. (17) Invariant £ already given, 8" degree in coefficients. (18) Quadratic, 8% degree in coefficients. (19) Cubic, 9% degree in coefficients. (20) Linear, 11 in coefficients. (21) Invariant Z already given, 12% degree in coefficients. (22) Linear, 13" in coefficients.

(23) Invariant J, 18™ in coefficients. For (18), (19), (20), (22) we refer to Prof. Cayley’s Ninth Memoir on Quantics, Phil. Trans., 1871, p. 17. 233. Prof. Cayley* has been led to consider in the theory of the quintic a new canonical: form, which is obtained as follows: Taking for convenience the quintic to be (a, b, c, d, e, f Xa, Y)’s * It has been already mentioned (p, 134) that the method of discussing covariants by means of their leading terms or sowrces was introduced by Prof. M. Roberts See Quarterly Journal, vol, Iv.

CRITERIA

OF REALITY

OF ROOTS.

239

using small Roman letters for the coefficients, suppose in the first instance that a, b,c, d, e, f denote the leading coefficients of the first six covariants of Art. 232 respectively, thus a=2,

¢=ac — b?,

6 =ae — 4bd + 3c?,

d= ace—ad?—b’e—c?+4 2hed, f=a?d — 3abe + 2b},

e = a’f—d5abe + 2acd + 8b7d— 6bc?,

where —f? =a" (ad— bc) + 4c° identically, so that any rational and integral function containing f can always be expressed as

a function linear in regard to f. function

This being so, we have the

(a, b, ¢, d, e, f )(@ — by, ay)§ = x5 + 102%y? (ac — b*) + 10x?y? (a2d — 3abc + 2b’) + 5zy* (a3e — 4a?bd + 6ab’c — 3b‘) + y5 (atf — 5atbe + 10a?b’d — 10ab%c + 4b),

and forming the values of a’) —3c* and a’e —2c/, this is found to be =(1, 0, ¢, f,ab — 8c’, we —2¢f La, y)’. The last-mentioned function, considering therein a, 0, c, e, f as denoting not the leading coefficients, but the covariants themselves, and (a, y) as variables distinct from those of the quintic and its covariants, is the canonical form in question. Using in like manner d to stand for the covariant, we have between the covariants a, 6, c, d, e, f the foregoing identical equation

—f’=a' (ad— be) + 4c’, which is to be used to reduce functions of the covariants so as to

render them linear in regard to f. 234. Criteria for the reality of roots of quintics. It ought to have been stated earlier that the sign of the discriminant of any quantic enables us at once to determine whether it has an even or odd number of pairs of imaginary roots. Imagine the quantic resolved into its real quadratic factors, then (Art. 110) the discriminant of the quantic is equal to the product of the discriminants of all the quadratics, multiplied by the square of the product of the resultants of every pair of factors. These resultants are all real, and their squares positive; therefore, in considering the sign of the discriminant, we need only attend to the discriminants of the quadratic factors. But the square of the difference of the roots of a quadratic is positive when the roots are real, and negative when they are

240

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imaginary. It follows then that the product of the squares of the differences of the roots of any quantic is positive when it has an even number of pairs of imaginary roots, and negative when it has an odd number. We have been accustomed to write the discriminant giving the positive sign to the term which is a power of the product of the two extreme coefficients. This will have the same sign as the product of the squares of differences of the roots when the order of the quantic is of the form 4m or 4m+1, and the opposite sign when the order is We see then, in the case of of the form 4m+2 or 4m+3. the quintic, that if the discriminant be positive, there will be either four imaginary roots or none; and if the discriminant be negative, there will be two imaginary roots. It remains then further to distinguish the cases when all the roots are real, and where only one is so. 235. In order to discriminate between these remaining cases, there are various ways in which we may proceed. ‘The following* are, in their simplest forms, the criteria furnished by Sturm’s theorem. Let J be the quartic invariant as before, and

H=b'—ac, S=ae—4bd+3c, T=ace+2bcd—ad’—eb*—c’, M=a’'e’ — a’df+ 3abcf— 38abde + 4acd? — 4ac’e — 20°f + 5b’ce + 207d? — 8be"d + 8c’, then the leading terms in the Sturmian functions are proportional to a, a, H, 5H8+9aT, —- HJ +128M+4S8*—216T7"%, the last of course being the discriminant; and the conditions furnished by Sturm’s theorem to discriminate the cases of four and no imaginary roots, are that when all the roots are real the three quantities H, 5HS+9aT, — HJ + &c. must all be positive. * These values are given by Mr. M. Roberts, Quarterly Journal, vol. v. p. 175. The reader who may use Prof. Cayley’s tables of Sturmian functions (Philosophical Transactions, vol. OXLVII. p. 735) must be cautioned that the fourth and fifth functions are there given with wrong signs.

M is already written as (8) in Art. 232, and is connected with D in Art. 226 by the equation D=S?—3M. In fact the expression for the discriminant R there given is 92 = 25J* — 192 (D,D, — 4D,D, + 3D,), where the covariant whose source is D is written Dyx4+4D,o7y+&c., and D=D,, C=2D,, 3B+A=—10J, 3B—A=48D,, CLIN) IO! = De

STURM’S

FUNCTIONS

FOR A QUINTIC.

241

236. We may apply these conditions to the canonical form (c—a)a* + 5eaty + 10cx*y? + 10cxy* + Scxy* + (c—b) 9°, in which case the equality of all but two of the coefficients renders the direct calculation also easy. We easily find then that the constants are c—a, c—a, ac, —a’c’; and the fourth being essentially negative, we need not proceed further, and we learn that the equation just written has always imaginary roots. We find then that when the invariant L of a quintic is positive, the roots of the equation cannot be all real. For Z being, with sign changed, the discriminant of the canonizant, when L is positive, the roots of the canonizant are all real, and the quintic can be brought to the canonical form by a real transformation. When JZ is negative, two factors of the canonizant are imaginary, and the canonical form is

a(—22)°+ {o- d y(- 1)} {a@t+y v(- 1)} which, expanded, is

+ {e+ dW(-1)} fe—y v(-1)},

dy’ + 5cy*x — 10dy’x” — 10cy’x* + 5dyx* + (c — 16a) x*. Writing for brevity c’+ d’=7’, I find for this form the Sturmian constants to be d, d, 7°, r*, 7° (—4a’d? + 20acr’ + 5r*), and it would seem that the discriminant being positive, the roots are all real if d and — 4a°d’ + 20acr’ + 5r* are both positive.* 237. In practice the criteriat furnished by Sturm’s theorem are more convenient than any other, because the functions to

be calculated are of lower order in the coefficients. It is, however, theoretically desirable to express these criteria in terms of the invariants, and this is what has been effected by different * I give this result, though suspecting its accuracy, because it seems to me to disagree with the theory derived from the other methods. + It may be noticed that there is no difficulty in writing down a multitude of criteria which might indicate the existence of imaginary roots; for any symmetric function of squares of differences of roots © (a — 8)’, &c. must be positive if all the roots are real. We can without difficulty write down such functions which are also invariants ;and which, if negative, show that the equation has imaginary roots, But then these may also be positive when the roots are imaginary, and the problem is te find some criterion or system of criteria, some one of which must fail to be satisfied when the roots are not all real.

242

HIGHER

BINARY

QUANTIGS.

methods by Hermite and by Sylvester. We proceed briefly to explain the principles of Sylvester’s method, which is We have seen already that when the highly ingenious. given, then a, b, ¢ of the canonical are DZ K, J, invariants form may be determined by a cubic equation; and we can infer that to every given system of values of J, K, Z will correspond some quintic. But to every system of values of J, K, L will not correspond a real quintic. In fact, we have seen, Art. 230, that the J, K, L of every quintic with real coefficients, are such that the quantity @ is essentially positive, where G is JK*+8LK*—2J°LK’ —72JdL’K— 432L° + J°L’. For G has been shown to be the perfect square of a real function of the coefficients of the general quintic, viz. a’d°f*+ &e., this being the eliminant of the quintic and its canonizant, and therefore necessarily real. We may in the above substitute for K its value in the discriminant from the equation J*— 128K= D, and so write G,

JD —4(J* + 2°L) D’ + (67* — 29-2"L) J*D — 4(J*— 61.2 L— 9.2" 14) JD + (J*—2"L)? (J*— 97.2"), If now, to assist our conceptions, we take J, D, ZL for the coordinates* of a point in space, then G=0 represents a surface; and points on one side of it, making G positive, answer to real quintics, while points on the other side, making G negativet, answer to quintics with imaginary coefficients. 238. Now, in the next place, we say that if the coefficients in an equation be made to vary continuously, the passage from real to imaginary roots must take place through equal roots. For, let any quantic ¢(«) become by a small change of coefficients $ (aw) +ey (x), where ¢ is infinitesimal, and let a be a real root of the first, a+ a root of the second; then we * Sylvester takes L in the usual direction of x, J of y, and D of z. t Points for which G = 0 answer to real quintics, and it is easy to see that in this case the equation admits of linear transformation

to the recurring form,

For we

have proved that when G = 0 two of the coefficients of the canonical form are equal. The equation is therefore of the form aa + ay +6 (a#+y)5=0.

METHOD

OF SYLVESTER.

243

have ¢(a+h) +e (a) =0; whence, since 6(a)=0,

we have

hG' (a)+e (a) =0, which gives a real value for k.

The con-

secutive root a +h is therefore also real. But if ’ (a) vanishes as well as ¢ (a), the lowest term in the expansion of $ («+h) will be A’, and the value of h may possibly be imaginary. When, therefore, the original quantic has equal roots, the corresponding roots of the consecutive quantic may be imaginary. It follows then, that if we represent systems of values of J, D, L by points in space, in the manner indicated in the last article, two points will correspond to quintics having the same number of real roots, provided that we can pass from one to the other without crossing either the plane D or the surface @. If points lie on opposite sides of the plane PD, we evidently cannot pass from one to the other without having, at an intervening point, D=0, at which point a change in the character of the roots might take place. If two points, both fulfilling the condition @ positive, be separated by sheets of the surface G, we can not pass continuously from one of the corresponding quintics to the other; because when on crossing the surface we have G negative, the corresponding quintic has imaginary coefficients. But when two points are not separated in one of these ways, we can pass continuously from one to the other, without the occurrence of any change in the character of the corresponding quintics.

Now

Sylvester’s

method

consists

in shewing, by a dis-

cussion of the surface G', that all points fulfilling the condition G positive, which he calls facultative points, may be distributed in three blocks separated from each other either by the plane D or the surface G. And since there may evidently be guintics of three kinds, viz. having four, two, or no imaginary

roots, the points in the three blocks must correspond respectively to these three classes. I have not space for the elaborate investigation of the surface G, by which Sylvester establishes this; but the following is sufficient to enable the reader to convince himself of the truth of his conclusions. 239. One of the three blocks we may dispose of at once, viz. points on the negative side of the plane D, which we have

244

HIGHER

BINARY

QUANTICS.

seen (Art. 234) correspond to quintics having two imaginary roots. Next with regard to points for which D is positive. We have seen, in the last article, that a change in the character of the roots takes place only when D=0; our attention is therefore directed to the section of G by the plane D. We see at once, by making D=0 in the value of G (Art. 237), that the remainder has a square factor, and consequently that the surface G touches D along the curve J*—2"L, and cuts it along J*—27.2°L. Now, if a surface merely cut a plane, the line of section is no line of separation between points on the same side of the surface. If, for example, we put a cup on a table, there is free communication between all the points inside the cup and between all those outside it. But if a plane touch a surface, as, for instance, if we’ place a cylinder on a table, then while there is still free communication between the points inside the cylinder, the line of contact acts as a boundary line, cutting off communication as far as it extends, between points outside the cylinder on each side of the boundary. Now Sylvester’s assertion is, that if we take the negative quadrant, viz. that for which both J and JL are negative, and if we draw in the plane of ay the curve J*—2"Z, then all facultative points in that quadrant, lying above the space included between the curve and the axis L=0, form a block completely separated from the rest, and correspond to the case of five real roots, 240. In order to see the character of the surface, I form the discriminant of G considered as a function of K, which I find to be —L' (J*+27L)*. Consequently, when both J and Z are negative, the discriminant is negative, and the equation in A has only two real roots. T’o every system of values, therefore, of J and LZ correspond two values of K, and consequently two values of D and the surface is one of two sheets. Now I say that it is the space between these sheets for which G is positive. In fact, since G is JD*+ &c., it may be resolved into its factors J (D—a)(D—£) {(D-—¥)? +8}; and since J is supposed to be negative in the space under consideration, D must evidently be intermediate between @ and 8 in order that G should be positive.

DISCUSSION

OF METHOD

OF SYLVESTER.

245

Now the last term of the equation being (J°—2"Z)? (J°-27.2"L), if J* be nearly equal to 2"Z, will be of opposite sign to Z, or in the present case will be positive. And the coefficient of D* being negative, we see that on both sides of the line J?=2"Z the values of D are, one positive and the other negative, that is to say, the two sheets of the surface are one above and the other below the plane D. But I say it is the upper sheet which touches D along J*X—2"Z. This may be seen immediately by looking at the sign of the penultimate term in the equation for D, by which we see that when the last term vanishes, the two roots are @ and negative. The theory then already explained shows that the curve J*’=2"Z acts as a boundary line cutting off communication in that direction between facultative points on the upper side of D. But, again, communication in the other direction is cut off by the plane Z=0. For when we make J positive, the discriminant becomes positive, and the equation in D has either four real or four imaginary roots. But the first Sturmian constant is proportional to Z(J*+12Z), which, when J is negative, and Z positive and small, is negative. Immediately beyond the plane ZL, therefore, the equation to determine D has four imaginary roots, or the surface does not exist. The facultative points, therefore, lying as they do within the surface or between its sheets, are cut off by the plane Z, on which the sheets unite, from communication with points beyond it. Thus the isolation of the block under consideration has been proved. I need not enter into equal detail to prove that all other facultative points have free communication ¢nter se. The line of contact 2" —J® is no line of separation in the quadrant where J and L are both positive. For then it is seen, as before, that it is the points owtside the two sheets which are facultative, and not the points between the surface and touching plane. The result of this investigation is, that in order to have all the roots real, we must have the quantity 2" —J* positive,* * Sylvester

has inadvertently

stated

his

condition

to be

that

2"L—J®*

is

negative. It is easy to see, however, that what he has proved is, that this quantity must be positive. For the block which he has described lies on the side of the curve

2NL—J3 next to the axis

1 = 0. But when L is 0 and J negative, 2".L—J?

is positive.

246

HIGHER

BINARY

QUANTICS.

and Z negative, which also infers J negative. If either condition fails, our roots are imaginary. It is supposed that in both cases D is positive.

241. We have seen that the cylinder parallel to the axis of z and standing on the curve 2" —J* does not meet G above the plane D; the two values of z being one 0, the other negative. Any other surface then standing on the same curve and not meeting G would serve equally well as a wall of separation between the two classes of facultative points. For, all the points between the cylinder and this surface would be nonfacultative, and therefore irrelevant to the question. Sylvester has thus seen that we may substitute for the criterion 2"L —J°, 2°L—J*°+pJD, provided that the second represent a surface not meeting G above the plane D. And on investigating within what limits ~ must be taken, in order to fulfil this condition, he finds that # may be any number between 1 and — 2. He avails himself of this to give criteria expressed as symmetrical functions of the roots. In the first place

= (a—- 8)’ (B- 9)’ (y—a)? (8—«)* is an invariant (Art. 136), and being of the same order and weight as J can only differ from it by a numerical factor, which factor must be negative, since this function is essentially positive; and J we have seen is essentially negative when the roots are all real. And secondly, the symmetric function

= (a—B)* 8-4)” (y—a)’ (e-a)*

(e—B)*(e—y)*(8-a)*(8- B)*(8-9),

(the relation of which to the other may be seen by writing it in the form D*S (a— 8)? (8 —+)*(y—a)?(S—«)*, where D is the discriminant), is also an invariant, and of the twelfth order. It must therefore be of the form aJ°+@JD+yL. Now, by using the quintic* a (x*— a’) (x*— 0’), the symmetric function * It was observed, Art. 229, that the characteristic of this form is that Hermite’s invariant J vanishes, hence it may be safely used in calculating any invariant function whose order is divisible by 4 and is below 36, since such forms cannot contain J, but though this form may be safely used in this case, it cannot always be safely used. For when a linear factor of a quintic is also a factor in the sextic covariant of the remaining quartic, a relation must exist between the invariants.

INVARIANTS OF PRODUCT OF QUARTIC BY A FACTOR.

247

may easily be calculated and identified with the invariants; and the result is that its value is proportional to 2"Z—J*+41JD. Since then the numerical multiplier of JD is within the prescribed limits, it may be used as a criterion, and Prof. Sylvester’s result is, that the two symmetrical functions mentioned are such that not only are both positive, as is evident, if the roots are all real, but also if both are positive, and D positive, the roots must be all real. It ought to be possible to verify this directly by examining the form of these functions in the case of an equation with four imaginary roots.

242. I have also tried to verify these results by examining the invariants of the product of a linear factor and a quartic, (aa + By) (a* + 6ma*y’ + y*); these being necessarily covariants of the quartic (Art. 201). The coefficients of the quintic are then 5a, 8, 3ma, 3mB8, a, 58; and I find for the J of the quintic, 48 (8SH— 3TU), or 48 times (5m + 27m*) (a* + B*) + (8 — 18m? — 54m’) a’. Now the roots of the quartic are all real when m is negative, and when 9m’ is greater than 1. On inspection of the value given for J, we see that when m is negative every term but one is negative. Giving then m its smallest negative value — 4, J is negative, viz. — 144 (a”—8”)?; and J is d fortiord negative for every greater negative value of m. Or we may see the same thing by supposing 8 = 0, when we have only to look at the coefficient of the highest power of a in 8SH—3TU, which is — 8 (b’—ac) S—3Ta. constants A, B, C, viz.

But now if we call the three Sturmian

A=08'—ac, B=28A+3Ta, C=S8*- 27T”, the value given for J becomes —6AS—B, which is essentially negative when the roots are all real. The invariant L, according to my calculation, is 54 (8SH—37TU)> — 6400 (S* —277”) (4H° — SHU* + TU*)

+150 (S* —27 7") U? (8SH+15TU) — 4050U?S* (28H—3TU),

whence 2"Z —J° differs only by a positive constant multiplier from — 128 (S*—277") (4H* -3SHU’* + TU*) + 3(S*—277") U? (8SH+15TU) —81U’S' (2SH- 38TU).

248

HIGHER

Writing

BINARY

QUANTICS.

the coefficient of the highest

0=a'd—3abc+2b*,

power of a in this is 128 C6? + 81a?,S* + 45a’ CB — 540° OSA.

All the terms of this but one are positive when the roots are all real, but as there is one negative term, is it not obvious, on the face of the formula, that the whole will be positive when the roots are all real. Still less that if this formula be positive and J negative, the roots are necessarily all real. Therefore, although no doubt Prof. Sylvester’s rule may be tested by the process here indicated, to do so requires a closer examination of this formula than I am able to give.* 243.

Prof.

Cayley in

his

Eighth

Memoir

on

Quantics

(Phil. Trans. 1867), proceeded by a method a little different from that described above. Adopting as coordinates 2"L— J* D ial iy 5 at i ae Yar J;

then from the foregoing equation 162? =JK‘*+8LK*-2°LK* — WILK — 4322° + J*L’, where, K= 4, (J”— D), we deduce without much difficulty 2.2" — = —3a°—a? + y (72x + 205a + 125) + y? (— 29a” — 17)

+y'(-2@—-9)+y'.2 = > (x, y) suppose 5 2

or, since

z=J, wehave

z2(2, y)= TS = positive,

and the surface G'=0 may be replaced by z¢ (a, y)=03; that is, by the plane z= 0 and the cylinder ¢ (x, y)=0. The configuration of the regions into which space is divided by this surface depends only on the form of the curve ¢(a#, y)=0 (Prof. Sylvester’s “ Bicorn”’), which is the section of the cylinder by the plane z =0, and the discussion as to the reality of the roots may be then effected by means of the plane curve alone; the results, of course, agree with those obtained above. * The verification, however, is easy in the particular case « (x! + 6ma?y? + y4), We have then J = 48m (5 + 27m?), L = 12m (5 — 9m?)4; 24 — J® proportional to m (1 — 9m?) (50 + 45m? + 64874 + 729m*). Thus, when m is negative, and 9m?> 1, we have J and L negative and 2"L—J® positive. The latter is positive for imaginary roots only when m is positive, but in this case J is positive. The imaginary roots must, therefore, be detected by one criterion or other.

HERMITE’S

TYPICAL

FORM.

249

244. It has been already mentioned (Art. 231), that M. Hermite has made use of the fact, that the quintic as well as every equation of odd degree is reducible to a forme-type, in which the 2 and y are linear covariants and the coefficients are invariants. It follows immediately, that by applying Sturm’s theorem to the forme-type, the conditions for reality of roots may be expressed by invariants. Hermite extends his theorem to equations of even degree above the fourth, by the method indicated in Art. 248. Writing J*-3K=M, JK+9L=N; and Q a numerical multiple of Hermite’s J, such that

Q = JK? M’* -2MNK(J’+ 12K) + JN? (J? +72K) —48N*, then the coefficients of the forme-type are

A=QM,

B=JKM’— MN (J* + 18K) +307N’, C= Q(JM-12N), D=J*KM’* —JMN (J’ + 30K) + N? (42?+ 1441), EL= Q(J°?M —24JN), F=J*KM’—-J?MN(J?+ 42K) + NI (54d? + 288K) — 1152.N*, Thus the first Sturmian constant B’ —AC is found to be 36N? {(MK—5JN)’ — 16MN*}. The Sturmian constants being essentially unsymmetrical, there seems no reason to expect that the discussion of these forms would lead to any results of practical interest. The coefficients of the forme-type, as M. Hermite remarked, satisfy the relations AJ*-20J+ H=0, BJ? -2DJ+ F=—-1152N", AE-4BD+3C*?=—-12%N’, AF-3BE+2CD=0, BF-4CE+ 3D? =12'JN’. Thus then the quadratic covariant is N° (a*— Jy’); and operating with this on the quintic, we get the canonizant in the

as

NCAT =O (BID, OJ B, DJ = F¥a, y)*;

the coefficients inside the parentheses being all further divisible by NV. Hence we have ACE+2BOD— AD*— EB’ — C?=-4.12°N°Q,

250

HIGHER

BINARY

QUANTICS.

and the second Sturmian constant is got immediately by substituting in the formula of Art. 235, the values just found for

B’—~ AC, AE-4BD +30", ACE+2BCD - &.,* 245. The Tschirnhausen transformation consists in taking a new variable y=at Batya +...F Av"; then there are n values ofy corresponding to the n values of a, and the coefficients of the new equation in y are readily found in terms of those of the given equation by the method of symmetric functions, the first for example being as,+ 8s,+s,+&e. The coefficient of y"” is evidently a linear homogeneous function of a, 8, &c., that of y”* a quadratic, of y”~ a cubic function, and so on. In the case of the quintic, the transformation is y=atPxe+rya°+ dx*, and we have four constants a, 8, y, 6 at our disposal. Mr. Jerrard pointed out that the coefficient of y° being a quadratic function of a, 8, y, 6 was (Art. 165) capable of being written as the algebraic sum of four squares, say ?-—u’+v'—w*. It can therefore be made to vanish, by assuming two linear relations between a, B, y, 6; t—u=0, v—w=0. If we combine with these two that linear relation which makes the coefficient of y* vanish, we have three relations enabling us to express three of the constants a, 8, y, 6 linearly in terms of the fourth. We can then by solving a cubic make the coefficient of y* also vanish, or else by solving a biquadratic make the coefficient of y vanish. In this way Mr. Jerrard showed, that by the solution of equations of inferior orders, a quintic may be reduced to either of the trinomial forms y°+by=c, or y°+by’=c. The actual performance of the * The coefficients of the forme-type of the quintic were given by M. Hermite (Cambridge and Dublin Mathematical Journal, 1854, vol. 1x. p. 193), and re-caleulated by me before I found out the key for the translation of Hermite’s notation into Sylvester’s, which is A= J, J, =— K, J, =JK+9L. The discussion of the invariantive characteristics of the reality of the roots of a quintic was originally commenced by M. Hermite in the same classical paper, and was resumed by him in his valuable memoir presented to the French Academy, t. 62, 1866. His result, in our notation, is that the roots are all real, when the discriminant being positive, we have also positive K, 2"Z—J*4+JD, and K (JL+ K*)-18L?. It seems to me that this result is superseded by the greater simplicity of Prof, Sylvester’s criteria,

TSCHIRNHAUSEN’S

TRANSFORMATION.

251

transformations would be a work of great labour, but M. Hermite showed how, by somewhat altering the form of substitution, we can avail ourselves of the help of invariants. If we have to transform the equation aa” + baz™" + cx"* + &e., Hermite assumes

y =ar+ (ax+ 6) at (ax*+ bu +c) B+ (ax*+ bx*+ cx +d) y+&c, then in the first place the transformed equation will be divisible by a; and secondly, if the given equation be linearly transformed, and if the corresponding substitution for the transformed equation be Y=An'+(AX+B) a +(AX’?+ BX+ C) PP’+ &e.,, then he has shewn that the expressions for a’, 6’, &c. in terms

of a, 8, &c. involve only the coefficients of linear transformation, and not those of the given equation. It is not so with respect to the first coefficient A, which we have therefore designated by a special letter. But the theory of linear substitutions will be directly applicable to all functions of the coefficients of the transformed equation which do not contain >. Such, for

example, will be all symmetric functions of the differences of the roots of the new equation, since, on subtracting y, =ar + (ax, +b)a+&e,

y,=ar+(ax,+b)a+ &e.,

d disappears. Or, what comes to the same thing, if we take » such that the coefficient of y"* in the new equation shall vanish, then the theory of linear substitutions is applicable to all the coefficients of the transformed. I give Cayley’s proof of Hermite’s theorem, and, after his example, take, to fix the ideas, the quartic (a,b, c,d, eum, 1)".

Then, as we have used binomial coefficients, we shall write the equation of transformation

y =anr+4 (aa + 4b) y— (aa?+ bx + 6c) 8 + (aa*+ 4bx*+ bea + 4d) a Adding the 4 values of y, and employing Newton’s formule for the sums of powers of the roots, we see that the coefficient

of y"* in the transformed equation will vanish if ad + 3by — 3c8 + da=0.

202

HIGHER

BINARY

QUANTICS.

This reduces the value of y to

(ax +b) y — (ax* + dbx + 3c) B+ (ax® + 4bx” + box + 3d) a. In general it will be observed, that in this substitution all the terms have the binomial coefficients corresponding to the order of the given equation, except the terms not involving a, which have the binomial coefficients answering to the order one lower. 246. Now what is asserted is, that all the coefficients of the transformed equation will be invariants of the system (a, b, ¢, d, eax, y)*, (a, B, vikx, y)’

and of course if we regard y as constant, the whole transformed function will be such an invariant. This will be proved by shewing that it is made to vanish by either of the operations d d

Se

d d

d d

d d

wilh

ergeae ee sd

é

+43).

Let the general substitution be y = V, and let V,, V,, &e. be what V becomes when we substitute for « each of the roots of the given equation, the transformed in y is the product of the factors y— V,, y— V,, &c., and it is sufficient to prove that each of these factors is reduced to zero by this differentiation. We may, as in Art. 64, write oea part of a i operation d

qe? and in order to laaa we must find e

;

Operating

on the given aa we get dz (Qbne, dh a.1) a + (a,b, c, dXa, SP UKO Hieeet

The part then of the differential of V which depends on the variation of x is — {ay - (2ax + 4b) 8 + (3ax’ + 8bax + 6c) a}, and the part got by directly operating on the a, b, &c. which explicitly appear in V is ay — (4ax + 6b) B+ (4ax” + 12b2 + 9c) a.

INVARIANTS

IN QUADRATIC

TRANSFORMATION.

253

Adding, we have 2)

dt

ee

renee

-(« CM eee)

dg

which proves that the effect of the first operation on V is zero. In like manner, for the second operation, we have, by performing on the original equation,

(Gn. ed Yet)

i

elonerdreveal \i=0:

But the original equation i be written

x(a, b,c, dfx,1)°+ (8, c,d, efx, 1) = Hence aad

The part of

due to the variation of x is

therefore

ax'y — (2ax* + 46x’) B + (3ax* + 8ba° + 6cx”) a. The remaining part is (4ba + 3c) y — (46x + 12cx + 6d) B + (4ba* + 12x” + 12dax + 3e) a.

Adding, the coefficient of « vanishes in virtue of the original equation, and the remaining part is found to be

dV

=(yTe 294in) which completes the proof of the ee 247. When this transformation is applied to a cubic, if we consider a, 8 as variables, the coefficients of the transformed equation in y will be covariants of the given equation. The transformed in fact has been calculated by Prof. Cayley, and found to be 7°+ 3Hy+/J, where H is the Hessian (ac—b”) a’+ &e., and J is the covariant (Art. 142), (a’d— 3abc + 2b") a’ + Ke. Prof. Cayley has also calculated the result of transformation as applied to a quartic. Take the two quantics, as in Art. 212, (a, b, ¢, d, eXa, Y)

(a, B, ya, y)"s

and let O denote the skew invariant of the same article, p. 203;

let S and Z denote the two invariants of the quartic; also let

>’=6A, then the transformed function in y is + 18THA. y' +6 (2+ SA)y?+4Cy + SP’ - 35°- 6SZA

254

HIGHER

BINARY

QUANTICS.

Prof. Cayley has also calculated the S and 7 of the transformed equation. In making the calculation, it is useful to observe that since the square of J, from which C was derived (p. 204), can be expressed in terms of the other invariants, so also may the square of C; the actual expression derived from his being in our notation

— 0? = Tp’— S3G*- 9 (23 + GA) TAD+ (2+4+3G9A)?S + 54770 The result then is that the new S is S¢?+397A’+ 187Aq, and the new 7' is T¢°+ S’A¢G’+ 9A’°ST¢ + A® (547” — S”). Finally, he has observed that these are the S and 7 of Ud+6HA, as may be verified by the formule of Art. 210. It follows, then, that tke effect of the Tschirnhausen transformation is always to change a quartic into an equation having the same invariants as one of the form U+)H, and, therefore, reducible by linear transformation to the latter form. The foregoing results in a different notation are reproduced, and the corresponding results for the quintic are obtained in Prof. Cayley’s Memoir on Tschirnhausen’s Transformation, Phil. Trans., vol. CLI. (1862). 248. The following is the form in which M. Hermite presented his theory, and applied it to the case of the quintic. Let u be a quantic (2, y)"; u,, uv, its differentials with regard to x and y; let ¢ be a covariant, which we take of the degree n—2 in order that the equation we are about to use may be homogeneous in x and ¥; then the coefficients of the transformed equation, obtained by putting 2=If, are all invariants of w. 1

The equation in z is got by eliminating « and y between zu,—yp=0, and u=0, or, what comes to the same thing, zu, +xp=0, which follows from the other two. If we linearly transform « and y, the new equation in z is got, in like manner, by eliminating between 2U,— Yb=0,2U,+ Xb=0. But, if C=AX+ wY, y=NX+p'Y, A=Dp’— 2p, we have AX=px-py, AY=dAy-Na, and Art. 130, U,=Au,+u,, U, = wu, + p’u,, and, since ¢ is a covariant, we have ®=A‘d. Making these substitutions, the

equation in 2, corresponding to the transformed equation, is got

HERMITE’S

THEORY.

255

by eliminating between 2 (Au, + Nu,) — Ad (Ay — V’x) = 0,

@ (wu, + w'u,) + A

(wa — wy) = 0.

Multiply the first by yu’, the second by 2’, and subtract, and we have Azu,—A'y#=0. In like manner, multiplying the first by p, the second by 2, and subtracting, we get Azu, + A‘xp =0. In other words, we have the two original equations, except that z is divided by A‘. Consequently, the equations in z corresponding to the original equation, and to the same linearly transformed, only differ in having the powers of z multiplied by different powers of the modulus of transformation A, and therefore the several coefficients of the powers of z are invariants.

The actual form of the equation in z will be

Se +e

pe ar ‘+

Fe

+ &e. = 0.

It is easy to see that the discriminant will appear in the denominator; and the coefficient of z"* will vanish, since, if ¢ be any function of the order n—2, the sum of the results of substituting all the roots of U in eevanishes.

In fact, when

1

the terms of this sum are brought to a common denominator, the numerator is the sum of ¢a multiplied by the differences of all the roots except a, and this is a function of the order n —2 in a, which vanishes for n— 1 values of a, a= 8, a=y, &c., and must therefore be identically nothing. In applying this method to the quintic (#, 1)*, Hermite substitutes

2U,= 4b, + Bb, + 7b, + 8,

where ¢,, ¢,, $,) $, are four covariant cubics of the orders 3, 5, 7, 9 respectively in the coefficients. ¢, is the canonizant. ¢, is the covariant cubic of the fifth order, the Jacobian of S and 7 whose leading term or source, whence all the other terms can be derived, is printed in full as (11) Art. 232; on inspection we see that this source vanishes if both a and 6 vanish; consequently, if the given quintic has two equal roots, their

256

HIGHER

BINARY

QUANTICS.

We can form a common value satisfies this covariant. covariant cubic of the seventh order from ¢, in the same way that ¢, was formed from ¢,, and by adding ¢,, multiplied by J and a numerical coefficient, can obtain ¢,, such that its source vanishes when a and 0 vanish; and, in like manner ¢, can be made to possess the same property.

When this substitution is made, the coefficient of 2° is a quadratic function of a, 8, y, 6. Hermite finds for its actual value (a result which may be verified by working with the special form, note, p. 248),

{Fo? + 6K Day — D (F'+10SK) y"} + D {KB + 2FB8 — (9KD + 10AF) 8°}, where F=9 (16L—JK), which vanishes when the quintic has two distinct pairs of equal roots. By breaking up into factors each of the parts into which this coefficient has been divided, the two linear relations between a, y; 8, 6, which will make it to vanish, can readily be obtained; as also by another process which I shall not delay to explain. The discussion of this coefficient is also the basis of Hermite’s later investigations as to the criteria for reality of the roots. He avails himself of a principle of Jacobi’s (Crelle, vol. L.), that if a, B, y, &c. be the roots of a given equation, and if the quadratic function (¢+au+a’vt...a"*w)’ + (£+ But 6v + &.)’ + &e., be brought by real substitution to a sum of squares, the number of negative squares will be equal to the number of pairs of imaginary roots in the equation. Hermite shews, by an easy extension of this principle, that the number of pairs of imaginary roots of the quintic is found by ascertaining the number of negative squares, when the coefficient of 2° just written is resolved into a sum of squares. And since the same process is applicable to every equation whose degree is above the fourth, he concludes that the conditions for reality of roots in every equation above the fourth can be expressed by invariants. 249. It does not enter into the plan of these Lessons give an account of the researches to which the problem

to of

RESOLVENTS

OF QUINTIC.

PASS

resolving the quintic has given rise.* The following, however, finds a place here on account of its connection with the theory of invariants. Lagrange, as is well known, made the solution of a quintic to depend on the solution of a sextic; and it can easily be proved that functions of five letters can be formed capable of six values by transposition of letters. Let 12345 denote any cyclic function of the roots of a quintic; such, for example, as the product

(a — 8)" (8 — 9)’ (y - 8)’ (8—«)* (e- a)’, where evidently 23451 and 15432 would denote the same as 12345; then it can easily be seen that there can be written down in all twelve such cyclic functions. But, further, these distribute themselves into pairs; and by so grouping them we can form a function capable of only six values; for instance,

12345 +13524, 12435414523,

13245412534,

13425 +14532,

14235 + 12543, 14325413542. The actual formation of the sextic having these values for its roots is in most cases a work of extreme labour. M. Hermite, however, pointed out that when the function 12345 is the product of the squares of differences written above,t all the coefficients of the corresponding sextic are invariants, and that the calculation therefore is practicable. I have thought it desirable actually to form the equation, because, in the development of the theory of sextics, it will be necessary to ascertain the invariant characteristics of sextics whose solution depends on that of a quintic; and it may be useful to be in possession of more than one of the sextics which spring out of the discussion of a quintic.t I take the simple example x° + 2mz*y’+ ay’, of which, * Among

the most remarkable

of recent

investigations

in this subject is the

application to it of the theory of elliptic functions by M. Hermite and M, Kronecker. + In the method of Messrs. Harley and Cockle, the function 12345 is aB+ By + yO + de + éa, and the sextic chosen is that whose roots are 12345 — 13524, &c. This has been calculated by Prof. Cayley (Philosophical Transactions, 1861, p. 263), and the result is very simple, two terms of the sextic are wanting; but the coefficients are not invariants. t The form arrived at by M. Kronecker and M. Brioschi is (x — a)> (w — 5a) + 10 (@ — a)? —¢ (aw — a) + 50? — ac = 0. By the help of the formule given further on, the invariants of this equation can be calculated, and a, b, ¢ eliminated.

258

HIGHER

BINARY

QUANTICS.

since two pairs of roots are equal with opposite signs, the functions of the differences can easily be formed. I find then that the sextic is the product of t? +.2° (m+ m°) t + 2” (m° — 2m* + 5m’), by the square of t? + 2° (mm? + 3m) t + 2° (m° + 5m* + 19m — 25). But if we first multiply the quintic by five, its invariants are D=2°.5°(1—m’),L =4m(5—m’)*. J =2'm (5+3m’), To avoid fractions I write J=2A, D=250B, J*-—2"L=500; and then forming the sextic, and expressing its coefficients in terms of the invariants, I obtain + 4At? + (6A? —25B) t+ (44° +2C—30AB) # +0 (A*+4AC- 17A°B+ £255?)

+#(24°C—44°B-7BC+110AB’) + C?-4ABC+ 20A4°B’, which is a perfect square, as it ought to be, when D=0.* 250. M. Hermite has studied in detail the expression of the invariants in terms of the roots. He uses the equation transformed so as to want the first and last terms ; that is to say, so that one root is 0 and another infinite; and the calculation is thus reduced to forming symmetric functions of the roots of a cubic. I had been led independently to try the same transformation on the problem discussed in the last Article, but found that, even when thus simplified, the problem remained a difficult one. It would be necessary to form for a cubic the sextic whose roots are the six values of

a’ (8B — 9)" (a— 4) + Bey? (a— BY, and then to identify the result with combinations of the forms assumed by the invariants of the quintic when a and / vanish. M. Hermite expresses his own invariant J as follows. Let

a, = (4— 8) (a— ¢) (8-4) +(a—y) (2-8) (B— 8), a, = (%—8)(a) (e - 8) +(a—8) (a— 2) (8-4), a, = (a — 8) (a — 8) (© — y) + (ay) (a— 2) (8-8);

the continued product of these is symmetrical with respect to all the roots except a; and if we multiply this product by the similar products obtained for the other four roots we get J. * Though the form with which I have worked is a special one, I believe that the result is general; because it seemed to me that the coefficients only admitted of being expressed in terms of the invariants in one way.

CONCOMITANTS

IN TERMS

These factors are of course

a? ge2) on

)

:

1, y+s 1

=4

OF

THE

ROOTS.

259

the values of the determinants

2a-—(y+e),

2a—(8+8),

a(y +e)— 2ye

Fe

a(6+8)—285

which express, p. 193, that one of the roots is self-conjugate of the involution determined by the other four, which is the case when J vanishes, as remarked, Art. 229. Determining the numerical constant by a special form, such as az*+5ery*+fy°=0, we find the product of these fifteen factors by a’ to be 10°°Z. 251. From the roots of formed by omitting each in the equi-anharmonic functions also T,., Tz, Ty, Ts, Te their see by comparing terms in a quintic ax*+ 10cxz*y’ = 0, that

=

a quintic five sets of four can be turn, let Sa, Sg, S,, Ss, Se denote of these sets of four, see Art. 199; harmonic functions; it is easy to simple case, for instance, for the we have in terms of the roots

8 = Sa (x-ay)’+ Sp(a-By)*+

8, (x-yy)’+ Ss(a-8y)"+

8. (a-ey)?

— ip DP=Tql2-ay)+ Tye-By)"+ Ty e—ny) Tece-By)"+ Te (ay) = 100H = R=4(5) -8(4) (6)(2): In the above formula, as at p. 17, 7 denotes the determinant formed with the unreduced second differential coefficients ;but if we suppose that, as in the Higher Plane Ourves, each coefficient has been cleared of the numerical factor six, we must write in the above for H, 2167; also, since P, Q, involve the second differential coefficients in the 2nd, 4th, 6th degrees respectively, these will be 67, 6*, 6° times the corresponding P, Q, &. Making these substitutions, the eliminant becomes

uH’+ 3PH’ +3QHu+ fu’. To reduce this further observe that a (‘)=— 3H; for (¢)

expanded is — (w,,w,, — v’,,) a,” + &c., but a,’a (Art. 298) is w,, &e., 22° 33

332

APPLICATIONS

OF SYMBOLICAL

METHODS.

on making which substitutions, the truth of what has been stated

appears.

In a similar

manner

it is easily seen

that

a (3)=— Hb, from which at once follows that a*b’ (5)=— Hu.

We have then P= 3Hu. In order to calculate Q, it appears from what we have just stated that

ab (2)(5)=a (4)x8 (5)=9H, and at (5)(5)=—H0 (;)D=k( px)? +1 (pr)? +m (py)’. Thus, if the required even quantic f/ be symbolically expressed by a,, we findf expressed by means of k, J, m by replacing p by a in this symbol and raising it to the power n:

FD" = {(aw’)fhe-+ (ad’)41 + (apf) x {(an’Pk + (an”)'l+ (ap’’)’m} x {(an'Pk + (an)70+ (ap)?m}. When this product is expanded the coefficients of its terms ktl’m™ are evidently invariants, thus the expression is typical (p. 249). The quadric relation which subsists between 4, /, m enables us to go back from this ternary to a binary form: introducing in so doing only the invariants of the three quadratics. If we retain k, 2, m as three variables the symbol a,” defined by a, =(ax)’y, + (aA)*y, + (am)*y, is evidently that of a unique curve of the ” order, which is determined by the 2n points of f,which are now given on a conic. The additional relations of this curve to the conic by which its uniqueness is secured arise from the fact that if the conic be written symbolically 8 =8,?, then their covariant (488’)’a,"” vanishes identically* ; * F, Lindemann, Bulletin de la Soc. Math. de France, t. v. 1877. For n> 3 this specializes the curve, thus he notices that the conditions for m= 4 limit the ternary quartic to admit of being the sum of five fourth powers, cf. Note, p, 151.

336

APPLICATIONS

OF SYMBOLICAL

METHODS.

or again, that there is an infinity of triangles conjugate to this conic which are inscribed in the polar conic of y relative to the curve ,"=0; in other words, that all the conical polars of this curve are apolar to the conic [Exe

314. The exposition, however, can be made to involve invariants of lower degrees by using as its basis instead of the quadratics their Jacobians. In fact, as in Art. 193, we can see that R,,.u,=D,J,,+ DyJy, + 43° DaFiy12 =U, Boag — Up May + Uylt 1234 41° 23 42° 31 2° 341 3° 412? or, in the present notation, that k (ax) +1 (ar)? +m (ap) = (ak) « + (al)? + (am)? whence /D"={(ak’)’« + (al) + (am’)’ p}

x {(all’ n+ (al. + (amp)

x {(ak)*« + (al)? + (am'")*u}. If n be even, we can multiply out and replace all powers of kK, A, # by functions of the same order in &, 1, m without denominator. If m be odd, the product of all factors but one gives us even powers of «, A, w which can be so treated, and the last factor when expressed by &, l, m introduces a new factor Dupon f. Thus the coefficients are the n™ transvectants of f on ktm, where a+ 8+y=4n, and are accordingly of much lower order than the others, which were its transvectants of the same order on «“r"u% where a+B+y= 4n. It is in this manner that Clebsch developes the expression for the sextic by its quadric covariants when £ does not vanish. Each covariant symbolically contained in

B'f=a, (alr + (am)'p + (an)'y} {(al "+ (amn’hu+ (an’yv}

is found as a linear function of 2, m, n by means of the invariants A, I,, I,, D’, and, on expanding, the products of the second order of A, #, v are replaced by quadric functions of 1, m, n without denominator.

315. In a similar manner Dr. F. Lindemann (Math, Ann. XXIII, p. 133) has given the typical exposition of forms of the order 3p by means of four independent cubic covariants in connection with the formule of Art. 219d, and the consequent geometrical reference to a twisted cubic without the canonical reduction employed in Arts. 219c, d.

TYPE FORM

OF BINARY

QUANTIC

OF ORDER

3p.

337

In fact, now denoting the SU: Py,hey 1’,m,;, n,*, and the forms a ft aie Tass a0 ioe,Ne, phe, Vay tS easily seen that for any other cubic a,° we have (Gove (mn)(nl)(lm)(al)(am)(an), and writing (k«)*= (A) = &c. = A, that

Aa, = (ax)'k,? + (ad), + (au) + (av)’n,’

= (ake? + (al)’r,? + (am)'y,? + (an) ve.

Whence we find the 3p* f by raising this to the degree p Ara’ = {(an’)*k+ (an )l+ (ap’)*m + (av’)n} x {(ak”’)*k + (ar) + Cap”)? + (av’’)*n}

or

x {(ax"?))*k + (ar) 91+ (ap'”’)?m + (av)*n},

= f(a’ et (AlN + (an’iu + (an’)} x (ah e+ aI (amin + (an) x {(ae)* K+

(al?)

ak (am”)*p aj (an®)%y Io

Either form expanded is a function of degree p in four variables with invariant coefficients. Taking the point z,=/,°, z,=1%, &c. as variable with a its locus is a twisted cubic, and aR coordinates of the osculating planes determined by a are u,=K«?, u,=r,;, &e. The equations of three quadrics through this curve or oF three quadrics touched by all its osculating planes can be easily found from the obvious relation A (an Pah

n tl r+

min, +n, v's

and the others which follow from it as successive polars. When found these equations enable us to return to a binary system introducing only invariants of the cubics. Thus the equation of the surface, whose intersections with the twisted cubic are the 3p points of the binary quantic, is either «”=0 where a,=(ax)’z, + (ad)2, + (au)'%, + (av)'2,) or B?=0 where £,=(ak)’u,+(al)’u,+ (am)’u,+ (an)*uy this tangential form involving invariants of lower order. This surface 4?=0

is unique, and if we denote by w*=0,

Ue? =0, ux’? =0, the three quadrics touched by all osculating planes of the twisted cubic, it is shown Np ye Lindemann that this is because the three equations a4"=0, aoa,?"=0 ? a,’a,’* = 0 are identically true, and Herefoter oP O18 conjugate,

or apolar, to all the quadrics which can be inscribed in the developable surface of the twisted cubic. XX

( 338)

NOTES. HISTORY

OF DETERMINANTS.

(Page 1).

Tux following historical notices are taken from Baltzer’s Theory of Determinants ; and from the sketch prefixed to Spottiswoode’s Elementary Theorems relating to Determinants. The first idea of determinants is due to Leibnitz, as Dirichlet has pointed out. In Leibnitz’s letter to L’H6pital, 28 April, 1693 (Leibnitz’s Mathematical Works, published by Gerhardt, vol. 11.. p. 239), is to be found the first example of the formation of these functions, and of their application to the solution of linear equations; the double suffix notation (p. 7) is entployed, and he expresses his conviction of the fertility of his idea. But nowhere else in his writings is there to be found any proof that he sought to draw any new fruits from his discovery; and the method was lost until re-discovered by Cramer in 1750. Cramer, in his /ntroduction a IAnalyse des lignes Courbes (Appendix), has exhibited the determinants arising from linear equations in the case of two and three variables, and has indicated the law according to which they would be formed in the case of a greater number. The rule of signs by the method of displacements (p. 6) is given by Cramer. The equivalence of the other method by permutations of suffixes was afterwards proved by Bezout and Laplace. In the Histoire de Académie Royale des Sciences, Année 1764 (published in 1767), Bezout has investigated the degree of the equation resulting from the elimination of unknown quantities from a given system of equations, and has at the same time noticed several cases of determinants, without however entering upon the general law of formation, or the properties of these functions. The Histoire de

VAcadéemie, An. 1772, part 11. (published in 1776), contains papers by Laplace and Vandermonde relating to determinants of the second, third, fourth, &c, orders. The former, in discussing a system of simultaneous differential equations, has given the

law of formation, and shown that when two rows or columns are interchanged, the sign of the determinant is changed, and that when two are identical, the determinant vanishes, The latter employs a notation in substance identical with that which, after Mr. Sylvester, we have called the umbral notation, and explained p. 8. In his Memoir on Pyramids (Mémoires de lAcadémie de Berlin, 1773), Lagrange made an extensive use of determinants of the third order, and demonstrated that the square of such a determinant can itself be expressed as a determinant. The next impulse to the study was given by Gauss, Disquisitiones Arithmetice, 1801, who showed, in the case of the second and third orders. that the product of two determinants is a determinant, and very completely discussed the case of determinants of the second order arising

from quadratic functions, z.e. of the form }?—ac. In 1812 Binet published a memoir on this subject (Journal de Ecole Polytechnique, tome 1X., cahier 16), in which he establishes the principal theorems for determinants of the second, third and fourth orders, and applies them to geometrical problems.

The

next volume of the same

series contains a paper, written at the same time, by Cauchy, on functions which only change sign when the variables which they contain are transposed. The second part of this paper refers immediately to determinants, and contains a large number of very general theorems. Cauchy introduced the name “determinants,” already applied by

NOTES.

339

Gauss to the functions considered by him, and called by him “determinants of quadratic forms.” In 1826 Jacobi took possession of the new calculus, and the volumes of Crelle’s Journal contain brilliant proofs of the power of the instrument in the hand of such a master. By his memoirs in 1841, De formatione etproprietatibus determinantium and De determinantibus functionalibus (Crelle, vol. XxX1I.), determinants first became easily accessible to all mathematicians. Of later papers on this subject, perhaps the most important are Cayley’s papers on Skew Determinants (Credle, vol. XXXII, and XxXXvill.). Of elementary treatises on this subject, I have to mention Spottiswoode’s Elementary Theorems relating to Determinants, London (1851) ; Brioschi, La teorica det determinanti, Pavia, 1854; and Baltzer, Theorie und Anwendung der Determinantem,

Leipzig,

1857;

1881.

fifth edition,

French

translations

both of Brioschi’s and Baltzer’s works have been published.

COMMUTANTS.

(Page 8).

In connection with the umbral notation may be explained what is meant by commutants, which are but an extension of the same idea. It is easy to see what, according to the rule of the umbral notation, is meant by &, n, ee sticy Pee

&, 1,

if we write for brevity &, 7, for pene

We

&, En, n’,

compound the partial constituents in

each column in order to find the factors in the product we want to form, and take the sum with proper signs of all possible products obtained by permuting the terms in the lower row. Thus the first example denotes €?.n?— £n.£y, which is the Hessian: the

second denotes &.£??.n4—£4. En. En? &e., which is the ordinary cubinvariant

of a

quartic. Again, since multiplication is performed by addition of indices, it will be readily

understood that we can equally form commutants where the partial constituents are combined by addition instead of by multiplication, Thus, considering the quantics (42) Gy; ag Kx, Y)?,

(Usy gy Any My ao Ka,y)',

the invariants in the last two examples may be written 1, 0,

2

1, 0,

2, 1, 0,

expanded are dgtty— 0, ; U%Ay%)—442,2, + ke. All these commutants with only two rows may be written as determinants, but it is a natural extension of the above

two rows, such as &, n,

notation

to form commutants

1, 0,

&, &n, n?.

&, 1;

1, 0,

(Eb Ep UP

&, 1;

1, 0,

&, En, 17

E, 1,

1, 0,

£7, &n, 17.

with more

than

These all denote the sum of

a number of products, each product consisting of as many factors as there are columns in the commutant and each factor being formed by compounding the constituents of the same column; and where we permute in every possible way the constituents in

each row after the first. namely,

the

Thus the first and second examples denote the same thing,

quadrinvariant

of

a

quartic

expressed

in

either

of

the

forms

E44 E59. En +8E'n?.En? or a4ao—4a,0, + 34,4, while the third example £%.£4n4.n8— &ec., denotes the cubinvariant of an octavic given at length, Art. 155. We have seen that the two invariants of a binary quartic can be expressed as commutants, but it will be found impossible to express in the same way the discriminant of a cubic. Thus the leading term in it being a,’a)? or £,&3n3n3, we are naturally led to expect that it might be the commutant €, », €, n, but this commugn,

& 1,

& 1, & 1

340

NOTES.

tant, instead of giving the discriminant, will be found to vanish identically. It may, however, be made to yield the discriminant by placing certain restrictions on the permutations which are allowable.. For further details I refer to the papers of Messrs. Cayley and Sylvester in the Cambridge and Dublin Mathematical Journal, 1852,

ON RATIONAL

FUNCTIONAL

DETERMINANTS.*

(Page 14).

The determinants considered Ex. 5, 6, are particular cases of the important form

$ (x), W (z) -.. (Y), WY) where ¢$ (x), W(x) denote rational integral functions of x, and ¢ (y), ¢ (2), &., the same functions of y, z, &c. respectively. Such a determinant may be briefly denoted by its top line | ¢(z), W(x)... |. Thus the determinant Ex. 5 may be written | 1, x, 2...2%1 |. This last determinant we have seen has for its value

(Cro (a, y 2...); by which notation Prof. Sylvester denotes the, continued product of the differences (x—y) {a —z) (a@—w)... x (y—z) (y—w) ... x (2@-w) &e. This alternate product is of the nature of a square root: its square we know is a symmetrical function of x, y, z, &c., and is unaltered by any permutation of these variables; but itself has two values corresponding to the different arrangement of the variables, its sign being altered if we permute any two of the variables. The function | 1, z,...2*1| was suggested by Cauchy as a symbolic representation of a determinant, viz. expanding it as the sum of a series of terms +y!z?w% ..., and

changing the exponents into suffixes, or the term Ho Ly---Cny development of the determinant

into x,y,2.w;..., we

have the

Yor Yire-Yn-a

It may be further remarked in passing, that any rational function of the variables x, y, &e., which, however the variables are permuted, has only two values, must be of the form P+ Q¢4, where P, Q are symmetric functions of the variables. Returning now to the general determinant | p(«), W(z)... |, it obviously contains (as a factor, for on the supposition x= y, it vanishes as having two rows the same, and is therefore divisible by 2—y; and similarly with regard to every other difference. Let us then in particular examine | z*, 74, x7... |, which we may call | a, B, y, -.. | im order to find the value of the remaining factor. If a be the least of these exponents, we may divide each row by «*, y%, ... respectively, so that we can at once reduce the investigation to that of the case where a=0. In the following we employ a method given by Jacobi, De Functionibus Alternantibus, Crelle 22, (1841); depending on the consideration of the determinant al ee eels For convenience we work with the case of three

z—a’a—b’ae—c”™’ | variables, but it will be seen that the process is perfectly general. Consider then the equation which is obviously true 1

1

il

z—a’ x—b’ x—c

Thatta hin tere

SCG 1

z—a’

AP age 1

F(a, v2) Gab,0) (w—a)(x—6) (a—c) (y—a) (y—8)(y—c) @—a)(e—4) (z—e)’

1

¢—b’ z-c! * This note is, in substance, Professor Cayley’s.

NOTES.

341

and expand each side by descending ie of a y, 2 We have

Lead =+5ptser &e.; “L-QG 2 whence, on the left-hand side, the term ete ames

Oy

the reciprocal of ya? is

cal

@FA, DBA, cha at}, bY-1, cy

In order to expand the right-hand side, observe first that if H,, H, &c. have the same meaning as at Art, 272, then

1

ahs

es

(2—a) («@—6)(c@—c)

a

age

ot * 2

as is easy to see by multiplying together the expansions for — v4 (a, y, 2) =—

1, w, x lly, ¥

&ce.

We

have also

{.

ihe Yh ee

Hence, the right-hand side is ¢ (a, b, c) multiplied by 1 pt

Le, atpztte,

yth&.,

eats § = toa the.

?

and the term multiplying the reciprocal of «*y®z7 is 2 (a, 5, c) multiplied by

Haz, Ha», Hy Ags

Hp 2, He,

Hy -25 Hy -25 Hy

We have thus

| a%1, bt), #1 aB, BA, eB Beats

.

Tihs Villers Vis -¢ (a, b,c) |Hp-s, Heo, He.

Coa

|;

H,, 3) H,- 2) Hy

which we may write (a—1, 8-1, y-1)=¢(a,}, c) H (a—3, B—3, y—8). We

may

verify this equation

by writing

a=1,

B=2,

y=3,

observing

that

Gia, b, c)=—(0, 1, 2), that H_,, H_, vanish and that A; is 1. If we write a=1, and for f, y write

8+1, y

+1,

have -»

Hy,

Hg

(0, 8, y)=CH (2, B-2, y-2)=% iHB» Be » He )

Dey-y H,Lye »

H,

But since H_,, H_, vanish, and H,=1, the last determinant reduces to Thus we have finally (0, 8, y) =— (Hp oH, i 1— Ap. y- 2).

He» H g Hy-2 i

As an example, taking

B=1, we get (0, 1, y)=-Hy- x, a formula teen includes that of Ex. 6, p. 15. We may also consider determinants involving the square roots of rational functions

Nip @)}, 21th @)}; Ne Mh y he (yh

?

but these, although presenting themselves ini the theories of Elliptic functions, have been but little studied.

HESSIAN.

and Abelian

(Page 17).

The name was given by Sylvester after Professor Otto Hesse, who made much use of the functions in question, which he called functional determinants. They are a particular case of those studied under the same name by Jacobi (Crelle

342

NOTES.

vol, Xx1I.), the constituents of which are the differentials of a series of n homogeneous functions in » variables. It is so convenient to have short distinctive names for the functions of which we have repeatedly occasion to speak, that I have followed Sylvester in calling the former Hessians, the latter Jacobians, see Art. 88, SYMMETRIC

FUNCTIONS.

(Page 56).

The rules for the weight and order of symmetric functions are Prof. Cayley’s, The formula, Art. 59, I have taken from Serret’s Lessons on Higher Algebra. The differential equation, Art. 60, is an anticipation of the differential equation for invariants, of which I speak, Art. 148.

Brioschi

(see M. Roberts, Quarterly Journal,

vol. Iv. p. 168), remarked that the operation {ydz} (Art. 65), expressed in terms of the alate d: roots, is Eps“1 ier &e.

ELIMINATION.

(Page 66).

The name ‘eliminant’ was introduced I think by Professor De Morgan; I believe I have done wrong in using a second appellation when a name to which there was no objection was already in use. The older name ‘resultant’ was employed by Bezout,

Histoire de VAcadémie de Paris, 1764. The method of elimination by symmetric functions is due to Euler (Berlin Memoirs, 1748). The reduction of the resultant to that

ofa linear system was made simultaneously by Huler (Berlin Memoirs, 1764) and Bezout (Paris Memoirs, 1764). The theorem as to the degree of the resultant is Bezout’s, The method used in Art. 74 of forming symmetric functions of the common values

of a system of two or more equations is Poisson’s (see Journal del Ecole Polytechnique, Cahier x1.). Sylvester gave his mode of elimination in the Philosophical Magazine for 1840, and called it ‘dialytical,’ because the process as it were dissolves the relations which connect the different combinations of powers of the variables and treats them as simple independent quantities. Cayley’s statement of Bezout’s method is to be found, Crelle, vol. LI11., p.366. Sylvester’s results in Art. 91 are to be found in the Cambridge and Dublin Mathematical Journal for 1852, vol. vit., p. 68; and Cayley’s general theory (Art. 92, &c.) in the same Journal, vol. III., p. 116. It was noticed by Lagrange, that when two equations have two sets of common roots, the differential of the resultant with respect to the last term vanishes (see Berlin Memoirs, 1770). Sylvester showed, in January, 1853, that the same was true of all the differentials, Cambridge and Dublin Mathematical Journal, vol. VII1., p. 64. He showed at the same time, that the common roots were given by the ratios of the differentials. The proof in Art. 99 is, I believe, my own. The theorem, Art. 99, is Jacobi’s Crelle, vol. xv., p.105. In this

part I have made some use of the Treatise on Elimination by Fad de Bruno.

The

theorem of Art, 102 is Prof. Cayley’s.

DISCRIMINANTS.

(Page 98).

The word ‘discriminant’ was introduced by Sylvester in 1852,

Cambridge and

Dublin Mathematical Journal, vol. v1., p. 52. The word ‘determinant’ had been previously used, and had come to have a perplexing variety of significations. The theorem referred to, Note, Art. 111, was the basis of my investigations (Cambridge and

Dublin Mathematical Journal, 1847 and 1849) on the nature of cones circumscribing surfaces having multiple lines. If the equation of a surface be })+ 0,24 b,07+ &ce., and if xy be a double line, y must be contained by 4) in the second and 3, in the first degree. The discriminant with respect to x is a tangent cone which has y? for a factor.

NOTES.

343

BEZOUTIANTS. (Page 107). It has been shown (Art. 85) that the resultant of two equations of the nt degree is expressed by Bezout’s method as a symmetrical determinant. This may be considered (Art. 118) as the discriminant of a quadratic function which Sylvester has called

the Bezoutiant of the system.

When the quantics are the two differentials of the

same quantic, then if we resolve the Bezoutiant into a sum of squares (Art. 165), the number of negative squares in this sum will indicate the number of pairs of imaginary roots in the quantic. The number of negative squares is found by adding (as in

Art. 46) » to each of the terms in the leading diagonal of the matrix of the Bezoutiant, and then determining by Des Cartes’ rule the number of negative roots in the equation for X. The result of this method is to substitute for the leading terms in Sturm’s functions, terms which are symmetrical with respect to both ends of the quantic; that is to say, which do not alter when for x we substitute its reciprocal (see Sylvester’s

Memoir, Philosophical Transactions, 1853, p. 518).

LINEAR

TRANSFORMATIONS.

(Page 108).

The germ of the principle of invariance may be traced to Lagrange, who, in the Berlin Memoirs, 1773, p. 265, established the invariance of the discriminant of the quadratic form az?+2bry+cy?, when for x is substituted 2+)y. Gauss, in his Disquisitiones Arithmetice (1801), investigated very completely the theory of the general linear transformation as applied to binary and ternary quadratic forms, and, in particular, established the invariance of their discriminants. This property of invariance was shown to belong to discriminants generally by the late Professor Boole, who, in a remarkable paper, Cambridge Mathematical Journal, 1841, vol. 111., pp. 1, 106, applied it to the theory of orthogonal substitutions. He there showed how to form simultaneous invariants of a system of two functions of the same degree by performing on the d d discriminant of one of them the operation a’ ee b , at &c. Boole’s9 paper led to Cayley’s proposing to himself the problem to determine & priori what functions of the coefficients of an equation possess this property of invariance, He found that it was not peculiar to discriminants, and he discovered other functions of the coefficients of an equation at first called by him ‘hyper-determinants,’ possessing the same property. Cayley’s first results were published in 1845 (Cambridge Mathe-

matical Journal, vol. tv., p 193). From this discovery of Cayley’s, the modern algebra which forms the subject of the bulk of this volume may be said to take its rise. Among the first invariants distinct from discriminants, which were thus brought to light, were the quadrinvariants of binary quantics, and in particular the invariant S of a quartic. Mr. Boole next discovered the other invariant 7’of a quartic, and the expression of the discriminant in terms of S and 7 (Cambridge Mathematical Journal, vol. IV., p. 208), It is worthy of notice that both the functions S and 7’ had been used by Hisenstein (Credle, 1844, xxvi1, p. 81) in his expression for the general solution of a quartic, but their property of invariance was unknown to him, as well as the expression for the discriminant in terms of them. Cayley next (1846) published the symbolical method of finding invariants, explained in Lesson x1v. (Cambridge and Dublin Mathematical Journal, vol. 1., p. 104, Crelle, vol. Xxx.). The next important paper was by Aronhold, 1849 (Crelle, vol. XXx1X., p. 140), in which the existence of

the invariants S and 7 of a ternary cubic was demonstrated.

arly in 1851 Mr. Boole

reproduced, with additions, his paper on Linear Transformations (Cambridge and Dublin Mathematical Journal, vol. v1., p. 87), and Sylvester began his series of papers in the same Journal on the Calculus of Forms, after which discoveries followed

344

NOTES.

in rapid succession. I can scarcely pretend to be able to assign to their proper authors the merits of the several steps; and, as between Messrs. Cayley and Sylvester, perhaps these gentlemen themselves, who were in constant communication with each other at the time, would now find it hard to say how much properly belongs to each. To Mr. Boole (Cambridge and Dublin Mathematical Journal, vol. VI., p. 95, January, 1851) is, I believe, due the principle that in a binary quantic the operative symbols Fe

£ may be substituted

for « and y.

The

principle was

extended to quantics in general by Sylvester, to whom is to be ascribed the general statement of the theory of contravariants, Cambridge and Dublin Mathematical Journal, (1857), vol. VI, p. 291; although particular applications of contravariants had previously been made in Geometry in the theory of Polar Reciprocals, and in the theory of ternary quadratic forms by Gauss (Disquisitiones Arithmetice, Art. 267), who gives the reciprocal under the name of the adjunctive form, and establishes its

invariance under what he calls the “transformed

substitution.”

Sylvester also re-

marked that we might not only replace contravariant by operative symbols, but also du du by the actual differentials &c. To Boole I would ascribe the principle da’ dy’ (Art. 126) that invariants of Seenaais are covariants of the quantic (1842), Cambridge Mathematical Journal, vol. 111., p. 110, though Boole’s methods were generalized by Sylvester, Cambridge and Dublin Mathematical Journal, vol. v1. p. 190. Some of the first steps in the general theory of covariants may thus be ascribed to Boole,

though a remarkable use of such a function had been made by Hesse in determining the points of inflexion of plane curves. I had myself been led to study the same functions both for curves and surfaces, in ignorance of what Hesse had done (Cambridge and Dublin Mathematical Journal, vol. 11., p. 74). The discovery of evectants (Art. 134) is Hermite’s, Cambridge and Dublin Mathematical Journal, vol. V1., p. 292. In Cayley’s first paper he gave a system of partial differential equations satisfied by invariants of functions linear in any number of sets of variables. The partial differential equations (Art. 149) satisfied by the invariants and covariants of binary quantics were, as far as I know, first given in print by Sylvester (Cambridge and Dublin Mathematical Journal, vol. vit., p. 211). Sylvester there acknowledges himself to have been indebted to an idea communicated to him in conversation by Cayley ; and he also speaks of having heard it said that Aronhold also was in possession of a system of differential equations. These are not made use of in Aronhold’s paper (Crelle, vol. XXxXIX.) already referred to, but he refers, Credle, vol. LXIL., to a communication made by him in 1851 to the Philosophical Faculty at Konigsberg, which, if it ever appeared in print, I have not seen.

Very probably there may be other

parts of the theory to which Aronhold may justly lay claim. After the publication in Crelle, vol. XXx., of Cayley’s paper, in which the symbolical method of forming. invariants was fully explained, Aronhold worked at the theory in Germany simultaneously with the labours of Cayley and Sylvester in England; and the mastery of the subject exhibited by his papers leads me to suppose that of some of the principles he must be able to claim independent if not prior discovery. The method in which the subject is introduced (Art. 121) is taken from his paper (Credle, vol. Lx11). I refer in a note on next page to the valuable paper by Hermite (Cambridge and Dublin Mathematical Journal, vol. 1X., p. 172), in which the theorem of reciprocity was established, which had at first suggested itself to Sylvester, but was hastily rejected by him, and in which the whole theory of quintics received important additions. Mixed concomitants are Sylvester's (Cambridge and Dublin Mathematical Journal, vol. vir p. 80). The theorem, Art. 135, is Cayley’s and Sylvester's. The application of symmetric functions to the invariants of binary quantics was, I believe, first made in the Appendix tomy Higher Plane Curves (1852). The method (Art. 138) of thence finding

NOTES.

345

conditions for systems of equalities between the roots is Cayley’s (Philosophical Transactions, 1857, p. 703). With regard to the subject generally, reference must be made to the important series of papers by Sylvester, beginning in the sixth volume of the Cambridge and Dublin Mathematical Journal; to a series of papers

on Quantics published by Cayley in the Philosophical Transactions ; and to Aronhold’s Memoir on Invariants (Crel/e, vol. LX11.). The name ‘invariant,’ as well as much of the rest of the nomenclature, is Sylvester’s.

CANONICAL

FORMS.

(Page 150).

The name is Hermite’s; the theory explained in this Lesson is Sylvester’s, see a paper (Philosophical Magazine, November, 1851) published separately, with a supplement, in the same year, with the title An Essay on Canonical Forms.

COMBINANTS.

(Page. 161).

The theory of combinants is Sylvester’s, Cambridge and Dublin Mathematical Journal (1858), vol. ViIl., p. 63. In the case of the resultant of two equations it had, I think, been previously shown by Jacobi, that the resultant of Aw+ pv, A’utp’'v was the resultant of wu, v multiplied by a power of (Au’—A’p). Sylvester’s results Arts, 185, 188, 189, are given in the Comptes rendus, vol. LVII1., p. 1074-9.

APPLICATIONS In Lesson

Prof.

TO BINARY

xvii the discussion

Cayley’s.

QUANTIOS.

(Page 175).

of the quadratic, cubic, and quartic, is mainly

See his Memoirs on Quantics in the Philosophical Transactions, 1854

The second form of the resultant of two quadratics, p. 180, is, as elsewhere stated, Dr. Boole’s. Sylvester proved (Philosophical Magazine, April, 1853) that every invariant of a quartic is a rational function of Sand 7. The theorem, Art. 206, that the quartic may be reduced to its canonical form by real substitutions, is Legendre’s (Traité des Fonctions Elliptiques, chap. 11). The discussion of the systems of quadratic and cubic, two cubics, and two quartics, was, I believe, for the most part new, when it appeared in the second edition in 1866. The form for the resultant of two cubics, obtained by him by a different method, was published by Clebsch (Credle, vol. LX1v.), but had been previously in my possession by the method given in Art. 213. On the connection p. 174 between concomitants of binary systems and those of a larger number of variables, R. Sturm’s paper (Borchardt, Lxxxvi. pp. 116-46) should be referred to, Also for the reduction of the system of two quartics, p. 224, announced by Sylvester, see Stroh, Math. Ann. xx1t, 293, who cites d’Ovidio as having also effected it. In Lesson xviiI the canonical form of the quintic ax + by5 +cz5, which so much facilitates its discussion, was given by Sylvester in his Essay on Canonical Forms, 1851. The invariants J and K were calculated by Prof. Cayley. The value of the discriminant and its resolution into the sum of products (p. 280) was given by me in 1850 (Cambridge and Dublin Mathematical Journal, vol. v. p. 154). Some most important steps in the theory of the quintic were made in Hermite’s paper in the Cambridge and Dublin Mathematical Journal, 1854, vol. 1X., p. 172, where the number of independent invariants was established; the invariant J was discovered ; attention was called to the linear covariants ; and thé possibility demonstrated of expressing by invariants the conditions of the reality of the roots of all equations of odd degrees. The theory of the quintic was further advanced by Sylvester's “Trilogy ” (Philo-

sophical Transactions, 1864, p. 579); and in Hermite’s series of papers in the first volume of the Comptes rendus for 1866 already referred to. The values of the invariants A, B, C of the sextic were given by Prof, Cayley in his papers on Quantics,

346

NOTES.

and the existence of the invariant E pointed out. The rest of what is stated in the text about the sextic in Arts 252—6, 260—1 is nearly as it appeared for the first time in the second edition. Arts, 257—9 are from Clebsch “ Theorie, &c.,” p. 297, and Arts, 262 to the end of the Lesson are from the sources indicated in the foot-notes. The term “apolar” is due to Th. Reye, whose investigations on ‘‘ Moments of Inertia, &c.,” Borchardt, Journal LXX11, led to his “Erweiterung der Polarentheorie algebraischer Flaichen,” vol. LX XVII, p. 97, which he opens by remarking, that the polar theory of surfaces of the n“* order has hitherto dealt only with polars of points taken singly or in groups, But in regard to any such surface F;, there also corresponds to any surface of class /, for &