Tables of Complex Hyperbolic and Circular Functions: Second Edition, Revised and Englarged [2nd ed., rev. a. enl. Reprint 2014] 9780674289406, 9780674288409

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TABLES OF

COMPLEX HYPERBOLIC AND CIRCULAR FUNCTIONS BY

A. E. K E N N E L L Y , Sc.D., A.M. P R O F E S S O R OF E L E C T R I C A L E N G I N E E R I N G IN H A R V A R D U N I V E R S I T Y

SECOND REVISED

EDITION

AND

ENLARGED

CAMBRIDGE HARVARD UNIVERSITY PRESS LONDON:

HUMPHREY

MILFORD

OXFORD UNIVERSITY PRESS I 9 2 7

COPYRIGHT, 1 9 1 4 AND

1921

BY HARVARD UNIVERSITY PRESS

PRINTED A T THE HARVARD UNIVERSITY PRESS CAMBRIDGE, MASS., U. S. A .

PREFACE THE tables in this book present hyperbolic and circular functions of a complex variable, both in polar and rectangular coordinates.

Such complex functions

have not hitherto been published, except over a very restricted range. They have important applications in electrical engineering. For instance, it is possible with their help to find in a few minutes the potential, current and power, at any point of an alternating-current line-conductor of known constants and terminal conditions; whereas the same problem, to a like degree of precision, without aid from these functions, and by older methods, would probably occupy hours of labor and cover several sheets of computing-paper. Although the principal application of these functions at the present time is in dealing with alternating-current lines, especially those of either great length or high frequency; yet it seems likely that other uses will develop for them. The author desires to acknowledge his indebtedness, for suggestions and help, to a number of workers, both in mathematical and practical fields; and particularly to Messrs. C. L. Bouton, W. Duddell, E. V. Huntington, F. B. Jewett, John Perry, H. J. Ryan, and Ε. B. Wilson. Α. Ε. K . HARVARD UNIVERSITY

January, 1914.

P R E F A C E TO T H E SECOND EDITION IN preparing the second edition of this book, six new tables have been computed. These are actually extensions of the tables I to V I already incorporated. It has been considered advisable to add the new material in new tables at the end of the volume rather than to recast the original tables in such a manner as to include the new matter. The new matter has been found necessary in certain departments of electrical engineering to which complex hyperbolic functions may be advantageously applied. Α. Ε. K. HARVARD UNIVERSITY

June, 1920.

TABLE OF CONTENTS

PAGE

I. II.

SINH (P /A) = R¿Y

HYPERBOLIC SINES,

COSH (Ρ /Δ)

HYPERBOLIC COSINES,

HI.

HYPERBOLIC T A N G E N T S ,

IV.

CORRECTING FACTOR.

V.

CORRECTING FACTOR.

.

.

= R [Y_

TANH (P ¿S)

. Δ 4 5 ° TO 90 O

.

"

"

"

"

"

"

"

"

« «

«

«

2Q

« «

«

«

2Ó

= R ¿Γ

U

O

FUNCTIONS OF SEMI-IMAGINARIES. F (P / 4 5 O ) VII. VIII. IX. X. XI. ΧΗ. ΧΙΗ.

HYPERBOLIC SINES,

SINH (X + IQ)

COSH (X + IQ)

HYPERBOLIC COSINES, HYPERBOLIC T A N G E N T S , HYPERBOLIC SINES,

= U +

SINH (X +

IQ)

IQ)

HYPERBOLIC T A N G E N T S ,

/(4

XIV. XV. XVI.

XVH. XVIH. XIX.

SEMI-EXPONENTIALS.

74 90 106

= R [_7

+ I Q )

E*

/ ß

X

122

= U + IV

138

=

139

R

L Y

\

— AND LOGIO (—)

R E A L HYPERBOLIC FUNCTIONS,

14

58

= R[J_

/ (4 + IQ)

8

42

= W + IV

TANH {X + IQ)

FUNCTIONS OF 4 + IQ.

IV

= R ¿Y_

COSH (X + I Q )

HYPERBOLIC COSINES,

. . . . . .

32

= U + IV

TANH (X +

2

140

F (X + IO)

= U + IO

144

SUBDIVISIONS OF A D E G R E E

150

EXPLANATORY T E X T

151

HYPERBOLIC SINES,

SINH (P ¿S)

HYPERBOLIC COSINES, HYPERBOLIC T A N G E N T S ,

= R¡J_

COSH ( P /Δ)

=

TANH (P /D)

R

¿

Y

= R ¿Y

S O° TO 4 5 °

214

"

"

"

216

" "

"

"

218

"

XX.

CORRECTING FACTOR.

" "

"

"

220

XXI.

CORRECTING FACTOR.

« «

«

«

2 2 2

XXII.

FUNCTIONS OF SEMI-IMAGINARIES.

XXIII.

HYPERBOLIC FUNCTION FORMULAS

/ (P / 4 5 O )

224 226

TABLES OF COMPLEX AND CIRCULAR

HYPERBOLIC

FUNCTIONS

TABLE I.

HYPERBOLIC

SINES,

sinh (P ¿S)

= r [y 0.4

0-3

48.094 49.094

0.20000 0.19995 0.19990 0.19986 0.19982

45-383 46.382 47-381 48.380 49-378

0.30001 0.29985 0.29969 0.29954 0.29939

45.860 46.858 47.856 48.854 49.852

0.40005 0.39968 0.39931 0-39893 0.39856

0.099970 0.099965 0.099960 0.099955 0.099950

50.094 51-093 52.092 53-091 54.090

0.19976 0.19972 0.19968 0.19963 0.19959

50.376

0.29923 0.29907 0.29892 0.29877 0.29862

50.848 51.842 52-834

0.39820 0.39784 0.39748 0.39712 0.39676

51-506

55-089 56.088 57-086 58.085 59-083

0.19951 0.19946 0.19941 0.19937

0.29847 0.29833 0.29819 0.29804 0.29790

55-8O9 56.799 57-787 58.773

59.760

0.39641 0.39607 0.39572 0.39538 0.39505

56.440

56-352

59

0.099944 0.099939 0.099933 0.099928 0.099922

60 61 62 63 64

0.099917 0.099912 0.099907 0.099902 0.099897

60.082 61.081 62.079 63.077 64-075

0.19934 0.19929 0.19925 0.19921 0.19918

60.331 61.324 62.317 63.309 64.3o1

0.29777 0.29764 0.29751 0.29738 0.29725

60.746 61.731 62.715 63.698 64.680

65

0.099893 0.099889 0.099885 0.099881 0.099877

65-073 66.071 67.069 68.066 69.064

0.19914 0.19911 0.19908 0.19904 0.19901

65-293 66.284 67.275 68.265 69-255

0.29712 0.29700 0.29689 0.29678 0.29667

0.099873 0.099869 0.099865 0.099861 0.099858

70.062 71.059 72.057 73-054 74-051

0.19898 0.19895 0.19892 0.19889 O.IC

70.245 7I-235 72.225 73-214 74-203

79

0.099855 0.099852 0.099850 0.099847 0.099845

75-048 76.045 77.042 78.039 79.036

0.19885 0.19883 0.19881 0.19878 0.19876

80 81 82 83 84

0.099843 0.099841 0.099839 0.099838 0.099837

80.033 81.030 82.026 83.023 84.020

85 87

88 89

0.099836 0.099835 0.099834 0.099833 0.099832

85.017 86.014 87.011 88.008 89.004

90

0.099831 90.000

45

46 47

48 49

5° 51

52 53 54 55

56

57

58

66 67 68 69 70 71 72

73 74 75

76 77

78

86

o. 10000 0.099993 0.099987 0.099981 0.099975

45·°96

46.095 47-095

0-I9955

51-374

52-371

53-367

54-362 55-357

53-826 54-8i8

46.532

47-529

48.526

49-52°

50.513

52-497 53-486 54-472 55-458 57-421

0.5

0.50016 0.49944 0.49872 0.49799 0.49727

47.391 48.388

0.49656 0.49585 0.49514 0.49444

52-357 53-342 54-325 55-304

0-49374

49-385

50.378 51-368

56.281

0-49305 57-254

60.354

0.49238 0.49172 0.49106 0.49041

60.160 61.123

0.39473 0.39441 0.39409 0.39379 0.39349

61.330 62.302 63.273 64.243 65.212

0.48978 0.48916 0.48855 0.48795 0.48738

62.085 63.042 63.998 64.950 65.900

65.661 66.641 67.621 68.599

0.39320 0.39293 0.39266 0.39240 0.39214

66.179 67.144 68.108 69.070 70.030

0.48681 0.48627 0.48575 0.48523 0.48473

66.848 67.794 68.738 69.679 70.617

0.29657 0.29647 0.29637 0.29628 0.29620

70.555 71-532 72.508 73-483

0.39190 70.990 0.39167 71.948 o - 3 9 I 4 5 72-905 0.39123 73.861 0.39104 74.817

0.48426 0.48380 0.48338 0.48296 0.48257

72.489 73.422

75-192 76.180 77.168 78.156 79.144

0.29612 0.29604 0.29596 0.29590 0.28584

75-432

75-771

78.351 79.322

0.39084 0.39066 0.39050 0.39034 0.39018

76.724 77.676 78.628 79-578

0.48220 0.48184 0.48152 0.48121 0.48093

76.212 77-I38 78.062 78.986 79.909

0.19875 0.19873 0.19872 0.19871 0.19870

80.131 81.118 82.106 83-093 84.080

0.29578 0.29573 0.29569 0.29566 0.29563

80.294 81.266 82.238 83.209 84.180

0.39004 0.38993 0.38983 0.38973 0.38965

80.528 81.477 82.425 83.373 84.321

0.48067 0.48044 0.48023 0.48004 0.47987

80.830 81.750 82.669 83-587 84-505

0.19869 0.19868 0.19868 0.19867 0.19867

85.067 86.054 87.041 88.028 89.014

0.29561 0.29559 0.29557 0.29555 0.29553

85-150 86.120 87.090 88.060 89.030

0.38958 0.38952 0.38948 0.38946 0.38944

85.268 86.215 87.162 88.108 89.054

0 . 4 7 9 7 2 85.422

0.38942 90.000

0.47943 90.000

57-347 58-342 59-336

0.19867 90.000

69-577

74-458

76.406 77-379

0.29552 90.000

58.400 59-378

59-195

71-554

74-354

75-284

0.47960 86.338 0.47952 87-254 0 . 4 7 9 4 7 88.170 0.47945 89.085

Note.

sinh (o ¿S) - o /S.

Examples,

sinh (0.3 /65 o ) = 0.29712 /65°.66i = 0.29712 /65°·39,·4θ°'. sinh - 1 (0.39018 /79°-578) = 0.4 /79 e .

[2]

58.226

TABLE I.

HYPERBOLIC

0.6

SINES,

sinh (p ¿5) = r / Γ . 0.8

0.7

CONTINUED

0.9

45 46 47 48 49

0.60042 0.59918 0-59793 0.59667 0.59542

48.440 49-437 50.434 51.426 52-414

0.70094 0.69894 0.69695 0.69497 0.69299

49.676 50.679 51.676 52.666 53-652

0.80184 0.79885 0.79587 0.79291 0.78996

51.108 52.112 53-io9 54099 55·ο82

0.90327 0.89904 0.89482 0.89060 0.88640

52.728 53-735 54-734 55-725 56.707

1.00553 0-99975 0.99394 0.98816 0.98242

54-531 55-540 56.550 57-543 58-525

50 51 52 53 54

0.59418 0.59295 0.59174 0.59053 0.58932

53-398 54-379 55-355 56.326 57-293

0.69102 0.68907 0.68713 0.68521 0.68331

54-632 55.606 56.574 57-537 58-493

0.78703 0.78412 0.78124 0.77838 0-77555

56.058 57.026 57·987 58.940 59-886

0.88224 0.87810 0.87400 0.86993 0.86590

57-679 58.642 59-595 60.538 61.472

0.97672 0.97105 0.96543 0.95986 0.9543S

59-495 60.453 61.399 62.333 63.255

55 56 57 58 59

0.58814 0.58698 0.58583 0.58469 0.58357

58.256 59-215 60.171 61.122 62.069

0.68144 0.67959 0.67776 0.67595 0.67419

59-445 60.391 61.331 62.265 63-193

0.77275 0.76999 0.76727 0.76459 0.76195

60.824 61.755 62.678 63.593 64.502

0.86192 0.85800 0.85414 0.85034 0.84660

62.396 63.312 64.218 65.114 66.000

0.94890 0.94353 0.93825 0.9330S 0.92795

64.166 65.065 65·952 66.827 67.691

60 61 62 63 64

0.58249 0.58142 0.58037 0.57934 0.57834

63013 63-953 64.889 65.821 66.749

0.67247 0.67078 0.66912 0.66749 0.66591

64.117 65-036 65-95I 66.859 67.762

0.75938 0.75686 0.75439 0.75197 0.74962

65.405 66.300 67.189 68.070 68.944

0.84295 0.83937 0.83587 0.83244 0.82909

66.878 67.747 68.607 69.458 70.300

0.92295 0.91805 0.91325 0.90856 0.90400

68.544 69-385 70.215 7x-°33 71.841

65 66 67 68 69

0.57737 0-57643 0.57553 0.57465 0.57379

67.674 68.596 69-5I5 70.430 71-342

0.66437 0.66288 0.66145 0.66005 0.65870

68.661 69-554 70.444 7I-329 72.209

0.74733 0.74512 0.74298 0.74091 0.73891

69.812 70.674 71-529 72-379 73-223

0.82585 0.82270 0.81967 0.81672 0.81387

71.136 71.962 72.780 73-590 74-392

0.89957 0.89527 0.89111 0.88708 0.88320

72-637 73-424 74.201 74.968 7S-723

70 71 72 73 74

0.57297 0-57219 0.57145 0.57074 0.57006

72.251 73-157 74-o6i 74-962 75.860

0.65740 0.65616 0.65498 0.65385 0.65278

73-085 73-957 74-825 75-689 76-55°

0.73698 0.73S13 0.73337 0.73169 0.73009

74.061 74.894 75.722 76.544 77-36I

0.81114 0.80853 0.80602 0.80363 0.80137

75-I87 75-975 76.756 77-530 78.298

0.87947 0.87589 0.87247 0.86921 0.86612

76.469 77.207 77·936 78.656 79.368

75 76 77 78 79

0.56941 0.56881 0.56824 0.56772 0.56724

76.756 77.649 78.540 79.429 80.317

0.65176 0.65081 0.64992 0.64909 0.64832

77.408 78.263 79.114 79.962 80.808

0.72858 0.72716 0.72583 0.72459 0.72345

78.174 78.982 79-787 80.588 81.385

0.79924 0.79723 0.79535 0.79359 0.79197

79-059 79-815 80.566 81.312 82.053

0.86320 80.072 0.86045 80.769 0.85788 81.458 0.85549

82.141

80 81 82 83 84

0.56679 0.56638 0.56602 0.56570 0.56542

81.203 82.087 82.970 83.852 84.732

0.64761 0.64697 0.64640 0.64589 0.64545

81.652 82.493 83.332 84.169 85.005

0.72241 0.72146 0.72061 0.71985 0.71919

82.179 82.969 83.757 84.543 85.327

0.79048 0.78913 0.78792 0.78685 0.78592

82.789 83.522 84.251 84.976 85.699

0.85125 0.84940 0.84774 0.84628 0.84501

83.489 84.156 84.817 85-474 86.128

85 86 87 88 89

0.56518 0.56498 0.56483 0.56473 0.56466

85.612 86.490 87.368 88.246 89.123

0.64507 0.64476 0.64452 0.64435 0.64425

85-839 86.673 87.506 88.338 89.169

0.71863 0.71817 0.71781 0-71755 0.71740

86.109 86.889 87.668 88.446 89.223

0.78513 0.78448 0.78397 0.78361 0-78339

86.420 87.138 87-855 88.571 89.286

0.84393 0.7430S 0.84236 0.84186 0.84156

86.779 87.426 88.071 88.715 89.358

90

0.56464 90.000

0.64422 90.000 Examples,

0.71736 90.000

0.78353 90.000

sinh (1.0 /90 o ) = 0.84147 /90 o . s i n h - 1 (0.87947 /76°·469) = i . o /70 e .

[3]

0.85328 82.818

0.84147 90.000

TABLE I.

HYPERBOLIC

SINES,

sinh (p ¿8)

= r ¿y. 1.4

1-3

45 46 47 48 49

1.1089 1.1012 1.0935 1.0858 1.0782

0 56-519 57-543 58.555 59-553 60.536

SO 51 52 53 54

1.0706 1.0630 1.0556 1.0482 1.0409

61.506 62.461 63-401 64.327 65-239

1.1641 1.1543 1.144Ó 1-1350 1.1256

63.712 64.666 65.603 66.523

56 57 58 59

1.0336 1.0265 1.0195 1.0126 1.0058

66.136 67.019 67.888 68.742 69-581

1.1162 1.1070 1.0979 1.0890 1.0802

68.310 69.178 70.028 70.860 71-675

60 61 62 63 64

0.99920 0.99269 0.98633 0.98013 0.97409

70.406 71.218 72.016 72.800 73-570

1.0716 1.0632 1.0550 1.0470 1.0392

65 66 67 68 69

0.96821 0.96251 0.95698 0.95165 0.94650

74.327 75-071 75.801 76.519 77-225

1.0316 1.0242 1.0171 1.0102 1.0035

70 71 72 73 74

0.94156 0.93682 0.93229 0.92798 0.92388

77.918 78.600 79.271 79-931 80.580

0.99712 0.99099 0.98514 0-97957 0.97428

75 76 77 78 79

0.92001 0.91637 0.91296 0.90978 0.90685

81.220 81.850 82.471 83.084 83.689

80 81 82 83 84

0.90416 0.90172 0.89953 0.89759 0.89590

8S 86 87 88 89

0.89447 0.89330 0.89238 0.89173 0.89134

po

0

ss

1.2138 1.2037 1.1937 1.1838 I.I739

58.692 59.726 60.748 6I-7S3

62.741

67-425

72.474 73-255

74.019 74-767 75-497

1-3205 1.3078 1-2951 1.2824 1.2699

61.034 62.092 63.128 64.142 65.137

1-2574 1.2450 1.2327 1.2206 1.2086

66.113 67.068 68.004 68.919 69.814

1.1967 1.1850 I-I735 1.1622

71-542

1.15" · 1.1403 1.1296 1.1192 1.1091 1.0992

70.688 72.376 73-188 73.980 74-752

75-502 76.232 76.942 77.632

CONTINUED

1.4297 1.4138 1-3979 1.3822

i-S

63.568 64.639 65.689 66.717 67.723

1.5418 1.5222 1.5027 1.4834 1.4642

66.262 67-35S 68.426 69.474 70.496

1-3509 1-3355 1.3202 1-3051 1.2902

68.707 69.668 70.605 71.519 72.409

I-445I 1.4262 I.4075 1.3889 1.3706

71.492 72.462 73.405 74.321 75.210

1-2754 1.2609 1.2466 1.2325 1.2187

73.27s 74.117 74.936 75.730 76.500

I-352S 1-3347 1.3172 1.3000 1.2831

76.072 76.907 77-713 78.491 79.240

1.2052 1.1919 1.1790 1.1664

77.246 77.967 78.663

79.964 80.657 81.321 81.956 82.562

1.3665

1.1541

79-983

1.2665 1-2503 1.2345 1.2191 1.2040

1.1421 1-1305 1.1193 1.1084 1.0979

80.607 81.207 81.783 82.33S 82.865

1.1894 1.1752 1.1615 1.1482 1.1354

83.140 83.690 84.211 84.704 85.169

1.1231 1.1113 1.1000 1.0893 1.0791

85.607 86.018 86.402 86.761 87.096

1.0694 1.0603 1.0518 1.0439 1.0366

87.406 87.693 87.958 88.201 88.425 88.630 88.818 88.991 89.149 89.294 89.429 89-554 89.673 89-785 89.893

79-335

1.0895 1.0802 1.0712 1.0625 1.0540

80.191 80.783

80.164 80.771 81.365 81.94Ó

1.0459 1.0382 1.0308 1.0237 1.0170

81.912 82.450 82.971 83.476

1.0879 1.0783 1.0691 1.0603 1.0520

0.96927 0.96456 0.96015 0.95605 0.95226

82.514 83.069 83.614 84.148 84.672

1.0107 1.0047 0.99916 0-99397 0.98917

83.966 84.440 84.900 85.346 85.780

1.0441 1.0367 1.0298 1.0233 I.OI73

85·59ΐ

84.286 84.877 85.462 86.041 86.616

0.94878 0.94562 0.94279 0.94029 0.93811

85.187 85-693 86.191 86.683 87.169

0.98477 0.98077 0.97715 0.97400 0.97124

86.202 86.614 87.016 87.408

87.346 87-651

87-793

I.0II9 1.0069 1.0024 0.99845 0.99502

88.227 88.499

1.0299 1.0238 1.0183 1-0135 1.0093

87.186 87·753 88.879 89.440

0.93626 0.93474 0-93356 0.93272 0.93221

87.649 88.124 88.596 89.065

0.96890 0.96698 0.96548 0.96441 0.96377

88.171 88.544 88.912 89.277 89.639

0.99210 0.98971 0.98785 0.98652 0.98572

88.762 89.018 89.269 89-515 89.758

1.0057 1.0027 1.0004 0.99880 0.99782

0.89121 90.000

0.93204

88.317

Examples,

76.211 76.909 77-59°

78.257 78.907 79-543

89-533

78.301 78.950 79-58O

81-357

0.96356 90.000

83·373

83.858 84.322 84.76S 85.188

85-975

86.342 86.692 87.026

87.944

0.98545 90.000

0.99749 90.000

sinh (1.1 /45 o ) = 1.1089 / s ó ' . s i g = 1.1089 / Φ ° · 3 ΐ ' · ° & " · sinh-i (1.1084 /82°.335) = 1-4 Z68L

[4]

TABLE I.

HYPERBOLIC

1.6

45 46 47 48 49

1-6575 1.6338 1.6103 1.5868 1-5635

0 69.117 70.241 71-339 72.409 73-451

50 51 52 53 54

1.5404 I-5I75 1.4949 1-4725 1.4504

74·465 75-449 76.402 77-325 78.218

55 56 57 58 59

1.4285 Ϊ.4070 1-3859 1-3651 1-3447

60 61 62 63 64

0

SINES,

sinh (P ¿S) = r /y. 1.8

1-7

CONTINUED

ι.9

1.7776 1-7493 ι.7210 1.6929 1.6651

72.133 73-288 74.418 75-515 76.582

1.9029 1.8693 I-8359 1.8027 1.7697

75.292 76.486 77.651 78.784 79.883

2-0343 1-9947 1-9554 1.9165 1.8779

78.590 79.829 81.037 82.210 83.346

2.1726 2.1266 2.0809 2.0356 1.9907

82.016 83-304 84.560 85.778 86.958

77.618 78.620 79-590 80.527 81.429

I-7370 1.7046 1.6726 1.6410 1.6098

80.946 81.974 82.966 83.921 84.839

1.8396 1.8016 1.7643 1.7273 1.6909

84.444 85-503 86.524 87-505 88.444

1.9462 1.9022 1.8588 I.8I60 1-7737

88.098 89.196 90-253 91.267 92.236

79.079 79.910 80.709 81.475 82.209

1-6375 1.6102 1-5831 1-5563 1.5290 1-5039 1.4782 1-453° 1.4282 I-4039

82.296 83.128 83-924 84.683 85.406

I-579I 1.5488 I-5I90 1.4898 1.4611

85.718 86.558 87.358 88.116 88.834

1.6550 1.6196 1.5848 ΐ·55Θ6 1-5171

89.342 90.197 91.008 91-773 92-493

1.7320 1.6911 1.6509 1.6114 1.5726

93.160 94.039 94.869 95-650 96.382

1-3247 1-3051 1.2860 1.2674 1.2492

82.910 83-578 84.212 84.813 85.380

1.3800 I-3567 1-3339 1.3117 ι.2901

86.091 86.738 87-347 87.917 88-447

I-4330 1-4055 1-3787 I-3S25 1.3270

89.510 90.142 90.730 91.274 9 I -774

1.4843 1-4523 1.4210 1.3904 1.3606

93.167 93.792 94.368 94.893 95-368

1-5347 1.4976 1.4614 1.4260 I.39I5

97.062 97.688 98.261 98.778 99.238

65 66 o6o hn°.\f

= 354,060 /5i°-47'·

sech (19.30 /45 o ) = cosech (19.3 /45 o ) = 2.3667 X io"« \78i°.55'. [40]

TABLE V I FUNCTIONS OF SEMI-IMAGINARIES. ρ

Sinh and cosh 0 /

/ (p / 4 5 o ) = r ¿y.

T a n h and coth 0

CONTINUED

Sech and cosech 0 t

19-55 19.60 19.65 19.70 19-75

504,230 522,380 541,220 560,650 599.830

792.03 794-05 796.06 798.08 800.09

I.OOO I.OOO 1.000 1.000 1.000

0.00 0.00 0.00 0.00 0.00

1.9832 Χ ί ο - 6 I -9I53 " 1.8478 " 1-7837 " 1.6671 "

792.03 794-05 796.06 798.08 800.09

19.80 19.85 19.90 19-95 20.00

601,730 623,390 645,820 669,070 693,!5θ

802.11 804.12 806.14 808.15 810.17

I.OOO I.OOO I.OOO ι.000 I.OOO

0.00 0.00 0.00 0.00 0.00

1.6619 1.6041 1.5484 1.4946 1.4426

" " " " "

802.11 804.12 806.14 808.15 810.17

20.05 20.10 20.15 20.20 20.25

718,090 743,93° 770,710 798,440 827,160

812.18 814.20 816.21 818.23 820.24

I.OOO I.OOO I.OOO I.OOO I.OOO

0.00 0.00 0.00 0.00 0.00

1.3926 1-3442 1-2975 1.2525 1.2090

" " " " "

812.18 814.20 816.21 818.23 820.24

20.30 20.35 20.40 20.45 20.50

856,940 887,770 919,730 952,820 987,120

822.26 824.27 826.29 828.30 830.32

I.OOO 1.000 I.OOO I.OOO I.OOO

0.00 0.00 0.00 0.00 0.00

1.1669 1.1264 1.0873 1.0496 1.0130

" " " " "

822.26 824.27 826.29 828.30 830.32

Note.

Negative quantities are in h e a v y type.

Example,

sinh (20.0 /45 e ) = cosh (20.0 /45 e ) = 693,150 /8IO°.I7' = 693,150 /qo°.iy'.

[41]

TABLE V I I .

HYPERBOLIC

SINES,

χ = 0.05 O.O 0.05

O.I · 0.15 0.2 0.25

0.00 0.00 0.00 0.00 0.00

0.00 0.07846 0.15643

0-23345 0.30902

0.05002 0.00 0.04987 0 . 0 7 8 5 6 0.04945 0 . 1 5 6 6 3 0.04864 0 . 2 3 3 7 4 0.04757 0.30940 0.04621 0.04457 0 . 3 8 3 1 6

s i n h (x +

ig) = u +

χ = 0.15

iv

X =

0.2

0 . 1 0 0 1 7 0.00 0.09986 0.07885 0.09893 0 . 1 5 7 2 2 0.09740 0.23461 0.09526 0 . 3 1 0 5 6

0.15056 0.15010 0.14871 0.14640 0.14319

0.00 0.07934 0.15820 0.23608 0.31250

0.20134 0.20072 0.19886 0.19577 0.19148

0.00 0.08003 0.15957 0.23813 0.31522

0.09254 0.38460 0.08925 0 . 4 5 6 2 6 0.08541 0 . 5 2 5 1 1 0 . 0 8 1 0 4 0.59073 0.07617 0.65270

0.13910 0.13415 0.12838 0.12181 0.11449

0.38700 0.45911 0.52839 0.59441 0.65677

O.I 8601 0.17939 0.17167 C.I6288 O.I53IO

0.39036 0.46310 0.53298 0-59958 0.66248

0.07083 0 . 7 1 0 6 5 0.06505 0 . 7 6 4 2 1 0.05888 0 . 8 1 3 0 7 0.05234 0.85691 0.04547 0.89547

0.10646 0.09778 0.08850 0.07867 0.06835

0.71508 0.76898 0.81814 0.86225 0.90105

0.14237 0.13076 0.11834 0 . 1 0 5 20 0.09I4I

0.72130 0.77567 0.82525 0.86975 0.90889

0.00 0.00 0.00 0.00 0.00 '

0.52250 0.58778 0.64944

0.04265 0.04047 0.03804

0.00 0.00 0.00 0.00 0.00

0.70711 0.76041 0.80902 0.85264 0.89101

0.03537 0.03249 0.02940 0.02614 0.02271

0.58850 0.65023 0.70796 0.76133 0.81000 0.85367 0.89208

0.00 0.00 0.00 0.00 0.00

0.92388 0.95106 0.97237 0.98769 0.99692

0.01914 0.01546 0.01168 0.00783 0.00392

0.92503 0.95225 0.97359 0.98892 0.99816

0.03833 0.03095 0.02338 0.01567 0.00786

0.92850 0.95582 0.97724 0.99263 1.00191

0.05762 0.04653 0.03515 0.02355 O.OII8I

0.93429 0.96178 0.98333 0.99882 1.00815

0.07705 0.06222 0.04700 0.03150 0.01580

0.94242 0.97014 0.99188 I.0075I 1.01692

ι.00000 0.99692 0.98769 0.97237 0.95106

0.00 1.00125 0.00392 0 . 9 9 8 1 6 0.00783 0.98892 0.01168 0-97359 0.01546 0 . 9 5 2 2 5

0.00 0.00786 0.01567 0.02338 0.03095

1.00500 1.00191 0.99263 0.97724 0.95582

0.00 0.0II8I 0.02355

I.OII27 1.00815 0.99882

1.2

0.00 0.00 0.00 0.00 0.00

0.03515 0.04653

0.98333 0.96178

0.00 0.01580 0.03150 0.04700 0.06222

1.02007 1.01692 I.0075I 0.99188 0.97014

1.25 1-3 I-3S 1-4 ι-45

0.00 0.00 0.00 0,00 0.00

0.92388 0.89101 0.85264 0.80902 0.76041

O.OI9I4 0.02271 0.02614 0.02940 0.03249

0.92503 0.89208 0.85367 0.81000

0.03833 0.04547 0.05234 0.05888 0.06505

0.92850 0.89547 0.85691 0.81307 0.76421

0.05762 0.06835 0.07867 0.08850 0.09778

0.93429 0.90105 0.86225 0.81814 0.76898

0.07705 0.09I4I 0.10520 0.11834 0.13076

0.94242 0.90889 0.86975 0.82525 0.77567

1-5 1-55

0.00 0.00 0.00 0.00 0.00

0.70711 0.64944 0.58778 0.52250

0.03537 0.03804 0.04047 0.04265 0.04457

0.70796 0.65023 0.58850

0.71065 0.65270

0.10646 O.II449 0.12181 0.12838

0.71508 0.65677 0.59441 0.52839

0.14237 O.I53IO 0.16288 0.17167

0.72130 0.66248 0.59958 0.53298

0.45454

0.07083 0.07617 0.08104 0.08541 0.08925

0.13415

0.459II

O.I7939 0.46310

0.38268 0.30902

0.38316 0.04757 0.30940 0.04864 0-23374 0.04945 0 . 1 5 6 6 3 0.04987 0 . 0 7 8 5 6

0.09254 0.09526 0.09740 0.09893 0.09986

0.38460 0.31056 0.23461 0.15722 0.07885

O.I39IO 0.38700 O.I43I9 0.31250

ι-95

0.00 0.00 0.00 0.00 0.00

0.14640 0 . 2 3 6 0 8 0.14871 0 . 1 5 8 2 0 0.I50I0 0.07934

0.18601 0.39036 0.19148 0 . 3 1 5 2 2 0.19577 0 . 2 3 8 1 3 0.19886 0-I5957 0.20072 0.08003

2.0

0.00 0.00

0.05002

O.IOOI7

0.00

0.15056

0.20134

°-3 0-3S 0.4

0-4S °-5 0-55 0.6 0.65 0.7

°-75 0.8 0.85 0.9

°-95 1.0 1.05

1.1 I.I5

1.6 1.6S

1-7 I-7S

1.80

1.8s 1.9

0.38268

0.45399

0.45399

0.23345 0.15643 0.07845

0.45454 0-52313

0.76133

0.52313

0.04621

Note. Examples,

0.00

0.59073 0.52511 0.45626

0.00

Negative quantities are in heavy type. sinh (0.1 + i 0.5) = 0.07083 + i 0.71065. s i n h (0.1 + i 1 . 2 ) =

[42]

— 0 . 0 3 0 9 5 + i 0.95582.

0.00

TABLE

VII.

H Y P E R B O L I C

» - 0.25

SINES,

χ = 0.3

s i n h (x

+

0.00 0.08202 0.16353 0.24403 0.32303

0.35719 0.00 0.35609 0.08331

0.40003

0.46825 0.53891 0.60625 0.66985

0.28134 0.27133 0.25965 0.24636 0.23156

0.17862 0.16406 0.14848 0.13199 0.11468

0.72932 0.78429 0.83443 0.87942 0.91900

0-75 0.8 0.85 0.9 0.95

0.09667 0.07801 0.05897 0.03952 0.01982

1.0 I.os 1.1 1-IS 1.2

0.00 0.01982 0.03952 0.05897 0.07801

0.25261 0.25183 0.24950 0.24563 0.24025

0.25 0-3 0-3S 0.4 0-4S

0.23338 0.22508 0.21539 0.20437 0.19208

o-S °-55 0.6 0.65 0.7

u +

iv.

CONTINUED

χ = 0.4

χ = 0.4s

0.34732 0.24789 0.33971 0.32814

0.41075 0.40949 0.40570 0.39940 0.39065

0.00 0.08482 0.16912 0.25237 0.33407

0.46534 0.46391 0.45961 0.45249 0.44257

0.00 0.08654 0.17254 0.25748 0.34084

0.54619 0.61444 0.67889

0.33000 0.40636 0.31826 0.48208 0 . 3 0 4 5 5 0-55483 0.28897 0.62416 0.27161 0.68964

0.37949 0.4I37I 0.36598 0.49080 0.35022 0 . 5 6 4 8 6 0.33231 0 . 6 3 5 4 4 0.31234 0.70210

0.42992 0.41462 0.39677 0.37647 0.35385

0.42209 0.50074 0.57630 0.64831 0.71632

0.21533 0.19777 0.17899 0.15911 0.13825

0.73917 0.79488 0.84570 0.89130 0.93140

0.25257 0.23198 0.20995 0.18663 0.16216

0.75086 0.80746 0.85908 0.90540 0.94614

0.29045 0.26676 0.24143 0.21462 0.18648

0.76443 0.82206 0.87461 0.92177 0.96324

0.3290s 0.30222 0.27352 0.24314 0.21126

0.77992 0.83871 0.89232 0.94044 0.98275

0.95290 0.98093 1.00292 1.01871 1.02823

0.11654 0.09410 0.07109 0.04764 0.02389

0.96577 0.99418 1.01646 1.03247 1.0421 2

0.13669 0.11038 0.08338 0.05588 0.02803

0.98105 1.00991 1-03254 1.04880 1.05860

O.I57I9 0.12693 0.09589 0.06426 0.03223

0.99878 1.02816 1.051 20 1.06776 1.07774

0.17808 0.14380 0.10863 0.07280 0.03651

I.0I90I 1.04899 1.07250 1.08939 1.09957

1.03141

1.02823 1.01871 1.00292 0.98093

0.00 0.02389 0.04764 0.07109 0.09410

1.04534 1.04212 1.03247 1.01646 0.99418

0.00 0.02803 0.05588 0.08338 0.11038

ι.06188 1.05860 1.04880 1-03254 1.00991

0.00 0.03223 0.06426 0.09589 0.12693

1.08107 1.07774 1.06776 1.05120 1.02816

0.00 0.03651 0.07280 0.10863 0.14380

1.10297 1.09957 1.08939 1.07250 1.04899

1-25 1-3 1-35 1.4 i-45

0.09667 0.95290 0.11468 0.91900 0 . 1 3 1 9 9 0.87942 0.14848 0.83443 0.16406 0.78429

0.11654 0.13825 0.15911 0.17899 0.19777

0.96577 0.93140 0.89130 0.84570 0.79488

0.13669 0.16216 0.18663 0.20995 0.23198

0.98105 0.94614 0.90540 0.85908 0.80746

0.I57I9 0.18648 0.21462 0.24143 0.26676

0.99878 0.96324 0.92177 0.87461 0.82206

0.17808 0.21126 0.24314 0.27352 0.30222

I.0I90I 0.98275 0.94044 0.89232 0.83871

1-5 I-5S 1.6

0.72932 0.66985 0.60625 0.53891 0.46825

0.21533 0.23156 0.24636 0.25965 0.27133

0.73917 0.67889 0.61444 0.54619 0.47457

0.25257 0.27161 0.28897 0.3045S 0.31826

0.75086 0.68964 0.62416 0.55483 0.48208

0.29045 0.76443

1-7

0.17862 0.19208 0.20437 0.21539 0.22508

0.32905 0.3538s 0.37647 0.39677 0.41462

0.77992 0.71632 0.64831 0.57630 0.50074

1-75 1.8 1.8 s 1.9 ι-95

0.23338 0.24025 0.24563 0.24950 0.25183

0.39471

0.28134 0.28962 0.29611 0.30077 0.30358

0.40003 0.32303 0.24403 0.16353 0.08202

0.33000 0.40636 033971 0.32814 0 . 3 4 7 3 2 0.24789 0.35279 0.I66II 0.35609 0.08331

0.37949 0.39065 0.39940 0.40570 0.40949

0.42992 0.42209

2.0

0.25261 0.00

0.35719 0.00

0.41075 0.00

1.6S

0.00 0.08092 0.16135 0.24078 0.31872

=

« = 0.35

0.30452 0-30358 0.30077 0.29611 0.28962

o 0.05 O.I 0.15 0.2

iq)

0.39471

0.31872 0.24078 0.16135 0.08092

0.47457

0.30452 0.00

0.35279

O.I66II

0.31234 0.70210 0 . 3 3 2 3 1 0.63544 0 . 3 5 0 2 2 0.56486 0.36598 0.49080

Note.

Negative quantities are in heavy type.

Examples,

sinh (0.4 + i o) = 0.41075 + i o. sinh (0.4 + i 1) = O. + i 1.08107.

[43]

0.4I37I 0.33407 0.25237 0.16912 0.08482

0.44257 0.45249

0.34084 0.25748

0.45961 0.17254 0.46391 0.08654 0.46534 0.00

TABLE V I I .

HYPERBOLIC

* = o.s

s i n h (x + iq)

SINES,

« = 0.55

χ = 0.6

= u + iv.

CONTINUED

χ = 0.65

X = 0.7

o 0.05 O.I 0.15 0.2

0.52110 0.51949 0.51468 0.50670 0.49559

0.00 0.08847 0.17640 0.26324 0.34846

0.57815 0.00 0.57637 0.09063 0.57103 0.18070 0.56218 0.26965 0.54986 0.35695

0.63665 0.00 0.63469 0.09301 0.62882 0.18545 0.61906 0.27674 0.60549 0-36633

0.69675 0.69460 0.68817 0.67750 0.66265

0.00 0.09563 0.19066 0.28452 0.37663

0.75858 0.75625 0.74925 0.73763 0.72146

0.00 0.09848 0.19635 0.29301 0.38787

0.25 °-3 0-35 0.40 0-4S

0.48143 0.43152 0.46430 0.51193 0.44431 0.58918 0.42158 0.66280 0.39624 0.73233

0.53414 0.44204 0.5I5I4 0.52441 0.49296 0.60354 0.46773 0.67895 0.43963 0.75018

0.58819 0.45366 0.56726 0.53819 0.54284 0.61940 0.51506 0.69680 0.48412 0.76990

0.64371 0.62081 0.59408 0.56368 0.52981

0.46641 0.55332 0.63682 0.71639 0.79154

0.70084 0.67590 0.64680 0.61371 0.57683

0.48033 0.56984 0.65582 0.73777 0.81517

0.5 0-S5 0.6 0.65 0.7

0.36847 0.79735 0.33842 0.85745 0.30629 0.91227 0.27227 0.96146 0.23657 1.00472

0.40882 0.81678 0.37548 0.87835 0.33983 0.93450 0.30208 0.98489 0.26248 1.02920

0.45018 0.83825 0.90144 0.37422 0.95906 0.33265 1.01078 0.28904 1.05626

0.49268 0.86182 0.45250 0.92678 0.40954 0.98602 0.36405 I.039I9 0.31632 1.08595

0.53640 0.88754 0.49266 0.95444 0.44589 I.OI545 0.39636 I.0702I 0.34439 1.11836

°-75 0.8 0.8 s 0.9 °-95

0.19942 0.16103 0.12165 0.08152 0.04088

I.04I79 1.07244 1.09647 I.II374 1.12415

0.22125 0.17866 0.13497 0.09044 0.04536

1.06717 1.09857 1.12319 1.14088 I-I5I54

0.24364 0.19674 0.14862 0.09959 0-04995

1.09523 1.12744 I.I527I 1.17087 I.I8I8I

0.26663 0.2I53I 0.16265 0.10900 0.05467

1.12602 I.I59I2 1.18512 1.20379 1.21504

0.29030 0.23442 0.17709 0.11867 0.05952

1.15962 1.19374 1.22049 1.23972 1.25130

1.0 i.os 1.1 i.iS 1.2

0.00 0.04088 0.08152 0.1216s 0.16103

1.12763 1.12415 I.II374 1.09647 1.07244

0.00 0.04536 0.09044 0.13497 0.17866

1.15510 I.I5154 1.14088 I.I23I9 1.09857

0.00 0.04995 0.09959 0.14862 0.19674

1.18547 I.I8I8I 1.17087 I.I527I 1.12744

0.00 0.05467 0.10900 0.16265 0.2I53I

1.21879 1.21504 1.20379 1.18512 I.I59I2

0.00 0.05952 0.11867 0.17709 0.23442

1.25517 1.25130 1.23972 1.22049 1.19374

1-2S 1-3 1-35 1.4 1-45

0.19942 0.23657 0.27227 0.30629 0.33842

ι .04179 1.00472 0.96146 0.91227 0.85745

0.22125 1.06717 0.26248 1.02920 0.30208 0.98489 0.33983 0.93450 0.37548 0.87835

0.24364 1.09523 0.28904 1.05626 0.33265 1.01078 0.37422 0.95906 0.41347 0.90144

0.26663 1.12602 0.31632 1.08595 0.36405 1-03919 0.40954 0.98602 0.45250 0.92678

0.29030 1.15962 0.34439 1.11836 0.39636 I.0702I 0.44589 ι.01545 0.49266 0.95444

1-5 1-55 1.6

0.36847 0.39624 0.42158 0.44431 0.46430

0.79735 0.73233 0.66280 0.58918 0.51193

0.40882 0.43963 0.46773 0.49296

0.81678 0.75018 0.67895 0.60354 0.52441

0.45018 0.83825 0.48412 0.76990 0.51506 0.69680 0.54284 0.61940 0.56726 0.53819

0.49268 0.52981 0.56368 0.59408 0.62081

0.86182 0.79154 0.71639 0.63682 0.55332

0.53640 0.57683 0.61371 0.64680 0.67590

0.48143 0-49559 0.50670 0.51468 0.51949

0.431S2 0.34846 0.26324 ο. 17640 0.08847

0.44204 0.3569s 0.26965 0.18070 0.09063

0.58819 0.60549 0.61906 0.62882 0.63469

0.64371 0.66265 0.67750 0.68817 0.69460

0.46641 0.37663 0.28452 0.19066 0.09563

0.70084 0.48033 0.72146 0.38787 0.73763 0.29301 0.74925 0.19635 0.75625 0.09848

0.00

0.63665 0.00

1.6s 1-7

1-75 1.8

1.8s 1.9 ι-95

0.52110 0.00

0.51514 0.53414 0.54986 0.56218 0.57103 0.57637 0.57815

0-41347

0.45366 0.36633 0.27674 0.18545 0.09301

0.69675 0.00

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

sinh (0.65 + i 0.75) = 0.26663 + i 1.12602. sinh (0.55 + i 1.40) = - 0 . 3 3 9 8 3 + i 0.93450.

[44]

0.88754 0.81517 0.73777 0.65582 0.56984

0.75858 0.00

TABLE VII. :

HYPERBOLIC

SINES,

sinh (x + iq) = u + iv.

χ = 0.85

0.75

CONTINUED

X = 0.9

X = 0.9s

0.82232 0.81978 0.81219 0.79960 0.78207

0.00 0.10158 0.20253 0.30224 0.40079

0.888II 0.885.37 0.87717 0.86357 0.84464

0.00 0.10493 0.20922 0.31222 0.41329

0.95612 0.95317 0-94435 0.92970 0.90932

0.00 0.10855 0.21643 0.32298 0-42753

1.02652 1.02335 1.01388 0.99816 0.97628

0.00 0.11244 0.22418 0.33455 0.44285

1.09948 1.09610 1.08595 1.06911 1.04567

0.00 0.11661 0.23250 0.34695 0.45927

0.75972 0.73269 0.70II4 0.66527 0.62529

0.49545 0.58777 0.67647 0.76100 0.84083

0.82050 0.79I3I 0.75724 0.71849 0.67532

0.51182 0.60718 0.69881 0.78613 0.86859

0.88334 0.85191 0.81522 0.77351 0.72704

0-52945 0.62811 0.72289 0.81322 0.89853

0.94838 0.91463 0.87525 0.83047 0.78057

0.54842 0.65061 0.74879 0-84235 0.93071

I.OI579 0.97965 0.93747 0.88950 0.83605

0.56875 0.67473 0.77655 0.87358 0.96523

0.58146 0.53405 0.48335 0.42966 0.37332

0.91548 0.98449 1.04742 1.10390 1.15356

0.62799 0.57678 0.52202 0.46403 0.40319

0.94571 1.01700 1.08201 1-14035 1.19166

0.67608 0.62095 0.56199 0.49957 0.43407

0.97830 1.05205 1-1193° 1.17965 1.23274

0.72586 0.66667 0.60337 0.53635 0.46603

1.01334 1.08973 1.15939 1-22191 1.27689

0.77745 0.71406 0.64621 0.57448 0.49916

1.05092 1-13013 1.20238 1.26722 1-32425

0.75 0.8 0.85 0.9 0.95

0.31469 0.25411 0.19197 0.12864 0.06452

1.19613 1.23132 1.25891 ι · 2 7874 1.29069

0.33986 0.27444 0.20732 0.13893 0.06968

1-23563 1.27198 1.30048 1.32097 I -3333 I

0.36589 0.29546 0.22320 0-I4957 0.07502

1.27822 1.31582 1-3453° 1.36650 1.37927

0.39283 0.31721 0.23964 0.16058 0.08054

1.32400 1.36294 1.39349 1.41544 1.42867

0.42076 0.33976 0.25667 0.17200 0.08627

1.37309 1.41348 1.44516 1.46793 1.48164

1.0 ι.05 1.1 115

1.2

o.oo 0.06452 0.12864 0.19197 0.25411

1.29468 1.29069 1.27874 1.25891 ι.23132

0.00 0.06968 0.13893 0.20732 0.27444

1-33743 I-3333I 1.32097 ι .30048 1.27198

0.00 I-38353 0.07502 1.37927 0 . 1 4 9 5 7 1.36650 0.22320 I-34530 0.29546 1.31582

0.00 0.08054 0.16058 0.23964 0.3I72I

1.43309 1.42867 1.41544 1.39349 1.36294

0.00 1.48623 0.08627 1.48164 0.17200 1.46793 0.25667 1.44516 0.33976 1.41348

1-25 1-3 1-35 1.4 1-45

0.31469 0.37332 0.42966 0.4833s 0.53405

1.19613 I-I535 6 1.10390 1.04742 0-98449

0.33986 0.40319 0.46403 0.52202 0.57678

1-23563 1.19166 1-14035 1.08201 1.ο 1700

0.36589 0.43407 0-49957 0.56199 0.62095

1.27822 1.23274 1.17965 1.11930 1.05205

0.39283 0.46603 0-53635 0.60337 0.66667

1.32400 1.27689 I.22I9I 1.15939 1-08973

0.42076 1-37309 0.49916 1-32425 0.57448 1.26722 0.64621 1.20238 0.71406 I.I30I3

ΐ·5 1-55 1.6 1.65 1-7

0.58146 0.62529 0.66527 0.70114 0.73269

0.91548 0.84083 0.76100 0.67647 0.58777

0.62799 0.67532 0.71849 0.75724 0.79I3I

0.94571 0.86859 0.78613 0.69881 0.60718

0.67608 0.72704 0-773SI 0.81522 0.85191

0.97830 0.89853 0.81322 0.72289 0.62811

0.72586 0.78057 0.83047 0.8752s 0.91463

I.OI334 0.93071 0.84235 0.74879 0.65061

0.7774s 1.05092 0.83605 0.88950 0.93747 0.97965

0.96523 0.87358 0.77655 0.67473

1-75 1.8 1.85 1.9 1-95

0.75972 0.78207 0.79960 0.81219 0.81978

0.49545 0.40079 0.30224 0.20253 0.10158

0.82050 0.84464 0.86357 0.87717 0.88537

0.51182 0.41329 0.31222 0.20922 0.10493

0.88334 0.90932 0.92970 0.94435 0.95317

0.52945 0.42753 0.32298 0.21643 0.10855

0.94838 0.97628 0.99816 1.01388 1.02335

0.54842 0.44285 0.33455 0.22418 O.II244

I.OI579 1.04567 1.06911 1.0859s 1.09610

0.56875 0.45927 0.34695 0.23250 O.II66I

2.0

0.82232

0.00

0.88811 0.00

o 0.05 O.I 0.15 0.2 0.25 0.3 0.35 0.4 0-45 0-5 0-5S 0.6 0.65 0.7

0.95612 0.00

1.02652 0.00

Note.

Negative quantities are in heavy type.

Examples,

sinh (0.8 + i 0.7) = 0.40319 + i 1.19166. sinh (0.8 + i 1.7) = —0.79131 + ¿0.60718.

[45]

1.09948 0.00

TABLE V I I . H Y P E R B O L I C

X = I.Ï

χ = ι .05

X = I.O

sinh (x + iq) = u + iv.

SINES,

CONTINUED

x = ι.IS

o

1.17520 0.00

1.25386

0.00

0.00

1.50946

0.00

1.17158

0.12107

1.24999

0.12583

1.41640

0.13632

1.50481

0.14206

O.I

1-33565 0.00 1-33153 O.I309I

1.42078

0.05

1.16073

0.24139

1.23842

0.25089

0.26101

1.40329

0.27179

1.49088

0.28325

o.is

I.3I920

1.14273

0.36023

1.21921

0.37440

1.2987s

0.38951

1.38152

0.40559

1.46776

0.42269

0.2

1.11768

0.47684

1.19249

0.49560

1.27028

0.51560

1-35124

0.53689

1-43558

0.55952

0.25

1.08574 .0.59051

1.15841

0.61375

1.23398

0.63852

1.31263

0.66488

1.39456

0.69291

0-3 0-35

1.04711

0.70055

I.II7I9

0.72811

1.19007

0.75749

1.26592

0.78877

1.34494

0.82202

1.00202

0.80626

1.06909

0.83798

1.13883

0.87180

I.2II4I

0.90780

1.28703

0.94607

0.4

0.95076

0.90700

I.OI439

0.94269

1.08056

0.98073

I.I4943

I.02I23

1.22118

1.06428

0.45

0.89363

1.00215

0.95344

1.04158

1.01564

1.08362

1.08037

1.12836

1.14781

1.71593

o-S 0-55

0.83099

1.09112

0.88661

1.13405

0.94445

1.22854

1-06735

1.28033

1.17337

0.81432

1.21953

0.86743

0.92272

1.37684

0.69077

1.24838

0.73700

1.29750

0.78508

1.32114 1.40560

0.98032

0.6

0.88724

1.46485

0.65

0.61404

1.31569

0.65514

1.36746

0.69787

0.78869

O.S33S3

I-37490

0.56924

1.42899

0.60637

0.74235 0.64502

1.48139

0.7

1.17982 1.2687s 1.34986 1.42265 1.48666

1.00464

0.76323

1.5480s

0.68528

1.54384 1.61331

0.7S

0.44973

1.42562

0.47983

1.48171

1.46756

0.38746

1.52530

0.54371 0.43904

1.67283

0.36316

1-54151 1.58685

0.57765

0.8

0.51113 0.41274

1.60517 1.65238

0.46645

1.72204

0.85

0.27435

1-50045

0.29271

1.55948

0.31180

1.62242

0.33167

1.68941

0.35238

1.76063

0.9

0.18384

1.52408

0.19615

1.58405

0.20894

1.64798

0.22226

1.71602

0.23613

1.78836

0.9S

0.09221

1.53832

0.09838

1.59885

0.10479

1-66337

0.III47

1.73206

0.11843

1.80507

1.0

0.00

1.54308

0.00

1.60379

0.00

1.66852

1.81066

0.09221

1.53832

0.09838

1.59885

0.10479

166337

1.73206

0.11843

1.80507

0.18384

1.52408

0.19615

1.58405

0.20894

1.64798

1.71602

0.23613

1.78836

0.2743s

1.50045

0.29271

1.55948

0.31180

1.62242

1.68941

0.35238

1.76063

1.2

0.36316

1.46756

0.38746

1.52530

0.41274

1.5868s

0.11147 0.22226 0.33167 0.43904

I-7374I

0.00

i.os 1.1 1-iS

1.65238

0.46645

1.72204

I-2S 1-3 I-3S

0.44973 0.53353

1.42562

0.47983

1.48171

0.5III3 1-54151

0.54371

1.60517

0.57765

1.67283

1.37490

0.56924

1.42899

0.60637

1.48666

0.64502

1.54805

0.68528

1.61331

0.61404

1-31569

0.65514

1.36746

0.69787

1.42265

0.74235

148139

0.78869

1.54384

0.73700

1.29750

0.78508

1.34986

0.83511

1.40560

0.88724

1.46485

0.83511

1.4

0.69077

1.24838

I-4S

0.76323

I-I7337

0.81432

I.21953

0.86743

1.26875

0.92272

1.32114

0.98032

1.37684

1-5 1-55

0.83099

1.09112

0.88661

1.13405

0.94445

1.17982

1.00464

1.22854

1-06735

1.28033

0.89363

1.00215

0.95344

1.04158

1.01564

1.08362

1.08037

1.12836

1.14781

I-I7593

1.6

0.95076

0.90700

I.OI439

0.94269

1.08056

0.98073

1.14943

1.02123

1.22118

1.65

1.00202

0.80626

1.06909

0.83798

1.13883

0.87180

1.21141

0.90780

1.28703

1 . 1 1 7 1 g 0.72811

1.19007

0.75749

1.26592

0.78877

1.34494

1.06428 0.94607 0.82202

0.66488

1-39456

0.69291

1-7

1.04711

0.70055

I-7S

1.08574

0.59051

1.15841

0.61375

1.23398

0.63852

1.31263

1.8

1.11768

0.47684

1.19249

0.49560

1.27028

0.51560

1.35124

0.53689

1.43558

0-55952

1.8s

1.14273

0.36023

I.2I92I

0.37440

1.2987s

0.38951

1.38152

0.40559

1.46776

0.42269

1.16073

0.24139

1.23842

0.25089

I.3I920

0.26101

1.40329

0.27179

1.49088

0.28325

1.17158

0.12107

1.24999

0.12583

I.33I53

0.I309I

1.41640

0.13632

1.50481

0.14206

1.25386

0.00

1.33565

0.00

1.42078

0.00

1.50946 0 . 0 0

1.9

i-95 2.0

1.17520 0.00

Note.

Negative quantities are in h e a v y type.

Examples,

sinh (1.0 + i 1.0) = o + i 1.54308. sinh (1.0 + i 1 . 5 ) = —0.83099 + i

[46]

1.09112.

TABLE V I I .

HYPERBOLIC

X = 1.2s

SINES,

* = 1.3

s i n h (x + iq) = u + iv.

* = 1.35·

CONTINUED

χ = 1.4

o 0.0s O.I 0.15 0.2

1.60192 1.59698 1.58220 I-SS766 I-52352

0.00 0.14816 0.29541 0.44084 0.58355

0.25 o-3 °-3S 0.4 0-4S

1.47998 1.42732 1.36586 1.29598 ι.21811

0.72267 0-85733 0.98670 1.10999 1.22643

1.56910 0.75424 I-5I327 0.89478 1.44811 1.02980 1.37402 1.15848 1.29146 1.28001

1.66215 1.60300 1-53398 I-45550 1.36804

0.78769 0.93446 1.07548 1.20986 1.33678

1-75934 1.69675 1.62369 1.54061 ι .44804

0.82311 0.97649 1.12384 1.26427 1.39690

1.86094 1-79473 I-7I745 1.62958 1.53166

0.86060 1.02095 1.17502 1.32184 1.46051

0.5 0-5S 0.6 0.65 0.7

1.13273 1.04036 0.94158 0.83700 0.72726

1-33532 1-43597 1-52777 1.61014 1.68260

1.20094 1.10301 0.99829 0.88740 0.77105

I-39365 1.49870 I-5945I 1.68048 1.75610

ι.27215 1.16842 1.05748 0.94002 0.81677

1-45546 1-56517 1.66523 I-75502 I-83399

1.34655 1.23674 1-11932 0.99500 0.86454

1.52092 1-63556 1.74012 1-83394 1.91646

1-42431 1.30817 1.18396 1-05245 0.91446

I-590I7 1.71007 1-81935 1-91745 2.00373

0.7S 0.8 0.85 0.9 0-9S

0.61303 0.49502 0-37396 0.25060 0.12569

1.74467 1.79600 1.83624 1.86517 1.88260

0.64994 0-52483 0.39648 0.26569 0.13325

1.82089 1-87445 1.91646 1.94665 1.96484

0.68848 0.55595 0.41999 0.28144 0.14116

1.90165 1-95759 2.00146 2.03299 2.05199

0.72875 1.98717 0.58846 2.04562 0.44455 2.09147 0.29790 2.12442 0.14427 2.14427

0.77083 0.62244 0.47022 0.31510 0.15804

2.07766 2.13878 2.18671 2.22II5 2.24I9I

1.0 i°S 1.1 1-iS 1.2

0.00 0.1256g 0.25060 0.37396 0.49502

.2 1.88260 1-86517 1.83624 1.79600

0.00 0.13325 0.26569 0.39648 0.52483

ι.97091 1.96484 1.94665 1.91646 1.87445

0.00 0.14116 0.28144 0.41999 0.55595

2.05833 2.05199 2.03299 2.00146 1.95759

0.00 0.14427 0.29790 0.44455 0.58846

2.15090 2.14427 2.12442 2.09147 2.04562

0.00 0.15804 0.31510 0.47022 0.62244

2.24884 2.24I9I 2.22II5 2.18671. 2.13878

1-25 Ι·3 1-35 1.4 i-45

0.61303 0.72726 0.83700 0.94158 1.04036

1.74467 1.68260 1.61014 1.52777 1-43597

0.64994 0.77105 0.88740 0.99829 1.10301

1.82089 1.75610 1.68048 I-5945I 1.49870

0.68848 0.81677 0.94002 1.05748 1.16842

1.90165 1-83399 1-75502 1.66523 1-56517

0.72875 0.86454 0.99500 I.II932 1.23674

1.98717 1.91646 1-83394 ι.74012 I-63556

0.77083 0.91446 I-05245 1.18396 1.30817

2.07766 2.00373 1-91745 1-81935 1.71007

1-5 i-SS 1.6 1.6S 1-7

1.13273 1.21811 1.29598 1.36586 1.42732

1-33532 1.22643 1.10999 0.98670 0-85733

1.20094 1.29146 1.37402 1.44811 I-5I327

I-39365 1.28001 1.15848 1.02980 0.89478

ι.27215 1.36804 1-45550 1-53398 1.60300

1-45546 1.33678 1.20986 1.07548 0.93446

1-34655 1.44804 1.54061 1.62369 1.69675

1.52092 1.39690 1.26427 1.12384 0.97649

ι.42431 1.53166 1.62958 1-71745 1-79473

ι.59017 1.46051 1.32184 1.17502 1.02095

1-75 1.8 1.8S 1.9 1-95

1.47998 1-52352 1-55766 1.58220 1.59698

0.72267 0-58355 0.44084 0.29541 0.14816

1.56910 1.61526 1.65146 1.67747 1-69315

0.75424 0.60905 0.46010 0.30832 0.15464

1.66215 I.71104 1.74938 1.77694 1-79354

0.78769 0.63606 0.48051 0.32199 0.16150

1-75934 1.81110 1.85169 1.88086 1.89843

0.82311 0.66466 0.50212

1.86094 1.91569 1.95862 1.98947 2.00806

0.86060 0.69493 0.52498 0.35180 0.17644

2.0

1.60192

0.00

1.69838 0.00

Note. Examples,

0.00 0.15464 0.30832 0.46010 0.60905

1.79909 1-79354 1.77694 1.74938 1.71104

0.00 0.16150 0.32199 0.48051 0.63606

1.90430 1.89843 1.88086 1.85169 1.81110

* = I-45

1.69838 I-693I5 1.67747 1.65146 1.61526

1.79909 0.00

0.00 0.16876 0.33647 0.50212 0.66466

2.01427 2.00806 1.98947 1.95862 1.91569

0.00 0.17644 0.35180 0.52498 0.69493

0.33647 0.16876

1.90430 0.00

N e g a t i v e quantities are in h e a v y type. sinh (1.35 + i o ) = 1.70909 +

io.

sinh (1.4 + i 1.15) = - 0 . 4 4 4 5 5 + » 2.09147·

[47]

2.01427 0.00

TABLE VII. H Y P E R B O L I C SINES, χ = 1.5

* = I-5S

sinh {x + ig) = u + iv. χ-

χ = 1.6

o

2.12928 0.00

2.24961

0.05

2.12272

0.18457

2.24268 0.19316

2.10307

0.36800

2.22191

0.00

2-37557 o-oo

CONTINUED

1.65

X = 1.7

2.50746 0.00

2.64563 0.00

0.20223 0.40320 0.60170 0.79648

2.49973

0.21180

2.63747

0.22I9I

2.47659 0.42230

2.61306

0.44245

2.43818 0.63019

2.57253

0.66026

2.38474 0.83420

2.51614

0.87400

0.15

2.07045

0.54916

2.18745

0.57471

0.2

2.02507

0.72693

2.13951

0.76076

2.36824 2.34632 2.30993 2.25930

0.25

1.96720 0.90023

2.07837

0.94211

2.19473

0.98636

2.31660

1.03306

2.44424

1.08235

1.89720

2.00442

1.11766

2.11664

I.I70I5

2.23417

1.22556

2.35727

1.28403

O.I

0.38512

°-3 0-35

1.81551

1.22913

1.91811

1.28632

2.02550

1.34672

2.13797

1.41050

2.25577

1.47779

0.4

1.72263

1.38271

1.81997

1.44704

1.92187

I.5I500

2.02858

1.58674

2.14036

1.66244

1.61912

1.52777

1.71062

1.59885

1.80640

1-67393

1.90669

1.75320

2.0II75

1.83684

I-77305

1.99992

0-45 °-5 0-55 0.6 0.65 0.7

0-75

1.06797

1-50563

1-66341

1.59071

1.74080

1.67978

1.82254

1.38286

1.78879

1.46101

1.87201

1.54281

1.95992

1.25156

I.903I4

1.32229

1.99169

1.39632

2.08522

I.II255

2.00576

1.17542

2.09908

1.24123

2.19765

0.96667

2.09601

1.02130

2.19353

1.07848

2.29654 2.38127

1.71820

2.15067

1-47385

2.18396

1.55506

2.28816

1.31015

2.30173

1.38234

2.4II54

1-13837

2.40529

1.20109

2.52005

2.49404

I.OI244

2.61302

0.81754

2.68989

2.62494

0.61761

2.75017

2.66629

0.41387

2.79350

2.69121

0.20757

2.81960

0.00

2.69951

0.00

2.82832

0.19673

2.69121

0.20757

2.81960

2.54573

0.39225

2.66629

0.41387

2.79350

2.17334

0.86089

2.27446

0.90909

0.65798

2.23727

0.69517

2.34137

0.49707

2.28742

0.52516

2.39384

0-73409 2-45I3I

0.33309

2.32345

0.35192

2.43155

0-95

0.37163

2-54573

0.16706

2.34516

0.17650

2.45427

0.18639

2.56952

1.0

0.00

2.35241

0.00

2.46186

0.00

2.57746

1.05

0.16706

2.34516

0.17650

2.45427

0.18639

2.56952

0.33309

2.32345

0.35192

2.43155

0.37163

1.1 1-I5

1.87074

2.05274

2.56740

0.81484

0.8 0.85 0.9

0.55456

1.90885

1.62847

2.50625

0.95957 0.77485 0.58536 0.39225 0.19673

0.49707

2.28742

0.52516

2.39384

0.55456

2.50625

0.58536

2.62494

0.61761

2.75017

1.2

0.65798

2.23727

0.69517

2.34137

0.73409

2.45131

0.77485

2.56740

0.81754

2.68989

1-25 1-3 I-3S 1-4 1-45

0.81484

2.17334

0.86089

2.27446

0.90909

2.38127

0-959S7 2.49404

I.OI244

2.61302

0.96667

2.09601

1.02130

2.19353

1.07848

2.29654

2.52005

2.00576

1.17542

2.09908

1.24123

2.19765

2.40529 2.30173

I.20I09

I.II255

1.13837 1.31015

1.38234

2.4II54

1.25156

1.90314

1.32229

1.99169

1.39632

2.08522

2.28816

1.78879

1.46101

1.87201

1.54281 1.95992

2.18396 2.05274

1-55506

1.38286

1.47385 1.62847

1.71820

2.15067

1-5 1-55

1.50563

1.66341

I.5907I

1.74080

1.67978

1.82254

1.90885

1.87074

1.99992

1.52777

1.71062

1.59885

1.80640

I-75320

2.0II75

1.83684

1.6

1.72263

1.38271

1.81997 1-44704

1.92187

1-67393 1.51500

1-77305

1.61912

2.14036

1.66244

1.81551

1.22913

I.9I8II

1.28632

2.02550

1.34672

2.13797

1.58674

1-65

1.41050

2.25577

1.47779

1-7

1.89720

1.06797

2.00442

1.11766

2.11664

1.17015

2.23417

1.22556

2.35727

1.28403

1-75

1.96720

0.90023

2.07837

0.94211

2-19473 0.98636

2.31660

1.03306

2.44424

1.08235

0.79648 0.60170 0.40320 0.20223

2.38474

0.83420

2.51614

0.87400

2.43818

0.63019

2.57253

0.66026

2.47659

0.42230

2.61306

0.44245

2.49973

0.21180

2-63747

0.22I9I

0.00

2.50746 0.00

2.64563

0.00

1.8

2.02507

0.72693

2.I395I

0.76076

1.85

2.07045

0.54916

2.18745

0.57471

2.10307

0.36800

2.22I9I

0.38512

1-95

2.12272

0.18457

2.24268

0.19316

2.25930 2.30993 2.34632 2.36824

2.0

2.12928

0.00

2.24961

0.00

2.37557

1.9

1.90669 2.02858

Note.

Negative quantities are in heavy type.

Examples,

sinh (1.7 sinh (1.7

+ +

i 0.7) i 1.7)

= 1.20109 +

i

= —2.35727 +

[48]

2.52005.

i

1.28403.

TABLE VII. H Y P E R B O L I C

SINES,

sinh (x + iq) = u + iv. CONTINUED

χ = 1.75

χ = I.Ç

1.85

1-95 3-44321 0.00 3-43259 0.28131

o o.os O.I 0-I5 0.2

2.79041 2.78181 2.75606 2.71331 2.65384

0.00 0.23257 0.46370 0.69198 0.91599

2.94217 2.93310 2.90595 2.86088 2.79817

0.00 0.24381 0.48612 0.72542 0.96026

3.06311 0.50975 3.01560 0.76069 2.94950 1.00694

O.25 O.3 °-35

2.57800 I.1343S 2.25749 1.74231

0.4S

2.12185

1.92509

1.18918 1.41076 1.62365 1.82653 2.01814

2.86522 1.24698 2.76327 1-47934 2.64429 1.70258 2.50900 1 -9 I 53 I 2-35824 2.11624

3-OI939

0.4

2.71821 2.62149 2.50862 2.38027 2.23725

1.30791 2.91196 1.55162 2.78657 1.78576 2.64400 2.00889 2.48513 2.21964

3.18m 3.06792 2.93582 2.78561 2.61823

°-S o-55 0.6 0.65 0.7

1.97312 1.81223 1.64016 1.45799 1.26682

2.09600 2.25399 2.39808 2.52739 2.64111

2.08043 1.91079 1.72937

2.19294 2-30413 2.01413 2.47780 1.82289 2.63620 1.62042 2-77835 1.40796

2.31094 2.12250 1.92098 1.70761

2.43471 2-53532

1-53728 1-33572

2.19731 2.36294 2.51400 2.64956 2.76878

1-48372 3-04522

2.23618 2.72642 2.02387 2.90071 1.79907 3-05713 1.56318 3.19469

0-7S 0.8 0.85 0.9 0.95

1.06784 2-7385S 0.86229 2.81911 0.65141 2.88229 0.43652. 2.92769 0.21893 2-9S5°5

1.12592 0.90918 0.68684 0.46026 0.23084

2.87093 2.95538 3.02161 3.06921 3.09789

I.I868I

3.01049 0.95835 3.09904 0.72398 3.16850 0.48627 3.21841 0.24332 3-24848

1.25067 1.00992 0.76294 0-51125 0.25642

3.15757 3.25045 3.32330 3.37565 3.40719

1.31766 3-31255 1.06401 3.41000 0.80380 3.48641 0.53864 3-54134 0.27015 3-57443

1.0

0.00

3.10747 3.09789 3.06921 3.02161

0.00

0.24332 0.48627 0.72398

3-41773

2.92769 2.88229 2.81911

0.00 0.23084 0.46026 0.68684 0.90918

0.00

0.21893 0.43652 0.65141 0.8622g

2-95505

0.00

1.05

1.06784 1.26682

2.73855

1.12592

2.64111

1.1 1-iS 1.2

1-25 1-3 1-35

2.48627 1.34571 2.37922 1.54878

2.96419

2.95538

0.9583s

2.87093

1.33572 2.76878 1.53728 2.64956 1.72937 2.51400 I.9I079 2.36294

I.I868I 3.01049 1.40796 2-90337 1.62042 2.77835 1.82289 2.63620 2.0I4I3 2.47780

1.25067 1.48372 1.70761 1.92098 2.12250

2.08043 2.23725 2.38027 2.50862 2.62149

2.19731 2.01814 1.82653 1.62365 1.41076

2.19294 2.35824 2.50900 2.64429 2.76327

1.18918 0.96026 0.72542 0.48612 0.24381

2.86522

0.91599 2-7I33I 0.69198 2.75606 0.46370 2.78181 0.23257

2 . 718 2 1 2.79817 2.86088 2.90595 2.93310

2.79041 0 . 0 0

2.94217

0.00

1-45799 2-52739

1.97312 2.12185 2.25749

2.09600 1.92509

1.6s 1-7

2.37922 2.48627

1.54878

I-7S 1.8 1.8S

2.57800 2.65384

1-5 1-55 1.6

1.9

ι -95 2.0

3-25853

0.25642 0.51125 0.76294 1.00992

2.39808

ι-45

2-9°337

3.26816 3.25809 3.22793 3.17787 3.10821

3-24848 3.21841 3.16850 3.09904

1.64016 1.81223

1.4

3.IOI29 0.00 3-09173 0.25566

2-25399

I-7423I I-3457I I-I3435

o.oo 0.26815 0.53465 0.79785 1.05614

2.41670 2.59887 2.76501 2.91409

3.40719

3-37565 3-32330 3-2504S

0.27015 0.53864 0.80380 1.06401

1.37210 1.62777 1-87341 2.10749 2.32858

3-58548 3-57443 3-54134 3.48641 3.41000

3-31255

2.59887

1.31766 1.56318 1.79907 2.02387 2.23618

2.31094 2.41670 2.48513 2.21964 2.64400 2.00889 2.78657 1.78576 2 . 9 1 1 9 6 1.55162

2.43471 2.61823 2.78561 2.93582 3.06792

2-53532

1.24698 2.94950 1.00694 3.01560 0.76069 3.06311 0-50975 3-09173 0.25566

3.01939 3.10821

3.18m 3-27468 3-34807 3.40081

ι.37210 1.10797 0.83701 0.56089 0.28131

3.10129 0.00

3.26816 0.00

2.30413 2.11624

ΐ·9 Ι 53ΐ 1.70258

1-47934

3-15757

3.40081 0.56089 3-34807 0.83701 3.27468 1.10797

3.04522 2.91409 2.76501

1.30791 1.05614

3 . 1 7 7 8 7 0.79785 3.22793 0.53465 3.25809 0.26815

Note.

Negative quantities are in heavy type.

Examples,

sinh (1.85 + i 0.75) = 1.18681 + i 3.01049. sinh (1.85 + i 1.35) = -1.62042 + i 2.77835.

[49]

3-43259

3.19469

3-057I3

2.90071 2.72642

2.32858 2.10749

1-87341

1.62777

3.44321 0.00

TABLE VII.

HYPERBOLIC

SINES,

sinh (x + iq) = u + iv.

» = 2.05

CONTINUED

χ = 2.15

3.62686 3.61568 3.58221 3.52666 3-4493S

0.00 0.29518 0.58854 0.87827 1-16258

3-8I958 3-80781 3-77256 3.71404 3.63264

0.00 0.30978 0.61765 0.92172 1.22010

4.02186 4.00946 3-97235 3.91074 3.82501

0.00 0.32516 0.64831 0.96747 1.28066

4.23419 4.22113 4.18206 4.11720 4.02695

0.00 0.34135 0.68060 1.01564 1-34443

4-457" 4-44337 4.40223 4-33396 4.23896

0.00 0.35839 0.71458 1.06636 1.41156

3.3S078 3-23156 3.09241 2.93420 2.75789

1-43973 1.70800 1-96574 2.21136 2-44335

3-52883 3-40327 3-23673 3.0901ι 2.90443

1.51096 1.79250 2.06299 2.32076 2.56423

3-7I57I 3-5835I 3.42920 3-25376 3-05825

1.58596 1.88148 2.16540 2-43597 2.69152

3.91188 3.77269 3.61024 3-42553 3.21970

1.66493 1.97516 2.27322 2.55726 2-82553

4.11783 3-97I3I 3.80031 3-60588 3.38921

1.74806 2.07379 2.38672 2.68495 2.96661

2.56458 2-35546 2.13182 1.89503 1.64656

2.66027 2.86080 3.04368 3.20780 3-35214

2.70085 2.48062 2.24509 1-99573 1-73405

2.79188 3.00232 3.19426 3-36649 3-51798

2.84389 2.61199 2.36399 2.10142 1.82589

2.93048 3-I5I37 3-35283 3-5336I 3.69261

2.99402 2.74988 2.48879 2.21236 1.92228

3-07639 3-30828 3-5I977 3-70956 3.87647

3-15165 2.89466 2.61982 2.32883 2.02349

3.23000 3-47347 3-69552 3.89478 4-07003

0.75

0.8 0.85 0.9 0.9S

1.38794 1.12076 0.84667 0.56737 0.28456

3.47581 3.57806 3.65825 3-7I587 3.75059

1.46169 1.18032 0.89166 0.59751 0.29968

3-64777 3-75507 3-83923 3.89971 3-93615

I-539IO 1.24282 0.93888 0.62916 0.31555

3.82885 3.94148 4.02981 4.09329 4-I3I54

1.62035 1.30844 0.98845 0.66237 0.33221

4.01950 4-13773 4.23046 4.29711 4-33726

1.70566 1-37732 1.04049 0.69724 0.34970

4.22019 4-34433 4.44170 4.51167 4-55382

1.0 1.05 1.1 1-I5 1.2

0.00 0.28456 0.56737 0.84667 1.12076

3.76220 3-75059 3-7I587 3-65825 3.57806

0.00 3-94832 0.29968 3-93615

0.00 0.31555 0.62916 0.93888 1.24282

4.14431 4-I3I54 4.09329 4.02981 3.94148

0.00 0.33221 0.66237 0.98845 1.30844

4-35067 4-33726 4-29711 4.23046 4-13773

0.00 0.34970 0.69724 1.04049 1-37732

4.56791 4-55382 4.51167 4.44170 4-34433

1.25 1-3 1-35 1.4 i-45

1.38794 1.64656 1.89503 2.13182

3-4758I 3-35214 3.20780 3.04368 2.35546 2.86080

1.46169 3-64777

I - 5 3 9 I 0 3.82885 1.82589 3.69261 2.10142 3-5336I 236399 3-35283 2.61199 3-I5I37

1.62035 1.92228 2.21236 2.48879 2.74988

4.01950 3.87647 3-70956 3-5I977 3-30828

1.70566 2.02349 2.32883 2.61982 2.89466

4.22019 4.07003 3.89478 3-69552 3-47347

i-S I-5S 1.6 1.65 1-7

2.56458 2.75789 2.93420 3.09241

2.66027 2.44335 2.21136 1.96574

2.70085 2.79188 2.90443 2.56423 3.0901ι 2.32076

2.84389 2.93048 3.05825 2.69152

3-38921 3.60588 3.80031

3.23156

1.70800

3.07639 2.82553 3-42553 2.55726 3.61024 2.27322 3.77269 1.97516

I-7S 1.8 1.85 1.9 1-95

3·35Ο78 1-43973 3-44935 1.16258 3.52666 0.87827 3.58221 0.58854 3.61568 0.29518

3.80781

3-77256 0.61765 0.30978

2.0

3.62686

3.81958

o 0.05 O.I 0.15 0.2 0.25

0-3 0.35

0.4

0.45

0.5 0-55 0.6 0.65 0.7

0.00

0.59751 3.89971 0.89166 3-83923 1.18032 3-75507 1-73405 3-51798 1-99573 3-36649 2.24509 3.19426 2.48062 3.00232

3-23673 2.06299 3-40327 1.79250 3.52883 1.51096 3.63264 ι.22010 3.71404 0.92172

0.00

3.25376 2-43597 3.42920 2.16540 3.58351 1.88148

2.99402 3.21970

3-15165 3.23000 2.96661 2.68495 2.38672 3-97I3I 2.07379

1.66493 1.34443 1.01564 0.68060 0.34135

4.11783 4.23896

3-97235 0.64831 4.00946 0.32516

3.91188 4.02695 4.11720 4.18206 4.22113

4.02186

4.23419

0.00

4.45711

3.71571

1.58596

3.82501 1.28066 3.91074 0.96747

0.00

Note.

Negative quantities are in heavy type.

Examples,

sinh (2.2 + i 1.0) = o + i 4.56791. sinh (2.2 + i 1^5) = —3.15165 + i 3.23000.

[50]

1.74806 1.41156 4-33396 1.06636 4.40223 0.71458 4-44337 0.35839 0.00

TABLE V I I .

HYPERBOLIC

χ - 2.3

χ = 2.25

χ -

2-35

CONTINUED

2.4

χ = 2.45

5.46623 0.00

5-75103

0-43599

I-55659

5-39893 0.86930 5-3I52I 1.29724 5.19869 1.71719

0.45799 5.68022 0.91316 5 - 5 9 2 I 3 1.36269 546955 1.80383

4.17986 2.17760 3 - 9 9 9 8 7 2.50620 3 - 7 9 5 2 3 2.81935 3-56719 3-11512

4.56116 1.92766 4.39887 2.28685 4.20946 2.63194 3 - 9 9 4 0 9 2.96081 3 - 7 5 4 I O 3-27141

4 - 7 9 9 6 S 2.02457 4.62887 2.40182 4 - 4 2 9 5 5 2.76426 4.20292 3.10966 3-95038 3-43588

5.05014 4.87045 4.66073 4.42228 4-15656

2.12655 2.52280 2.90349 3.26629 3.60894

5-3I325 5.12420 4-90356 4.65268

3.31716 3.39168 3.04677 3 - 6 4 7 3 4 2.75740 3.88050 245113 4.08975 2.12975 4 - 2 7 3 7 7

3.49096 3-56185 3.20630 3-83034 2.90188 4.07520 2 . 5 7 9 5 6 4.29494 2.24134 4.48820

3-6735I

¿(..02290 3-05360 4.28008 2.71443 4.51087 2 - 3 5 8 5 3 4.71384

3-86521

3-92935

3.21297 4 - 4 9 5 6 7 2.85610 4.73808 2.48162 4-95127

4.06659 4.12761 3 - 7 3 4 9 9 4.43873 3-38036 4.72249 3.00490 4-97713 2.61091 5.20109

I-79523 4 - 4 3 1 4 5 1.44965 4.56181 1-09513 4.66404 0.73386 4 - 7 3 7 5 1 0.36807 4.78178

1.88930 4 - 6 5 3 7 8 1.52560 4.79058 1.15251 4-89805 0.77231 4 - 9 7 5 2 1 0.38735 5.02169

I.I 4.88776 1.60537 5-03153 1.21277 5.14429 0.81269 5 - 2 2 5 3 3 0.40760 5.27416

2.09184 5 - 1 3 3 9 4 1.68916 5.28496 1.27607 5-40341 0.85511 5 - 4 8 8 5 3 0.42888 5 - 5 3 9 8 I

2.20082 1.77716 1-34255 0.89966 0.45122

0.00 4-79657 0.36807 4.78178 0.73386 4 - 7 3 7 5 1 1-09513 4.66404 1.44965 4.56181

0.00

0.00

5.29047

0.00

0.38735 5.02169 0.77231 4 . 9 7 5 2 1 1.15251 4.89805 1.52560 4-79058

0.40760 0.81269 1.21277 1.60537

5.27416 5.22533 5.14429 5.03153

5-55695 0.42888 5 - 5 3 9 8 I 0.85511 5 - 4 8 8 5 3 1.27607 5-40341 1.68916 5.28496

0.00 S-83732 0.45122 5 - 8 I 9 3 3 0.89966 5 - 7 6 5 4 5

1.79523 4 - 4 3 1 4 5 2.12975 4 - 2 7 3 7 7

3.04677 3 - 6 4 7 3 4

1.88930 4 - 6 5 3 7 8 2.24134 4.48820 2 - 5 7 9 5 6 4.29494 2.90188 4.07520 3.20630 3-83034

1.98808 2.35853 2.71443 3.05360 3-37395

4.88776 4.71384 4.51087 4.28008 4.02290

2.09184 5 - 1 3 3 9 4 2.48162 4.95127 2.85610 4.73808 3.21297 4 - 4 9 5 6 7 3.55003 4-22554

2.20082 5-39298 2.61091 5.20109 3.00490 4 - 9 7 7 1 3 3.38036 4.72249 3-73499 4-43873

3·3ΐ7ΐ6 3.39168 356719 3.11512 3 - 7 9 5 2 3 2.81935 3 9 9 9 8 7 2.50620 4.17986 2.17760

3.49096 3-75410 3.99409 4.20946 4.39887

3-56185 3-27141 2.96081 2.63194 2.2868s

3-67351 3-74093 3-95038 3-43588 4.20292 3.10966 4 - 4 2 9 5 5 2.76426 4.62887 2.40182

3.86521 3 - 9 2 9 3 5 4.15656 3.60894 4.42228 3.26629 4.66073 2.90349 4.87045 2.52280

4.06659 4-373" 4.65268

1-83557

1.48222 1.11974 0.75035 0.37633

1.92766 I.55659 1.17592 0.78799 0.39522

4 . 7 9 9 6 5 2.02457 4.94083 1-63485 5-05I56 1-23504 5.I3II4 0.82761 5.17909 0.41509

2.12655 1-7I7I9 1.29724 0.86930

1-95

4.33407 4.46157 4·56ΐ55 4.63341 4.67671

4.56116 469533 4.80056 4.87618 4.92174

5.05014 5.19869 5.31521 5-39893 5.44938 0 - 4 3 5 9 9

5.3I325 2-23385 5 · 4 6 9 5 5 1-80383 5-59213 1.36269 5.68022 0.91316 5 - 7 3 3 3 0 0.45799

2.0

4.69117

0.00

4.93696 0.00

5.46623

0.00

5.75103

o-35

0.4

0.45

0.5 o-SS

0.6 0.6s 0.7 0.75

0.8 0.8s 0.9 0-95

1.0 1-OS

1.1 i-iS 1.2

1.25 1-3 ι-35 1.4

ι.45 1-5 1-55

1.6 1.65 1-7 1-75

1.8 1.8s 1.9

4-33407

1-83557

2-45113 4.08975 2.75740 3.88050

4.93696 4.92174 0.39522 4.87618 0.78799 4.80056 1.17592

* =

= u + iv.

S-44938

0.25 0.03

0.00 0.37633 0.75035 1.11974 1.48222

s i n h (x + iq)

0.00 5-I7909 0.41509 5-I.3II4 0.82761 5-05156 1-23504 4.94083 1.6348s

o 0.05 O.I o-iS 0.2

4.69117 4.67671 4-6334I 4-56I5S 4-46I57

SINES,

4-09533

Note.

5.03722

I

I

5- 95 °

3-74093

3-37395

5-I95IO

0.00

3-55003

4-22554

Negative quantities are in heavy type.

Examples, sinh (2.4 + i 0.05) = 5.44938 + i 0.43599. sinh (2.4 + i 1.95) = -5.44938 + i 0.43599.

[51 J

5-73330

4-373"

2.23385 2.65009 3.04999 3-43I09 3-79104

5-39298 5-55I62 5.67603 5-76545

5-81933

134255 5.67603 1.77716 5-55I62

490356 5.12420

4.12761 3.79104 3-43I09 3-04999

2.65009

0.00

TABLE

VII.

H Y P E R B O L I C

2-5

SINES,

X = 2-55

0.15 0.2 0.25 °·3 0-35 °·4 °·45

5.58966 2-34673 5-39°77 2.78401 5-15865 3.20411 4.89472 3.60448 4.60062 3.98260

5.67082 5.42664 5.14900 4.83961

2.46547 2.92488 3.36624 3.78686 4.18413

6.18512

°·5 °·55 0.6 o.6s 0.7

4.27814 4-33619 3.92929 4.66304 3-55022 4.96113 3.16122 5.22864 2.74674 5-46392

4.50039 4.13342 3.74097 3-3 2 545 2.88943

4-5556O 4.89898 5.21217 5-49321 5.74039

4.73389 4.78641

0-7S

2.31531 1.86961 1.41239 0.94646 0.47469

2.43559

1.96674 1.48577 0-99563 0-49935 0.00 0-49935 0.99563 1.48577 1.96674

0.05

O.I

0.8 0.8s 0.9 °·95

1.0 1-°S 1.1 I-I5

5-66550 5.83215 5.96287 6.05680 6.11339

0.00

6.13229 6.11339 6.05680 1.41239 5.96287

0.47469 0.94646

6.36451 6.34489 6.28615 6.18866 6.05301

X — 2.6

6.05020 6.03155 5.97572 5.88304 5.75408

o

0.00 0.48113 0.95930 1.4315S 1.89498

s i n h (x +

5·

0.00 0.50548 1.00784 1.50399 1.99087

6.69473 6.67409 6.61231 6.50976 6.36706

0-53109 1.05891 1.58019 2.09174

iq)

= u +

7.04169 7.01998 6.95500 6.84713 6.69705

3.93506 5.47624 3-49799 5-77I53 3.03934 6.03123

4-97923 4-57321 4.13900 3.67927 3.19686

5.02919 5.40827 5-7540I 6.06427 6-337*4

5-23702 5-28454 4.80998 5.68287 4-35329 6.04606 3.86976 6.37218 3-36237 6.65891

5.95218 6.12727 6.26458 6.36327 6.42273

2.56196 2.06878 1.56285 1.04729 0.52526

6.25374 6.43770 6.58199 6.68567 6.74814

2.69474 2.17600 1.64385 1.10156 0-55249

6.57095 6.76424 6.91583 7.02478 7-09042

2.83425 6.90458 2.28866 7.10769 1.72896 7.26697 1-15859 7.38146 0.58109 7-45043

6.44259 6.42273 6.36327 6.26458 6.12727

0.00 0.52526 1.04729 1.56285 2.06878

6.76901 6.74814 6.68567 6.58199 6.43770

0.00 0.55249 1.10156 1.6438s 2.17600

7.11234 7.09042 7.02478 6.91583 6.76424

0.00 7-47347 0.58109 7-45043 1-15859 7.38146 1.72896 7.26697 2.28866 7.10769

2.56196 3-03934 3-49799 3-93506

6.25374 6.03123 5-77153 5.47624

2.69474 3.19686 3.67927 4.13900 4-57321

6.57095 6.33714 6.06427 S-7540I 5-40827

2.83425 3.36237 3.86976 4-35329 4.80998

4-78641 4.39612 3.97872 3.53680

5 02919 4-61910 4-18053 3.71619 3.22894

5.23702 5-28454

2.72178 2.19784 1.66034 1.11262

2.59039 5-96505 3.07306 5.70819 3-53680 5-41615 3.97872 5.09071 4.39612 4.34788

5.14719

2.31531 5-66550 2 . 7 4 6 7 4 5-46392

2-43559 5-95218

3.16122 3.SS622 3.92929

5.22864 4.96113 4.66304

3-32545 5-49321 4.13342 4.89898

4.34788

4.27814 4.60062 4.89472

4-336I9

4.50039 4.83961 5.14900 5.42664 5.67082

4-55560 418413 3.78686 3.36624 2.92488

4·73389 5.09071 5-41615 5.70819

5.96505 3-07306

4·97923 5-35454 5.6968s 6.00403 6.27419

2.46547 1.99087 1 -50399 1.00784 0.50548

6.18512 6.36706 6.50976 6.61231 6.67409

6.50567 6.69705 6.84713 6.95500 7.01998

I-3S 1.4

1-45

1-5 1-55 1.6 1.6S

1-7

3.98260 3.60448 5-1S86S 3.20411 5-39077 2.78401

3.74097

5-74039

5-2I2I7

1.8 1.8S 1.9 i-95

5-75408 1.89498 5 . 8 8 3 0 4 I-43I55 5-97572 0-95930 6 . 0 3 1 5 5 0.48113

5.88004 6.05301 6.18866 6.2861s 6.34489

2.0

6.05020 0.00

6.36451 0.00

I-7S

5.58966

2-34673

7.40626 0.00 7·38343 0.58636 7·3ΐ5θ8 1.16911 7.20163 1.74465 7-04377 2-30943 6.84249 2.85997 6-59903 3-39288 6.31488 3.90488 5-99179 4-39279 5-63I77 4.85363

1.86961

2.88943

0.00 0.55803 1.11262 1.66034 2.19784

X = 2.7

6.50567 2.72178 6.27419 3.22894 6.00403 3.71619 5-69685 4-18053 4.61910 5-35454

1.25 1-3

CONTINUED

χ = 2.65

1.2

5-83215

iv.

5.14719

2.59039

2.09174 1.58019 1.05891 0.53109

6.69473 0.00

0.55803

7.04169. 0.00

Note.

Negative quantities are in heavy type.

Examples,

s i n h (2.7 + i 0.7) = 3.36237 + i 6.65891. s i n h (2.5 + i 1.25) = - 2 . 3 1 5 3 1 + ¿5.66550.

[52]

6.90458 6.65891 6.37218 6.04606 5.68287

5-63177 4-85363

5.99I79 4-39279 6.31488 3.90488 6.59903 3.39288 6.84249 7-04377 7.20163 7·3ΐ5ο8 7·38343

2.85997 2.30943

1.74465 1.16911 0.58636

7.40626 0.00

TABLE V I I . χ -

HYPERBOLIC

2.75

X =

SINES,

2.8

sinh ( * + ig) =u

X = 2.8s

+ iv.

CONTINUED

X = 2.9

X - 2.9s

o 0.0s O.I o.is 0.2

7-7893S 7·76534 7-69345 7-57413 7.40811

0.00 0.61616 1.22852 I-8333I 2.42680

8.19192 8.16666 8.09106 7-96557 7.79098

0.00 0.64750 1.29101 1.92656 2.55023

8.61497 8.58841 8.50891 8.37694 8.19332

0.00 0.68046 I-35673 2.02463 2.68005

9-05956 9.03163 8.94802 8.80924 8.61616

0.00 0.71512 1-42583 2.12776 2.81656

9.52681 9.49744 9.40952 9.26358 9-06053

0-75I57 1.49851 2.23621 2.96012

0.25 °-3 0.3S 0.4 0-4S

7.19642 6.94036 6.64151 6.30172 5-92307

3-00532 3-5653I 4-10333 4.61604 5.10030

7-56834 7.29905 6.98476 6.62740 6.22918

3.15818 3.74666 4.31204 4.85083 5-35972

7-959I9 7-67599 7-34547 6.96966 6.55088

3-31894 3-93738 4-53153 5-09775 5-63254

8.36994 8.07213 7-72455 7-32934 6.88894

3.48800 4-13793 4.76236 5-35742 6.91945

8.80162 8.48845 8.12294 7-70735 7.24424

3.66578 4.34884 5.00509 5-63049 6.22116

0.5 °-5S 0.6 0.65 0.7

5-50790 5.05878 4-57847 4.06993 3-53629

5-55311 5.97168 6-35344 6.69602 6.99732

5-79256 5.32022 4.81509 4.28026 3-71905

5-83556 6.27542 6.67660 7.03661 7-35323

6.09170 5-59498 5-06375 4-50131 3.91112

6.13261 6.59486 7.01646 7-39479 7-72754

6.40608 5-88371 5-32508 4-7336I 4.11295

6.44498 6.93078 7-37385 7.77146 8.12115

6.73647 6.18717 5.63048 4-97774 4-32508

6.77348 7.28404 7.74969 8.16757 8.53508

0-7S 0.8 0.8s 0.9 0-9S

2.98086 2.40704 1.81839 1.21852 0.61115

7-25548 7.46891 7.63629 7-7S6S9 7.82907

3-I349I 2-53144 1.91237 1.28150 0.64273

7.62452 7.84881 8.02470 8.15112 8.22728

3.29681 2.66217 2.01112 1.34768 0.67592

8.01263 8.24834 8.43318 8.56604 8.64608

3.46694 2.79956 2.11491 1-41723 0.71081

8.42078 8.66850 8.86275 9.00237 9.08649

3-64575 2-94395 2.22399 1.49032 0.74747

8.84998 9.11031 9-31447 9.46121 9.54962

1.0 1.05 1.1 1-iS 1.2

0.00 0.61115 1.21852 1.81839 2.40704

7.85328 7.82907 7-75659 7.63629 7.46891

0.00 0.64273 1.28150 1.91237 2.S3144

8.25273 8.22728 8.15112 8.02470 7-84881

0.00 0.67592 1.34768 2.01112 2.66217

8.67281 8.64608 8.56604 8.43318 8.24834

0.00 0.71081 1-41723 2.11491 2.79956

9.11458 9.08649 9.00237 8.86275 8.66850

0.00 0.74747 1.49032 2.22399 2-94395

9-57915 9-54962 9.46121 9-3I447 9-II°3I

I-2S 1-3 I-3S 1-4 1-45

2.98086 3-53629 4.06993 4-S7847 5.05878

7-25548 6.99732 6.69602 6-35344 5.97168

3-I349I 37190S 4.28026 4.81509 5.32022

7.62452 7-35323 7.03661 6.67660 6.27542

3.29681 3.91112 4-50131 506375 5.59498

8.01263 7-72754 7-39479 7.01646 6.59486

3.46694 4.11295 4.73361 5-32508 5.88371

8.42078 8.12115 7.77146 7-37385 6.9307S

3·64575 4.32508 4-97774 5.63048 6.18717

8.84998 8.53508 8.16757 7.74969 7.28404

1-5 I-5S 1.6 1.6S 1-7

5.50790 592307 6.30172 6.64151 6.94036

5-553« 5.10030 4.61604 4-I0333 3-5653I

5.79256 6.22918 6.62740 6.98476 7.29905

5-83556 5-35972 4-85083 4.31204 3.74666

6.09170 6.55088 6.96966 7-34547 7.67599

6.13261 5-63254 5-0977S 4-53153 3 93738

6.40608 6.88894 7·32934 7-72455 8.07213

6.44498 5-9I945 5-35742 4.76236 4-13793

6.73647 7.24424 7-70735 8.12294 8.48845

6-77348 6.22116 5-63049 5.00509 4.34884

I-7S 1.8 1.8S 1.9 I-9S

7.19642 7.40811 7-57413 7.6934S 7 76534

300532 2.42680 1-83331 1.22852 0.61616

7.56834 7.79098 796557 8.09106 8.16666

3-15818 2.55023 1.92656 1.29101 0.64750

7.95919 8.19332 8.37694 8.50891 8.58841

3-31894 2.68005 2.02463 I-35673 0.68046

8.36994 8.61616 8.80924 8.94802 9.03163

3.48800 2.81656 2.12776 1-42583 0.71512

8.80162 9.06053 9.26358 9.40952 949744

3-66S78 2.96012 2.23621 1.49851 O.75157

2.0

7.78935 0.00

8.19192 0.00

8.61497 0.00

9.05956 0.00

Note.

Negative quantities are in heavy type.

Examples,

sinh (2.9 + i 0.9) = 1.41723 + 19.00237. sinh (2.8 + i 1.4) = —4.81509 + i 6.67660.

[53]

9.52681 0.00

T a b l e VII.

HYPERBOLIC

χ = 3·°

o.o

0.05

O.I

°·Ι5

0.2

0.2S

°·3

10.01787 9.98699 9-89454 9·74ιο8 9-52757

0.78990 1-57493

2.35025

3·ιιιο8

ΙΟ-53399 ο - ο ο 10.50150 0.83020 10.40430 1-65529 10.24290 2.47017 10.01840 3-26982

9.73216 9-38586

4.80383

8.0Ι0ΙΙ

6.21956 6.87204

4.04931

5-52874

sinh

(x

+

χ = 3.10

3-°5

ο.οο

3-85273 9-25531 4.57062

8.92599

SINES,

ο.οο

ig) = u + iv.

x

Continued

= 3·!5

X = 3·20

11.64661 11.61070 11.50320 11.32480 11.07660

ο.οο ο.9ΐ7χ4 1.82863 2.72885 3·6ΐ224

12.24588 Ι2.208ΐ0 12.09510 11.90750 11.64650

ο.οο 0.96399 1.92205 2.86826 3-79678

10.76010

4-47336 5.30690 6.10773 6.87089

11.31370 10.91120 ΙΟ·44 ι 3° 9-9°712 9.31184

4.70190 5·578Ο2 6.41976 7-22191 7-97954

11.07645 11.04230 ιο·94οιο 10.77040 10.53430

0.87258 1-73979 2.59626 3-43673

10.23330 9.86919 9-44423 8:96104 8.42260

4.25602 5.04906 5.81097 6·537°5 7.22284

7-83223

7.86409 8.45686 8-99749 9.48264 9·9°933

8-23539 7·56387 6.8457ο

8.26570 8.88873 9-45697 6.08533 9.96690 5-28745 10.41540

8.65915 8.68797 7·953°6 9-34284 7-19795 9-940ΙΙ 6.39846 10.47610 5-55952 Ι0-94750

4.45696 Ι0.79970 3.59900 11.1173° 2.71885 11.36650 1.82193 ιΐ·5455° 0.91378 11.6534°

4.68630 3.78419 2.85874 1.91568 0.96080

11.35140 11.68530 ΙΙ-94720 12.13540 12.24880 12.28665 12.24880 12.13540 ΙΧ-94720 11.68530

10-37720 9.93036

5.26034 5.91762 6.53842

8.98171 8.52218

7.11891 7-65551 8.Ι4491 8.58409 8·97°35 3-83368 9.30131 3-09569 9 - 5 7 4 9 2 2.33863 9.78949 1.56714 9 - 9 4 3 7 1 0.78599 10.03660

7.44866 6.84128

4-78233

7.48215 8.04612 8.56049 9-02209 9.42805

4.03119 3.25518 2.45911 1.64788 0.82649

9-77589 10.06350 10.28900 10.45110 10.54870

4.23878 10.27490 3.42281 10.57720 2.58575 10.81420

ι.ο Ι.05 Ι.Ι 1-IS

ο.οο 10.06766 0 . 7 8 5 9 9 10.03660 1.56714 9-94371 2-33863 9-78949 3.09569 9-57492

ο.οο 0.82649 1.64788 2-45911 3.25518

10.58135 10.54870 10.45110 10.28900 10.06350

0.00

0 . 8 6 9 0 5 11.08720 1.73274 10.98460 2-58575 10.81420

3 . 4 2 2 8 1 ΙΟ-577 20

ο.οο 11.68946 0 . 9 1 3 7 8 11.65340 1.82193 11-5455° 2 . 7 1 8 8 5 11.36650 3-59900 1 1 . 1 1 7 3 °

ο.οο 0.96080 1.91568 2.85874 3.78419

I-2S Ι·3 I-3S

3.83368 4-54806 5-23433 5-88836 6.50609

9.30131 8-97035 8.58409 8.Ι4491 7-65551

9-77589 9.42805 9.02209 8.56049 8.04612

4 . 2 3 8 7 8 ΙΟ.2749° 5 . 0 2 8 6 0 9·9°933 5-78743 9.48264 6.51058 8.99749 7-19358 8.45686

4 . 4 5 6 9 6 ΙΟ·7997° 5-28745 10.41540 6.08533 9-96690 6.84570 9 - 4 5 6 9 7 7·56387 8.88873

4 . 6 8 6 3 0 11.3514° 5-55952 ι ο · 9 4 7 5 ° 6 . 3 9 8 4 6 10.47610 7-19795 9-940ΙΙ 7-95306 9·34284

7.08371 7.61765 8.10463

6.53842 5-9Ι762

°·35

°·4

ο·45 °·5

0-SS 0.6

0.6s 0.7 °·75 0.8

0.8S O.Ç) 0-9S

1.2

1.4 I-4S I-S 1-55 1.6 I.6S

ΐ·7 ΐ·75 1.8

I.8S ΐ·9 ΐ·95 2.0

8.54164 8.10463 7.61765 7.08371 6.50609 5.88836 5-23433 4.54806

8.54164 8.92599 9*25531 9-52757 9.74x08 9-89454 9.98699 10.01787

6.19173

5.50400

4-03119 4-78233

5.50400

6.19173 6.84128

7-44866

7.11891

5.26034 4-57002

8.0Ι0ΙΙ

8-52218 8.98171 9.38586

1-57493 0.78990

9-73216 10.01840 10.24290 10.40430 10.50150

ο.οο

10.53399

3-85273 3·ΙΙΙΟ8

2-3502S

Note. Examples,

7.48215 6.87204 6.21956

7-19358 6.51058 5-78743 5.02860

1.73274 10.98460

0.86905 11.08720

7-83223

ΙΙ.Ι2Ι50

7.86409 7.22284 6-53 70S

9.42230 8.85615

7.59170

8.26570

5.04906

8-23539 8.85615 9.42230 9-93036 10.37720

4.25602 3-43673 2.59626 1-73979 0.87258

10.76010 11.07660 11.32480 11.50320 11.61070

4-47336 3.61224 2.72885 1.82863

0.83020

10.23330 Ι0-53430 10.77040 10.94010 11.04230

0.91714

11.3X370 11.64650 11.90750 12.09510 12.208X0

4.70190 3.79678 2.86826 1.92205 0.96399

°·°°

11.07645

°·°°

11.64661

ο.οο

12.24588

ο-οο

5-52874

4.80383 4.04931

3.26982 2.47017

1-65529

8.42260 8.96104 9-44423 9-86919

5-81097

7.59170

6.87089

9.90712

8.68797 7-97954

7.22191

6.Ι0773

10.44130 Χ0.9ΙΙ20

6.41976 5.57802

5.30690

Negative quantities are in heavy type. sinh (3.0 + i 0.95) = 0.78599 + i 10.03660. s i n h (3.0 + i 1.05) = — 0.78599 + i 10.03660.

[54]

8.659X5 9.31184

TABLE

VII.

HYPERBOLIC SINES, x = 3-3°

X = 3-25

sinh

(x + iq) = u + iv.

x = 3-35

x = 3-4°

CONTINUED

x = 3-45

o.o 0.05 O.I 0.15 0.2

12.87578 12.83610 12.71726 12.52002 12.24560

0.00 1.01326 2.02028 3.01484 3.99082

13-53788 13-49615 13.37120 13.16388 12.87530

0.00 1.06507 2.12356 3.16897 4.19483

14.23382 14.18950 14.05851 13.84054 13-53717

0.00 1.11953 2.23215 3.33101 4-40933

14.96536 14.91923 14.78111 14.55188 14.23291

0.00 1.17679 2.34632 3.50139 4.63487

15-73432 15.68581 15.54061 15-29959 14.96423

0.00 1.23699 2.46636 3-68053 4.87198

0.2s 0-3 0-35 0.4

11.89566 11.47240 10.97841 10.41673 9.79082

4.94218 5.86309 6.74784 7-59099 8-38733

12.50736 12.06234

5-t948s 6.16381

10.95235 10.29428

7.97905 8.81610

I3-I5°33 12.68244 12.13633 11.51541 10.82349

5-46075 6.47795 7.45549 8.38705 9.26691

13.82620 I3-33423 12.76006 12.19723 II-37975

5-73977 6.80928 7.83682 8.81604 9-74090

14.53662 14.01940 13.41572 12.72933 11.96447

6.03341 7.15765 8.23775 9.26706 10.23924

°-5 °-55 0.6 0.65 0.7

9.10455 8.36217 7.56820 6.72758 5.84548

9.13197 9.82031 10.44810 11.01147 11.50695

9-57273 8.79215 7-95737 7.07352 6.14670

9-5988I 10.32233 10.98222 11.57440 12.09520

10.06483 9.24413 8.36643 7.43715 6.46202

10.08964 10.85016 11-54379 12.16623 12.71369

10.58212 9.71923 8.79642 7.81938 6.79414

10.60570 11.40512 12.13423 12.78852 13.36397

11.12585 11.14830 10.21863 11.98861 9.24840 12 75501 8.22116 13.44278 7.14324 14.04764

0.7s 0.8 0.85 0.9 °-95

4.92735 3.97883 3.00579 2.01422 1.01022

11.931S0 12.28247 12.55773 12.75556 12.87474

5.18072 4.18343 3.16036 2.11780 1.06217

12.54145 12.91035 13.19970 13.40764 13.53290

5-44705 4.39850 3.32282 2.22666 1.11677

13-18275 13.57054 13.87465 14.09323 14.22498

5.72700 4.62455 3.49359 2.34110 1.17417

13.85702 14-26465 14-58432 14.81410 14.95250

6.02127 14.56595 4.86217 14.99442 3 . 6 7 3 1 1 15.33045 2.46139 15-57196 1.23450 15.71746

1.0 I °S 1.1 1-IS 1.2

0.00 1.01022 •2.01422 3.00379 3.97883

12.91456 12.87474 12.75556 12.55773 12.28247

0.00 1.06217 2.11780 3.16036 4.18343

I3-57476 13.53290 13.40764 13.19970 12.91035

0.00 1.11677 2.22666 3.32282 4.39850

14.26891 14.22498 14.09323 13.87465 13-57054

0.00 I-I74I7 2.34110 3-49359 4.62455

14.99874 14-95250 14.81410 14-58432 14.26465

0.00 1.23450 2.46139 3.67311 4.86217

15.76607 15.71746 15-57196 i5-33°45 14.99442

I-2S 1-3 I-3S 1.4 i-45

4-92735 5.84548 6.72758 7.56820 8.36217

11.9315° 11.50695 11.01147 10.44810 9.82031

5.18072 6.14670 7.07352 7-95737 8.79215

12.54145 12.09520 11.57440 10.98222 10.32233

5.44705 6.46202 7-43715 8.36643 9.24413

13.18275 12.71369 12.16623 11-54379 10.85016

5.72700 6.79414 7.81938 8.79642 9.71923

13.85702 I3-36397 12.78852 12.13423 11.40512

6.02127 7.14324 8.22116 9.24840 10.21863

14.56595 14.04764 13.44278 12.75501 11.98861

1-5 1-55 1.6 1.6s 1-7

9.10455 9.79082 10.41673 10.97841 11.47240

9.13197 8.38733 7-59099 6.74784 5-86309

9-57273 10.29428 10.95235 11.54294 12.06234

9-5988I 8.81610 7-97905 7-09279 6.16381

10.06483 1 0 . 0 8 9 6 4 10.82349 9 . 2 6 6 9 1 11.51541 8.38705 12.13633 7.45549 12.68244 6 . 4 7 7 9 5

10.58212 1 0 . 6 0 5 7 0 11.37975 9-74090 12.10723 8 . 8 1 6 0 4 12.76006 7 . 8 3 6 8 2 13-33423 6 . 8 0 9 2 8

11.12585 11.14830 11.96447 1 0 . 2 3 9 2 4 12.72933 9 . 2 6 7 0 6 13-41572 8 . 2 3 7 7 5 14.01940 7 . 1 5 7 6 5

1-75 1.8 1.8s 1.9 1-95

11.89566 12.24560 12.52002 12.71726 12.83610

4.94218 3.99082 3.01484 2.02028 1.01326

12.50736 12.87530 13.16388 13.37120 13.49615

5.19485 4-19483 3.16897 2.12356 1.06507

I3-I5033 13-53717 13.84054 14-05851 14.18950

13.82620 14.23291 14-55I88 14.78111 14.91923

5-73977 4.63487 3-50139 2.34632 1.17679

14.53662 14.96423 15-29959 15.54061 15.68581

6.03341 4.87198 3-68053 2.46636 1.23699

2.0

12.87578

0.00

13.53788

0.00

14.23382 0 . 0 0

14.96536

0.00

15-73432

0.00

0.45

11.54294 7-09279

Note. Examples,

5-46075 4-40933 3-33IOI 2.23215 1.11953

Negative questions are in heavy type. sinh ( 3 . 4 0 + i o) = 1 4 . 9 6 5 3 6 + i o. sinh ( 3 . 4 5 + i 1.45) = — 1 0 . 2 1 8 6 3 + i 11.98861.

L 55 1

TABLE V I I .

HYPERBOLIC

χ = 3-5°

SINES,

s = 3-55

sinh (x + iq) = u+

χ = 3.60

iv.

CONTINUED

X = 3-65

x = 3-7°

0.0 0.05 0.1 °-IS 0.2

16.54263 16.49163 16.33896 16.08555 IS-73300

0.25 °-3 0-3S 0.4 0.45

15.28340 6.34215 14.73960 7.52391 14.10490 8.65928 13-38330 9.74126 12.57910 10.76320

16.06840 6.66674 15.49665 7-90898 14.82933 9-Ι0246 14.07066 10.23982 13.22520 11.31405

16.89356 7·°°8ΟΟ 16.29246 8.31383 15.59090 9-56840 14.79326 10.76400 13.90436 11.89320

17.76096 7-36678 17.12900 8-73946 16.39143 10.05827 15.55281 11.31505 14.61830 12.50208

18.67280 7-74399 18.00840 ç.i86g6 17.23295 10.57330 16.35127 11.89444 15.36878 13.14223

°-5 o-SS 0.6 0.65 0.7

11.69740 10.74357 9.72361 8.64360 7.51020

11.71820 12.60210 13.40770 14.13065 14.76650

12.29820 ΙΙ.29540 ίο.22293 9.08745 7-89594

12.31852 13-24705 Ι4·°939° Ι4-85386 15-52225

12.92978 11-87545 10.74789 9-55412 8.30143

Ι2.94910 I3-925IS 14-81535 15.61421 16.31680

Ι3-59365 12.48520 11.29972 10.04469 8.72766

13-61203 14-63806 15-57383 16.41360 17.15216

14.29155 13.12620 11.8799° 10.56037 9.17573

14-30902 15.38760 16.37127 Ι7·254°4 18.03040

0.7 5 0.8 0.85 0.9 0-95

6-33059 Ι 5 · 3 Ι Ι 3 ° 5 . I I I 9 5 15.76170 3.86180 16.11491 2.58783 16.36878

6.65574 5-37451 4.06015 2.72075 1.36458

16.09492 16.56840 Ι6·9397° 17-20653 17-36731

6.99754 5.65052 4.26865 2.86047 1.43466

16.91880 i7-4!65O 17.80680 18.08732 18.25632

7-35683 5.94065 4.48783 3-00735 1.50832

17-78497 18.30814 18.71843 Ι9·ΟΙ33Ι 19.19100

7-73453 6.24563 4.71823 3.16174 1.58576

18.69565 19-24560 19.67690 19-98688 20.17362

1.0 i.os 1.1 1-I5 1.2

ο.οο 16.57282 1.29792 16.52173

Ο.ΟΟ

17.42102

0.00

18.31278

0.00

19-25033

0.00

20.23601

1.36458

17.36731

18.25632 18.08732 17.80680 17.41650

1.50832

Ι9-Ι9100

3-00735 1 9 0 1 3 3 1

1.58576 2 0 . 1 7 3 6 2 3.16174 Ί9.98688

4.48783

4 . 7 r 8 2 3 19-67690

16.91880 16.31680

8.72766

17.15216

9.55412 15-61421

10.04469

16.41360

I-2S 1-3 I-3S 1.4 ι-45 1-5 i-SS 1.6 1.6s 1-7 1-75 1.8 1.8s ι.g ΐ·95 2.0

1.29792

0.00 1.30029 2.59256 3-86885 5.12129

17.39230 17-33870 17-17817 16.91175 16.54105

16.52173

ο-οο 1.36684 2.72525 4·°6686 5-38339

18.28546 18.22900 ι8.ο6Ο33 17.78022 Ι7·3905°

3·86Ι8Ο I Ó . H 4 9 1 5 . Ι Ι Ι 9 5 I5-7ÓI70

4.06015

5-37451 16.56840

1.43466 2.86047 4.26865 5.65052

6-33059 IS-3II30 7-51020 14.76650

6.65574 16.09492 7-89594 15-52225

6-99754 8.30143

2.58783

8.64360 9.72361

2.72075 17-20653

16.36878

14.13065 Ι3-40770

9.08745 10.22293

Ι6·9397°

14-85386 14-0939°

ο.οο 1.43680 2.86475 4·275°3 5-65896

10.74789 11.87545

1481535 Ι3-925Ι5

10.74357 Ι2.6Ο2ΙΟ

11.29540 Ι3-24705

11.69740

12.29820 13.22520 14-07066 14.82933

12.31852 11.31405 10.23982 9·ιο246

12.92978 13-90436 14.79326

12.94910 11.89320 10.76400

Ι5-59090

9-56840

15-49665

7-90898

16.29246

8.31383

6.66674

16.89356 17.39050 17.78022

7-°ο8οο 5-65896 4·275°3

11.71820

12.57910 10.76320 Ι3-38330 9-74126 14.10490 14.73960

8.65928 7-52391

15.28340

6.34215

16.06840

15-73300

5-!2I29

16.54105

5-38339

16.08555 16.33896 16.49163

3-86885 2.59256 1.30029

16.91175 17.17817

4·Ο6680 2.72525

ι 8.00033

2.86475

Ι7-33870

1.36684

18.22900

16.54263

ο.οο

17-39230

ο.οο

18.28546

19.22434 19.16506 18.98765 18.69316 18.28342

°·ο° 1.51036 3·οιι42 4-4939° 5-94868

18.71843

Ο.ΟΟ 1.58770 3·ι656ι 4-7240Ι 6.25327

5.94065 18.30814

6.24563

7-35683 17-78497

7-73453 18.69565 9-17573 18.03040

1 1 . 2 9 9 7 2 15-57383 12.48520

14.63806

13-593Ô5 13-61203 14-61830

12.50208

15.55281 11.31505 ι 6 . 3 9 χ 4 3 Ί0.05827 17-12900 8.73946

19-24560

10.56037 11.87990 13.12620

Ι7·254°4 16.37127 15.38760

14.29155 15.36878 16.35127

Ι4·3°9°2 13.14223 11.89444

17-23295 JO.57330 18.00840

ç.i86ç6

18.67280 19-22208 19.65290 19.96246

7-74399 6.25327 4-72401 3-ι656ι

17.76096

7-36678

1.43680

18.28342 18.69316 18.98765 19.16506

5-94868 4-4939° 3·οιι42 1.51036

20.14900

1.58770

ο.οο

19-22434

ο-οο

20.21129

ο-οο

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

sinh (3.70 + i 0.5) = 14.29155 + i 14.30902. sinh (3.70 + i ±5) = - 1 4 . 2 9 1 5 5 + i 14.30902.

[56]

20.21129 20.Ι4900 19.96246 19.65290 19.22208

TABLE V I I .

?

HYPERBOLIC

SINES,

χ = 3.80

x = 3-75

sinh (x + ig) = u + iv.

CONTINUED

χ = 3.90

x = 3-85

3-95

o.o 0.05 0.1 0.15 0.2

21.24878 21.18327 20.98716 20.66167 20.20879

0.00 1.66900 3.32772 4.96592 6.57350

22.33941 22.27052 22.06437 21.72216 21.24603

0.00 1.75448 3.49815 5.22025 6.91017

23.48589 23.41348 23.19673 22.83696 22.33640

0.00 1-84435 3- 6 7733 5.48764 7.26411

24.69110 24.61500 24.38710 24.00888 23.48262

0.00 1.93883 3-86571 5-76875 7.63623

25.95806 25.87805 25.63849 25.24084 24.68760

0.00 2.03816 4-06375 6.06429 8.02744

0.25 0.3 0.35 0.4 0.45

19.63131 8.14055 18.93280 9.65741 18.11756 11.11473 17.19062 12.50353 16.15770 13.81524

20.63891 I9-90455 19.04746 18.07296 16.98701

8.55748 10.15203 11.68400 13.14392 14.52281

21.69813 20.92608 20.02501 19.00048 17.85880

8.99580 10.67203 12.28246 13.81716 15.26668

22.81160 21.99993 21.05272 19.97556 18.77526

9.45662 11-21871 12.91164 14-52497 16.04874

23.98212 23.12881 22.13290 21.00052 19.73867

9.94109 11.79366 13-57315 15.26910 16.87098

0.5 0-55 0.6 0.65 0.7

15.02516 I3-79998 12.48971 11.10246 9.64674

15.04177 16.17556 17.20963 18.13760 18.95373

15.79634 14.50828 13.13076 11.67230 10.14188

15.81216 17.00402 18.09105 19.06655 19.92448

16.60702 15.25286 13.80465 12.27134 10.66233

16.62208 17.87500 19.01770 20.04315 20.94503

17.45924

17.47355

18.35512 18.36873

16.03558 14.51307 12.90106 11.20952

18.79065 19.99190 21.06988 22.01797

16.85842

0.75 0.8 0.85 0.9 0.95

8.13156 6.56624 4.96043 3·3 2 4θ4 1.66716

19.65301 20.23113 20.68452 21.01038 21.20670

8.54892 6.90325 5.21503 3.49465 1.75273

20.65958 21.26731 21.74391 22.08646 22.29283

8.98766 7.25754 5.48267 3.67400 1.84268

21.71778 22.35664 22.85766 23.21775 23.43470

9.44887 7.62997 5.76402 3.86254 1.93724

22.83030 23.50188 24.02856 24.40710 24.63516

9-93373 8.02149 6.05979 4.06074 2.03665

23-9999I 24.70590 25.25957 25.65749 25.89724

1.0 1.05 1.1 1.15 1.2

0.00 1.66716 3.32404 4.96043 6.56624

21.27230 21.20670 21.01038 20.68452 20.23113

0.00 1-75273 3.49465 5.21503 6.90325

22.36178 22.29283 22.08646 21.74391 21.26731

0.00 1.84268 3.67400 5.48267 7.25754

23.50717 23.43470 23.21775 22.85766 22.35664

0.00

24.71135

1.93724

24.63516

0.00 2.03665 4.06074 6.05979 8.02149

25.97731 25.89724 25.65749 25.25957 24.70590

1.25

8.13156 19-65301

1.3

9-64674

20.65958 19.92448 19.06655 18.09105 17.00402

8.98766 10.66233 12.27134 13.80465 15.25286

21.71778 20.94503 20.04315 ig.01770 17.87500

9.44887 11.20952 12.90106 14-51307 16.03558

22.83030 22.01797 21.06988 19.99190 18.79065

9.93373 11.78474 13.56305 I5-25776 16.85842

23.99991 23.14597 22.14931 21.01610 I9-7533I

18.35512 18.36873 19.73867 16.87098 21.00052 15.26910

3.86254 24.40710 5.76402 24.02856 7.62997 23.50188

1.35 1.4 1-45

11.10246 18.13760 12.48971 17.20963 13-79998 16.17556

8.54892 10.14188 11.67230 13.13076 14.50828

1.5 1.55 1.6 1.65 1.7

15.02516 16.15770 17.19062 18.11756 18.93280

15.04177 13.81524 12.50353 11.11473 9.65741

15-79634 16.98701 18.07296 19.04746 19.90455

15-81216 14.52281 13.14392 11.68400 10.15203

16.60702 17.85880 19.00048 20.02501 20.92608

16.62208 15.26668 13.81716 12.28246 10.67203

17.45924 18.77526 19-97556 21.05272 21.99993

17-47355 16.04874 14-52497 12.91164 11.21871

1-75. 1.8 1.85 1.9 I.95

19-63131 20.20879 20.66167 20.98716 21.18327

8.14055 6.57350 4.96592 3.32772 1.66900

20.63891 21.24603 21.72216

8.55748 6.91017 5.22025

22.06437

3-49815

21.69813 22.33640 22.83696 23-19673

8.99580 7.26411 5.48764 3-67733

22.81160 23.48262 24.00888 24.38710

9.45662 7.63623 5.76875 3.86571

23.41348

1.84435

24.61500

1.93883

2.0

21.24878

0.00

22.33941

23.48589

0.00

24.69110

0.00

18.95373

22.27052

1.75448

0.00

Note.

Negative quantities are in heavy type.

Examples,

sinli (3.90 + i 0.75) = 9.44887 + i 22.83030. sinh (3.95 + i i.OS) = -25.87805 + i 2.03816.

[57]

19.75331

15.25776 21.01610 13-56305 22.14931 11.78474 23.14597

22.13290 23.12881

13.57315 11.79366

23.98212 24.68760 25.24084 25-63849 25.87805

9.94109 8.02744 6.06429 4-06375 2.03816

25.95806

0.00

TABLE V I I I .

HYPERBOLIC χ

COSINES.

: 0.15

= 0.05

o

ι.0000

0.00

I.OOI25

0.00

0.05

0.99692

0.00

0.99816

0.00392

1.00500 0.00 I.00I9I 0.00786

0.1

0.98769 0.00 0.97237 0.00

0.98892

0.00783

0.99263

0.97385 O.OII68 0.95225 0.01546 0.92503

0.89208 0.02271

o-iS

0.95106

0.00

cosh (χ + iq) = u + iv

I.OII27

0.00

1.02007

1.00815

O.OII8I

1.01692

0.01580

0.01567

0.99882 0.02355

1-00751

0.03150

0.97724

0.02338

0.95582

0.03095

0-98333 0-03515 0.96178 0.04653

0.99188 0.04700 0.97014 0.06222

0.92850 0.03833 0.89547 0.04547

0.93429 0.05762

0.94242

0.90105 0.06835

0.09I4I

0.00

0.2 0.92388 0.00

c.oo

O.OI9I4

0.07705

0.25 0.3

0.89101

0-3S

0.85264 0.00

0.85367

0.02614

0.85691

0.05234

0.86225 0.07867

0.86975 0.10520

0.4

0.80902 0.00

0.81000 0.02940

0.81307

0.05888

0.81814 0.08850

0.82525

0.11834

°-45

0.76041

0.00

0.76133

0.03249

0.76421

0.06505

0.76898 0.09778

0.77567

0.13076

o-S O.S5

0.70711

0.00

0.70796

0.03537

0.71065

0.07083

0.71508

0.10646

0.72130

0.64945

0.00

0.65023

0.03804

0.65270 0.07617

0.65677

0.11449

0.6 0.65 0.7

0.58779

0.00

0.58850 0.04047

0.59027

0.08104

0.59441

0.52313

0.04265

0.525II

0.08541

0.45439

0.04457

0.45626 0.08925

0.75

0.38268 0.00

0.38316

0.04621

0.38460 0.09254

0.38700 O.I39IO

0.8

0.30902

0.00

0.30940 0.04757

0.31056

0.09526

0.31250

O.I43I9

0.31522

0.19148

0.85

0.23345

0.00

0.23374

0.04864

0.23461

0.09740

0.23813

0.19577

0.9

0.15643

0.00

0.15663

0.04941

0.15722

0.09893

0.23608 0.14640 0.15820 0.14871

0-95

0.07846

0.00

0.07856

0.04987

0.07885

0.09986

0.07934 O.I50IO

0.15957 0.19886 0.08003 0.20072

1.0

0.00

0.00

0.00

0.05002

0.00

O.IOOI7

0.00

0.15056

0.00

1.05

0.07846

0.00

0.07856

0.04987

0.07885

0.09986

0.07934

0.I50I0

0.08003

0.20072

1.1 1-I5

0.15643

0.00

0.15663

0.04941

0.15722

0.09893

0.15820

0.14871

0.19886

0.2334s

0.00

0.23461

0.09740

0.23608

0.14640

1.2

0.30902

0.00

0 . 2 3 3 7 4 0.04864 0.30940 Ö.04757

0.15957 0.23813

0.31056

0.09526

0.31250

0.I43I9

0.31522

0.19577 0.19148

1.25

0.38268

0.00

0.38316

0.04621

0.38460

0.09254

0.38700

O.I39IO

0.39036

0.I860I

1-3 1-35

0.45399

0.00

0.45439

0.04457

0.45626

0.08925

0.45911

O.I34I5

0.46310

0.17939

0.52250

0.00

0.52313

0.04265

0.525II

0.08541

0.53298

0.17167

1.4

0.58779

0.00

0.59027

0.08104

0.59958

0.16288

i-45

0.64945

0.00

0.58850 0.04047 0.65023 0.03804

0.52839 0.12838 0.59441 0.12181

0.65270

0.07617

0.65677

O.I 1449

0.66248

O.I53IO

1-5 1-55

0.70711

0.00

0.70796

0.03537

0.71065

0.07083

0.71508

0.10646

0.72130

0.14237

0.76041

0.00

0.76133

0.03249

0.76421

0.06505

0.76898

0.09778

0.77567

0.13076

1.6

0.80902

0.00

0.81000

0.02940

0.81307

0.05888

0.81814

0.08850

0.82525

0.11834

1.65

0.85264

0.00

0.85367

0.02614

0.85691

0.05234

0.86225

0.07867

0.10520

1-7

0.89101

0.00 •

0.89208

0.02271

0.89547

0.04547

0.90105

0.06835

0.8697s

1-75

0.52250 0.00 0.45399 0.00

0.12181 0.52839 0.12838 0.45911 O . I 3 4 I 5

0.14237 0.66248 O.I53IO 0.59958 0.16288 0.53298 0.17167 0.46310 0 . 1 7 9 3 9 0.39036 0.I860I

0.9088g

0.20134

0.09141

0.92388

0.00

0.92503

O.OI9I4

0.92850

0.03833

0.93429

0.05762

0.94242 0.07705

1.8

0.95106

0.00

0.95225

0.01546

0.95582

0.03095

0.96178

0.04653

0.97014

0.06222

ï.85

0.97237

0.00

0.97385

O.OII68

0.97724

0.02338

0.03515

0.99188

0.04700

1.9

0.98769

0.00

0.98892

0.00783

0.99263

0.01567

0.98333 0.99882

0.02355

I.0075I

0.03150

1-95

0.99692

0.00

0.99816

0.00392

I.00I9I

0.00786

I.008I5

0.0II8I

1.01692

0.01580

1.0000

0.00

1.00125 0.00

I.OII27

0.00

1.02007

0.00

1.00500 0.00

Note.

Negative quantities are in heavy type.

Examples,

cosh (o + i 0.75) cosh (0.2 + i 1 . 5 )

= 0.38268 +

i o. i

= —0.72130 +

[58]

0.14237

TABLE V I I I .

HYPERBOLIC

χ = 0.25 o 0.05 O.I 0.15

0.2 0.25 °-35 0.4

0.45

χ

COSINES,

- 0.3

x

c o s h (x + iq)

= u + iv.

0.4

= 0.3s

CONTINUED

X = 0.4s

ι.03141 1.02823 1.01871 1.00292 0.98093

0.00 0.01982 0.03952 0.05897 0.07806

1.04534 1.04212 1.03247 1.01646 0.99418

0.00 0.02389 0.04764 0.07109 0.09410

1.06188 1.05860 1.04880 1.03254 1.00991

0.00 0.02803 0.05588 0.08338 0.11038

1.08107 1.07774 1.06776 1.05120 1.02816

0.00 0.03223 0.06426 0.09589 0.12693

1.10297 1.09957 1.08939 1.07250 1.04899

0.00 0.03651 0.07280 0.10863 0.14380

0.95290 0.91900 0.87942 0.83443 0.78429

0.09667 0.11468 0.13199 0.14848 0.16406

0.96577 0.11654 0.93140 0.13825 0.89130 0.15911 0.84570 0.17899 0.79488 0.19777

0.98105 0.94614 0.90540 0.85908 0.80746

0.13669 0.16216 0.18663 0.20995 0.23198

0.99S78 0.96324 0.92177 0.87461 0.82206

0.15719 0.18648 0.21462 0.24143 0.26676

1.01901 0.98275 0.94044 0.89232 0.83871

0.1780S 0.2II2Ó 0.24314 0.27352 0.30222

0.75086 0.68964 0.62416 0.55483 0.48208

0.25257 0.27161 0.28897 0.30455 0.31826

0.76443 0.70210 0-63544 0.56486 0.49080

0.29045 0.31234 0.33231 0.35022 0.36598

o. 77992 0.71632 0.64831 0.57630 0.50074

0.32905 0.35385 0.37647 0.39677 0.41462

0.5 °-5S 0.6 0.65 0.7

0.72932 0.17862 0.66985 0.19208 0.60625 ° - 2 ° 4 3 7 0.53891 0.21539 0.46825 0.22508

0.73917 0.67889 0.61444 0.54619 0.47457

0.75 0.8 0.85 0.9 °-95

0.39471 0.31872 0.24078 0.16135 0.08092

0.23338 0.24025 0.24563 0.24950 0-25183

0.40003 0.28134 0.32303 0.28962 0.24403 0.29611 0.16353 0.30077 0.08202 0.30358

0.40636 0.33000 0.32814 0.33971 0.24789 0.34732 0.16611 0.35279 0.08331 0.35609

0.41371 0-37949 0.33407 0.39065 0.25237 0.39940 0.16912 0.40570 0.08482 0.40949

0.42209 0.34084 0.25748 0.17254 0.08654

0.42992 0.44257 0.45249 0.45961 0.46391

1.0 ι-OS 1.1 i-iS 1.2

0.00 0.08092 0.1613s 0.24078 0.31872

0.25261 0.25183 0.24950 0.24563 0.24025

0.00 0.08202 0.16353 0.24403 0.32303

0.00 0.35719 0.08331 0.35609 0.16611 0.35279 0.24789 0.34732 0.32814 0.33971

0.00 0.08482 0.16912 0.25237

0.41075 0.40949 0.40570 0.39940 0.39065

0.00 0.08654 0.17254 0.25748 0.34084

0.46534 0.46391 0.45961 0.45249 0.44257

1-25 1-3 1-35 1-4 ι . 45

0.39471 0.46825 0.53891 0.60625 0.66985

0.23338 0.22508 0.21539 0.20437 0.19208

0.40003 0.28134 0.47457 0.27133 0.54619 0.25965 0.61444 0.24636 0.67889 0.23156

0.40636 0.48208 0.55483 0.62416 0.68964

0.33000 0.31826 0.30455 0.28897 0.27161

0.37949 0.36598 0.35022 0.33231 0.31234

0.42209 0.50074 0.57630 0.64831 0.71632

0.42992 0.41462 0.39677 0.37647 0-35385

1.5 1-55 1.6 1.6S 1-7 .

0.72932 0.78429 0.83443 0.87942 0.91900

0.17862 0.16406 0.14848 0.13199 0.11468

0.73917 0.79488 0.84570 0.89130 0.93140

0.21533 0.19777 0.17899 0.15911 0.13825

0.75086 0.80746 0.85908 0.90540 0.94614

0.25257 0.23198 0.20995 0.18663 0.16216

0.41371 0.49080 0.56486 0.63544 0.70210 0.76443 0.82206 0.87461 0.92177 0.96324

0.20945 0.26676 0.24143 0.21462 O.I δ

0.77992 0.83871 0.89232 0.94044 0.98275

0.32905 0.30222 0.27352 0.24314 0.21126

1-75 1.8 1.8S 1.9 i-95

0.95290 0.98093 1.00292 1.01871 x.02823

0.09667 0.07806 0.05897 0.03952 0.01982

0.96577 0.11654 0.99418 0.09410 1.01646 0.07109 1.03247 0.04764 1.04212 0.02389

0.98105 1.00991 1.03254 1.04880 1.05860

0.13669 0.11038 0.08338 0.05588 0.02803

0.99878 1.02816 1.05120 1.06776 1.07774

0.15719 0.12693 0.09589 0.06426 0.03223

1.01901 1.04899 1.07250 1.08939 1.09957

0.17808 0.14380 0.10863 0.07280 0.03651

1.03141

0.00

1.04534 0.00

1.06188 0.00

1.08107

0.00

1.10297

0.00

0.21533 0.23156 0.24636 0.25965 0.27133

0.30452 0.30358 0.30077 0.29611 0.28962

0.33407

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

cosh (0.3 + i 0.9) = 0.16353 + i 0.30077. cosh (0.45 + i 1.7) = —0.98275 + i 0.21126.

[59]

TABLE V I I I .

HYPERBOLIC

χ = 0.5

χ = 0.55

COSINES,

χ - 0.6

0.00 0.04088 0.08152 0.12165 0.16103

1.1551° 1.151S4 1.14088 1.12319 1.09857

0.00 0.04536 0.09044

0.15 0.2

1.12763 I.I24I5 I.II374 1.09647 1.07244

0.25 0-3 0.3s 0.4 0.4s

1.04179 1.00472 0.96146 0.91227 0.85745

0.19942 0.23657 0.27227 0.30629 0.33842

1.06717 1.02920 0.98489 0.93450 0.87835

o-S o-SS 0.6 0.65 0.7

0-7973S 0.73233 0.66280 0.58918 ο·5ΙΙ93

0.36847 0.39624 0.42158 0.44431 0.46430

0.81678 0.75018 0.67895 0.60354 0.52441

0-75 0.8 0.85 0.9 0-9S

0.43152 0.34846 0.26324 0.17640 0.08847

0.48143 0.49559 0.50670 0.51468 0.51949

0.44204 0.3569S 0.26965 0.18070 0.09063

1.0 1.05 1.1 1-iS 1.2

0.00 0.08847 0.17640 0.26324 0.34846

0.52110 0.51949 0.51468 0.50670 0.49559

0.00 0.09063 0.18070 0.26965

0.57815 0.57637 0.57103 0.56218 0.35695 0.54986

0.00 0.09301 0.18545 0.27674

1.25 1-3 1-35 1.4 1-45

0.43152 0.51193 0.58918 0.66280 0.73233

0.48143 0.46430 0.44431 0.42158 0.39624

0.44204 0.52441 0.60354 0.67895 0.75018

0.53414 0-51514 0.49296 0.46773 0.43963

i-S 1-55 1.6 1.65 1-7

0.79735 0.85745 0.91227 0.96146 1.00472

0.36847 0.33842 0.30629 0.27227 0.23657

0.81678 0.87835 0.93450 0.98489 1.02920

1-75 1.8 1.85 1.9 1-95

1.04179 1.07244 1.09647 1.11374 1.12415

0.19942 0.16103 0.12165 0.08152 0.04088

2.0

1.12763

0.00

o 0.05 O.'X

cosh (x + iq) = u +

iv.

CONTINUED

χ '— 0.65

X = 0.7

0.17866

1.18547 1.18181 1.17087 1.15271 1.12 744

0.00 0.04995 0.09959 0.14862 0.19674

1.21879 1.21504 I.20379 1.18512 I.I59I2

O-OO 0.05467 0.10900 0.16265 0.2I53I

ι.25517 1.25130 1.23972 1.22049 1.19374

0.00 0.05952 0.11867 0.17709 0.23442

0.22125 0.26248 0.30208 0.33983 0.37548

1.09523 1.05626 1.01078 0.95906 0.90144

0.24364 0.28904 0.33265 0.37422 0.41347

1.12602 1.08595 I.039I9 0.98602 0.92678

0.26663 0.31632 0.36405 0.40954 0.45250

1.15962 1.11836 I.0702I I.OI545 0.95444

0.29030 0.34439 0.39636 0.44589 0.4926Ò

0.40882 0.43963 0.46773 0.49296

0.83825 0.76990 0.69680 0.61940

0.51514

0.45018 0.48412 0.51506 0.54284 0.53819 0.56726

0.86182 0-79 I 54 0.71639 0.63682 0.55332

0.49268 0.52981 0.56368 0.59408 0.62081

0.88754 0.81517 0.73777 0.65582 0.56984

0.53640 0.57683 0.61371 0.64680 0.67590

0.53414 0.54986 0.56218 0.57103 0.57637

0.45366 0.36633 0.27674 0.18545 0.09301

0.58819 0.60549 0.61906 0.62882 0.63469

0.46641 0.37663 0.28452 0.19066 0.09563

0.64371 0.66265 0.67750 0.68817 0.69460

0.48033 0.38787 0.29301 0.19635 0.09848

0.70084 0.72146 0.73763 0.74925 0.75625

0.63665 0.63469 0.62882 0.61906 0.36633 0.60549

0.00 0.09563 0.19066 0.28452 0.37663

0.69675 0.69460 0.68817 0.67750 0.66265

0.00

0-75858

0.45366 0.53819 0.61940 0.69680 0.76990

0.58819 0.56726 0.54284 0.51506 0.48412

0.46641 0.55332 0.63682 0.71639 0.79154

0.64371 0.62081 0.59408 0.56368 0.52981

0.48033 0.70084 0.56984 0.67590 0.65582 0.64680

0.40882 0.37548 0-33983 0.30208 0.26248

0.83825 0.90144 0.95906 1.01078 1.05626

0.45018 0.41347 0.37422 0.33265 0.28904

0.86182 0.92678 0.98602 1.0391g 1.08595

0.49268 0.45250 0.40954 0.36405 0.31632

0.88754 0.53640 0.95444 0.49266

1.06717 1.09837 1.12319 1.14088 1.15154

0.22125 0.17866 0.13497 0.09044 0.04536

1.09523 1.12744 I.I527I 1.17087 I.I8I8I

0.24364 0.19674 0.14862 0.09959 0.04995

1.12602 1.15912 1.18512 1.20379 1.21504

0.26663 0.21531 0.16265 0.10900 0.05467

1.15962 1.19374 1.22049 1.23972 1.25130

0.29030 0.23442 0.17709 0.11867 0.05952

1.15510

0.00

1.18547

0.00

1 . 2 1 8 7 9 O.OO

1.25517

0.00

0.13497

Note.

Negative quantities are in heavy type.

Examples.

cosh (0.6 + i 0.95) cosh (0.6 + i 1.05)

[60]

0.09301 + i 0.63469. —0.09301 4" ì 0.63469.

0.09848 0.75625 0.19635 0.74925 0.29301 0.73763 0.38787 0.72146

0.73777 0.81517

1.01545

0.61371 0.57683

0.44589

1.07021 0.39636 1 . 1 1 8 3 6 0-34439

TABLE

VIII.

HYPERBOLIC

COSINES,

c o s h (x

χ = 0.85

χ = 0.75 o 0.0s O.I 0.15 0.2

1.29468 1.29069 1.27874 1.25891 ι.23132

0.00 0.06452 0.12864 0.19197 0.25411

1-33743 I-3333I 1.32097 1.30048 1.27198

0.00 0.06968 0.13893 0.20732

0.2s °-3 0-3S 0.4 0.45

1.19613 I-IS35Ó 1.10390 1.04742 0.98449

0.31469 0-37332 0.42966 0.48335 0-53405

1.23563 1.19166 1-14035 1.08201 1.01700

0-5 0.5S 0.6 0.65 0.7

0.91548 0.84083 0.76100 0.67647 0.58777

0.58147 0.62529 0.66527 0.70114 0.73269

0.75 0.8 0.85 0.9 0.9S

0.40008 0.30224 0.20253 0.10158

1.0 1.05 1.1 1-iS 1.2

0.00 0.10158 0.20253 0.30224 0.40008

+

iq)

=

u

+

iv.

χ = 0.9

0.00 0.07502 0.14957 0.22320 0.29546

I-43309 1.42867 I.4I544 1-39349 1.36294

0.00 0.08054 0.16058 0.23964

0.27444

I-383S3 1.37927 1.36650 1.34530 1.31582

0.33986 0.40319 0.46403 0.52202 0.57678

1.27822 1.23274 1.17965 1.11930 1.05205

0.36589 0.43407 0.49957 0.56199 0.62095

1.32400 1.27689 1.22194 1.15939 1.08973

0.94571 0.86859 0.78613 0.69881 0.60718

0.62799 0.67532 0.71849 0.75724 0.79131

0.97830 0.89853 0.81322 0.72290 0.62811

0.67608 0.72704 0.77351 0.81522 0.85191

0.75972 0.78207 0.79960 0.81219 0.81978

0.51182 0.41329 0.31222 0.20922 0.10493

0.82050 0.84464 0.86357 0.87717 0.88537

0.52945 0.42753 0.32298 0.21643 0.10855

0.82232 0.81978 0.81219 0.79960 0.78207

0.00 0.10493 0.20922 0.31222 0.41329

0.88811 0.88537 0.87717 0.86357 0.84464

I-2S 1-3 1-35 1.4 1-45

0-49545 0.75972 0.58777 0.73269 0.67647 0.70114 0.76100 0.66527 0.84083 0.62529

0.51182 0.60718 0.69881 0.78613 0.86859

1-5 I-5S 1.6 1.6s 1-7

0.91548 0.98449 1.04742 1.10390 1.15356

0.58147 0.53405 0.48335 0.42966 0.37332

1-75 1.8 1.8s 1.9 I-9S

1.19613 1.23132 1.25891

2.0

CONTINUED

χ = 0.9s

0.31721

1.48623 1.48164 1.46793 1.44516 1.41348

0.00 0.08627 0.17200 0.25667 0.33976

0.39283 0.46603 0.53635 0.60337 0.66667

1.37309 1.32425 1.26722 1.20238 1.13013

0.42076 0.49916 0.57448 0.64626 0.71406

1.01334 0.93071 0.84235 0.74879 0.65061

0.72586 0.78057 0.83047 0.87525 0.91463

1.05092 0.96523 0.87358 0.77655 0.67473

0.77745 0.83605 0.88950 0.93747 0.97965

0.88334 0.90932 0.92970 0.94435 0.95317

0.54842 0.44285 0.33455 0.22418 0.11244

0.94838 0.97628 0.99816 1.01388 1.02335

0.56875 0.45927 0.34695 0.23250 0.11661

1.01579 1.04567 1.06911 1-08595 1.09610

0.00 0.10855 0.21643 0.32298 0.42753

0.95612 0.95317 0.94435 0.92970 0.90932

0.00 1.02652 0.11244 1.02335 0.22418 1.01388

0.00 0.11661 0.23250 0.34695

1.09948 1.09610 1.08595 1.06911

0.45927

1.04567

0.82050 0.79131 0.75724 0.71849 0.67532

0.52945 0.62S11 0.72290 0.81322 0.89853

0.88334 0.85191 0.81522 0.77351 0.72704

0.54842 0.65061 0.74879 0.84235 0.93071

0.94838 0.91463 0-87525 0.83047 0.78057

0.56875 0.67473

0.94571 1.01700 1.08201 1.14035 1.19166

0.62799 0.57678 0.52202 0.46403 0.40319

0.97830 1.05205 1.11930 1.17965 1.23274

0.67608 0.62095 0.56199 0.49957 0.43407

I-OI334 0.72586 1.08973 0.66667 115939 0.60337 Ι.22194 0.53635 1.27689 0.46603

1.05092 1.13013 1.20238 1.26722 1.32425

1.27874 1.29069

0.31469 0.25411 0.19197 0.12864 0.06452

1.23563 1.27198 1.30048 1.32097 1.33331

0.33986 0.27444 0.20732 0.13893 0.06968

1.27822 1.31582 1-34530 1.36650 1.37927

0.36589 0.29546 0.22320 0.14957 0.07502

1.32400 1.36294 1-39349 1.41544 1.42867

0.39283 0.31721 0.23964 0.16058 0.08054

137309 0.42076 1.41348 0.33976 1.44516 0.25667 1.46793 0.17200 1.48164 0.08627

1.29468

0.00

1.33743 0.00

1.38353

0.00

I.43309

0.00

1.48623 0.00

0.49545

0.33455

0.99816

0.44285 0.97628

Note.

Negative quantities are in heavy type.

Examples,

cosh (0.9 + i 1.0) = 0 + 11.02652. cosh (0.9 -+- i 1.10) = —0.22418 + i 1.01388.

[61]

1.01579 0.97965 0.7765S 0.93747 0.87358 0.88950 0.96523 0.83605 0.77745 0.71406 0.64626 0.57448 0.49916

TABLE V I I I .

HYPERBOLIC

COSINES,

cosh (x + ig) = u + iv.

x = ι .05

χ =

1.60379 1.59885 1-58405 1-55948 I-52530

0.00 0.09838 0.19615 0.29271 0.38746

1.66852 1.66337 1.64798 1.62242 1.58685

0.47983 0.56924 0-65514 0.73700 0.81432

I.54I5I 0.51113 1.48666 0.60637 1.42265 0.69787 1.34986 0.78508 1.26875 0.86743

0.83099 0.89363 0.95076 1.00202 1.04711

1.48171 1.42899 1.36746 1.29750 I-2I953 ι-13405 1.04158 0.94269 0.83798 0.72811

0.88661

1.06909 1.11719

1.17982 1.08362 0.98073 0.87180 0.75749

1.08574 1.11768 1.14273 1.16073 1.17158

0.61375 0.49560 0.37440 0.25089 0.12583

1.15841 1.19249 I.2I92I 1.23842 1.24999

1-°S 1.1 i-iS 1.2

0.00 1.17520 0.I2I07 1 . 1 7 1 5 8 0.24139 1 . 1 6 0 7 3 0.36023 1.14273 0.47684 1.11768

0.00 0.12583 0.25089 0.37440 0.49560

I-2S 1-3 I-3S 1.4 1-45

0.59051 0.70055 0.80626 0.90700 1.00215

1.08574 I.047II 1.00202 0.95076 0.89363

1-5 1-55 1.6 1.6s 1-7

I.09II2 I-I7337 1.24838 1.31569 1.37490

I-7S 1.8 ι .85 1.9 I-9S

1.42562 1.46756 1.50045 1.52408 1.53832

2.0

1.54308 0.00

o 0.05 O.I o-iS 0.2

1-54308 1-53832 1.52408 1-50045 1.46756

0.25 0-3 0-3S 0.4 0-4S o-S °·55 0.6 o.6s 0.7

1.42562 1.37490 1-31569 1.24838 I-I7337 1.09112 1.00215 0.90700 0.80626 0.70055

0.75 0.8 0.85 0.9 0-9S

0-5905I 0.47684 0.36023 0.24139 0.12107

1.0

0.00 0.09221 0.18384

0.00 0.10479 0.20894 0.31180 0.41274

CONTINUED

1.15

I-7374I 0 . 0 0 1.73206 0.11147 1.71602 0.22226 1.68941 0.33167 1.65238 0.43904

1.81066 1.80507 1.78836 1.76063 1.72204

0.00 0.11843 0.23613 0.35238 0.46645

1.60517 1.54805 1.48139 1.40560

0.54371 0.64502 0.74235 0.83511 1-32114 0 . 9 2 2 7 2

1.67283 1.61331 I-54384 1.46485 1.37684

0.57765 0.68528 0.78869 0.88724 0.98032

0.94445 1.01564 1.08056 1.13883 1.19007

1.22854 1.12836 1.02123 0.90780 0.78877

1.00464 1.08037 I-14943 1.21141 1.26592

1.28033 I-I7593 1.06428 0.94607 0.82202

1-06735 1.14781 1.22118 1.28703 1-34494

0.63852 0.51560 0.38951 0.26101 O.I 3091

1.23398 1.27028 1.29875 1.31920 I -33 I 53

0.66488 0.53689 0.40559 0.27179 0.13632

1.31263 I-35I24 1.38152 1.40329 1.41640

0.69291 0.55952 0.42269 0.28325 0.14206

1-39456 1-43558 1.46776 1.49088 1.50481

1.25386 1.24999 1.23842 I.2I92I 1.19249

0.00 0.I309I 0.26101 0.38951 0.51560

I-33S65 I-33I53 1.31920 1.29875 1.27028

0.00 0.13632 0.27179 0.40559 0.53689

1.42078 1.41640 1.40329 1.38152 1-35124

0.00 0.14206 0.28325 0.42269 0.55952

1.50946 1.50481 ι .49088 1.46776 1-43558

0.6137s 0.72811 0.83798 0.94269 1.04158

1.15841 I.II7I9 1.06909 I.OI439 0.95344

0.63852 1 . 2 3 3 9 8 1.19007 0.75749 0.87180 1 . 1 3 8 8 3 0.98073 1 . 0 8 0 5 6 1.08362 1 . 0 1 5 6 4

0.66488 0.78877 0.90780 1.02123 1.12836

1.31263 1.26592 1.21141 1.14943 1.08037

0.69291 0.82202 0.94607 1.06428 I-I7593

1-39456 1-34494 1.28703 1.22118 1.14781

0.83099 0.76323 0.69077 0.61404 0.53353

1.13405 I.2I953 1.29750 1.36746 1.42899

0.88661 0.81432 0.73700 0.65514 0.56924

1.17982 1.26875 1.34986 1.42265 1.48666

0.94445 0.86743 0.78508 0.69787 0.60637

1.22854 1.32114 1.40560 1.48139 1.54805

1.00464 0.92272 0.83511 0.74235 0.64502

1.28033 1.37684 1.46485 1.54384 1.61331

1.06735 0.98032 0.88724 0.78869 0.68528

0.44973 0.36316 0.27435 0.18384 0.09221

1.48171 1.52530 1.55948 1.58405 1.59885

0.47983 0.38746 0.29271 0.19615 0.09838

1.54151 1.58685 1.62242 1.64798 1.66337

0.51113 0.41274 0.31180 0.20894 0.10479

1.60517 1.65238 1.68941 1.71602 1.73206

0.54371 0.43904 0.33167 0.22226 0.11147

1.67283 1.72204 1.76063 1.78836 1.80507

0.57765 0.46645 0.35238 0.23613 0.11843

0.27435

0.36316 0.44973

0-53353 0.61404 0.69077 0.76323

0.95344 1.01439

1.60379 0.00

Note. Examples,

1.66852 0 . 0 0

I-7374I 0 . 0 0

Negative quantities are in heavy type. cosh (1.2 + io) = 1.81066 + i o . cosh (1.1 + i i.i) = —0.26101 + i 1.31920.

[62]

1.81066 0 . 0 0

TABLE V I I I .

HYPERBOLIC χ

χ = 1.25

COSINES,

- 1.3

cosh (x + iq) = u + in.

χ = 1-35

X = 1.4

CONTINUED

χ= ms

o 0.0s O.I o-i 5 0.2

1.88842 1.88260 1.86517 1.83624 1.79600

0.00 0.12569 0.25060 0-3739Ó 0.49502

I.9709I 1.96484 1.94665 1.91646 1.87445

0.00 1-13325 0.26569 0.39648 0.52483

2.05833 2.05200 2.03299 2.00146 1-95759

0.00 0.14116 0.28144 0.41999 0.55595

2.15090 2.14427 2.12442 2.09147 2.04562

0.00 0.14941 0.29790 0.44455 0.58846

2.24884 2.24191 2.22115 2.18671 2.13878

0.00 0.15804 0:31510 0.47022 0.62244

0.25 °·3 °·35 °·4 0-4S

1.74467 1.68260 1.61014 1.52777 1-43597

0.61303 0.72726 0.83700 0.94158 1.04036

1.82089 1.75610 1.68048 I-5945I 1.49870

0.64994 0.77105 0.88740 0.99829 1.10301

1.90165 1-83399 1-75502 1.66523 1-56517

0.6

0.81677 0.94002 1.05748 1.16842

1.98717 ι .91646 1-83394 1.74012 1-63556

0.72875 0.86454 0.99500 1-11932 1.23674

2.07766 2.00373 I-9I745 1-81935 ι.71007

0.77083 0.91446 1-05245 1.18396 1.30817

o-S ο-55 0.6 0.65 0.7

I-33S32 1.22643 1.10999 0.98670 0.85733

1-I3 2 73 1.21811 1.29598 1.36586 1.42732

I-39365 1.28001 1.15848 1.02980 0.89478

1.20094 1.29146 1.37402 1.44811 1-51327

1-45546 1.33678 1.20986 1.07548 0.93446

1.27215 1.36804 I-45S50 I-53398 1.60300

1.52092 1.39690 1.26427 1.12384 0.97649

1-34655 1.44804 1.54061 1.62369 1.69675

I-590I7 1.46051 1.32184 1.17502 1.02095

142431 1.53166 1.62958 1-71745 1-79473

°-75 0.8 0.85 0.9 0-9S

0.72267 0.58355 0.44084 0.29541 0.14816

1.47998 1-52352 1-55766 1.58220 1-59698

0.75424 0.60905 0.46010 0.30832 0.15464

1.56910 1.61526 1.65146 1.67747 I-693I5

0.78769 0.63606 0.48051 0.32199 0.16150

1.66215 1.71104 1.74938 1.77694 1-79354

0.82311 0.66466 0.50212 0.33647 0.16876

1-75934 1.81110 1.85169 1.88086 1.89843

0.86060 0.69493 0.52498 0.35180 0.17644

1.86094 1.91569 1.95862 1.98947 2.00806

1.0 1.05 1.1 1-iS 1.2

0.00 0.14816 0.29541 0.44084

1.60192 1.59698 1.58220 1-55766 0.5835s 1-52352

0.00 0.15464 0.30832 0.46010 0.60905

1.69838 1-69315 1.67747 1.65146 1.61526

0.00 0.16150 0.32199 0.48051 0.63606

1.79909 1-79354 1.77694 1.74938 1.71104

0.00 0.16876 0.33647 0.50212 0.66466

1.90430 1.89843 1.88086 1.85169 1.81110

0.00 0.17644 0.35180 0.52498 0.69493

2.01427 2.00806 1.98947 1.95862 1.91569

I-2S !-3 1-35 1.4 1-45

0.72267 0.85733 0.98670 I.I 0999 1.22643

1.47998 1.42732 1.36586 1.29598 ι.21811

0.75424 0.89478 1.02980 1.15848 1.28001

1.56910 1-51327 1.44811 1.37402 1.29146

0.78769 0.93446 1.07548 1.20986 1.33678

1.66215 1.60300 1-53398 I-45550 1.36804

0.82311 0.97649 1.12384 1.26427 1.39690

1-75934 1.69675 1.62369 1.54061 1.44804

0.86060 1.02095 1.17502 1.32184 1.46051

1.86094 1-79473 I-7I745 1.62958 ι·53ι66

I -S

i-SS 1.6 1.6s 1-7

1.33532 1-43597 1-52777 1.61014 1.68260

1-13273 1.04036 0.94158 0.83700 0.72726

1.39365 1.20094 1.49870 1.10301 1 . 5 9 4 5 1 0.99829 1.68048 0.88740 1.75610 0.77105

1.45546 1-56517 1.66523 1-75502 1.83399

ι.27215 1.16842 1.05748 0.94002 0.81677

1.52092 1-63556 1.74012 1.83394 1.91646

I-3465S 1.23674 1-11932 0.99500 0.86454

I.59017 I-4243I I.71007 I.30817 I.81935 I.18396 I.91745 I.05245 2.OO373 O.91446

I-7S 1.8 1.8s 1.9 I-9S

1.74467 1.79600 1.83624 1.86517 1.88260

0.61303 0.49502 0.37396 0.25060 0.12569

1.82089 1.87445 1.91646 1.94665 1.96484

0.64994 0.52483 0.39648 0.26569 0.13325

1.90165 1-95759 2.00146 2.03299 2.05200

0.68848 0.55595 0.41999 0.28144 0.14116

1.98717 2.04562 2.09147 2.12442 2.14427

0.72875 0.58846 0.44455 0.29790 0.14941

2.07766 2.13878 2.18671 2.22115 2.2419I

2.0

I.88842 O.OO

I.9709I O.OO

2.05833 O.OO

2.15090 O.OO

Note.

Negative quantities are in heavy type.

Examples,

cosh (1.4 + i 1.9) = —2.12442 + i 0.29790. cosh (1.4 + i 1.4) = —1.26427 + i 1.54061.

[63]

0.77083 O.62244 0.47022 O.3151O 0.15804

2.24884 O.OO

TABLE V I I I .

HYPERBOLIC

χ = 1.5 0 0.0s O.I

2.35241 2.34516 2-3234S 2.28742 2.23727

COSINES,

« = 1-55

0.00 0.16706

2.46186 0.00 2.45427 0.17650

0-33309 0.49707 0.65798

2.43155 0-3SI92 2·39384 0.52516 2-34137 0.69517

0.81484 0.96667 1.11255 1.25156 1.38286

2.27446 2-19353 2.09908 1.99169 1.87201

0.86089 1.02130

0-4S

2.17334 2.09601 2.00576 1-90314 1.78879

0.5 0-55 0.6 0.65 0.7

1.66341 I-S2777 1.38271 ι.22913 1.06797

1-50563 1.61912 1.72263 1.81551 1.89720

1.74080 1.59885 1.44704 1.28632 1.11766

0.75 0.8 0.85 0.9 °-95

0.90023 0.72693 0.54916 0.36800 0.18457

1.96720 2.02507 2.07045 2.10307 2.12272

1.0 1.05 1.1

0.00 0.18457 0.36800 0.54916 0.72693 0.90023 1.06797 1.22913 1.38271

o.iS 0.2 0.25 0-3 °-35 0.4

1-IS

1.2

1-25 1-3 1-35 1.4 1 -45 I-S

i-SS 1.6 1.6S 1-7

1-52777

2.49404 0.95957

1-17542 1.32229 1.46101

2.40529 2.30173 2.18396 2.05274

1-13837 I.3IOI5 1.47385 1.62847

1.59071 1.71062 1.81997 1.91811 2.00442

1.82254 1.67978 1-67393 1.80640 I.5I500 1.92187 1.34672 2.02550 I.I70I5 2.11664

1.90885 1.75320 1.58674 I.4I050 1.22556

1.77305 1.90669 2.02858 2.13797 2.23417

1.99992 1.83684 1.66244 1.47779 1.28403

1.87074 2.01175 2.14036

0.94211 2.07837 0.76076 2.13951 0.57471 2.18745 0.38512 2.22191 0.19316 2.24268

0.98636 0.79648 0.60170 0.40320 0.20223

2.19473 2.25930

1.03306 0.83420 0.63019 0.42230 0.21180

2.31660 2.38474 2.43818 2.47659 2.49973

1.0823s 0.87400 0.66026 0.44245 0.22I9I

2.44424 2.51614

2.12928 2.12272 2.10307 2.07045 2.02507

0.00 0.19316 0.38512 0.57471 0.76076

2.24961 2.24268 2.22191 2.18745 2.13951

0.00 0.20223 0.40320 0.60170

2.37557 2.36824 2.34632 2.30993

0,00 0.21180 0.42230 0.63019 0.83420

2.50747 2.49973 2.47659 2.43818 2.38474

0.00 0.22I9I

1.96720 1.89720

0.94211 1.11766 1.28632 1.44704 1.59885

2.07837 2.00442 I.9I8II 1.81997 1.71062

0.98636 I.I70I5 1.34672 I.5I500

1.03306 1.22556 I.4I050 1.58674 1.75320

2.31660 2.23417 2.13797 2.02858 1.90669

1.08235 2.44424 1.28403 2.35727 1.47779 2.25577 1.66244 2.14036 1.83684 2.01175

1.74080 1.87201 1.99169 2.09908

I.5907I 1.46101 1.32229 1.17542

1.82254 1.95992 2.08522

1.90885 1.77305 2.05274 1.62847

1.99992 1.87074 2.15067 1.71820 2.28816 1-55506 2 . 4 I I 5 4 1.38234 2.52005 1.20109

1-81551 1.72263 1.61912

1.66341 1-50563 1.78879 1.38286 ι.90314 1.25156 2.00576 I - I I 2 5 5 2.09601 0.96667

2.19353

I.02I30 0.86089 0.69517 0.52516 0.35192 0.17650

2.0

2.35241 0.00

2.46186 0.00

2-17334 0.81484 2.23727 0.65798 2.28742 0.49707

Note. Examples,

2.30993 2.34632. 2.36824

0.79648 2.25930

1-67393

2.19765 2.29654

2 . 1 9 4 7 3

2.11664 2.02550 1.92187 1.80640 1.67978 1.54281 1.39632 1.24123 1.07848

2.38127 0.90909 2.45I3I 0.73409 2.50625 0.55456

0.00 0.19673 0.39225 0.58536 0.77485

χ = 1.7

2.38127 0.90909 2.29654 1.07848 2.19765 1.24123 2.08522 1.39632 1.95992 1.54281

2.32345 0.33309 2.34516 0.16706

1.9

χ = 1.65 2.69952 2.69121 2.66629 2.62494 2.56740

I-9S

1.8s

1.6

CONTINUED

2.57746 0.00 2.56952 0.18639 2.54573 0.37162 2.50625 0.55456 2.45I3I 0.73409

2.27446 2.34137 2.39384 2.43155 2.45427

I-7S 1.8

χ -

c o s h ( * + iq) = u + iv.

2.18396 1.47385 2.30173 I.3IOI5 2.40529 1.13837

2.54573 0.37162 2.56952 0.18639

2.49404 2.56740 2.62494 2.66629 2.69121

2.57746 0.00

2.69952 0.00

0.95957 0.77485 0-58536 0.39225 0.19673

N e g a t i v e quantities are in h e a v y t y p e . cosh ( 1 . 6 + i o . 4 ) = 2.08522 + i 1 . 3 9 6 3 2 . cosh ( 1 . 7 + i 1^2) = — 0 . 8 7 4 0 0 + i 2.51614.

[64]

2.82832 2.81960 2.79350 2.75017 2.68989

0.00 0.20757 0.41387 0.61761 0.81754

2.61302 I.OI244 2.52005 1.20109 1-38234 2.28816 1-55506 2.15067 1.71820 2 . 4 1 1 5 4

2.25577 2.35727

2-57253 2.61306 2.63747 2.64563 2.63747 2.61306

0.44245 0.66026 2.57253 0.87400 2.51614

2.61302 2.68989 2.75017 2.79350 2.81960

I.OI244 0.81754 0.61761 0.41387 0.20757

2.82832 0.00

TABLE V I I I .

HYPERBOLIC

χ = ι.7s

COSINES,

c o s h (x + iq)

χ = 1.85

χ = 1.8

= u +

iv.

CONTINUED

χ = i.g

ι-95

o 0.05 O.I 0.15 0.2

2.96419 2-95505 2.92769 2.88229 2.81911

0.00 0.21893 0.43652 0.65141 0.86229

3.10747 3.09789 3.06921 3.02161 2-9S538

0.00 0.23084 0.46026 0.68684 0.90918

3-25853 3-24848 3.21841 3.16850 3.09904

0.24332 0.48627 0.72398 0.95835

3.41773 3.40719 3-3756S 3-32330 3-25045

0.00 0.25642 0.51125 0.76294 1.00992

3-58548 3-57443 3-54134 3.48641 3.41000

0.27015 0.53864 0.80380 ι .06401

0.25 °-3 0-3S 0.4 0.45

2·73855 2.64111 2-52739 2.39808 2-25399

1.06784 1.26682 1-45799 1.64016 1.81223

2.87093 2.76878 2.64956 2.51400 2.36294

1.12592 1-33572 1-53728 1.72937 1.91079

3.01049 2.90337 2.77835 2.63620 2.47780

I.18681 1.40796 1.62042 1.82289 2.01413

3-15757 3.04522 2.91409 2.76501 2.59887

1.25067 1.48372 1.90761 1.92098 2.12250

3-31255 3.19469 3-057I3 2.90071 2.72642

1.31766 1-56318 1.79907 2.02387 2.23618

o-5 0-5S 0.6 0.65 0.7

2.09600 1.92509 1-74231 1.54878 ι - 3 4 5 71

1.97312 2.12185 2.25749 2.37922 2.48627

2.19731 2.01814 1.82653 1.62365 1.41076

2.08043 2.23725 2.38027 2.50862 2.62149

2.30413 2.11624 I-9I53I 1.70258 1-47934

2.19294 2.35824 2.50900 2.64429 2.76327

2.41670 2.21964 2.00889 1.78576 1-55162

2.31094 2.48513 2.64400 2.78657 2.91196

2.53532 2.32858 2.10749 1.87341 1.62777

2.43471 2.61823 2.78561 2.93582 3.06792

0-7S 0.8 0.85 0.9 0-9S

I-I3435 0.91599 0.69198 0.46370 0.23257

2.57800 2.65384 2.71331 2.75606 2.78181

1.18918 0.96026 0.72542 0.48612 0.24381

2.71821 2.79817 2.86088 2.90595 2.93310

1.24698 1.00694 0.76069 0.50975 0.25566

2.86522 2.94950 3.01560 3.06311 3-09I73

1.30791 1.05614 0.79785 0-53465 0.26815

3-OI939 3.10821 3-I7787 3-22793 3.25809

1.37210 1.10797 0.83701 0.56089 0.28131

3.18m 3.27468 3-34807 3.40081 3-43259

1.0 1.05 1.1 1-iS 1.2

0.00 0.23257 0.46370 0.69198 0.91599

2.79041 2.78181 2.75606 2.71331 2.65384

0.00 2.94217 0.24381 2.93310 0.48612 2.90595 0.72542 2.86088 0.96026 2.79817

0.00 0.25566 0.50975 0.76069 1.00694

3.10129 2.09173 3.06311 3.01560 2.94950

0.00 0.26815 0.53465 0.79785 1.05614

3.26816 3.25809 3.22793 3-I7787 3.10821

0.00 0.28131 0.56089 0.83701 1.10797

3-44321 3-43259 3.40081 3-34807 3.27468

1-25 1-3 1-35 1.4 1-45

1.13435 1.34571 1.54878 1.74231 1.92509

2.57800 2.48627 2.37922 2.12185

1.18918 1.41076 1.62365 1.82653 2.01814

2.71821 2.62149 2.50862 2.38027 2.23725

1.24698 1-47934 1.70258 1-9I53I 2.11624

2.86522 2.76327 2.64429 2.50900 2.35824

1.30791 1.55162 1.78576 2.00889 2.21964

3-OI939 2.91196 2.78657 2.64400 2-48513

1.37210 1.62777 1.87341 2.10749 2.32858

3.18111 3.06792 2.93582 2.78561 2.61823

1-5 1-55 1.6 1.6S 1-7

2.09600 2.25399 2.39808 2.S2739 2.64111

1.97312 1.81223 1.64016 1-45799 1.26682

2-I973I 2.36294 2.51400 2.64956 2.76878

2.08043 1.91079 1-72937 1-53728 1-33572

2-30413 2.47780 2.63620 2.7783S 2.90337

2.19294 2-01413 1.82289 1.62042 1.40796

2.41670 2.59887 2.76501 2.91409 3.04522

2.31094 2.12250 1.92098 1.90761 1.48372

2.53532 2.72642 2.90071 3.05713 3.19469

2.43471 2.23618 2.02387 1.79907 1.56318

1-75 1.8 1.8S 1.9 I-9S

2·73855 2.81911 2.88229 2.92769 2.95505

1.06784 0.86229 0.65141 0.43652 0.21893

2.87093 2.95538 3.02161 3.06921 3.09789

1.12592 0.90918 0.68684 0.46026 0.23084

3.01049 3.09904 3.16850 3.21841 3.24848

1.18681 0.95835 0.72398 0.48627 0.24332

3-15757 3.25045 3.32330 3.37565 3.40719

1.25067 1.00992 0.76294 0.51125 0.25642

3-31255 3.41000 3.48641 3.54134 3-57443

1.31766 1.06401 0.80380 0.53864 0.27015

2.0

2.96419 0.00

2.25749

3.10747 0.00

3-25853 0.00

3.41773 0.00

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

cosh (1.8 + i 0.2) = 2.95538 + i 0.90918. cosh (1.8 + i 2¿)

= —3.10747 + i o .

[65]

3.58548 0.00

TABLE VIII.

HYPERBOLIC

COSINES,

cosh (x + ig) = u+

χ = 2.05

iv.

CONTINUED

χ = 2.is

o 0.0s O.I 0.15 0.2

3.76220 3-75°59 3-71587 3-65825 3.57806

0.00 0.28456 0-56737 0.846Ò7 1.12076

3-94832 3-936I5 3.89971 3-83923 3-75507

0.00 0.29968 0.59751 0.89166 1.18032

4-I443I 4-I3I54 4.09329 4.02981 3.94148

0.31555 0.62916 0.93888 1.24282

4-35o67 4-33726 4-297" 4.23046 4-13773

0.33221 0.66237 0.98845 1.30844

4-56791 4-55382 4.51167 4.44170 4-34433

0.34970 0.69724 1.04049 I-3773 2

0.25 °-3 °-35 0.4 0-4S

3-47581 3-35214 3.20780 3.04368 2.86080

1.38794 1.64656 1.89503 2.13182 2-35546

3-64777 3-5I798 3-36649 3.19426 3.00232

1.46169 I-7340S 1-99573 2.24509 2.48062

3-82885 3.69261 3-5336I 3-35283 3-I5I37

I-539IO 1.82589 2.10142 2.36399 2.61199

4.01950 3-87647 3-70956 3-51977 3-30828

1.62035 1.92228 2.21236 2.48879 2.74988

4.22019 4.07003 3.89478 3-69552 3-47347

1.70566 2.02349 2.32883 2.61982 2.89466

0.5 0-55 0.6 0.65 0.7

2.66027 2-44335 2.21136 1.96574 1.70800

2.56458 2.75789 2.93420 3.09241 3-23156

2.79188 2.56423 2.3207Ò 2.06299 1.79250

2.70085 2.90443 3.09011 3-23673 3-40327

2.93048 2.69152 2-43597 2.16540 i.r ~

2.8 3-05825 3-25376 3.42920 3-58351

3-07639 2-82553 2.55726 2.27322 1.97516

2.99402 3.21970 3-42553 3.61024 3.77269

3-23000 2.96661 2.68495 2.38672 2.07379

3-l5l6S 3.38921 3-60588 3.80031 3-97I3I

0.7S 0.8 0.85 0.9 0.9s

1-43973 1.16258 0.87827 0.58854 0.29518

3-35078 3-44935 3.52666 3.58221 3-61568

1.51096 ι.22010 0.92172 0.61765 0.30978

3-52883 3.63264 3-71404 3-772S6 3-80781

1.58596 1.28066 0.96747 0.64831 0.32516

3-7I57I 3.82501 3.91074 3-97235 4.00946

1.66493 1-34443 1.01564 0.68059 0.3413S

3.91188 4.02695 4.11720 4.18206 4-22113

1.74806 1.41156 1.06636 0.71458 0.35839

4-11783 4.23896 4-33396 4.40223 4-44337

1.0 I.os 1.1 1.1 5 1.2

0.00 0.29518 0.58834 0.87827 1.16258

3.62686 3-61568 3.58221 3.52666 3-44935

0.00 0.30978 0.61765 0.92172 1.22010

3-81958 3.80781 3-77256 3.71404 3.63264

0.00 0.32516 0.64831 0.96747 1.28066

4.02186 4.00946 3-9723S 3-9I074 3.82501

0.00 0.34135 0.68059 1.01564 1-34443

4.23419 4-22113 4.18206 4.1172a 4-02695

0.00 0.35839 0.71458 1.06636 1.41156

4-457" 4-44337 4.40223 4-33396 4.23896

1-25 ΐ·3 1-35 1.4 ι-45

1-43973 1.70800 I.96S74 2.21136 2.44335

3-35078 3-23156 3.09241 2.93420 2.75789

1.51096 1.79250 2.06299 2.32076 2.56423

3-52883 3-40327 3-23673 3.09011 2.90443

1.58596 1.88148 2.16540 2-43597 2.69152

3-7I57I 3-583SI 3.42920 3-25376 3-05825

1.66493 1.97516 2.27322 2.55726 2.82553

3.91188 3.77269 3.61024 3-42553 3.21970

1.74806 4-11783 3-97I3I 2.38672 3.80031 2.68495 3.60588 2.96661 3-38921

1-5 1-55 1.6 1.6s 1.7

2.66027 2.86080 3.04368 3.20780 3.35214

2.56458 2-35546 2.13182 1.89503 1.64656

2.79188 3.00232 3.19426 3-36649 3-51798

2.70085 2.48062 2.24509 1-99573 I-7340S

2.93048 3.I5I37 3-35283 3·5336ι 3.69261

2.84389 2.61199 2-36399 2.10142 1.82589

3.07639 3.30828 3.51977 3.70956 3.87647

2.99402 2.74988 2.48879 2.21236 1.92228

3.23000 3-47347 3-69552 3.89478 4.07003

3-15165 2.89466 2.61982 2.32883 2.02349

1-75 1.8 1.85 1.9 1-95

3.47S81 3.57806 3.65825 3.71587 3.75059

1-38794 1.12076 0.84667 0-56737 0.28456

3-64777 1.46169 3 . 7 5 5 0 7 1.18032 3.83923 0.89166 3.89971 0.59751 3.93615 0.29968

3.82885 3.94148 4.02981 4.09329 4.13154

I-539IO 1.24282 0.93888 0.62916 0.31555

4.01950 4-13773 4.23046 4.29711 4·33726

1.62035 1.30844 0.98845 0.66237 0.33221

4.22019 4-34433 4.44170 4.51167 4·55382

1.70566 1-37732 1.04049 0.69724 0.34970

2.0

3.76220 0.00

3.94832 0.00

4.14431 0.00

4.35067 0.00

Note.

Negative quantities are in heavy type.

Examples,

cosh (2.1 + io.8) = 1.28066 -+- i 3.82501. cosh (2.2 + i 1.25) = —1.74806 + i 4.11783.

[66]

O.OO

2.07379

4.56791 0.00

TABLE V I I I .

HYPERBOLIC

χ = 2.25

COSINES,

χ = 2.3

c o s h (x + iq)

= u + iv.

χ = 2.4

χ = 2.35

CONTINUED

χ = 2.4s

o 0.05 O.I °-IS 0.2

4-796S7 4.78178 4-737SI 4.66404 4.56181

0.00 0.36807 0.73386 I-09SI3 1.44965

5.03722 5.02169 4-97521 4.89805 4.79058

0.00 0.38735 0.77231 1-15251 1.52560

5.29047 5.27416 5.22533 5.14429 5.03153

0.00 0.40760 0.81269 ι.21277 1.60537

5·55695 5-5398I S-48853 5-4034I 5.28496

0.00 0.42888 0.85511 1.27607 1.68916

5-83732 5-81933 5-76545 5-67603 5-55I62

0.25 °-3 °-35 0.4 0-4S

4-43145 4-27377 4.08975 3-88050 3-64734

l-79523 2.12975 2.45113 2.75740 3.04677

4-6S378 4.48820 4.29494 4.07520 3-83034

1.88930 2.24134 2.57956 2.90188 3.20630

4.88776 4.71384 4.51087 4.28008 4.02290

2-35853 2.71443 3-05360 3-37395

5-13394 4.95127 4-738o8 4-49567 4-22554

2.09184 2.48162 2.85610 3-21297 3-5S003

o-S o-5S 0.6 0.65 0.7

3.39168 3-11512 2.81935 2.50620 2.17760

3-3I7I6 3-56719 3-79523 3-99987 4.17986

3-56I85 3-27141 2.96081 2.63194 2.28685

3.49096 3-754IO 3-99409 4.20946 4.39887

3-74093 3-43588 3.10966 2.76426 2.40182

3-6735I 3-95038 4.20292 4-42955 4.62887

3-92935 3.60894 3.26629 2.90349 2.52280

3.86521 4.15656 4.42228 4.66073 4.87045

5-39298 5.20109 4-97713 4.72249 4-43873 4.12761 3.79104 3-43I09 3-04999 2.65009

4.06659 4-3 7 3 1 1 4.65268 4-90356 5.12420

0.75 0.8 0.85 0.9 0-95

I-83557 1.48222 1.11974 0-75035 0.37633

4-33407 4.46157 4-56I55 4-6334I 4.67671

1.92766 I-55659 I.I7592 0.78799 0.39522

4.56116 4-69533 4.80056 4.87618 4.92174

2.02457 1.63485 1.23504 0.82761 0.41509

4.79965 4-94083 5.05156 5-13114 5.17909

2.12655 1.71719 1.29724 0.8Ö930 0-43599

5-05014 5.19869 5-3I52I S-39893 5-44938

2-23385 1.80383 1.36269 0.91316 0-45799

5-3I325 5-46955 5-S92I3 5.68022 5-73330

1.0 1-°S 1.1 I-IS 1.2

0.00 0.37633 0.75035 1.11974 1.48222

4.69117 4.67671 4.63341 4-56155 4.46157

0.00 0.39522 0.78799 1.17592 I-55659

4.93696 4.92174 4.87618 4.80056 4-69533

0.00 0.41509 0.82761 1.23504 1.63485

5-I95I° 5.17909 5.13114 5.05156 4-94083

0.00 0.43599 0.86930 1.29724 1.71719

5.46623 5-44938 5-39893 5-3I52I 5.19869

0.45799 0.91316 1.36269 1.80383

5-75I03 5-73330 5.68022 5-59213 5-46955

1-25 1-3 I-3S 1-4 I-4S

1-83557 2.17760 2.50620 2.81935 3.11512

4-33407 4.17986 3-99987 3-79523 3-567I9

1.92766 2.28685 2.63194 2.96081 3-27I4I

4.56116 4-39887 4.20946 3-99409 3-754IO

2.02457 2.40182 2.76426 3.10966 3.43588

4-79965 4.62887 4·42955 4.20292 3.95038

2.12655 2.52280 2.90349 3.26629 3.60894

5-05014 4.87045 4-66073 4.42228 4.15656

2.23385 2.65009 3.04999 3.43109 3.79104

5-3I325 5.12420 4.90356 4-65268 4-373"

ι-5 i-SS 1.6 1.6s 1-7

3.39168 3.64734 3.88050 4.08975 4.27377

3.31716 3-°4677 2.75740 2.45113 2.12975

3-56185 3.83034 4.07520 4.29494 4.48820

3.49096 3.20630 2.90188 2.57956 2.24134

3-74093 4.02290 4.28008 4.51087 4.71384

3-6735I 3.37395 3.05360 2.71443 2.35853

3.92935 4-22554 4-49567 4.73808 4.95127

3-86521 3-55003 3-21297 2.85610 2.48162

4.12761 443873 4.72249 4-97713 5.20109

4.0665g 3-73499 3-38036 3.00490 2.61091

I-7S 1.8 1.8S 1.9 i-95

4.43145 4.56181 4.66404 4-73751 4.78178

1-79523 1.44965 I-095I3 0.73386 0.36807

4.65378 4.79058 4.89805 4-97521 5.02169

1.88930 1.52560 1.15251 0.77231 0-38735

4.88776 1.9 5-03153 1-60537 5.14429 ι.21277 5-22533 0.81269 5.27416 0.40760

513394 5.28496 5.40341 548853 5.53981

2.09184 1.68916 1.27607 0.85511 0.42888

5.39298 •5.55162 5.67603 5.76545

2.20082 1.77716 I-3425S 0.8996^ 0.45122

2.0

4.79657 0.00

5.29047 0.00

5.55695

0.00

Note.

5.03722 0.00

N e g a t i v e quantities are in h e a v y type.

Examples. . cosh (2.4 + i 0.4) = 4.49567 + i 3.21297. cosh (2.4 + » ü > = - 3 · 9 2 9 3 5 + »3-86521

[67]

5.81933 5.83732

0.45122 0.89966 1-34255 1.77716 2.20082 2.61091 3.00490 3-38036 3-73499

0.00

T a b l e VIII.

HYPERBOLIC

χ = 2.5

COSINES,

s = 2.55

cosh (x + iq) = u+ χ =

Χ = 2.6

iv.

2.65

Continued Χ = 2·7

o 0.0s O.I 0.15 0.2

6.13229 6.11339 6.05680 5.96287 5-83215

0.00 0.47469 0.94646 1-41239 1.86961

6.44259 6.42273 6.36327 6.26458 6.12727

0.00 0.49935 0.99563 1.48577 1.96674

6.76901 6.74814 6.68567 6.58199 6.4377ο

ο.οο 0.52526 1.04729 1.56285 2.06878

7 - l i 234 7.09042 7.02478 6.91583 6.76424

0-55249 1.10156 1.64385 2.17600

7-47347 7·45°43 7.38146 7.26697 7.10769

0.00 0.58109 Ι-Ι5859 1.72896 2.28866

O.25 O.3 °-35 0.4 0-45

5-66550 S-46392 5.22864 4.96113 4.66304

2.31531 2.74674 3.16122 3-55622 3.92929

5·952ΐ8 5-74039 5-49321 5-2Ι2Ι7 4-

2-43559 2.88943 3·32545 3-74097 4-13342

6.25374 6.03123 5-77153 5.47624 5-Ι47Ι9

2.56196 3-03934 3-49799 3-93506 4-34788

6.57095 6-33714 6.06427 5-7540Ι 5.40827

2.69474 3-19686 3.67927 4-13900 4-57321

6.90458 6.65891 6.37218 6.04606 5.68287

2.83425 3-36237 3-86976 4·35329 4.80998

°-5 0-55

4-336I9 3.98260 3.60448 3-20411 2.78401

4.27814 4.60062 4.89472 5-I5865 5-39077

4-55560 4.18413 3-78686 3.36624 2.92488

4-50039 4.83961 5.14900 5.42664 5.67082

4.78641 4.39612 3.97872 3·5368Ο 3.07306

4-73389 5.09071 5·4ΐ6ΐ5 5.70819 5-96JOS

5.02919 4.61910 4.18053 3.71619 3.22894

4-97923 5-35454 5-69685 6.00403

5.23702 5-63Ι77 5-99179 6.31488

6.27419

5-28454 4.85363 4-39279 3.90488 3-39288

0-7S 0.8 0.85 0.9 0-95

2.34673 1.89498 1-43I55 0-95930 0.48113

5.58966 5-75408 5.88304 5-97572 6.03155

2.46547 1.99087 Ι·5°399 1.00784 0.50548

5.88004 6.05301 6.18866 6.28615 6.34489

2·59°39 2.09174 1.58019 1.05891 0.53109

6.18512 6.36706 6.50976 6.61231 6.67409

2.72178 2.19784 Ι.66Ο34 I.H2Ó2 °·558°3

6.50567 6.69705 6.84713 6.955°° 7-ΟΙ998

2.85997 2-3°943 1.74465 1.16911 0.58636

6.84249 7-04377 7.20163 7-3I50S 7·38343

1.0 i.os 1.1 1-iS 1.2

0.00 0.48113 0.95930 1.431 SS 1.89498

6.05020 6-03155 5-97572 5.88304 5-75408

0.00 0.50548 1.00784 1.50399 1.99087

6.36451 6.34489 6.28615 6.18866 6.0530Ι

0.53109 1.05891 1.58019 2.09174

6.69473 6.67409 6.61231 6.50976 6.36706

0.00 0.55803 1.11262 1.66034 2.19784

7.04169 7.01998 6.95500 6.84713 6.69705

0.00 0.58636 1.16911 1.74465 2.30943

1-25 1-3 1-35 1.4 i-45

2-34673 2.78401 3-20411 3.60448 3.98260

5-58966 5-39077 5-I586S 4.89472 4.60062

2.46547 2.92488 336624 3-78^86 4.18413

5.88004 5.67082 5.42664 5.14900 4-83961

2-59039 3.07306 3-53680 3-97872 4.39612

6.18512 5-96505 5.70819 5·4ΐ6ΐ5 5.09071

2.72178 3.22894 3.71619 4.18053 4.61910

6.50567 6.27419 6.00403 5.69685 5-35454

2.85997 3.39288 3.90488 4-39279 4.85363

7.40626 7·38343 7-3Ι508 7.20163 7-04377 6.84249 6-59903 6.31488 5-99179 5-63Ι77

i-S 1-55 1.6 1.6 S i-7

4-336I9 4.66304 4-96113 5.22864 5-46392

4.27814 3.92929 3-55622 3.16122 2.74674

4-55500 4.89898 5.2Ι2Ι7 5-49321 5-74039

4·5°°39 4-13342 3-74097 3-32545 2.88943

4.78641 5-Ι47Ι9 5.47624 5-77I53 6.03123

4-73389 4-34788 3·935°6 3-49799 3-03934

5.02919 5.40827 5.75401 6.06427 6.33714

4-97923 4-57321 4.13900 3.67927 3.19686

5.28454 5.68287 6.04606 6.37218 6.65891

5.23702 4.80998 4-35329 3-86976 3·36237

I-7S 1.8 1.85 1.9 ι-95

5.66550 583215 5.96287 6.05680 6.11339

2.31531 1.86961 1-41239 0.94646 0.47469

5-95218 6.12727 6.26458 6.36327 6.42273

2-43559 1.96674 1-48577 0.99563 0-49935

6.25374 6-43770 6.58199 6.68567 6.74814

2.56196 2.06878 1.56285 1.04729 0.52526

6.57095 6.76424 6.91583 7.02478 7.09042

2.69474 2.17600 1-64385 1.10156 0.55249

6.90458 7.10769 7.26697 7.38146 7-45043

2.83425 2.28866 1.72896 1-15859 0.58109

2.0

6.13229 Ο.ΟΟ

7.1x234

0.00

7.47347 0.00

0.6

o.6s 0.7

6.44259 °-°ο

6.76901 0.00

Note.

Negative quantities are in heavy type.

Examples,

cosh (2.7 + i i.oo) = o + i 7.40626. cosh (2.6 + i 1.2) = —2.09174 + 16.36706.

[68]

6-59903

TABLE V I I I .

HYPERBOLIC

χ -

χ = 2.75

0

7-85328 0.00

0.05

7.82907 0.61115 7·75659 1.21852 7.63629 1.81839 7.46891 2.40704

O.I 0.15 0.2 0.25

o-3 0-3S

7-25S48 2.98086 6.99732 3-53629 6.69602 4.06993

o-45 o-S o-SS

0.4

8.25273 8.22728 8.15112 8.02470 7.84881

COSINES,

2.80 0.00 0.64273 1.28150 I.91237 2.53144

c o s h ( * + ig)

χ -- 2.85 8.67281 8.64608 8.56604 8.43318 8.24834

0.00 0.67592 1.34768 2.01112 2.66217

7.62452 3.I349I

8.01263 3.29681 7-72754 3.91112

7-39479 4-50131

6-35344 4-57847 5-97168 5-05878

7.03661 4.28026 6.67660 4.81509 6.27542 5.32022

5 - 5 5 3 " 5-50790 2 5.10030 5·9 3θ7

5.83556 5-79256 5-35972 6.22918

7.35323 3-7I905

0.6 0.65 0.7

4.61604 6.30172 4-10333 6.64151 3-5653I 6.94036

4-85083 6.62740 4.31204 6.98476 3.74666 7.29905

°-75

3-°°532 7.19642

3-I58I8 7-56834 2.55°23 7.79098 1.92656 7-96557

2.42680 7.40811

7.01646 5-06375 6.59486 5·59498 6.13261

5-63254 5-09775 4-53I53 3-93738

3-31894 7·959ΐ9

8.19192 8.16666 8.09106

1.2

0.00 0.64750 I.29101 I.92656 2.55023

0.00 0.68046 1-35673 2.02463 2.68005

I-2S 1-3 1-35 1-4 1-4 5

3-00532 7.19642 3-5653I 6.94036 4 - I 0 3 3 3 6.64151 4.61604 6.30172 5.10030 5-92307

3.15818 3.74666 4-31204 4-85083

7-56834

i-S I-5S

5 - 5 5 3 « 5-50790 597168 5-05878 6 - 3 5 3 4 4 4-57847

1-IS

7.40811

5-35972

7.96557

7.79098

7.29905 6.98476 6.62740 6.22918

5-79256

0.00 0.71081

1-41723 2.11491 2.79956

8.42078 3.46694 8.12115 4-11295 7.77146 4·7336Ι

7-37385 5-32508

6.93078 5-88371

4.76236 7-72455 4-13793 8.07213

0.00 0.61616 1.22852 1.83331 2.42680

1.X

9.11458 9.08649 9.00237 8.86275 8.66850

7-34547 7·67599

I.29101 8.09106 0.64750 8.16666

7-78935 7.76534 7-6934S 7-57413

- 2.90

6.44498 6.40608 5·9ΐ945 6.88894

1-83331 7-57413 1.22852 7-6934S 0.61616 7-76534

0-9S 1.0 1

8.61497 8.58841 8.50891

8.37694 8.19332

3-31894 7.95919 3 - 9 3 7 3 8 7·67599

4-53153 7-34547 5.09775 6.96966 5-63254 6.55088

3.48800 2.81656 2.12776 1-42583 0.71512 0.71512 1.42583 2.12776 2.81656 3.48800 4-13793 4.76236 5-35742

5.91945

8.36994 8.61616 8.80924 8.94802 9.03163

905956

7.85328 0.00

8.25273 0.00

8.67281

0.00

9.11458 0.00

Examples,

4.1129s

1.41723

0.71081

Negative quantities are in heavy type. cosh (2.95 + i o) = 9.57915 + i o. c o s h (2.8 + i 1-2) = —2.55023 + i 7.79098.

[69]

8.80162 9.06053 9.26358 9.40952 9.49744

8.84998 3-64575 9.11031 2-94395 931447 2.22399 9.46121 1.49032 9.54962 0.74747

2.0

Note.

7-70735 8.12294 8.48845

3.46694 2.79956 2.II49I

I-9S

1.91237 1.28150 0.64273

°-75I57

6.73647 7.24424

6.73647 6.18717 5.63048

3.29681 2.66217 2.01112 1.34768 0.67592

1.9

3.66578 2.96012 2.23621 1.49851

5.63048 6.18717

6.77348 7.28404 7.74969 8.16757 8.53508

6.Î

8.01263 8.24834 8.43318 8.56604 8.64608

1.8s

6.77348 6.22116 5.63048 5.00509 4.34884

3·64575 4-32508 4-97774

6.40608 5.88371 5-32508 4-73361

7-72455 7-32934

7.62452 7.84881 8.02470 8.15112 8.22728

3-I349I 2.53144

8.84998 8.5350S 8.16757 7.74969 7.28404

8.80162 8.48845 8.12294

7 - 2 S S 4 8 2.98086 7.46891 2.40704 7.63629 1.81839 7 - 7 5 6 5 9 1.21852 7.82907 0.61115

I-7S 1.8

6.69602 4.06993 6.99732 3-53629

9.57915 0.00 9.54962 0.74747 Q.46I2I 1.49032 9.31447 2.22399 9.II03I 2.94395

3.66578 4.34884 5.00509 563048 6.22116

8.36994 8.07213

3.9U12

1-7

χ = 2.95

9.52681

7.35323 3-71905

1.6 1.65

5·59498 5-OÓ375 4-50131

CONTINUED

0.00 0.75157 1.49851 2.23621 2.96012

9.03163 8.94802 8.80924 8.61616

6.44498 6.93078 7·37385 7.77146 8.12115 8.42078 8.66850 8.86275 9.00237 9.08649

5.32022 4.81509 4.28026

6.09170

5-35742 7-32934

6.13261 6.59486 7.01646 7-39479 7-72754

5-83556 6.27542 6.67660 7.03661

iv.

6.09170 6.55088 6.96966

2.68005 8.19332 2.02463 8.37694 I-35673 8.50891 0.68046 8.58841

0.8 0.8s 0.9

χ

= u +

9-579IS

9-49744 9.40952 9.26358 9.06053

7-7073S 7.24424

4-97774 4.32508

0.00

TABLE V I I I .

HYPERBOLIC

χ = 3.0

COSINES,

z = 3-OJ

cosh {x + iq)

X = 3.10

= u + iv.

* = 3-15

CONTINUED χ

- 3.20

o 0.0s O.I 0.15 0.2

10.06766 10.03660 9-94371 9.78949 9-57492

ο.οο 0.78599 1.56714 2.33863 3-09569

10.58135 10.54870 ΙΟ-45110 10.28900 10.06350

ο.οο 0.82649 1.64788 2-459 1 1 3·255ι8

11.12150 11.08720 10.98460 10.81420 10.57720

ο.οο 0.86905 J. 73 2 74 2.58575 3·4228Ϊ

11.08946 11.6534° ιι.5455° 11.3665° ιι.ιΐ73°

ο.οο 0-9Ι378 1.82193 2.71885 3-59900

12.28665 12.24880 12.1354° 11.94720 11.68530

°·°° 0.96080 i^Sóe 2.85874 3·7?4ΐ9

0.25 °·3 °·35 °·4 °·45

9·3ΟΙ3Ι 8.97035 8.58409 8.Ι449 1 7-65551

3-83368 4· 548ο6 5-23433 5.88836 6.50609

9-77589 9.42805 9-02209 8.56049 8.04612

4.03120 4-78233 5.50400 6.84128

10.27490 9-90933 9.48264 8-99749 8.45686

4-23878 5.02860 5-78743 6.51058 7-19358

Ι0.79970 10.41540 9.96690 9-45697 8.88873

4.45696 5-28745 6.08533 6.84570 7·56387

11.3514° 10.94750 10.47610 9.94011 9.34284

4-6863° 5-55952 6.39846 7-19795 7-95306

°·5 0-55 ο.6 0.65 0.7

7.11891 6.53842 5.91762 5.26034 4.57062

7.08371 7.61765 8.10463 8.54164 8.92599

7.48215 6.87204 6.21956 5-52874 4-80383

7-44866 8.ΟΙ0ΙΙ 8.52218 8.98171 9-38586

7.86409 7.22284 6.53705 5-81097 5·°49°6

7.83223 8.42260 8.96104 9-44423 9.86919

°·75 0.8 0.85 0.9 °·95

3-85273 3·ΙΙΙΟ8 2.35025 1-57493 0.78990

9-25531 9-52757 9·74ιο8 9-89454 9.98699

4-0493 1 3.26982 2·47 Ο Ι 7 1.65529 0.83020

9-73216 10.01840 10.24290 10.40430 10.50150

4.25602 3.43673 2.59626 1.73979 0.87258

Ι.Ο ΐ·°5 Ι.Ι Ι-Ι5

ο.οο ίο.53399 0.83020 10.50150

1.2

0.00 10.01787 0.78990 9.98699 1-57493 9-89454 2-35025 9 · 7 4 ι ο 8 3·ΙΙΙΟ8 9-5275.7

ο.οο 0.87258 1.73979 2.59626 3-43673

I.2S Ι·3 Ι ·35 ΐ·4 1-45

3-85273 4.57062 5.26034 5-91762 6.53842

9-25531

4-04931 4.80383 5-52874 6.21956 6.87204

9.73216 9.38586 8.98171 8.52218 8.0Ι0ΙΙ

4.25602 10.23330 5.04906 9.86919 5-81097 9-44423 6-53705 8.96104 7.22284 8.42260

ι -5 Ι ·55 1.6 1.65 ΐ·7

7.11891 7-65551 8.Ι449 1 8.58409 8-97035

7.08371 6.50609 5.88836 5-23433 4.54806

7.48215 8.04612 8.56049

7-44866 6.84128

9.42805

6.19173 5.50400 4-78233

7.86409 8.45686 8.99749 9.48264 9-90933

7.83223 7-19358 6.51058 5-78743 5.02860

8.26570 8.88873 9-45697 9.96690 10.41540

8-23539 7-56387 6.8457° 6.08533 5-28745

8.68797 9.34284 9-94011 10.47610 ΙΟ·9475°

8.65915 7·953°6 7-19795 6.39846 5-55952

I-7S

3-83368 3-09569 2.33863 I-5Ó7I4 0.78599

9.77589 10.06350 10.28900 10.451 ί ο 10.54870

4-03Ι20 3-25518 2.459 1 1 1.64788 0.82649

ίο.27490

1.85 1.Q ΐ·95

9-30131 9-57492 9.78949 9-94371 10.03660

10.81420 10.98460 11.08720

4-23878 3.42281 2-58575 1-73274 0.86905

ΙΟ-79970 ΙΙ.ΙΙ730 11.36650 11-54550

4-45696 3-599°° 2.71885 1.82193

11.35140 11.68530 11.94720 12.13540 12.24880

4-68630 3-784Ι9 2.85874 1.91568 0.96080

2.0

10.06766

Ο.ΟΟ

10.58135

ο.οο

ΙΙ.Ι2Ι50

ο.οο

12.28665

ο.οο

1.8

6 . 1 9 1 73

1.65529 10.4043° 2.47017 10.24290 3.26982 10.01840

8.92599

8.54164 8.10463 7.61765

9-02209

10.57720

8.26570 8.23539 7.5917° 8.85615 6.87089 9-42230 6.10773 9-93036 5.30690 10.37720

8.68797 8.65915 7-97954 9.31184 7.22191 9.90712 6.41976 10.44130 5.57802 10.91120

10.23330 ιο.5343° 10.77040 ιο.94ΟΙ° 11.04230

4.47336 3.61224 2.72885 1.82863 ο·9ΐ7Ι4

10.76010 11.07660 11.32480 11.50320 11.61070

4.70190 3.79678 2.86826 1.92205 O.964OO

11.3137° 11.64650 11.90750 12.09510 Ι2.208ΐ0

11.07645 11.04230 Ι0.940Ι0 10.77040 ιο·5343°

ο.οο 0.91714 1.82863 2.72885 3-61224

11.64661 11.61070 11.50320 11.32480 11.07660

0.00 0.96400 1-92205 2.86826 3.79678

12.24588 Ι2.2θ8ΐΟ 12.09510 11.90750 11.64650

4-47336 10.76010 5.30690 ΙΟ-37720 6.10773 9·93°36 6.87089 9-4223° 7-59170 8.85615

11.65340

0.91378

11.68946

ο.οο

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

cosh (3.10 + i 0.5) = 7.86409 + i 7.83223. cosh (3.10 + i 1-55) = - 8 . 4 5 6 8 6 + i 7.19358.

[70]

4.70190 1 1 . 3 1 3 7 ° 5.57802 Ι0.9ΙΙ20 6.41976 ΙΟ·44 ι 3° 7.22191 9·9°712 7-97954 9-3H84

TABLE V I I I .

HYPERBOLIC

* = 3- 2 S o 0.05 0.1 0-15

COSINES,

« = 3-3°

c o s h (x + ìq) = u + iv.

CONTINUED

* = 3-4°

3-35

3-45

12.91456 12.87474 I 2 -75555 12.55773 12.28247

0.00 1.01022 2.01422 3-°°579 3· 97883

I3-57476 13-5329° 13.40764 13.19970 12.91035

0.00 1.06217 2.11780 3.16036 4.18343

14.26891 14.22498 14.09323 13.87465 I3-57054

0.00 1.11677 2.22666 3.32282 4-39850

14.99874 14.95250 14.81410 14.58432 14.26465

0.00 1.17417 2.34110 3-49359 4-62455

15.76607 15.71746 15.57196 I5-33045 14.99442

11.93150 11.50695 11.01147 10.44810 9.82031

4-92735 5.84548 6.72758 7.56820 8.36217

12.54145 12.09520 11.57440 10.98222 10.32233

5.18072 6.14670 7.07352 7.95737 8.79215

13.18275 12.71369 12.16623 11.54379 10.85016

5.44705 6.46202 7-437 I 5 8.36643 9.24413

13.85702 I3-36397 12.78852 12.13423 11.40512

5.72700 6.79414 7.81938 8.79642 9.71923

I4.56595 6.02127 14.04764 7-I4324 13.44278 8.22116 12.75501 9.24840 11.98861 10.21863

0.6 0.65 0.7

9.13197 9.10456 8.38733 9.79082 7.59099 10.41673 6.74784 10.97841 5.86309 11.47240

9.59881 8.81610 7.97905 7.09279 6.16381

9.57273 10.29428 10.95235 11.54294 12.06234

10.08964 9.26691 8.38705 7-45549 6.47795

10.06483 10.82349 11.51541 12.13633 12.68244

10.60570 9.74090 8.81604 7.83682 6.80928

10.58212 11.37975 12.10723 12.76006 I3-33423

11.14830 10.23924 9.26706 8.23775 7.15765

11.12585 11.96447 12.72933 1341572 14.01940

0.7S 0.8 0.85 0.9 0-95

4.94218 3.99082 3.01484 2.02028 1.01326

11.89566 12.24560 12.52002 12.71726 12.83610

5-I948S 4.19483 3.16897 2.21356 1.06507

12.50736 12.87530 13.16388 13.37120 13-49615

5.46075 4.40933 3.33101 2.23215 1.11953

13.15033 13.53717 13.84054 14.05851 14-18950

5-73977 4.63487 3.50139 2.34632 1.17679

13-82620 14.23291 14.55188 14.78111 14.91923

6.03341 4.87198 3.68053 2.46636 1.23699

14-53662 14.96423 15.29960 15.54061 15.68581

1.0 ι.OS 1.1 1-IS 1.2

0.00 1.01326 2.02028 3.01484 3.99082

12.87578 12.83610 12.71726 12.52002 12.24560

0.00 1.06507 2.12356 3.16897 4.19483

13-53788 13-49615 13.37120 13-16388 12.87530

0.00 1.11953 2.23215 3.33101 4-40933

14.23382 14-18950 14.05851 13.84054 i3-537I7

0.00 1.17679 2.34632 3.50139 4.63487

14.96536 I4-9 I 9 2 3 14.78111 14-55188 I4-2329I

0.00 1.23699 2.46636 3.68053 4.87198

I5-73432 15-68581 15.54061 15.29960 14.96423

1.25 1-3 I-3S 1.4 1-45

4.94218 5.86309 6.74784 7.39099 8.38733

11.89566 11.47240 10.97841 10.41673 9-79082

5.19485 6.16381 7.09279 7.97905 8.81610

12.50736 12.06234 11.54294 10.95235 10.29428

5.46075 6-47795 7-45549 8.38705 9.26691

13.15033 12.68244 12.13633 11.51541 10.82349

5-73977 6.80928 7.83682 8.81604 9.74090

13-82620 13.33423 12.76006 12.10723 11.37975

6.03341 7-15765 8.23775 9.26706 10.23924

14-53662 14.01940 13.41572 12.72933 11.96447

1-5 i-SS 1.6 1.6s 1-7

9.13197 9.82031 10.44810 11.01147 11.5069s

9-io4S6 8.36217 7.56820 6.72758 5-84S48

9.59881 10.32233 10.98222 11.57440 12.09520

9.57273 8.79215 7-95737 707352 6.14670

10.08964 10.06483 10.85016 9.24413 11-54379 8.36643 12.16623 7-43715 12.71369 6.46202

10.60570 10.58212 11.40512 9.71923 12.13423 8.79642 12.78852 7.81938 I3-36397 6.79414

11.14830 11.12585 11.98861 10.21863 12.75501 9.24840 13.44278 8.22116 14.04764 7-i43 2 4

1-75 1.8 1.8s 1.9 1-95

11.93150 12.28247 12.55773 12.75555 12.87474

4-92735 3.97883 3-°°579 2.01422 1.01022

12.54145 12.91035 13.19970 13.40764 13.53290

5.18072 4-18343 3.16036 2.11780 1.06217

13-18275 13-57054 13-87465 14.09323 14.22498

5.44705 4-39850 3-32282 2.22666 1.11677

13-85702 14.26465 14.58432 14.81410 14-95250

5.72700 4.62455 3-49359 2.34110 1.17417

14.56595 14.99442 15-33045 15-57196 15.71746

6.02127 4.86217 3·673" 2.46139 1.23450

2.0

12.91456

0.00

I 3-57476

0.00

14.26891

0.00

14.99874

0.00

15.76607

0.00

0.2 0.2s 0.3 0-3S 0.4 °-45 o-5

0.5s

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

cosh (3.45 + i 0.05) = 15.71746 + i 1.23450. cosh (3.25 + i 1.95) = —12.87474 + i 1.01022.

[71]

0.00 1.23450 2.46139 3-67311 4.86217

TABLE V I I I .

HYPERBOLIC

« = 3-5° O 0.05 0.1 0.15

16.57282 16.52173 16.36878 16.11491

COSINES,

* = 3-55

0.00 1.29792 2.58783 3.86180

0.2

15.76170

5.III95

O.25 0.3 O.35 O.4 O.45

I5.3II3O 14.76650 14.13065 13.40770 12.60210

6.33059 7.51020 8.64360 9·7236ΐ IO.74357

O.5 O.55 0.6 O.65

I I . 7 1 8 2 O II.6974O IO.7632O I 2 . 5 7 9 I O 9.74126 13.3833° 8.65928 I4.I0490

15-2834° 15.733°° 16.08555 16.33896 16.49163

ι.ο 1.05 1.1 1.15 1.2

0.00 1.3002p 2*59256 3.86885 5.12129

16.54263 16.49163 16.33896 16.08555 15.733°°

0.00 17.39230 1.36684 1 7 . 3 3 8 7 ° 2.72525 1 7 . 1 7 8 1 7 4.06686 1 6 . 9 1 1 7 5 5-38339 1 6 . 5 4 1 0 5

I.25 1-3 1-35 1.4 1.45

6.34215 7-S239I 8.65928 9.74126 10.76320

I5-2834° 14-7396° 14.10490 13.38330 12.57910

6.66674 7.90898 9.10246 10.23982 II.3I405

1.5 1.55 1.6 1.65 1.7

11.71820 12.60210 13.40770 14.13065 14.76650

11.69740 10.74357 9.72361 8.64360 7.51020

16.57282 O.OO

I7-78497

7-3S683

18.69565

7-73453

17.15216 16.41360

8.72766 10.04469

14.63806

12.48520

18.03040 17.25404 16.37127 15.38760

9.17573 10.56037 11.87990 13.12620

12.92978 13.90436 14.79326 15.59090 16.29246

13.61203 12.50208 11.31505 10.05827 8.73946

13-59365 14.61830 15.55281 16.39143 17.12900

14.30902 13.14223 11.89444 10.57330 9.18696

14-29155 15.36878 16.35127 17.23295 18.00840

12.94910 11.89320 10.76400 9.56840 8.31383

6-34215 5-12129 3.86885 2.59256 1.30029

2.0

6.99754 8.30143 9.55412 10.74789 11.87545

I2.31852 II.31405 IO.23982 9.IO246

0-75 0.8 0.85 Θ·9 0.95

6.33059 5.11195 3.86180 2.58783 1.29792

0.00 1-58576

16.91880 16.31680 15.61421 14.81535 13.92515

7.90898 15-49665

15-31130 15.76170 16.11491 16.36878 16.52173

20.23601 20.17362 19.98688 19.67690 19.24560

16.09492 6.65574 I5.52225 7.89594 14.85386 9.08745 I4.O939O IO.22293 I3.24705 II.2954O I2.2982O 13.22520 14.07066 14-82933

Χ = 3.70

0.00 1.50832 3.00735 4-48783 5·94°65

4.06015 5.3745I

16.93970 16.56840

* = 3-ÔS

CONTINUED

19.25033 19.19100 19.01331 18.71843 18.30814

0.00

6.66674 5.38339 4.06686

I-75 1.8 1.85 1-9 1.95

Χ = 3.60 18.31278 18.25632 18.08732 17.80680 17.41650

I7.42IO2 O.OO Ι7·3673Ι I.36458 17.20653 2.72075

7-52391 14-7396°

°·7

c o s h (x + iq) = u + iv.

1.43466 2.86047 4.26865 5-65052

IS-57383 11.29972

3.16174 4.71823 6.24563

7.00800 16.89356 5 . 6 5 8 9 6 17.39050 4.27503 17.78022 2.8647S 1 8 . 0 6 0 3 3 1.43680 18.22900

7.36678 17.76096 5.94868 18.28342 4.49390 18.69316 3.01142 18.98765 1.51036 19.16506

7.74399 18.67280 6.25327 19.22208 4.72401 19.65290 3.16561 19.96246 1.58770 20.14900

0.00 1.43680 2.86475 4-27503 5.65896

18.28546 18.22900 18.06033 17.78022 17-3905°

0.00 19.22434 1.51036 1 9 . 1 6 5 0 6 3.01142 1 8 . 9 8 7 6 5 4.49390 1 8 . 6 9 3 1 6 5.94868 1 8 . 2 8 3 4 2

0.00 1.58770 3.16561 4.72401 6.25327

16.06840 15.49665 14.82933 14.07066 13.22520

7.00800 8.31383 9.56840 10.76400 11.89320

16.89356 16.29246 15.59090 14.79326 13.90436

7.36678 1 7 . 7 6 0 9 6 8.73946 1 7 . 1 2 9 0 0 10.05827 1 6 . 3 9 1 4 3

12.31852 1 2 . 2 9 8 2 0 13-24705 1 1 . 2 9 5 4 0 14.09390 1 0 . 2 2 2 9 3 14.85386 9.08745 15.52225 7.89594

12.94910 13.92515 14.81535 15.61421 16.31680

12.92978 11.87545 10.74789

16.09492 16.56840 16.93970 17.20653

6.65574 5-37451 4.06015 2.72075

17.36731

1.36458

16.91880 17.41650 17.80680 18.08732 18.25632 18.31278

16.06840 16.54105 16.9II75

2.72525 17.17817 I.36684

17.42102

17.33870

0.00

11-31505 15-55281 12.50208

14.61830

7·74399 18.67280 9.18696 1 8 . 0 0 8 4 0 10.57330 17.23295 11.89444 16.35127 13.14223 1 5 . 3 6 8 7 8

14-30902 14-29155

8.30143

13.61203 13-59365 14.63806 1 2 . 4 8 5 2 0 ΐ 5 · 5 7 3 8 3 11.29972 16.41360 1 0 . 0 4 4 6 9 17.15216 8.72766

6.99754 5.65052 4.26865 2.86047 1.43466

17.78497 18.30814 18.71843 19.01331 19.19100

7-35683 5-94065 4-48783

7-73453

1.50832

.18.69565 *9·2456Ο 19.67690 19.98688 20.17362

0.00

19.25033

o.oo

20.23601

O.OO

9-554

12

3·°°735

Note.

Negative quantities are in heavy type.

Examples,

cosh (3.50 + i 0.70) = 7.52391 + i 14.73960. cosh (3.60 + i = - 1 3 . 9 2 5 1 5 + i 11.87545.

[72]

20.21129 20.14900 19.96246 19.65290 19.22208

15.38760 1 3 . 1 2 6 2 0 16.37127 11.87990 17.25404 1 0 . 5 6 0 3 7 18.03040 9-17573 6.24563 4.71823 3.16174 1.58576

TABLE V I I I .

HYPERBOLIC

COSINES,

χ = 3.80

χ = 3-75 o 0.05 0.1 O.I s 0.2

21.27230 21.20670 21.01038 20.68452 20.23113

0.00 1.66716 3.32404 4-96043 6.56624

22.36178 22.29283 22.08646 21-74391 21.26731

ο.οο 1-75273 3-49465 5·2Ι5°3 6.90325

0.25 0.3 0.35 0.4 0.45

19.65301 8.13156 i8.g5373 9.64674 18.13760 11.10246 17.20963 12.48971 16.17556 13.79998

20.65958 19.92448 19-06655 18.09105 17.00402

o-S o-SS

15.04177 13.81524 Ι2·5°353 II.II473 9.65741

15.02516 16.15770 Ι7·ι9°62 18.11756 18.93280

0.8 0.85 0.9

8.14055 6-5735° 4.96592 3.32772 1.66900

1.0 1-OS 1.1 I-IS ι. 25 1.3 1.35 1.4 1.45

0.6 0.65 0.7

0-7S

Q-95

1.2

1.5

c o s h (x + iq) = u + iv.

* = 3-8S

* = 3-95

3-9°

ο.οο 1.84268 3-67400 5-48267 7· 2 5754

24.71135 24.63516

ο-οο 1-93724

24.02856 23.50188

5-76402 7-62997

8.54892 10.14188 11.67230 13.13076 14.50828

21.71778 8.98766 2θ·945°3 10.66233

22.83030

15.81216 14.52281 13.14392 11.68400 10.15203

Ι5-79634 16.98701 18.07296 19.04746 19-90455

16.62208 15.26668 13.81716 12.28246 10.67203

19.63131 20.20879 20.66167 20.98716 21.18327

8-55748 6.91017 5.22025 3.49815 1.75448

20.63891 21.24603 21.72216 22.06437 22.27052

0.00 1.66900 3.32772 4.96592 6-57350

21.24878 21.18327 20.98716 20.66167 20.20879

ο.οο 22.33941 1.75448 22.27052 3.49815 22.06437

8.14055 9-65741 11.11473 12.50353 13.81524

19-63131 18.93280 18.11756 17.19062 16.15770

15-04177 15-02516

5.22025 21.72216 6.91017 21.24603 8-55748 10.15203 11.68400 13-14392 14.52281

2Ο.6389Ι Ι9·9°455 19-04746 18.07296 16.98701

Ι 5 · 8 ΐ 2 ΐ 6 Ι5-79634

23.50717 23.43470 23-21775 22.85766 22.35664

* -

20.04315

12.27134

19.01770 13.80465 17.87500 15.25286 16.60702 17.85880 19.00048 20.02501 20.92608

24·4 0 7 1 0

3-86254

9-44887

22.0Ι797 11.20952

21.06988 12.90106 19.99190 14-51307 18.79065 16.03558

17-47355 17-45924

16.04874 18.77526

ΐ4-5 2 497 19-97556

12.91164 21.05272

11.21871 21.99993

8.9958ο 21.69813 7.26411 22.33640 5.48764 22.83696 3-67733 23.19673 1-84435 23.41348

9.45662 7.63623 5.76875 3.86571 1.93883

ο.οο 1-84435 3-67733 5.48764 7.26411

23.48589

ο.οο 24-69110 1.93883 24.61500

23.19673 22.83696 22.33640

3.86571 24.38710

8.99580 10.67203 12.28246 13.81716 15-26668

21.69813 20.92608 20.02501 19.00048 17.85880

9-45662 22.8ΙΙ6Ο 11.21871 21.99993 12.91164 21.05272

23.41348

CONTINUED

22.8ΐΐ6θ 23.48262 24.00888 24-3871° 24.61500

5-76875 24-00888 7.63623 23.48262

14-52497 19-97556

16.04874 18.77526

16.62208 16.60702 17-87500 15.25286 19.01770 13-80465

17-47355 17-45924

20.04315

21.06988 12.90106

25-97731 25.89724 25-65749 25-25957 24.7059°

ο.οο 2.03665 4-ο6Ο74 6.05979 8.02Ι49

23-99991 9-93373 23-14597 11-78474

22.14931 13-56305

2I.OIÓIO 15.25776 19-75331 16.85842 18.36873 16.87098 15.26910 Ι3·573Ι5 11.79366

18.35512 19.73867 21.00052 22.13290 23.12881

9-94ιο9 8.02744 6.06429 4.06375 2.03816

23.98212 24-68760 25.24084 25.63849 25.87805

0.00

2.03816 4-OÔ375 6.06429 8.02744

25.95806 25.87805 25.63849 25.24084 24.68760

9-94109 11.79366 13-57315 15.26910 16.87098

23.98212 23.12881 22.13290 21.00052 19.73867 18-35512 16.85842 15.25776 13-56305 11.78474

17-20963 12.48971 18.13760 11.10246 18.95373 9·64674

17.00402 18.09105 19-06655 19.92448

14.50828 13-13076 11.67230 ί ο . 14188

20.94503

10.66233

22.01797 11.20952

18.36873 19-75331 2ΐ.οι6ιο 22.14931 23-14597

1-75 1.8 1.85 1.9 1.95

19-65301 20.23113 20.68452 21.01038 21.20670

8.13156 6.56624 4-96043 3-32404 1.66716

20.65958 21.26731 21.74391 22.08646 22.29283

8.54892 6.90325 5·2Ι5°3 3-49465 1-75273

21.71778 22.35664 22.85766

23.21775 23-43470

8.98766 7-25754 5-48267

3-6740° 1.84268

22.83030 23.50188 24-02856 24.40710 24.63516

9-44887 7-62997 5-76402 3-86254 1-93724

23-99991 24.70590 25-25957 25-65749 25.89724

9-93373 8.02149 6.05979 4-OÓ074 2.03665

2.0

21.27230

ο.οο

22.36178

ο.οο

23.50717

Ο-ΟΟ

24-71135

ο-οο

25-9773 1

ο.οο

Ι

·55

1.6 1.65 1.7

16.17556 13-79998

Note. Examples,

12.27134

18.79065 16.03558

19-99190 Ι4·5 307 Γ

Negative quantities are in heavy type. cosh (3.90 + i 0.25) = 22.83030 + 19.44887. cosh (3.75 + i 1.25) = - 8 . 1 4 0 5 5 + i 19.63131.

[73]

TABLE I X .

HYPERBOLIC

TANGENTS.

tanh (x + iq) = u + iv

0.05

» = 0.15

0 0.05 O.I 0.15 0.2

0.00 0.00 0.00 0.00 0.00

0.00 0.07870 0.15838 0.24008 0.32492

0.04996 0.05027 0.05121 0.05283 0.05522

0.00 0.07850 0.15798 0.23944 0.32402

0.09967 0.10028 0.10214 0.10535 0.11008

0.00 0.07792 0.15677 0.23755 0.32136

0.14889 0.14979 0.15254 0.15727 0.16422

0.00 0.07695 0.15479 0.23446 0.31698

0.19738 0.19855 0.20213 0.20828 0.21732

0.25 0-3 0.35 0.4 0.45 '

0.00 0.00 0.00 0.00 0.00

0.41421 0-S0953 0.61280 0.72654 0.85408

0.05850 0.06289 0.06865 0.07623 0.08624

0.41300 0.50792 0.61070 0.72378 0.85040

0.11657 0.12522 0.13659 0.15149 0.17113

0.40940 0.50317 0.60446 0.71557 0.83951

0.17377 0.18647 0.20310 0.22484 0.25339

0.40350 0.49538 0.59427 0.70222 0.82186

0.22970 0.24613 0.26758 0.29549

0.5 O-SS 0.6 0.65 0.7

0.00 0.00 0.00 0.00 0.00

1.0000 1.17085 1.37638 1.63185 1.96261

0.09967 0.11804 0.14392 0.18179 0.24007

0-995°3 1-16395 1.36649 1.61702 1.93900

0.98033 1-14365 1-33754 ι.5 7401 1.87150

0.29131 0.34258 0.41357 0.51496 0.66202

0.95663 1.11113 1.29164 1.50674 1.75880

0.75 0.8 0.85 0.9 0-9S

0.00 2.4I42I 0.00 3.07768 0.00 4.16530 0.00 6.31375 0.00 12.70620

0.33624 0.51109 0.87867 1.85674 5.79801

2-37365 2.99911 3.98246 5-72808 9.05499

0.19738 0-23313 0.28315 0.3S567 0.4657S 0.64333 0-95397 1.56000 2.91746 6.21808

2-25941 2.78504 3-5I76S 4.47780 4-83133

0.90034 1.28858 I-973I6 3-22989 5.28217

2.09061 2.48723 2.94167 3-277S8 2.71349

1.0 1-05 I.I I I - S 1.2

00 0.00 0.00 12.70620 0.00 6.31375 0.00 4.16530 0.00 3.07768

20.01667 5.79801 1.85674 0.87867 0.51109

0.00 9.05499 5.72808 3.98246 2.99911

10.03331 6.21808 2.91746 1.56000 0-95397

0.00 4.83133 4-47780 3.51765 2-78504

6.71659 5.28217 3.22989 1.97316 1.2Í

0.00 2.71349 3-27758 2.94167 2.48723

0.39543 0.48477 0.58044 0.68417 0.33192 0.79813 0.37995 0.92501 1.06819 0.44423 1.23185 0.53203 1.42088 0.65502 1.64005 0.83268 1.09837 1.89083 1.50982 2.16055 2 . 1 6 1 1 1 2.38860 3-I5925 2.37676 4.39854 1-67517 5.06649 0.00 4.39854 1.67517 3-I5925 2.37676 2 . 1 6 1 1 1 2.38860 1.50982 2.16055

1.25 1-3 1-35 1-4 1-45

0.00 0.00 0.00 0.00 0.00

2.4I42I 1.96261 1.63185 1.37638 1.17085

0.33624 0.24007 0.18179 0.14392 0.11804

2.37365 1.93900 1.61702 1.36649 1.16395

0.64333 0.46575 0.3S567 0.28315 0.23313

2.25941 1.87150 I-S740I 1.33754 1.14365

0.90034 0.66202 0.51496 0.41357 0.34258

2.09061 1.75880 1.50674 1.29164 X.XII13

1.09837 0.83268 0.65502 0-53203 0.44423

1.89083 1.64005 1.42088 1.23185 1.06819

1-5 1-55 1.6 1.65 1-7

0.00 0.00 0.00 0.00 0.00

r.0000 0.85408 0.72654 0.61280 0.50953

0.09967 0.08624 0.07623 0.06865 0.06289

0.99S03 0.85040 0.72378 0.61070 0.50792

0.19738 0.17113 0.15149 0.13659 0.12522

0.98033 0.83951 0.71557 0.60446 0.50317

0.29131 0.25339 0.22484 0.20310 0.18647

0.95663 0.82186 0.70222 0.59427 0.49538

0-37995 0.33192 0.29549 0.26758 0.24613

0.92501 0.79813 0.68417 0.58044 0.48477

1-75 1.8 i.8s 1.9 1-95

0.00 0.00 0.00 0.00 0.00

0.41421 0.32492 0.24008 0.15838 0.07870

0.05850 0.05522 0.05283 0.05121 0.05027

0.41300 0.32402 0.23944 0.15798 0.07850

0.11657 0.11008 0.10535 0.10214 0.10028

0.40940 0.32136 0.2375s 0.15677 0.07792

0.17377 0.16422 0.15727 0.15254 0.14979

0.40350 0.31698 0.23446 0.15479 0.07695

0.22970 0.21732 0.20828 0.20213 0.19855

0.39543 0.31098 0.23021 0.15207 0.07562

2.0

0.00

0.00

0.04996 0.00

0.09967 0.00

0.14889 0.00

Note.

Negative quantities are in heavy type.

Examples,

tanh (o + i 0.95) = o + i 12.70620. tanh (o + i 1.45) = o — i 1.17085.

[74]

0.00 0.07562 0.15207 0.23021 0.31098

0.19738 0.00

TABLE I X .

HYPERBOLIC

Χ = 0.25

TANGENTS,

* = 0.3

t a n h (x + iq) = u+

Χ = 0.35

iv.

X = 0.4

CONTINUED

« = 0.4s

0.00 0.07199 0.14464 0.21864 0.29471

0.33638 0.33322 0.34384 0.35346 0.36750

0.00 0.06975 0.14007 0.21153 0.28475

0-3 7995 0.38196 0.38808 0.39853 0.41376

0.00 0.06728 0.13503 0.20373 0.27384

0.42190 0.42405 0.43056 0.4416G

0.26907 0-30351

0.29131 0.29296 0.29799 0.30660 0.31921

0.3 o-35 0.4 0-4S

0.28402 0.30377 0.32947 0.36272 0.40582

0.38540 0.47162 0.56335 0.66200 0.76919

0.33640 0.35903 0.38833 0.42600 0.47444

0.37362 0.45623 0.54348 0.63638 0.73604

0.38658 0.36035 0.41161 0.43898 0.44383 0.52131 0.48497 0.60802 0.69969 0-53739

0.43438 0.46130 0.49575 0.53941 0.59450

0.34585 0.42022 0.49737 0.57764 0.66116

0.47964 0.50796 0.54397 0.58924 0.64580

0.33039 0.40033 0.47216 0.54593

0.5 o-55 0.6 0.65 0.7

0.46212 0.88682 0.536SS 1.01699 0.63656 1 . 1 6 1 8 0 0.77356 1.32269 0.96528 1.49862

0.53705 0.61869 0.72640 0.87037 1.06521

0.84355 0.95983 1.08513 1.21810 1.35360

0.60437 0.69041 0.80176 0.94684 1.13666

0.79671 0.89893 1.00519 1.11212 Ι.21222

0.66404 0.75200 0.86357 1.00528 1.18469

0.74770 0.83632 0.92478 1.00856 1.07919

0.71630 0.80407 0.91321 1.04844 1.21438

0.69779 0.77365 0.84608 0.91003 0.95708

0-7S 0.8 0.85 0.9 °-95

Ι.23914 1-63553 2.20223 2.95122 3.72316

1.68151 1.84484 1.91863 1.75011 1.11789

1.33091 1.47820 1.69121 1.56140 2.16210 Ι.54177 2.71602 1.31829 3.21905 0.79096

1.38412 1.70028 2.08309 2-49443 2.83603

1.29020 I-3I745 1.24667 1.01613 0.58484

1.40896 1 . 1 2 1 8 1 1.68068 1-11235 1.98936 1.01695 2.29855 0.79978 2.53928 0.44728

1-41397 1.64487 1.89366 2.12957 2.30471

0.97400 0.94186 0.83750 0.64107 0.35122

1.0 ι.05 1.1 1-I5 1.2

4.08299 3.72316 2.95122 2.20223 I-63553 1-23914 0.96528 0.77356 0.63656 0-53655

0.00 1.11789 1.75011 1.91863 1.84484

3.43274 0.00 3.21905 0.79096 2.71602 1.31829 2.16210 1.54177 1.69121 1.56140

2.97287 2.83603 2-49443 2.08309 1.70028

0.00 0.58484 1.01613 1.24667

2.63193 2.53928 2-2985S 1.98936 1.68068

0.00 0.44728 0.79978 1.01695 J.11235

2.37024 2.30471 2.12957 1.89366 1.64487

0.00 0.35122 0.64107 0.83750 0.94186

1.68151 1.49862 1.32269 1.16180 1.01699

1-33091 1.47820 1.06521 I-35360 0.87037 1.21810 0.72640 1.08513 0.61869 0.95983

1.38412 1.29020 1.13666 1.21222 0.94684 I.II2I2 0.80176 I.OO5I9 O.69O4I O.89893

1.40896 1.12181 1.18469 1.07919 1.00528 1.00856 0.86357 0.92478 0.75200 0.83632

1-41397 1.21438 1.04844 0.9I32I 0.80407

0.97400 0.95708 0.91003 0.84608 0.77365

o 0.05 O.I 0.15 0.2

0.24492 0.00 0-24635 0.07395 0.25069 0.14866 0.25814 0.22490

0.25

1-25 1-3 ι-35 1.4 ι-45

1.31745

0.00 0.06462 0.12961 0-I9534 0.45784 0.26216

0.62138

0.88682 0.76919 0.66200 0.56335 0.30377 0.47162

0.53705 0.47444 0.42600 0.38833 0-35903

0.84355 0.73604 0.63638 0.54348 0.45623

0.60437 0-53739 0.48497 0.44383 0.41161

0.79671 0.69969 0.60802 0.52131 0.43898

0.66404 0.59450 0.53941 0.495 75 0.46130

0.74770 0.66116 0.57764 0.49737 0.42022

0.71630 0.64580 0.58924 0-54397 0.50796

0.69779 0.62138 0.54593 0.47216 0.40033

1.8s

1.9 i-95

0.28402 0.26907 0.25814 0.25069 0.24635

0.38540 0.30351 0.22490 0.14866 0.07395

0.33640 0.31921 0.30660 0.29799 0.29296

0.37362 0.29471 0.21864 0.14464 0.07199

0.38658 0.36750 0.35346 0-34384

0.36035 0.28475 0.21153 0.14007 0.06975

0.43438 0.41376 0.39853 0.38808 0.38196

0.34585 0.27384 0.20373 0.13503 0.06728

0.47964 0.45784 0.44169 0.43056 0.42405

0.33039 0.26216 0.19534 0.1296t 0.06462

2.0

0.24492

0.00

0.29131

0.00

0-37995

0.00

0.42190 0.00

1-5 I-5S 1.6

1.6s 1-7

1-75 1.8

0.46212 0.40582 0.36272 0.32947

0.33822

0.33638 0.00

Note.

Negative quantities are in heavy type.

Examples,

tanh (0.4 + i 04) = 0.53941 + ¿0.57764. tanh (0.45 + i 1.75) = 0.47964 - i 0.33039.

[75]

TABLE I X .

HYPERBOLIC

χ = 0.5

TANGENTS,

» = o-SS

t a n h (x + iq) = u + iv.

χ — 0.6

X = 0.65

CONTINUED

X = 0.7

0.50052 0.00 0.50284 0.05889 0.50987 0.11796 0.52183 0.17737 0.53910 0.23725

0.53704 0.53941 0.54657 0.55872 0.57620

0.00 0.05590 0.11189 0.16804 0.22437

0.57167 0.57405 0.58125 0.59344 0.61094

0.00 0.05288 0.10576 0.15863 0.21144

0.60437 0.60674 0.61390 0.62602 0.64336

0.00 0.04984 0.09962 0.14925 0.19858

0.52227 0.31424 0.55151 0.37967 0.58846 0.44616 0.63452 0.51350 0.69149 0.58116

0.56223 0.59196 0.62928 0.67541 0.73188

0.29765 0.35856 0.41979 0.48093 0.54121

0.59953 0.62935 0.66653 0.71212 0.76736

0.28085 0.33731 0.39344 0.44869 0.50211

0.63419 0.26404 0.66377 0.31618 0.70039 0.36748 0.74493 0.41714 0.79836 0.46428

0.66631 0.69534 0.73105 0.77413 0.82533

0.24741 0.29540 0.34205 0.38662 0.42806

0.65 0.7

0.76159 0.84752 0.95230 1.07907 1.23020

0.64805 0.71229 0.77067 0.81812 0.84688

0.80050 0.88332 0.98247 1.09973 1-23587

0-59933 0.65320 0.69957 0.73362 0.74858

0.83365 0.91249 1.00521 1.11262 1.23436

0.55229 0.59707 0.63346 0.65676 0.66157

0.86173 0.50738 0.93607 0.54434 1.02195 0.57227 1.11962 0.58738 1.22793 0.58492

0.88535 0.95480 1.03389 1.12222 I.21827

0.46492 0.49522 0.51635 0.52508 O.51757

0-7S 0.8 0.8s 0.9 °-95

I.40579 0.84585 1.60095 0.80073 1.80225 0.69623 1.98505 0.52197 2.11599 0.28167

1.38926 1.55398 1.71785 1.86183 1.96180

0.73549 0.68387 0.58390 0.43071 0.22977

1.36782 0.64076 1.50695 0.58681 1.64135 0.49366 1.75601 0.35949 1.83417 0.19009

1.34386 1.46173 1.57271 1.66532 1.72736

I.31896 ι.41909 1.51148 1-58713 1-63711

0.48976 0.43803 0.36034 0.25755 0.13449

1.0 1.05 1.1 1-iS 1.2

2.16395 2-II599 1.98505 1.80225 1.60095

0.00 0.28167

1.99792 0.00

1.86202 0.00

1.74926 0.00

1.96180 1.86163 1.71785 1.55398

0.22977 0.43071 0.58390 0.68387

1.83417 1.75601 1.64135 1.50695

0.19009 0.35949 0.49366 0.58681

1.72736 1.66532 1.57271 1.46173

1.25 1-3 1-35 1.4 ι-45

1.40579 1.23020 1.07907 0.95230 0.84752

0.84585 0.84688 0.81812 0.77067 0.71229

1.38926

0.73549 0.74858 0.73362 0.69957 0.65320

1.36782 1.23436 1.11262 1.00521 0.91249

0.64076 0.66157 0.65676 0.63346 0.59707

1.34386 0.55951

1.23587 1.09973 0.98247 0.88332

1-5 i-SS 1.6

0.76159 0.69149 0.63452 0.58846 o-SSiSi

0.64805 0.58116

0.54121 0.48093 0.41979 0.35856

0.83365 0.76736 0.71212 0.66653 0.62935

0.55229 0.50211 0.44869

0.44616 0.37967

0.80050 0.73188 0.67541 0.62928 0.59196

0.52227 0.49964 0.48281 0.47119 0.46436

0.31424 0.24990 0.18651 0.12390 0.06181

0.56223 0.53910 0.52183 0.50987 0.50284

0.17737 0.11796 0.05889

0-59953 0.57620 0.55872 0.54657 0.53941

0.46211

0.00

0.50052

0.00

0.53704 0.00

o 0.05 O.I O.I s

0.2

0.2s °-3 0-3S 0.4 0-45 0-5 0-5S 0.6

1.6s

1-7 1-75 1.8 1.85

1.9 ι-95 2.0

0.46211 0.46436 0.47119 0.48281 0.49964

0.00 0.06181 0.12390 0.18651 0.24990

0.52197 0.69623 0.80073

051350

0-59933

0.29765 0.23725

O.I59IO 0.30300 0.42040 0.50588

1.65462 0.00 1.63711 0.13449 1.58713 1.51148 1.41909

0.25755 0.36034 0.43803

1.22793 1.11962 I.02I95 0.93607

0.58492 0.58738 0.57227 0.54434

1.31896 1.21827 1.12222 1.03389 0.95480

0.48976 0.51757 0.52508 0.51635 0.49522

0.39344 0.33731

0.86173 0.79836 0.74493 0.70039 0.66377

0.50738 0.46428 0.4I7I4 0.36748 0.31618

0.88535 0.82533 0.77413 O-73io5 0.69534

0.46492 0.42806 0.38662 0.34205 0.29540

0.28085 0.22437 0.16804 0.11189 0.05590

0 . 6 3 4 1 9 0.26404 0.61094 0.2II44 0.59344 0.15863 0.58125 0.10576 0.57405 p.05288

0.66631 0.64336 0.62602 0.61390 0.60674

0.25741 0.19858 0.14925 0.09962 0.04984

0.57167

0.60437

°·00

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

tanh (0.6 + i o.6) = 1.00521 + i 0.63346. tanh (0.6 + i = 0.83365 - i 0.55229.

[76]

0.55951 0.50588 0.42040 0.30300 O.I59IO

0.00

TABLE I X .

HYPERBOLIC

TANGENTS,

t a n h (x + ig)

* = 0.85

* = 0.75

=u+iv.

χ = 0.9

CONTINUED

χ = 0.95

0.63515 0.63 749 0.64456 0.65649 0.67352

0.00 0.04684 0.09354 0.13997 0.18592

0.66403 0.66633 0.67325 0.68490 0.70148

0.00 0.04388 0.08758 0.13089 0.17357

0.69107 0.69330 0.70002 0.7II32 0.72736

0.00 0.04099 0.08176 0.12206 0.I6I60

0.71629 0.71845 0.72494 0.73582 0.75123

0.00 0.03820 0.07614 0.11354 0.15008

0.73978 0.00 0:74185 0.03551 0.74807 0.07073 0.75850 0 . 1 0 5 3 7 0 . 7 7 3 2 1 0.13906

0-45

0.69595 0.72420 0.75872 0.80005 0.84871

0.23112 0.27516 0.31749 0.35735 0.39368

0.72325 0.75051 0.78364 0.82300 0.86893

0.21528 0.25559 0.29392 0.32949 0.36128

0.74832 0.77446 0.80603 0.84328 0.88638

0.20001 0.23683 0.27146 0.30314 0.33091

0.77130 0.79620 0.82611 0.86117 0.90143

0.18537 0.21893 0.25018 0.27837 0.30261

0.79231 0 . I 7 I 4 3 0.81592 o. 20198 0.84411 0.23013 0.87695 0.25520 0.91438 0.27634

°·5 °-55 0.6 0.65 0.7

0.90515 0.96963 1.04203 1.12161 1.20665

0.42510 0.44978

0.92167 0.98122 1.04722 1.11872 1.19395

0.38798 0.40796 0.41926 0.42057 0.40661

0.93541 0.99018 1.05015 I.II427 I.I808I

0.35357 0.36966 0.37751 0.37527 0.36108

0.94681 0.99700 1.05136 1.10885 1.16765

0.32181 0.33469 0.33985 0.33580 0.32108

0.95624 1.00211 1-05129 I.I027I

0-7S 0.8 0.85 0.9 0-95

1.29416 1.37961 1.45701

0.42977 0.38084 0.31065 o.22051 0.11463

1.24723 I.3IOI7 1.36562 I.4093I

0.33335 0.29108 0.23434 0.16461

1.56023

0.37806 0.33238 0.26920 0.19000 0.09840

1-43735

0.08499

1.22572 1.28006 1.32742 1.36438 1.38796

0.29458 0.25573 0.20483 0.14330 0.07380

1.20569 1.25279 1.29344

Ι·5Ι945

1.27012 I.3433I 1.40862 1.46062 1.49428

1.0 1.05 1.1

1-57443 1.56023 I-SI94S 1.45701 1.37961

0.00 0.11463 0.22051 0.31065 0.38084

1-50594 1.49428 1.46062 1.40862 I-34331

0.00 0.09840 0.19000 0.26920 0.33238

1.44703 1-43735 1.40931 1.36562 1.31017

0.00 0.08499 0.16461 0-23434 0.29108

1.39606 1.38796 1.36438 1.32742 1.28006

0.00 0.07380 0.14330 0.20483 0.25573

1.35175 1.34490 1.32493 1.29344 1.25279

0.00 0.06438 0.12528 0.17968 0.22532

0.42977

i-45

1.29416 1.20665 1.12161 1.04203 0.96963

0.45847 0.46934 0.46543 0.44978

χ.27012 1.19395 1.11872 1.04722 0.98122

0.37806 0.40661 0.42057 0.41926 0.40796

1.24723 1.18081 1.11427 1-05015 0.99018

0.33335 0.36108 0.37527 0.37751 0.36966

1.22572 1.16765 1.10885 1.05136 0.99700

0.29458 0.32108 0.33580 0.33985 0.33469

1.20569 1.15485 I.I027I I.05I29 1.00211

0.26087 0.28588 0.30064 0.30593 0.30285

1-5 1-55 1.6 1.6s 1-7

0.90515 0.84871 0.80005 0.75872 0.72420

0.42510 0.39368 0-35735 0.31749 0.27516

0.92167 0.86893 0.82300 0.78364 0.75051

0.38798 0.36128 0.32949 0.29392 0.25559

0.93541 0.88638 0.84328 0.80603 0.77446

0.35357 0.33091 0.30314 0.27146 0.23683

0.94681 0.90143 0.86117 0.82611 0.79620

0.32181 0.30261 0.27837 0.25018 0.21893

0.95624 0.91438 0.87695 0.84411 0.81592

0.29259 0.27634 0.25520 0.23013 0.20198

I-7S 1.8 1.8S 1.9 ι-95

0-69595 0.67352 0.65649 0.64456 0.63749

0.23112 0.18592 O.I3997 0.09354 0.04684

0.72325 0.21528 0 . 7 0 1 4 8 0.17357 0.68490 0.13089 0.67325 0.08758 0.66633 0.04388

0.74832 0.72736 0.71132 0.70002 0.69330

0.20001 0.16160 0.12206 0.08176 0.04099

0.77130 0.75123 0.73582 0.72494 0.71845

0.18537 0.15008 011354 0.07614 0.03820

0.79231 0.77321 0.75850 0.74807 0.74185

0.17143 0.13906 0.10537 0.07073 0.03551

2.0

0.63515

0.00

0.66403

0.69107

0.00

0.71629

0.00

0.73978

0.00

o 0.05 O.I 0.15 0.2 0.25 °-3 °-3S 0.4

·Ι5 1.2 Ι

1-2S 1-3 I-3S 1.4

0-46543 0.46934 0.45847

0.00

Note.

N e g a t i v e q u a n t i t i e s a r e in h e a v y t y p e .

Examples,

t a n h (0.95 + i o) = 0.73978 +

io.

t a n h (0.9 + i 1.9) = 0.72494 — i 0.07614.

[77l

1-15485

0.29259 0.30285 0-30593 0.30064 0.28588

0.26087 0.22532 0.17968 1-32493 0 . 1 2 5 2 8 1.34490 0.06438

Table IX.

HYPERBOLIC TANGENTS,

tanh (x + iq) = u + iv. χ

χ = ι .05

Continued

= ι.IS

°S ι 15 2

0.761S9 0.76357 0.76950 0.77943 0.79341

0.00 .0.03293 0.06556 0.09757 0.12858

0.78181 0.78368 0.78932 0.79873 0.8119s

0.00 0.03048 0.06065 0.09016 0.11867

0.80050 0.80227 0.80760 0.81648 0.82893

0.00 0.02816 0.05599 0.08317 0.10932

0.81775 0.81943 0.82444 0.83279 0.84447

0.00 0.02597 0.05160 0.07658 0.10054

0.8336s 0.83522 0.83992 0.8477s 0.85867

0.00 0.02390 0.04748 0.07041 0.09233

25 3 35 4 45

0.81151 0.83377 0.86022 0.89086 0.92554

0.15821 0.18598 0.21133 0.23361 0.25205

0.82901 0.84991 0.87464 0.90311 0.9351s

0.14575 0.17096 0.19377 0.21356 0.22966

0.84495 0.86450 0.88753 0.91392 0.94344

0.I340S 0.15692 0.17742 0.I950I 0.20906

0.8594s 0.87768 0.89907 0.92345 0.95058

0.12310 0.14383 0.16226 0.17789 0.19017

0.87264 0.88958 0.90938 0.93186 0.95674

0.11288 0.13166 0.14823 0.16213 0.17287

5 55 6 65 7

0.96403 1.00585 1.05030 1.09632 i.i45 I ( 5

0.26580 0.27392 0.27542 0.26933 0.25486

0.97045 1.00852 1.04864 1.08984 1.13084

0.24130 0.24767 0.24798 0.24144 0.22747

0.97574

0.21892 I.0I034 0.22389 1.04654 0.22331 1.08342 0.21658 1.11984 0.20326

0.98010 1.01151 1.04415 1.07718 1.10957

0.19852 0.20236 0.20115 0.19441 0.18182

0.98368 I.OI2I7 1.04160 I.07II9 1.10003

0.17996 0.18288 0.18123 0.17461 0.16281

75 8 85 9 95

1.18715 1.22812 1.26319 ι.29017 1.30721

0.23145 0.19904 0.15812 0.10993 0.05638

1.17009 0.20572 1.2058s 0.17623 1.23624 0.1395s 1.25949 0.09677 1.27410 0.04956

1.15445

0.18315 1.18575 0.15637 I.2I2I9 0.12347 ι.23231 0.08544 1.24492 0.04369

1.1401S 1.16763 1.19346 1.20821 1.21914

0.16330 0.13902 0.10950 0.07563 0.03863

1.12709 0.14580 1.15129 0.1-2380

0 05 1 15 2

1.31304 0.00 1.30721 0.05638 1.29017 0.10993 1.26319 0.15812 1.22812 0.19904

1.27908 0.00 1.27410 0.04956 1.25949 0.09677

1.24922 0.00 1.24492 0.04369 1.23 231 0.08544 I.2I2I9 0.12347

1.2058s

0.17623

1.1857s

1.22286 0.00 1.21914 0.03863 1.20821 0.07563 1.19346 0.10950 1.16763 0.13902

1.19954 1.19631 1.18680 I.I7I52 1.15129

0.00 0.03424 0.06709 0.09730 0.12380

25 3 35 4 45

1.18715 0.2314s 1.14516 0.25486 1.09632 0.26933 1.05030 0.27542 1.00585 0.27392

1.17009 1.13084 1.08984 1.04864 1.00852

0.20572 0.22747 0.24144 0.24798 0.24767

I-IS44S 0.18315 1.11984 0.20326 1.08342 0.21658 1.04654 0.22331 1.01034 0.22389

1.14015 1.10957 1.07718 1.04415 1.01151

0.16330 0.18182 0.19441 0.20115 0.20236

1.12709 1.10003 I.07II9 1.04160 I.OI2I7

0.14580 0.1628t 0.17461 0.18123 0.18288

5 SS 6 65 7

0.96403 0.92554 0.89086 0.86022 0.83377

0.26580 0.25205 0.23361 0.21133

0.97045

0.97574

0.18598

0.90311 0.87464 0.84991

0.24130 0.22966 0.21356 0.19377 0.17096

0.21892 0.20906 0.I950I 0.17742 0.15692

0.98010 0.95058 0.92345 0.89907 0.87768

0.19852 0.19017 0.17789 0.16226 0.14383

0.98368 0.95674 0.93186 0.90938 0.88958

0.17996 0.17287 0.16213 0.14823 0.13166

75 8 85 9 95

0.81151 0.79341 0.77943 0.76950 0.76357

0.15821 0.12858 0.09757 0.06556 0.03293

0.82901 O.I457S 0.8119s 0.11867 0.79873 0.09016 0.78932 0.06065 0.78368 0.03048

0.84495 0.82893 0.81648 0.80760 0.80227

0.13405 0.10932 0.08317

0.85945 0.84447 0.83279 0.82444 0.81943

0.12310 0.10054 0.07658 0.05160 0.02597

0.87264 0.85867 0.8477s

0.11288 0.09233 0.07041 0.04748 0.02390

o

0.76159 0.00

0.78181 0.00

0.80050 0.00

1.23624 0 . 1 3 9 5 s

0.9351s

0.94344 0.91392 0.88753 0.86450

0-15637

0.05599

0.02816

0.8177s O'OO

Note.

Negative quantities are in heavy type.

Examples,

tanh (1.2 + i 0:75) = 1.12709 + ¿0.14580. tanh (1.2 + i 1.25) = 1 . 1 2 7 0 9 — ¿ 0 . 1 4 5 8 0 .

[78]

1.17152 0.09730 1.18680 0.06709 1.19631 0.03424

0.83992 0.83522

0.83365 0.00

TABLE IX.

HYPERBOLIC TANGENTS, 1-3

χ = 1.25

tanh (x + iq) = u + iv.

x = 1-35

χ = 1.4

CONTINUED

χ - 1.4s

0.86172 0.00 0.86309 0.02017 0.86719 0.04003 0.87398 0.05927 0.88344 0.07756

0.8853s

2

0.84828 0.00 0.84975 0.02197 0.85414 0.04363 0.86145 0.06464 0.87162 0.08468

0.8740s 0.00

OS 1 IS

0-87533 0.87913 0.88544 0.89421

0.01849 0.03668 0.05428 0.07097

0.88653 0.89006 0.89591 0.90401

25 3 35 4 45

0.88461 0.90032 0.91861 0.93928 0.96207

0.10339 0.12039 0.13528 0.1476s 0.15706

0.89548 0.90791 0.92686 0.94585 0.96669

0.09458 0.10997 0.12336 0.13437 0.14262

0.9053s 0.91875 0.93425 0.95166 0.97069

0.08644 0.10036 0.II240 O.I 2 221 O.I294S

0.91429 0.92663 0.94087 0.95680 0.97417

0.07892 0.09I5I 0.10234 O.I 1108 0.11745

0.92240 0.94680 0.96137 0.97719

0.09312 0.10092 0.10654

5 SS 6 65 7

0.98661 1.01244 1.03897 1.06550 1.09121

0.16307 0.16528 0.16332 0.15691 0.14591

0.98903 I.OI243 1.03634 1.06013 1.08308

0.14773 0.14937 0.14722 0.I4I09 0.13088

0.99101 I.OI2I9

O.I3381 O.I3499

0.I2II7 0.I2I99 0.II972 0.1142s 0.10555

0.99396 1.01133

0.10971 0.11026

1.0288s 1.04607 1.06248

0.10801 0.10288 0.09488

75 8 85 9

1.11521 1.13656 1.15434 1.16772 1.17603

0.13034 0.11042 0.08662 0.05964 0.03041

1.10439 1.12328 1.1389s 1.15070 X.15799

0.11665 0.09862 0.07724 0.05311 0.02706

0.09371 0.07896 0.06167 0.04232 0.02153

1.07756 1.09078 1.10166 1.10976 1.11476

0.08411 0.07077 0.05521 0.03785 0.01925

95

1.17885 0.00 1.17603 0.03041 1 . 1 6 7 7 2 0.05964 1 . 1 5 4 3 4 0.08662 1.13656 0.II042

1.16047 0.00 1 . 1 5 7 9 9 0.02706 1 . 1 5 0 7 0 0.05311 1 . 1 3 8 9 s 0.07724 1.12328 0.09862

I . I I 5 2 I 0.13034 I.09I2I O . I 4 5 9 I 1.06550 0.15691 1.03897 0.16332 1.01244 0.16528

1.10439 0.11665 1.08308 0.13088 1.06013 O . I 4 I 0 9 1.03634 0.14722 I.OI243 0.14937

0.98661 0.96207 0.93928 0.91861 0.90032

0.16307 0.15706 0.14765 0.13528 0.12039

0.98903 0.96669 0.94585 0.92686 0.90791

0.14773 0.14262 0.13437 0.12336 0.10997

85 9 95

0.88461 0.87162 0.86145 0.85414 0.84975

0.10339 0.08468 0.06464 0.04363 0.02197

0.89548 0.88344 0.87398 0.86719 0.86309

0.09458 0.07756 0.05927 0.04003 0.02017

o

0.84828 0.00

o 05 -I IS 2 25

3 35 4 45 5 SS 6 65

7 75

8

0.86172 0.00

Note. Examples,

0.00 0.01693 0-03357 0.04965 0.06486

0.89569 0.00 0.89678 0.01549 0.9000s 0.03070 0.9054s 0.04537 0.91293 0.05923 0.07199

0-93375 0.08338

I.05510 I.07560

O.I2693 O.II749

0.99263 I.0II8I 1-03125 1.05042 1.0687s

1.09457

0.10450 0.08820 0.06897 0.04738 0.02412

1.08565 1.10052 I.II277 I.I2I92 1.12758

0.00

1.12950 0.00 1 . 1 2 7 5 8 0.02153 1 . 1 2 1 9 2 0.04232 1 . 1 1 2 7 7 0.06167 1.10052 0.07896

1 . 1 1 6 4 6 0.00 1 . 1 1 4 7 6 0.01925 1.10976 0.03785 1.10166 0.05521 1.09078 0.07077 1.07756 1.06248 1.04607 1.02885 1.01133

0.08411 0.09488 0.10288 0.10801 0.11026

1-0337S 0.13275

1-11131 1.12512

1-13551 I.I4I92 I.I44IO I.I4I92

1-13551 I.I25I2 1-11131

0.02412 0.04738 0.06897 0.08820 0.10450 0.II749 0.12693 0.1327s 0.13499

1.08565

0.09371

1.0687s 1.05042 .1.03125 I.0II8I

0.10555 O.II425 O.II972 O.I2I99

0.93425 0.91875

0.13381 0.12945 0.I222I 0.11240 0.10036

0.99263 0.97417 0.95680 0.94087 0.92663

0.I2II7 0.17745 O.I 1 1 0 8 0.10234 0.09I5I

0.99396 0.97719 0.96137 0.94680

0.10971 0.10654 0.10092 0.09312

0-93375

0-08338

0-90535 0.89421 0.88544 0.87913 0-87533

0.08644 0.07097 0.05428 0.03668 0.01849

0.91429 0.90401 0.89591 0.89006 0.88653

0.07892 0.06486 0.04965 0.03357 0.01693

0.92240 0.91293 0.90545 0.90005 0.89678

0.07199 0.05923 0.04537 0.03070 0.01549

0.8853s

0·°°

0.89569 0.00

1-09457 1.07560 1.055 IO 1-03375 I.OI2I9 0.99I0I 0.97069 0.95166

0.87405 0.00

Negative quantities are in heavy type. tanh (1.4 + i 0.8) = 1.10052 + i 0.07896. t a n h ( 1 . 3 + i 1.3) = 1.08308 — i 0.13088.

[79]

TABLE I X .

HYPERBOLIC

χ = ι.s

TANGENTS,

» = I-SS

t a n h (x + ig) = u + iv.

χ = 1.6

X = 1.65

CONTINUED

X = 1.7

0 0.05 O.I 0.15 0.2

0.90515 0.90616 0.90917 0.91415 0.92104

0.00 0.01415 0.02804 0.04143 0.05404

0.91379 0.91471 0.91749 0.92208 0.92842

0.00 0.01292 0.02560 0.03779 0.04927

0.92167 0.92253 0.92508 0.92929 0.93511

0.00 0.01178 0.02334 0.03445 0.04488

0.92886 0.92964 0.93199 0.93586 0.94119

0.00 0.01074 0.02127 0.03138 0.04086

0.93541 0.93613 0.93828 0.94183 0.94671

0.00 0.00979 0.01937 0.02857 0.03718

0.25 0-3 0-3S 0.4 0.45

0.92975 0.94017 0.95212 0.96542 0.97983

0.06563 0.07593 0.08468 0.09165 0.09660

0.93641 0.94595 0.95689 0.96903 0.98214

0.05978 0.06909 0.07697 0.08320 0.08758

0.94245 0.95118 0.96117 0.97223 0.98415

0.05442 0.06284 0.06993 0.07551 0.07938

0.94791 0.95589 0.96501 0.97509 0.98592

0.04951 0.05712 0.06351 0.06850 0.07193

0.95285 0.96015 0.96846 0.97763 0.98748

0.04502 0.05190 0.05766 0.06213 0.06517

o-S 0-5S 0.6 0.65 0.7

0.99506 1.01076 1.02657 1.04204 1-05675

0.09933 0.09965 0.09746 0.09268 0.08534

0.99595 1.01016 I.o 2441 1.03834 I.05I52

0.08992 0.09008 0.08796 0-08353 0.07680

0.99668 1.00954 1.02240 1.03492 1.04676

0.08139 0.08142 0.07940 0.07530 0.06915

0.99728 1.00891 1.02052 1-03179 1.04242

0.07367 0.07361 0.07169 0.06791 0.06230

0.99777 1.00829 1.01877 1.02892 1.03847

0.06667 0.06655 0.06474 0.06126 0.05614

0-75 0.8 0.8 5 0.9 0-95

1.07022 1.08200 1.09167 1.09886 1.10329

0.07554 0.06349 0.04947 0.03390 0.01723

1.06357 1.07408 1.08269 1.08909 1.09302

0.06790 0.05700 0.04438 0.03038 0.01544

1.05755 1.06693 1.07461 1.08030 1.08380

0.06107 0.05121 0.03984 0.02726 0.01385

1.05209 1.06049 1.06734 1.07241 I.07554

0.05495 0.04604 0.03579 0.02448 0.01243

1.04714 1.05466 1.06079 1.06533 1.06811

0.04948 0.04142 0.03218 0.02200 0.01117

1.0 i.os 1.1 US 1.2

1.10479 1.10329 1.09886 1.09167 1.08200

°·00 0.01723 0.03390 0.04947 0.06349

1.09436 1.09302 1.08909 1.08269 1.07408

0.00 0.01544 0.03038 0.04438 0.05700

1.08500 1.08380 1.08030 1.07461 1.06693

0.00 0.01385 0.02726 0.03984 0.05121

1.07659 I.07554 1.07241 1.06734 1.06049

0.00 0.01243 0.02448 0.04604

1.06906 1.06811 1.06533 1.06079 1.05466

0.00 0.01117 0.02200 0.03218 0.04142

1.25 1-3 1-35 1.4 1 -45

1.07022 1-05675 1.04204 1.02657 1.01076

0.07554 0.08534 0.09268 0.09746 0.09965

1.06357 I.05I52 1.03834 I.0244I I.OIOI6

0.06790 0.07680 0.08353 0.08796 0.09008

I.05755 1.04676 1.03492 1.02240 1.00954

0.06107 0.06915 0.07530 0.07940 0.08142

1.05209 1.04242 ι.03ι79 1.02052 1.00891

0.0549S 0.06230 0.06791 0.07169 0.07361

1.04714 1.03847 1.02892 1.01877 1.00829

0.04948 0.05614 0.06126 0.06474 0.06655

1-5 1-55 1.6 1.6S 1-7

0.99506 0.97983 0.96542 0.95212 0.94017

0.09933 0.09660 0.09165 0.08468 0.07593

0.08992 0.08758 0.08320 0.07697 0.06909

0.99668 0.98415 0.97223 0.96117 0.95118

0.08139 0.07938 0.07551 0.06993 0.06284

0.99728 0.98592 0.97509 0.96501 0.95589

0.07367 0.07193 0.06850 0.06351 0.05712

0.99777 0.98748 0.97763 0.96846 0.96015

0.06667 0.06517 0.06213 0.05766 0.05190

1-75 1.8 ι. 85 1.9 1-95 2.0

0.92975 0.92104 0.9I4I5 0.90917 0.90616

0.06563 0.05404 0.04143 0.02804 O.OI4I5

0.99595 0.98214 0.96903 0.95689 0-94595 0.93641 0.92842 0.92208 0.91749 0.91471

0.05978 0.04927 0.03779 0.02560 0.01292

0.94245 0.93511 0.92929 0.92508 0.92253

0.05442 0.04488 0.03445 0.02334 0.01178

0.94791 0.94119 0.93586 0.93199 0.92964

0.04951 0.04086 0.03138 0.02127 0.01074

0.95285 0.94671 0.94183 0.93828 0.93613

0.04502 0.03718 0.02857 0.01937 0.00979

0.90515 0.00

0.91379 0.00

0.92167 0.00

0.92886 0.00

Note.

Negative quantities are in heavy type.

Examples,

tanh (1.7 + i 0.7) = 1.03847 -f- i 0.05614. tanh (1.6 + i 1.6) = 0.97223 — i 0.07551.

[80]

0.03579

0.93541 0.00

TABLE I X .

HYPERBOLIC

TANGENTS,

χ =

x = 1-75 0

0.94138 0.00 0.94204 0.00891 0.94400 0 . 0 1 7 6 3 0.94725 0.02600 0 . 9 5 1 7 2 0.03382

0.94681 0.94741 0.94921 0.95218 0.95626

0-95733 0.96399 0.97156 0.97991 0.98884

0.04092 0.04714 0.05233 0.05634 0.05904

°·5 0-5S o.6

0.99818 1.00769 1.01714 1.02629 1.03488

ο·75 0.8 0.8s 0.9

(x

1.85

+

iq)

=u+iv.

χ = 1.9

CONTINUED

χ = 1.9s

0 . 9 5 1 7 5 0.00 0.95230 0.00737 0.95394 0 . 0 1 4 5 9 0.95666 0.02149 0.96038 0.02793

0.95624 0.95674 0.95825 0.96072 0.96412

0.00 0.00670 0.01326 0.01952 0.02537

0.96032 0.96078 0.96215 0.96441 0.96750

0 . 9 6 1 3 9 0.03718 0.96746 0.04280 0-97435 0.04748 0.98194 0.05107 0.99005 0.05348

0.96506 0.97059 0.97686 0.98376 0.99111

0.03376 0.03885 0.04307 0.04629 0.04843

0.96838 0.97342 0.97912 0.98538 0.99206

0.03065 0.03525 0.03905 0.04195 0.04386

0 . 9 7 1 3 8 0.02782 0.97597 0.03198 0 . 9 8 1 1 6 0.03541 0.98684 0.03801 0.99289 0.03972

0.06034 0.06017 0.05848 0.05528 0.05061

0.99851 1.00711 1.01565 1.02389 1.03162

0.05461 0.05440 0.05283 0.04989 0.04564

0.99878 1.00656 1.01427 1.02170 1.02866

0.04942 0.04919 0.04773 0.04504 0.04118

0.99900 1.00604 1.01300 1.01970 1.02596

0.04472 0.04448 0.04313 0.04067 0.03716

0.99918 1.0055s 1.01184 1.01788 1.02352

0.04047 0.04022 0.03897 0.03673 0.03354

1.04266 1.04940 1.05489 1.05895 1.06143

0.04457 0.03729 0.02895 0.01978 0.01004

1.03861 1.04466 1.04958 1.05320

0.04016 0.03358 0.02606 0.01780 0.00903

1.03494 1.04037 1.04478 1.04804 1.05003

0.03621 0.03026 0.02347 0.01602 0.00813

1.03162 1.03650 1.04046 1.04337 1.04516

0.03265 0.02727 0.02115 0.01443 0.00732

1.02862 1.03300 1.03656 1.03918 1.04078

0.02946 0.02459 0.01906 0.0I30I 0.00659

1.0 ι.OS 1.1 1-IS

1.06228 1.06143 1.05895 1.05489 1.04940

0.00 0.01004 0.01978 0.02895 0.03729

1.05619 1.05543 1.05320 1.04958 1.04466

0.00 0.00903 0.01780 0.02606 0.03358

1.05070 1.05003 1.04804 1.04478 1.04037

0.00 0.00813 0.01602 0.02347 0.03026

1.04576 0.00 1 . 0 4 5 1 6 0.00732 1.04337 0.01443 1.04046 0.02115 1.03650 0.02727

I . 0 4 I 3 I 0.00 1.04078 0.00659 1 . 0 3 9 1 8 0.0I30I 1.03656 0.01906 1.03300 0.02459

I-2S 1-3 1-35

1.04266 1.03488 1.02629 1.01714 1.00769

0.04457 0.05061 0.05528 0.05848 0.06017

1.03861 0.04016 1 . 0 3 1 6 2 0.04564 1.02389 0.04989 1 . 0 1 5 6 5 0.05283 1 . 0 0 7 1 1 0.05440

1.03494 0.03621 1.02866 0.04118 ι . 0 2 1 7 0 0.04504 1 . 0 1 4 2 7 0.04773 1.00656 0.04919

1.03162 1.02596 1.01970 1.01300 ι .00604

0.03265 0.03716 0.04067 0.04313 0.04448

1.02862 1.02352 1.01788 1.01184 1.00555

0.02946 0.03354 0.03673 0.03897 0.04022

0.99818 0.98884 0.97991 0.97156 0.96399

0.06034 0.05904 0.05634 0.05233 0.04714

0.99851 0.05461 0.99005 0.05348 0.98194 0.05107 0-9743S 0.04748 0.96746 0.04280

0.99878 0.99111 0.98376 0.97686 0.97059

0.04942 0.04843 0.04629 0.04307 0.03885

0.99900 0.04472 0.99206 0.04386 0.98538 0.04195 0 . 9 7 9 1 2 0.03905 0.97342 0.03525

0.99918 0.99289 0.98684 0.98116

0-97597

0.04047 0.03972 0.03801 0.03541 0.03198

0-95733

1 -95

0.95172 0.94725 0.94400 0.94204

0.04092 0.03382 0.02600 0.01763 0.00891

0.96139 0.03718 0.95626 0.03074 0 . 9 5 2 1 8 0.02364 0.94921 0.01604 0.94741 0.00811

0.96506 0.03376 0.96038 0.02793 0.95666 0.02149 0-95394 0.01459 0.95230 0.00737

0.96838 0.03065 0.96412 0.02537 0.96072 0.01952 0.95825 0.01326 0.95674 0.00670

0.97138 0.96750 0.96441 0.96215 0.96078

0.02782 0.02303 0.01773 0.01204 0.00609

2.0

0.94138

0.00

0.94681

°-95 I 7S 0.00

0.95624.0.00

0.96032

0.00

0.05

O.I

0.15 0.2 0.2s

°-3 °·35 °·4 0-4S

0.65 0.7

0-9S

1.2

1.4

I-4S 1-5 I-S5 1.6 1.65

1-7

1-75 1.8 1.8s 1.9

Note.

1-05543

0.00 0.00811 0.01604 0.02364 0.03074

tanh

0.00

Negative quantities are in heavy type.

Examples, tanh (1.85 + i 0.85) = 1.04478 + »0.02347. t a n h ( 1 . 9 5 + i 1 . 2 5 ) = 1.02862 — i 0.02946.

[81]

0.00 0.00609 0.01204 0.01773 0.02303

TABLE I X .

X =

o 0.05 O.I 0.15

HYPERBOLIC

TANGENTS,

t a n h ( * + iq)

χ = 2.05

2.0

= u + iv.

CONTINUED

χ = 2.15

0.96403 0.96445 0.96570 0-96775 0.97058

0.00 O.OOSS3 0.01094 0.01610 0.02090

0.96740 0.96778 0.96892 0.97079 0-97336

0.00 0.00502 0.00993 0.01461 0.01897

0.97045 0.97080 0.97184 0.97354 0.97588

0.00 0.00456 0.00901 0.01326 0.01721

0.97323 0.97354 0.97449 0.97604 0.97816

o-oo 0.00413 0.00817 0.01203 0.01561

0.97574 0.97603 0.97689 0.97830 0.98023

0.00 0.0037s 0.00741 0.01091 0.01415

0.4S

0.97411 0.97827 0.98299 0.98815 0.99364

0.02524 0.02900 0.03209 0.03444 0.03596

0.97657 0.98036 0.98464 0.98932 0.99430

0.02289 0.02630 0.02909 0.03120 0.03256

0.97880 0.98224 0.98613 0.99037 0.99489

0.02076 0.02384 0.02636 0.02826 0.02948

0.98081 0.98394 0.98747 0.99132 0.99541

0.01882 0.02161 0.02388 0.025S9 0.02669

0.98264 0.98548 0.98868 0.99217 0.99587

0.01706 0.01958 0.02163 0.02317 0.02416

o-5 o-55 0.6 0.6s 0.7

0-99933 1.00509 1.01077 1.01623 1.02131

0.03662 0.03638 0.03523 0.03318 0.03028

0-9994S 1.00466 1.00979 1.01471 1.01930

0.03314 0.03290 0.03184 0.02998 0.02734

0-99955 1.00426 1.00890 1-OI334 1.01748

0.02998 0.02976 0.02878 0.02709 0.02469

0.99963 1.00389 1.00808 1.01210 1.01583

0.02713 0.02691 0.02602 0.02448 0.02231

0.99970 1.00355 1.00734 1.01097 I.OI434

0.02455 0.02434 0.02353 0.02212 0.02015

0-7S o.S 0.8s 0.9 0-9S

1.02589 0.02658 1.02984 0.02218 1.03304 0.01719 Ι·°3539 0.01172 1.03683 0.00594

1.02343 1.02698 1.02986 1.03197 1.03327

0-02399 0.02001 0.01550 0.01057 0.00536

1.02120 1.02440 1.02699 1.02889 1.03005

0.02166 0.01806 O- o i 399 0.00954 0.00483

1.01919 1.02207 1.02440 1.02611 1.02715

0.01956 0.01631 0.01262 0.00861 0.00436

1.01736 1.01996 1.02206 1.02360 1.02454

0.01767 0.01472 0.0II40 0.00777 0.00394

1.0 1-OS 1.1 I-IS 1.2

1.03731 0.00 1.03683 0.00594 Ι ·°3539 0.01172 1.03304 0.01719 1.02984 0.02218

1.03370 1.03327 1.03197 1.02986 1.02698

0.00 0.00536 0.01057 0.01550 0.02001

1.03045 1.03005 1.02889 1.02699 1.02440

0.00 0.00483 0.00954 0.01399 0.01806

1.02751 ι.02715 1.02611 1.02440 1.02207

0.00 0.00436 0.00861 0.01262 0.01631

1.02486 1.02454 1.02360 1.02206 1.01996

0.00 0.00394 0.00777 0.0II40 0.01472

I.2S 1-3 I-3S 1.4 i-45

1.02589 0.02658 1.02131 0.03028 1.01623 0.03318 1.01077 0.03523 1.00509 0.03638

1.02343 1.01930 1.01471 1.00979 1.00466

0.02399 0.02734 0.02998 0.03184 0.03290

1.02120 1.01748 ι.01334 1.00890 1.00426

0.02166 0.02469 0.02709 0.02878 0.02976

1.01919 1.01583 1.01210 1.00808 1.00389

0.01956 0.02231 0.02448 0.02602 0.02691

1.01736 I.OI434 I.OIO97 I.OO734 I.OO355

0.01767 0.0201g 0.02212 O.O2353 O.O2434

i-S i-SS 1.6 ι.65 1-7

0-99933 0.99364 0.98815 0.98299 0.97827

0.03662 0.03596 0.03444 0.03209 0.02900

0.99945 0.99430 0.98932 0.98464 0.98036

0.03314 0.03256 0.03120 0.02909 0.02630

0.99955 0.99489 0.99037 0.98613 0.98224

0.02998 0.02948 0.02826 0.02636 0.02384

0.99963 0.99541 0.99132 0.98747 0.98394

0.02713 0.02669 0.02559 0.02388 0.02161

O.9997O O.99587 O.99217 O.98868 O.98548

O.O2455 O.O2416 O.O2317 O.O2163 O.OI958

1-75 1.8 ι.8S 1.9 i-95

0.97411 0.97058 0.96775 0.96570 0.96445

0.02524 0.02090 0.01610 0.01094 0.00553

0.97657 0.97336 0.97079 0.96892 0.96778

0.02289 0.01897 0.01461 0.00993 0.00502

0.97880 0.97588 0-97354 0.97184 0.97080

0.02076 0.01721 0.01326 0.00901 0.00456

0.98081 0.01882 0.97816 0.01561 0.97604 0.01203 0.97449 0.00817 0.97354 0.00413

O.98264 0.98023 O.9783O O.97689 0.97603

0.01706 O.OI415 O.OIO9I O.OO74I O.OO375

2.0

0.96403 0.00

0.97045

0.00

0.97323 0.00

0.97574 0.00

0.'2

0.2S 0-3 °-35 0.4

0.96740 0.00

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

tanh (2.2 + i o) = o 97574 + i o. tanh (2.15 + i 1.15) = 1.02440 — ¿0.01262.

[82]

TABLE I X .

HYPERBOLIC

χ = 2.25

TANGENTS,

χ = 2.3

»=

t a n h (x + ig) = u + iv. χ -- 2.4

2-35

CONTINUED χ = 2.45

0.97803 0.97829 0.97907 0.9803 s 0.98210

0.00 0.00340 0.00672 0.00989 0.01283

0.98010 0.98034 0.98105 0.98221 0.98380

0.00 0.00308 0.00610 0.00897 0.01163

0.98197 0.98219 0.98283 0.98389 0.98534

0.00 0.00280 0.00553 0.00813 0.01054

0.98367 0.98387 0.98446 0.98541 0.98673

0.00 0.00253 0.00501 0.00736 0.00955

0.98522 0.98340 0.98592 0.98680 0.98799

0.00 0.00230 0.00454 0.00667 0.00865

2S 3 35 4 45

0.98429 0.98687 0.98977 0.99294 0.99629

0.01547 0.01774 0.01960 0.02098 0.02187

0.98579 0.98813 0.99076 0.99363 0.99667

0.0I40I 0.01607 0.0177s 0.01900 0.01979

0.98714 0.98926 0.99165 0.99425 0.99700

0.01270 0.01456 0.01607 0.01720 O.OI79I

0.98836 0.99029 0.99245 0.99481 0.99730

0.0II50 O.OI3I9 0.01456 0.01557 0.01621

0.98947 0.991 21 0.99317 0.99531 0-99757

0.01042 0.01194 0.01318 0.01410 0.01468

5 55

0.99975 1.00324 1.00666 ι .00994 1.01298

0.02222 0.02202 0.02127 0.02000 0.01821

0.99980 1.00295 1.00604 1.00900 I.OII75

0.02010 0.01992 0.01924 0.01808 0.01646

0.99984 1.00269 1.00549 I.008I6 1.01064

0.01819 0.01802 0.01740 0.01634 0.01487

0.99986 1.00245 1.00498 1.00739 1.00963

0.01646 0.01630 0.01573 0.01478 0.0134s

0.99989 1.00222 1-00451 1.00693 1.00872

0.01489 0.01474 0.01423 0.01336 0.01216

1.01571 1.01805 1.ο 1994 I.02I33 1.02218

0.01596 0.01330 0.01029 0.00701 0.00355

X.OI42I 1.01632 1.01803 1.01928 1.02006

0.01442 O.OI2OI 0.00929 0.00633 0.00322

1.01286 1-01477 1.01631 I.OI744 1.01812

0.01303 0.01085 0.00839 0.00572 0.00290

1.01164 1.01336 1-01475 1.01576 1.01639

0.01178 0.00981 0.00758 0.00517 0.00262

1.01053 1.01208 1.01333 1.01425 1.01482

0.01064 0.00886 0.00685 0.00467 0.00237

o 05 -I IS 2

1.02247 1.02218 I.02I33 I.OI994 1.01805

0.00 0.00355 0.00701 0.0102g 0.01330

1.02031 0.00 1.02006 0.00322 1.01928 0.00633 1.01803 0.00929 1.01632 0.0I20I

1.01836 1.01812 1.01744 1.01631 I.OI477

0.00 0.00290 0.00572 0.00839 0.01085

1.01659 1.01639 1.01576 I-OI47S 1.01336

0.00 0.00262 0.00517 0.00758 0.00981

1.01500 1.01482 I.OI425 I-OI333 1.01208

0.00 0.00237 0.00467 0.00685 0.00886

25 3 35 4 45

I.OI57I 1.01298 1.00994 1.00666 1.00324

0.01596 0.01821 0.02000 0.02127 0.02202

I.OI42I I.OII75 1.00900 1.00604 1.00295

1.01286 1.01064 1.00816 1.00549 1.00269

0.01303 0.01487 0.01634 0.01740 0.01802

1.01164 1.00963 1.00739 1.00498 1.00245

0.01178 0.01345 0.01478 0.01573

1-01053 1.00872 ι .00693 1-00451 1.00222

0.01064 0.01216 0.01336 0.01423 0.01474

5 55

0-99975 0.99629 0.99294 0.98977 0.98687

0.02222 0.02187 0.02098 0.01960 0.01774

0.99980 0.02010 0.99667 0.01979 0.99363 0.01900 0.99076 0 . 0 1 7 7 5 0.98813 0.01607

0.99984 0.99700 0.99425 0.99165 0.98926

0.01819 O.OI79I 0.01720 0.01607 0.01456

0.99986 0.99730 0.99481 0.99245 0.99029

0.01646 0.01621 0.01557 0.01456 O.OI3I9

0.99989 0-99757 0-9953I 0.99317 0.99121

0.01489 0.01468 0.01410 0.01318 0.01194

85 9 95

0.98429 0.98210 0.98035 0.97907 0.97829

0.01547 0.01283 0.00989 0.00672 0.00340

0.98579 0.98380 0.98221 0.9810s 0.98034

0.98714 0.98534 0.98389 0.98283 0.98219

0.01270 0.01054 0.00813 0.00553 0.00280

0.98836 0.98673 0.98541 0.98446 0.98387

0.01150 0.00955 0.00736 0.00501 0.00253

0.98947 0.98799 0.98680 0.98592 0.98540

0.01042 0.00865 0.00667 0.00454 0.00230

o

0.97803 0.00

05 1 15 2

6

65 7 75

8

85 9 95

6

65 7 75

8

0.01442 0.01646. 0.01808 0.01924 0.01992

0.0I40I 0.01163 0.00897 0.00610 0.00308

0.98010 0.00

0.98197 0.00

0.01630

0.98367 0.00

Note.

Negative quantities are in heavy type.

Examples,

tanh (2.25 + i 0.25) = 0.98429 + ¿0.01547. tanh (2.45 + i 1.45) = 1.00222 — i 0.01474.

[83]

0.98522 0.00

TABLE I X .

χ =

OS 1 IS 2

o . 8661 0.98678 0.98726 0.98805 0.98913

HYPERBOLIC

2.s

TANGENTS,

* = 2-55

0.00 0.00208 0.00411 0.00605 0.00784

X = 2.6

= u +

χ =

iv.

2.6s

CONTINUED

X = 2.7

0.98788 0.00 0.98803 0.00189 0.00373 o. 0.00548 0.99016 0.00710

0.98903 0.98916 0.98956 0.99021 0.99109

0.00 0.00171 0.00337 0.00496 0.00643

0.99007 0.99019 0-99055 0.99113 0.99194

0.00 0.0015s 0.00306 0.00449 0.00582

0.99I0I 0.99II2 0.99144 0.99198 0.99270

0.00 0.00140 0.00277 0.00407 0.00527

25 3 35 4 45

0.00944 0.01082 0-99383 o . 0 1 1 9 3 0.99576 0.01276 0.99781 0 . 0 1 3 2 8

0.99138 0.99281 0.99442 0.99617 0.99802

0.00855 0.00979 0.01080 0.0115s 0.01202

0.99220 0.00774 0.99350 0.00887 0.99495 0.00978 0.99654 0.01046 0.99822 0.01088

0.99294 0.99412 0.99544 0.99688 0.99839

0.00701 0.00803 0.00886 0.00947 0.00985

0.99361 0.99468 0.99588 0.99718

5 55 6 65 7

0.99991 1.00202 ι .00409 1.00606 1.00789

0.01348 0.01334 0.01287 0.01208 0.01099

0.99993 1.00184 ι.003 71 1.00549 1.00714

0.01219 0.01207 0.01164 0.01093 0.00994

0.99994 0 . 0 1 1 0 3 1.00167 0.01092 1.00336 0.01053 1.00498 1.00647

0.99995 1.00151 1.00304 1.00450 1.00585

0.00998 0.00988 0.00952 0.00894 0.00812

0.99996 I.OOI37 1.00276 1.00408 1.00530

0.00903 0.00893 0.00862 0.00808 0.0073s

75 8 85 9 95

1.00953 1 . ο 1093 1 .0 1 2 0 6 1.01289 i . o 1340

0.00962 0.00801 0.00619 0.00422 0.00214

1.00862 1.00989 1.01091 1.01166 1.01211

0.00870 0.00726 0.00560 0.00381 0.00193

1.00780 1.00895 1.00987 1.01054 1.01096

0.00786 0.00654 0.00506 0.00345 0.00175

ι .00706 1.00809 1.00893 1.00954 1.00991

0.00711 0.00592 0.00457 0.00312 0.00158

1.00639 1.00732 1.00807 1.00862 1.00896

0.00643 0.00535 0.00413 0.00282 0.00143

0

1-OI357 1.01340 1.01289 1 .0 1 2 0 6 I . ο 1093

0.00

1.01227 1.01211 1.01166 1.01091 1.00989

0.00 0.00193 0.00381 0.00560 0.00726

I.OHIO 0.00 1.01096 0.00175 I . 0 I 0 5 4 0.00345 1.00987 0.00506 1.00895 0.00654

I.0I003 I.0099I 1.00954 1.00893 1.00809

0.00 0.00158 0.00312 0.00457 0.00592

1.00907 1.00896 1.00862 1.00807 1.00732

0.00

0.00214 0.00422 0.00619 0.00801

25 3 35 4 45

1.00953 1.00789 1.00606 1.00409 1.00202

0.00962 0.01099 0.01208 0.01287 0.01334

1.00862 0.00870 1 . 0 0 7 1 4 0.00994 1.00549 0.01093 1.003 71 0.01164 1.00184 0.01207

1.00780 1.00647 1.00498 1.00336 1.00167

0.00786 0.00898 0.00988 0.01053 0.01092

1.00706 1.0058s 1.00450 1.00304 1.00151

0.007II 0.00812 0.00894 0.00952 0.00988

1.00639 1.00530 1.00408 1.00276 I.OOI37

0.00643 0.00735 0.00808 0.00862 0.00893

5 55 6 65 7

0.99991 0.01348 0 . 9 9 7 8 1 0.01328 0 . 9 9 5 7 6 0.01276 0.01193 0.01082

0.99994 0.99822 0.99654 0-99495 0-99350

0.01103 0.01088 0.01046 0.00978 0.00887

0.99995 0.99839 0.99688 0.99544 0.99412

0.00998 0.00985 0.00947 0.00886 0.00803

0.99996

0.99383 0.99205

0.99993 0.01219 0.99802 0.01202 0.99617 0.01155 0.99442 0.01080 0.99281 0.00979

0-99855 0.99718 0.99588 0.99468

0.00903 0.00891 0.00857 0.00802 0.00727

75 8 85 9 95

0.99047 0.98913 0.98805 0.98726 0.98678

0.00944 0.00784 0.00605 0.00411 0.00208

0.99138 0.99016 0.98918 0.98846 0.98803

0.00855 0.00710 0.00548 0.00373 0.00189

0.99220 0.99109 0.99021 0.98956 0.98916

0.00774 0.00643 0.00496 0.00337 0.00171

0.99294 0.00701 0 . 9 9 1 9 4 0.00582 0 . 9 9 I I 3 0.00449 0.99055 0.00306 0.99019 0.00155

0.99361 0.00635 0.99270 0.00527 0.99198 0.00407 0 . 9 9 1 4 4 0.00277 0 . 9 9 1 1 2 0.00140

0.98661

0.00

0.98788

0.00

0.98903

0.00

0.99007

0.99101

OS 1 15 2

0.99047 0.99205

t a n h ( χ + ig)

Note. Examples,

0.00

N e g a t i v e q u a n t i t i e s a r e in h e a v y t y p e . t a n h (2.60 + i 0.35) = 0.99495 + ¿0.00978. t a n h (2.70 + i 1.35) = 1.00408 — i 0.00808.

[84]

0.0063 s 0.00727 0.00802 0.00857 0-99855 0.00891

0.00143 0.00282 0.00413 0.00535

0.00

TABLE

IX.

HYPERBOLIC TANGENTS, X — 2.80

* = 2.75

tanh

(x + iq) = u + iv.

* = 2.8s

χ = 2.90

CONTINUED

* =

2.9s

°5 -I IS 2

0.99186 0.99196 0.99225 0.99274 0.99340

0.00 0.00127 0.00251 0.00368 0.00477

0.99263 0.99272 0.99299 0.99343 0.99403

0.00 0.00115 0.00227 0.00334 0.00432

0-99333 0.99341 0.99365 0.99405 0.99459

° °° 0.00104 0.00206 0.00302 0.00391

0.99396 0.99404 0.99426 0.99462 0.99511

0.00 0.00094 0.00186 0.00273 0.00354

0.99454 0.99460 0.99480 °-995I3 0-99557

0.00 0.00085 0.00168 0.00248 0.0032I

25 3 35 4 45

0.99422 0.99519 0.99627 0.9974s 0.99869

0.00575 0.00658 0.00726 0.00775 0.00806

0.99477 0.99564 0.99663 0.99769 0.99882

0.00519 0.00596 0.00657 0.00702 0.00730

0.99527 0.99606 0.99695 0.99791 0.99893

0.00471 0.00539 0.00594 0.00635 0.00660

0.99572 0.99644 0.99724 0.99811 0.99904

0.00426 0.00488 0.00538 0.00575 0.00598

0.99613 0.99678 0-99750 0.99830 0.99913

0.00386 0.00442 0.00487 0.00520 0.0054I

5 55. 6 65 7

0.99997 ι.00125 1.00250 1.00369 1.00479

0.00817 0.00808 0.00779 0.00731 0.00664

0.99997 1.00113 1.00226 1.00334 1.00434

0.00740 0.00731 0.00705 0.00661 0.00601

0.99998 1.00103 1.00205 χ. 00303 1.00393

0.00669 0.00662 0.00638 0.00598 0.00544

0.99998 1.00093 1.00186 1.00274 1.00355

0.00606 0.00599 0.00577 0.00541 0.00480

0.99999 1.00084 I.OOI68 1.00248 1.00322

0.00548 O.00542 0.00522 0.00489 0.00445

75

1.00578 1.00662 1.00730 1.00780 1.00810

0.00581 0.00484 0.00374 0.00255 0.00129

1.00523 1.00599 1.00661 1.00706 1.00733

0.00525 0-00437 0.00338 0.00230 0.00117

1.00473 1.00542 1.00598 1.00638 1.00663

0.00475 0.00396 0.00306 0.00208 0.00105

1.00428 1.00490 1.00540 1.00577 1.00600

0.00430 0.00358 0.00276 0.00188 0.0009s

1.00387 I.00444 1.00489 1.00522 1.00542

0.00389 0.00324 0.00250 0.00170 0.00086

1.00821 I.008IO 1.00780 1.00730 1.00662

0.00 0.0012g 0.0025s 0.00374 0.00484

1.00742 1.00733 1.00706 ι.00661 1.00599

0.00 0.00117 0.00230 0.00338 0.00437

1.00671 1.00663 1.00638 1.00598 1.00542

0.00 0.00105 0.00208 0.00306 0.00396

1.00607 1.00600 1.00577 1.00540 1.00490

0.00 0.00095 0.00188 0.00276 0.00358

1.00549 1.00542 1.00522 1.00489 1.00444

0.00 0.00086 0.00170 0.00250 0.00324

•25 •3 •35 •4 •45

1.00578 1.00479 1.00369 1.00250 1.00125

0.00581 0-00664 0.00731 0.00779 0.00808

1.00523 1.00434 1.00334 1.00226 1.00113

0.00525 0.00601 0.00661 0.00705 0.00731

1.00473 1.00393 1.00303 1.00205 1.00103

0.00475 0.00544 0.00598 0.00638 0.00662

1.00428 1.00355 1.00274 1.00186 1.00093

0.00430 0.00480 0.00541 0.00577 0.00599

1.00387 1.00322 1.00248 1.00168 1.00084

0.00389 0.00445 0.00489 0.00522 0.00542

•5 •55 .6 •65 •7

0.99997 0.99869 0.9974s 0.99627 0.99519

0.00817 0.00806 0.00775 0.00726 0.00658

0.99997 0.99882 0.99769 0.99663 0.99564

0.00740 0.00730 0.00702 0.00657 0.00596

0.99998 0.99893 0.99791 0.9969s 0.99606

0.00669 0.00660 0.00635 0.00594 P.00539

0.99998 0.99904 0.99811 0.99724 0.99644

0.00606 0.00598 0.00575 0.00538 0.00488

0.99999 0.99913 0.99830 0.99750 0.99678

0.00548 0.00541 0.00520 0.00487 0.00442

•75 .8 •85 •9 •95

0.99422 0.99340 0.99274 0.99225 0.99196

0.00575 0.00477 0.00368 0.00251 0.00127

0.99477 0.00519 0.99403 0.00432 0-99343 0.00334 0.99299 0.00227 0.99272 0.00115

0.99527 0.99459 0.99405 0 ·99305 0.99341

0.00471 0.00391 0.00302 0.0020Ö 0.00104

0.99572 0.99511 0.99462 0.99426 0.99404

0.00426 0.00354 0.00273 0.00186 0.00094

0.99613 0.99557 0.99513 0.99480 0.99460

0.00386 0.00321 0.00248 0.00168 0.00085

.0

0.99186

0.00

0.99263 0.00

0-99333

°·00

0.99396 0.00

-8

85 9 •95 .0 •05 .1 •IS .2

Note.

N e g a t i v e quantities are in h e a v y type.

Examples,

tanh (2.9 + i 0.9) = 1.00577 + i 0.00188. tanh (2.9s + i 1.95) = 0.99460 — i 0.00085.

[ 85 ]

0.99454 0.00

TABLE I X .

HYPERBOLIC

* = 3-°S

χ = 3.0

o 0.0s O.I 0.15

TANGENTS,

tanh (x + iq)

χ = 3.10

=u+iv.

» = 3-iS

CONTINUED

χ = 3.20

0.99505 0.99512 0.99530 0.99559 0.99599

0.00 0.00077 0.00153 0.00224 0.00290

0.99552 0.99558 0.99574 0.99601 0.99637

0.00 0.00070 0.00138 0.00203 0.00263

0.99600 0.99615 0.99639 0.99672

0.00 0.00063 0.00125 0.00184 0.00238

0.99633 0.99638 0.99651 0.99673 0.99703

0.00 0.00057 0.00113 0.00166 0.00215

0.99668 0.99672 0.99684 0.99704 0.99731

0.00 0.00052 0.00102 0.00150 0.00195

0.4 o-45

0.99649 0.99708 0.99774 0.99846 0.99921

0.00349 0.00400 0.00441 0.00471 0.00489

0.99683 0.99736 0.99796 0.99861 0.99929

0.00316 0.00362 0.00399 0.00426 0.00443

0.99713 0.99761 0.99815 0.99874 0.99936

0.00286 0.00328 0.00361 0.00386 0.00401

0.99740 0.99784 0-99833 0.99886 0.99942

0.00259 0.00297 0.00327 0.00349 0.00363

0.99765 0.99805 0.99849 0.99897 0.99947

0.00234 0.00268 0.00296 0.00316 0.00328

o-S o-SS 0.6 0.65 0.7

0.99999 1.00076 ι.00152 1.00224 1.00291

0.00496 0.00490 0.00472 0.00443 0.00402

0.99999 ι .00069 1.00138 I.00203 1.00263

0.00449

0.99999 1.00063 1.00125 1.00184 1.00238

0.00406 0.00401 0.00387 0.00362 0.00329

0.99999 1.00057 1.00113 I.OOI66 1.00216

0.00367 0.00363 0.00350 0.00328 0.00298

0.99999 I.0005I 1.00102 I.00I5I 1.0019s

0.00332 0.00328 0.00316 0.00297 0.00269

1.00351 1.00401 1.00443 1.00473 1.00491

0.00352 0.00293 0.00226 0.00154 0.00078

1.00317 0.00318 1.00363 0.00265

1.00287 1.00329 1.00362 1.00387 1.00402

0.00288 0.00239 0.00185 0.00126 0.00064

1.00260 1.00297 1.00328 1.00350 1.00363

0.00260 0.00217 0.00167 0.00114 0.00058

1.0023s 1.00269 1.00297 I.003I7 1.00329

0.00236 0.00196 O.OOI5I 0.00103 0.00052

1.00407 1.00402 1.00387 1.00362 1.00329

0.00 0.00064 0.00126 0.00185 0.0023g

1.00368 1.00363 1.00350 1.00328 1.00297

0.00 0.00058 0.00114 0.00167 0.00217

1-00333 1.00329 I.003I7 1.00297 1.00269

0-00 0.00052 0.00103 0.00I5I 0.00196

1.00287 1.00238 1.00184 I.OOI25 1.00063

0.00288 0.00329 0.00362 0.00387 0.00401

1.00260 1.00216 I.OOI66 1.00113 1.00057

0.00260 0.00298 0.00328 0.00350 0.00363

1.00235 1.00195 I.00I5I ι.00102

0.00236 0.00269 0.00297 0.00316

1.00051

0.00328

0.2

0.25 °-3

0-35

0.75 0.8 0.85 0.9 0-9S

1.0 i.os 1.1 1-iS

1.00497 0.00 1.00491 0.00078 1.00473

0.00154

I.00400

0 -°°443

0.00427 0.0040I 0.00364

0.00205

1.00426 O.OOI39 I.00444 0.0007I I.00450 I.00444 1.00426 I.OO4OO

o.oo 0.00071 O.OOI39 0.00205

1.00363

0.00265

0-9959S

1.2

1.00443 0.00226 1.00401 0.00293

1.25 1-3 1-35 1.4 i-45

1.00351 1.00291 1.00224 1.00152 1.00076

0.00352 0.00402 0.00443 0.00472 0.00490

1.00263 1.00203 1.00138 1.00069

0.00364 0.00401 0.00427 0.00443

i-S 1-55 1.6

0.99999 0.99921 0.99846 0.99774 0.99708

0.00496 0.00489 0.00471 0.00441 0.00400

0.99999 0.99929 0.99861 0.99796 0.99736

0.00449 0.00443 0.00426 0.00399 0.00362

0.99936 0.99874 0.99815 0.99761

0.00406 0.00401 0.00386 0.00361 0.00328

0.99999 0.99942 0.99886 0.99833 0.99784

0.00367 0.00363 0.00349 0.00327 0.00297

0.99999 0.99947 0.99897 0.99849 0.99805

0.00332 0.00328 0.00316 0.00296 0.00268

0.99649 0.99599 0.99559 0.99530 0.99512

0.00349 0.00290 0.00224 0.00153 0.00077

0.99683 0.99637 0.99601 0.99574 0.99558

0.00316 0.00263 0.00203 0.00138 0.00070

0.99713 0.99672 0.99639 0.99615 0.99600

0.00286 0.00238 0.00184 0.00125 0.00063

0.99740 0.99703 0.99673 0.99651 0.99638

0.00259 0.00215 0.00166 0.00II3 0.00057

0.99765 0.99731 0.99704 0.99684 0.99672

0.00234 0.00195 0.00150 0.00102 0.00052

0.99505

0.00

0.99552 0.00

ι.6S 1-7 1-75 1.8 1.8S 1.9 1-95 2.0

ι.00317 0.00318

0.99999

0.99595 0.00

0.99633 0.00

Note.

Negative quantities are in heavy type.

Examples,

tanh (3.0 + i LOO) = 1.00497 + i o . tanh (3.0 + i 1.50) = 0.99999 — i 0.00496.

[86]

0.99668 0.00

TABLE I X .

HYPERBOLIC

TANGENTS,

x = 3-30

» = 3-25

t a n h (x + ig) = u+

* = 3-35

χ =

iv.

3-4°

CONTINUED

* = 3-45

0.99700 ο.οο Ο.997°4 0.00047 0.99715 0 . 0 0 0 9 3 0-99732 0.00136 0-997S7 0.00176

0-99732 0.00042 0.99742 0 . 0 0 0 8 4 0-99758 0.00123 0.99780 0.00160

0.99754 ο . ο ο 0-99757 0.00038 0.99766 0.00076 O.99781 Ο.ΟΟΙΙ2 0.99801 Ο.ΟΟΙ44

0-99777 0.99780 0.99788 0.99802 0.99820

ο.οο 0.00035 0.00069 ο.οοιοι 0.00131

0.002Ι2

0.99808 0.99840 0.99876 0.99916 0-99957

0.00192 0.00220 0.00242 0.00259 0.00269

0.99826 0.99855 O.99888 θ·99924 0.99961

Ο.ΟΟΙ74 Ο.ΟΟΙ99 0.002Ι9 0.00234 0.Ο0243

0.99843 0.99869 0.99899 0-99931 0.99965

Ο.ΟΟΙ57

°·35 ο·4 °·45

0.99787 0.98823 0.99863 0.99907 0-999S3

°·5

ι.οοοοο Ι.00047

0.0030Ι 0.00297 0.00286

ι.οοοοο 0.00272 Ι.00042 0.00269 1 . 0 0 0 8 4 0.00259 1 . 0 0 1 2 3 0.00243 1 . 0 0 1 6 0 0.00220

Ι.ΟΟΟΟΟ 1.00038 1.00076 Ι.ΟΟΙΙ2 Ι.ΟΟΙ45

0.00246

ο-55 0.6 0.65 0.7

Ι.ΟΟΟΟΟ 0.00223 I.0003S 0 . 0 0 2 2 0 1.00069 0.002Ι2 Ι.ΟΟΙΟΙ Ο.ΟΟΙ99 Ι.00Ι3Ι ο.οοιβο

0.75

Ι.002Ι3 0.002Ι3 Ι.00244 Ο.ΟΟΙ77 I . 0 0 2 Ó 8 Ο.ΟΟΙ37 Ι.00286 0 . 0 0 0 9 3 1.00298 0.00047

1.00192 1.00222 1.00242 1.00260 1.00269

Ο.ΟΟΙ93 O.OOIÓO Ο.ΟΟΙ24 0.00084 0.00043

Ι . Ο Ο Ι 9 9 Ο.ΟΟΙ45 Ι.00220 Ο . Ο Ο Ι Ι 2 Ι.00234 0 . 0 0 0 7 6 Ι.00243 0 . 0 0 0 3 9

0.00 0.00047

0.00

1.00246

0.00

0.00043

0.00039

0.00084

Ι.00243 Ι.00234

0.00137 Ι.00244 0 . 0 0 1 7 7

1.00273 1.00269 1.00260 1.00242 1.00222

Ο.ΟΟ 1 2 4 ο.οοι6ο

Ι.002 20 0 . 0 0 1 1 2 Ι.ΟΟΙ99 0.00145

Ι.002Ι2 Ι.ΟΟΙ99 ι.οοιέο

Ι.002Ι3 Ι.ΟΟΙ77 1.00136 ι.00091

0.00213 0.00244 0.00268 0.00286 Ι.00047 0 . 0 0 2 9 7

1.00192 1.00160 1.00123 ι .00084 1.00042

O.OOI93 0.00220 0.00243 0.00259

Ι.ΟΟΙ74 Ο.ΟΟΙ74 Ι.ΟΟΙ45 Ο.ΟΟΙ99 Ι . Ο Ο Ι Ι 2 0.00220 Ι.ΟΟΟ76 0.00234 Ι.ΟΟΟ38 0 . 0 0 2 4 3

1 . 0 0 1 5 8 0.00158 Ι . Ο Ο Ι 3 Ι 0.00180 Ι . Ο Ο Ι Ο Ι 0.00199 1 . 0 0 0 6 9 0.00212 Ι . 0 0 0 3 5 0.00220

Ο.ΟΟΙ43 Ι . Ο Ο Ι Ι 8 0.00163 1.00091 ο . ο ο ι β ο I.OOOÓ2 0 . 0 0 1 9 2 ι . 0 0 0 3 1 0.00200

ι . ο ο ο ο ο 0.00301 0-99953 0.00297 0 . 9 9 9 0 7 0.00286 0 . 9 9 8 6 3 0.00268 0.99823 0.00243

1.00000 0-99957

0.00272

o.gggió

0.00259 0.00242 0.00220

Ι.ΟΟΟΟΟ 0.00246 0.99961 0.00243 0.00234 0 . 9 9 9 2 4 0.99888 0.00219 0.99855 Ο.ΟΟΙ99

ι.οοοοο 0.99965 0-9993Ι 0.99899 0.99869

0.00223 0.00220 0.00212 0.00198 0.00180

Ι.ΟΟΟΟΟ 0.99968 0.99938 0.99908 0.99881

0.99826 0.99801 0.99781 0.99766 0-99757

0.00157 0.00131

0.99857

0.00076 0.00038

0.99843 0.99820 0.99802 0.99788 0.99780

0.00035

O.ΟΟΙ42 ο.οοιιβ 0 . 9 9 8 2 1 0.00091 0 . 9 9 8 0 9 0.00062 0.99801 0.00031

ο.οο

0-99777

ο·οο

ο.99799

o

0.05 0.1 0.15

0.2

0.25 °·3

0.8 0.8s 0.9 °·95 ι.ο Ι-°5

ι.ι

I-IS

1.2 I-2S

ΐ·3 I-3S 1-4 I-4S I-S I-5S 1.6 1.65

ΐ·7 I-7S 1.8

1.85 1.g

ι.00091 1.00136 Ι.ΟΟΙ77

ι.00301 1.00298 1.00286 I.002Ó8

0.99787 0-99757 0.99732

0.99715

0.00243 0.002Ó8

0.00286 0.00297

0.002Ó8

0.00244

0.00093

0.002Ι2 0.00176 0.00136

0.00093 0.00047

ΐ·95

Ο·997°4

2.0

0.99700 0.00

0.99728

0.99876 0.99840 0.99808 0.99780 0.99758 0.99742 0.99732

0.00

0.00269

0.00269

0.00192 ο.οοιβο

O.OOI23 0.00084

0.00042

0.99728 0.00

Ι.ΟΟΙ74

0-99754

0.00243 0.00234

Ο.ΟΟ 2.2 Ο Ο.ΟΟΙ99 Ο.ΟΟΙ74

0.00076

Ο.ΟΟΙ74 ο.οο 144 Ο.ΟΟΙΙ2

ο.οοιβο 0.00198 0.002Ι2 0.00220

1.00158 0.00158 ι . ο ο ι 8 ο Ο.ΟΟΙ3Ι Ι.ΟΟΙ99 0.00Ι03 Ι.002Ι2

Ι.00220 Ι . 0 0 2 23 Ι.00220

o.oooóg

0.00035 0.00 0.00035 0.00069 0.00103 0.00131

Ο.ΟΟΙΟΙ

0.00069

Note.

N e g a t i v e quantities are i n h e a v y type.

Examples,

t a n h (3.25 + t o . 7 5 ) = 1.00213 + ¿0.00213. t a n h (3-3° + i ι-50) = 1.00000 — i 0.00272.

[87]

0-99799 0.99801 0.99809 0.99821

ο.οο

0.99837

Ο.ΟΟΙίδ

0.0003Ι

O.OOOÓ2 0.00091

0 . 9 9 8 5 7 Ο.ΟΟΙ42 0.99881 0.00163 0 . 9 9 9 0 8 Ο.ΟΟΙ79 0.99938 0 . 0 0 1 9 2 0 . 9 9 9 6 8 0.00200 Ι.ΟΟΟΟΟ 0.00202 Ι . 0 0 0 3 Ι 0.00200 1.00062 Ο . Ο Ο Ι 9 2 ι.00091 ο.οοιβο 0.00163 I.OOII8 Ι.ΟΟΙ43

Ο.ΟΟΙ43 Ο.ΟΟΙΙ9

1.00163 0.00092 ι . ο ο ι 8 ο O.OOOÓ2 Ι1 .. 00 00 21 90 20 0 . 0 0 0 3 2 Ι.00202 Ι.00200 Ι.ΟΟΙ92 ι.οοιδο 1.00163

0.00 0.00032 0.00062 0.00092 Ο.ΟΟΙΙ9

Ι.ΟΟΙ43

0.00202 0.00200 0.00192 0.00179 0.00IÓ3

0.99837

°·00

TABLE I X

HYPERBOLIC

* = 3-5° o 0.05 O.I

TANGENTS,

t a n h (x + iq) = u+ iv.

3.60

* = 3-55

» = 3-65

CONTINUED

χ - 3.70

0.00 0.00028 0.00056 0.00083 0-99853 0.00107

0-99835 0.00 0.99837 0.00026 0.99843 0.00051 0.99853 0.00075 0.99867 0.00097

0.99851 0.00 0.99853 0.00023 0.99858 0.00046 0.99867 0.00068 0.99879 0.00088

0.99865 0.99867 0.99872 0.99880 0.99891

0.00 0.00021 0.00042 0.00061 0.00079

0.99878 0.99879 0.99884 0.99891 0.99901

0.00 0.00019 0.00038 0.00055 0.00072

0.99871 0.99893 0.99917 0.99944 0.99971

0.00129 0.00147 0.00162 0.00173 0.00180

0.99883 0.99903 0.99925 0.99949 0.99974

0.00117 0.00133 0.00147 0.00157 0.00163

0.99894 0.99912 0.99932 0.99954 0.99977

0.00105 O.OOI2I 0.00133 0.00142 0.00147

0.99904 0.99921 0.99939 0.99958 0.99979

0.00095 0.00109 0.00120 0.00128 0.00133

0.99914 0.99928 0.99944 0.99962 0.99981

0.00086 0.00099 0.00109 O.OOI 16 O.OOI 21

1.00000 ι 00028 1.00056 1.00083 1.00107

0.00182 0.00180 0.00174 0.00163 0.00148

1.00000 1.00026 I.0005I 1.00075 1.00097

0.00165 0.00163 0.00157 0.00147 0.00134

1.00000 1.00025 1.00048 1.00068 I.00088

0.00149 0.00148 0.00142 0.00133 O.OOI 21

1.00000 0.00135 1.00021 0.00133 1.00042 0.00129 1.00061 0.00120 1.00079 0.00109

I.000I9 1.00038 1.00055 1.00072

O.OOI 21 O.OOI 16 0.00109 0.00099

1.00129 1.00148 1.00163 1.00174 1.00180

0.00129 0.00107 0.00083 0.00056 0.00029

1.00117 1.00133 1.00147 1.00157 1.00163

0.00117 0.00097 0.00075 0.00051 0.00026

I.OOI06 I.OOI2I I.OOI33 I.OOI42 1.00148

0.00106 0.00088 0.00068 0.00046 0.00023

1.00096 I.00I08 I.00I20 I.OOI29 1.00134

0.00096 0.00080 0.00061 0.00042 0.00021

1.00086 1.00099 I.00I09 I.00II6 1.001 21

0.00086 0.00072 0.00056 0.00038 0.00019

1.0 1-OS 1.1 I-IS 1.2

1.00183 0 . 0 0 1.00180 0 . 0 0 0 2 9 I.OOI74 0 . 0 0 0 5 6

1.00165 1.00163

°· 00 0.00026

1.00149 0.00

1.00157 1.00147

0.00051 0.00075

I.OOI42 I.OOI33

ι.00133

0.00097

1.001 21 0.00088

1.00135 I.OOI34 I.OOI29 1.00120 I.00I08

0.00 0.00021 0.00042 0.00061 0.00080

I.OOI2 2 I.OOI2I I.00II6 I.00I09 1.00099

0.00 o.oooig 0.00038 0.00056 0.00072

1.25 1-3 I-3S 1.4 M5

I.OOI29 0 . 0 0 1 2 9 I.00I07 0 . 0 0 1 4 8 1.00083 0 . 0 0 1 6 3

1.00117 0.00117

I.00I06 0 . 0 0 1 0 6 1.00088 O.OOI 21

0.00096 0.00109 0.00120 0.00129 1.00021 0.00133

1.00086 1.00072 1.00055 1.00038 I.OOOI9

0.00086 0.00099 0.00109 0.00II6

0.15

0.2 0.2J

O.3 O.35 O.4 0-4S °-5 0-S5 0.6 0.65 0.7 0.7s 0.8 0.8s 0.9 0.95

i-S i-SS 1.6 1.6 S

ΐ·7 I-7S 1.8

0.99818 0.99820 0.99827 0.99837

1.00163 0.00083 1.00148

0.00107

1.00097

0.00134

1.00056 0.00174

1.00075 1.00051

0.00147 0.00157

1.00028

1.00026 0.00163

0.00180

0.00046 0.00068

1.00068 0.00133 1.00048 0.00142 1.00025 0.00148

1.00096 1.00079 1.00061 1.00042

1.00000 0.00182

1.00000 0.00165

1.00000 0.00149

0.99971 0.99944 0.99917 0.99893

0.00180 0.00173 0.00162 0.00147

0.99974 0.99949 0.99925 0.99903

0.00163 0.00157 0.00147 0.00133

0.99977 0 . 0 0 1 4 7 0.99954 0 . 0 0 1 4 2 0.99932 0 . 0 0 1 3 3 0.99912 O.OOI2I

1.00000 0.99979 0.99958 0.99939 0.99921

O.OOI3S 0.00133 0.00128 0.00120 0.00109

0.99871 0.0012g

0.99883 0.99867 0.99853 0.99843 0.99837

O.OOII7 0.00097 0.00075 0.00051 0.00026

0.99894 0.99879 0.99867 0.99858

0.00105 0.00088 0.00068 0.00046 0-99853 0.00023

0.99904 0.99891 0.99880 0.99872 0.99867

0.0009s 0.00079 0.00061 0.00042 0.00021

0.99851

0.99865 0.00

0.00107 0.00083 0.00056 0.00028

1.9 I-9S

0-99853 0.99837 0.99827 0.99820

2.0

0.99818 0.00

1.85

1.00148 0.00023

0.99835 0.00

0.00

Note.

Negative quantities are in heavy type.

Examples,

tanh (3.60 + i 0.80) = 1 . 0 0 1 2 1 + i 0.00088. tanh (3.70 + i 1 . 7 0 ) = 0.99928 — i 0.00099.

[88]

1.00000 O.OOI22

0.00I2I

1.00000 0.00122 0.99981 O.OOI 21

0.99962 O.OOII6 0.99944 0 . 0 0 1 0 9 0.99928 0 . 0 0 0 9 9 0.99914 0.99901 0.99891 0.99884 0.99879

0.00086 0.00072 0.00055 0.00038 0.00019

0.99878 0.00

TABLE I X .

HYPERBOLIC

TANGENTS,

χ = 3.80

χ = 3-75

t a n h (x + iq)

= u + iv.

x = 3-9°

» = 3-85

CONTINUED

* = 3-95

0.00 0.00017 0.00034 0.00050 0.00065

0.99900 0.00 0.99901 0.00016 0.99905 0.00031 0 . 9 9 9 1 1 0.00045 0.99919 0.00059

0.99909 0.99911 0.99914 0.99919 0.99927

0.00 0.00014 0.00028 0.00041 0.00053

0.99918 0.99919 0.99922 0.99927 Q-99934

0.00 0.00013 0.00025 0.00037 0.00048

0.99926 0.00 0.99927 0.00012 0.99930 0.00023 0.99934 0.00034 0.99940 0.00044

0-4S

0.999 2 2 °·99935 0.99950 0.99966 0.99983

0.00078 0.00089 0.00099 0.00105 0.00109

0.99929 0.99941 0-99955 0.99969 0.99984

0.00071 0.00081 0.00089 0.00095 0.00099

0.99936 0.99947 0-99959 0.99972 0.99986

0.00064 0.00073 0.00081 0.00086 0.00089

0.99942 0.99952 0.99963 0-99975 0.99987

0.00058 0.00066 0.00073 0.00078 0.00081

0.99948 0.99956 0.99966 0.99977 0.99988

0.00052 0.00060 0.00066 0.00071 0.00073

0.5 o-55 0.6 0.65 0.7

ι.00000 1.00017 1.00034 1.00050 ι .00065

O.OOI 1 1 0.00109 0.00105 0.00099 0.00090

ι . 00000 1.00016 1.00031 1.00045 1.00059

0.00100 0.00099 0.0009s 0.00089 0.00081

ι.00000 ι.00014 1.00028 1.00041 1.00053

0.00091 0.00089 0.00086 0.00081 0.00073

1.00000 1.00013 1.00025 ι .0003 7 1.00048

0.00082 0.00081 0.00078 0.00073 0.00066

1.00000 1.0001 2 1.00023 1.00034 ι .00044

0.00074 0.00073 0.00071 0.00066 0.00060

0-7S 0.8 0.8s 0.9 °·95

1.00078 1.00089 1.00099 i.ooios 1.00109

0.00078 0.00065 0.00050 0.00034 0.00017

1.00071 1.00081 1.00089 1.00096 1.00099

0.00071 0.00059 0.00045 0.00031 0.00016

1.00064 1.00073 1.00081 1.00086 1.00089

0.00064 0.00053 0.00041 0.00028 0.00014

1.00058 1.00066 1.00073 1.00078 1.00081

0.00058 0.00048 0.00037 0.00025 0.00013

1.00052 1.00060 1.00066 I.0007I 1.00073

0.00052 0.00044 0.00034 0.00023 0.00012

1.0 i.os 1.1

Ι.ΟΟΙΙΙ I.00I09 1.00105 1.00099 1.00089

0.00

0.00

0.00017 0.00034 0.00050 0.00065

1.00099 1.00096 1.00089 I.0008I

0.00016 0.00031 0.00045 0.00059

1.00090 ι .00089 1.00086 1.00081 1.00073

0.00 0.00014 0.00028 0.00041 0.00053

1.00082 0.00 1.00081 0.00013 1.00078 0.00025 1.00073 0.00037 1.00066 0.00048

1.00074 1.00073 I.0007I 1.00066 1.00060

0.00 0.00012

0.00078 0.00090 0.00099 0.00105 0.00109

1.00071 1.00059 1.00045 1.00031 ι.00016

0.00071 0.00081 0.00089 0.00095 0.00099

1.00064 1.00053 1.00041 1.00028 I.OOOI4

0.00064 0.00073 0.00081 0.00086 0.00089

1.00058 1.00048 1.00037 1.00025 1.00013

0.00058 0.00066 0.00073 0.00078 0.00081

1.00052 1.00044 1.00034 1.00023

ι -45

1.00078 1.00065 1.00050 1.00034 1.00017

Ι·5 1-55 1.6 I.6S ΐ·7

1.00000 0.99983 0.99966 0.99950 0-99935

O.OOI 1 1 0.00109 0.00105 0.00099 0.00089

ι.00000 0.99984 0.99969 0-99955 0.99941

0.00100 0.00099 0.00095 00089 0.00081

1.00000 0.99986 0.99972 0-99959 0.99947

0.00091 0.00089 0.00086 0.00081 0.00073

1.00000 0.99987 0-99975 0.99963 0.99952

0.00082 0.00081 0.00078 0.00073 0.00066

1.00000 0.99988 0.99977 0.99966 0.99956

O.OOO74 O.OOO73 O.OOO7I O.OOO66

Ι·75

0.99922 0.99911 0.99901 0.99895 0.99891

0.00078 0.00065 0.00050 0.00034 0.00017

0.99929 0.99919 0.99911 0.99905 0.99901

0.00071 0.00059 0.00045 0.00031 0.00016

0.99936 0.99927 0.99919 0.99914 0.99911

0.00064 0.00053 0.00041 0.00028 0.00014

0.99942 0-99934 0.99927 0.99922 0.99919

0.00058 0.00048 0.00037 0.00025 0.00013

0.99948 0.99940 0-99934 0.99930 0.99927

O.OOO52 O.OOO44 O.OOO34 O.OOO23

0.99889

0.00

0.99900 0.00

0.99909

0.00

0.99918

0.00

0.99926

0.00

o 0.05 O.I 0.15 0.2 0.2s 0-3 0-3S 0.4

Ι·Ι S 1.2 1-2S ΐ·3 1-35 1.4

1.8

1.85 1-9

1-95 2.0

0.99889 0.99891 0.99895 o.gggoi 0.999 1 1

Note. Examples,

N e g a t i v e q u a n t i t i e s a r e in h e a v y t y p e . t a n h (3.95 + i 0.95) = 1.00073 +

¿0.00012.

t a n h (3.95 + i ι .05) = 1.00073

¿0.00012.

[89]

—

O.OOO23 O.OOO34 O.OOO44

O.OOO52 0.00060 0.00066 O.OOO7I I.OOOI 2 O.OOO73

0.00060

0.00012

TABLE X . X = o.o ?

r

0 0.05 0.1 0.15 0.2

0.000 0.07846 0.15643 0.23345 0.30902

90 90 90 90 90

0.25 0-3 0-3S 0.4 0-45

0.38268 0-45399 0.52250 0.58778 0.64944

0-5 0-5S 0.6 0.65 0.7

y

χ r

HYPERBOLIC - 0.05 y

0.05002 0-09305 0.16424 0.23874 0.31304

o O.OOO 57-593 72.493 78.24S 81.259

90 90 90 90 90

0.38594 0.45672 0.52487 0.58989 0.65135

0.70711 0.76041 0.80902 0.85264 0.89101

90 90 90 90 90

0-75 0.8 0.85 0.9 0-95

0.92388 0.95106 0.97237 0.98769 0.99692

1.0 1.05 I.I I-IS 1.2

SINES, X = O.I

r

sinh (* + ig) = r [y χ = 0.2

Χ = O.I5 y

O.IOOI7 0.12724 0.18576 0.25403 0.32485

o 0.000 38.300 57.819 67.455 72.947

83.123 84.400 85.339 86.066 86.652

0-39558 0.46491 0.53201 0.59626 0.65713

0.70803 0.76202 0.81053 0.85407 0.89237

87.139 87.557 87.921 88.246 88.542

90 90 90 90 90

0.92523 0.95237 0.97366 0.98895 0.99817

1.00000 0.99692 0.98769 0.97237 0.95106

90 90 90 90 90

I-2S 1-3 I-3S 1.4 1-45

0.92388 0.89101 0.85264 0.80902 0.76041

1-5 I-5S 1.6 1.65 1-7

r

r

y

y

0.15056 0.16978 0.2I7I2 0.27779 0.3437s

o 0.000 27.861 46.771 58.195 65.382

0.20134 0.21608 0-25497 0.30827 0.36882

o 0.000 21.739 38.746 50.576 58.723

76-471 78.932 80.762 82.189 83.344

0.4II24 0.47831 0.54376 0.60676 0.66667

70.229 73.711 76.344 78.419 80.III

0.43422 0.49663 0-55995 0.62131 0.67994

64.522 68.825 72-147 74.802 76.988

0.7I4I7 0.76698 0.81520 0.85851 0.89662

84.308 85.134 85.858 86.505 87.093

0.72296 0.77527 0.82291 0.86583 0.90364

81.532 82.753 83.826 84.787 85.662

0.73521 0.78661 0.83370 0.87609 0.91347

78.835 80.431 81.839 83.103 84.257

88.814 89.070 89.313 89.547 89.775

0.92930 0.95632 0.97752 0.99275 I.OOI94

87.636 88.145 88.629 89.095 89.550

0.93607 0.96290 0.98396 0.99909 1.00822

86.471 87.231 87.953 88.649 89.329

0.94556 0.97213 0.99300 1.00800 ι.01704

85.326 86.331 87.287 88.209 89.110

ι.00125 0.99817 0.98895 0.97366 0.95237

90.000 90.225 90.453 90.687 90.930

1.00500 I.OOI94 0.99275 0.97752 0.95632

90.000 90.450 90.905 91.371 91.855

I.OII27 1.00822 0.99909 0.98396 0.96290

90.000 90.671 91.351 92.047 92.769

1.02007 1.01704 1.00800 0.99300 0.97213

90.000 90.890 91.791 92.713 93.669

90 90 90 90 90

0.92523 0.89237 0.85407 0.81053 0.76202

91.186 91.458 91.754 92.079 92.443

0.92930 0.89662 0.85851 0.81520 0.76698

92.364 92.907 93.495 94.142 94.866

0.93607 0.90364 0.86583 0.82291 0-77527

93.529 94.338 95.213 96.174 97-247

0.94556 0-9I347 0.87609 0.83370 0.78661

94-674 95-743 96.897 98.161 99.569

0.707II 0.64944 0.58778 0.52250 0-45399

90 90 90 90 90

0.70803 0-65135 0.58989 0.52487 0.45672

92.860 93-348 93-934 94.661 95.600

0.7I4I7 95.692 0.65713 96.656 0.59626 97.811 0.53201 99.238 0.46491 101.068

0.72296 98.468 0.66667 99.889 0.60676 101.581 0.54376 103.656 0.47831 106.289

0.73521 0.67994 0.62131 0.55995 0.49663

101.165 103.012 105.198 107.853 i l 1.17s

1-75 1.8 1.85 1.9 1-95

0.38268 0.30902 0.2334s 0-15643 0.07846

90 90 90 90 90

0.38594 96-877 0.31304 98-741 0.23874 101.755 0.16424 107.507 0.09305 122.407

0.39558 0.32485 0.25403 0.18576 0.12724

0.41124 0-34375 0.27779 0.21712

0.43242 0.36882 0.30827 0.25497

115.478 121.277 129.424 141-254

2.0

0.00

90

0.05002 180.000

O.IOOI7 180.000

Example.

103.529 107.053 112.545 122.181 141.700

109.771 114-618 121.805 133.229

0.16978 152.139

0.21608 158.261

0.15056 180.000

0.20134 180.000

sinh (0.15 + ¿0.15) = 0.27779 /s8°.iq.s = 0.27779 /58o· 11'.42".

[90]

TABLE X .

HYPERBOLIC

χ - 0.25

SINES,

χ - 0.3

s i n h (x + iq)

» = 0.35

= r ¿y.

CONTINUED

χ — 0.4

χ = 0.4s

0 0.05 O.I 0.15 0.2

0.25261 0.26452 0.29782 0-34396 °-399I3

o.ooo 17.814 32.890 44-943 52.992

0.30452 0.31447 0.34235 0.38370 0.43385

0.000 15.118 28.533 39-493 48.122

0.35719 0.36571 0.38994 0.42671 0.47231

0.000 13.169 25.214 35-516 44.008

0.41075 0.41818 0-43953 0.47246 0.51401

0.000 11.703 22.629 32.288 40.536

0.2S 0-3 0-3S 0.4 0-4S

o.4S8S4 0-5I954 0.58036 0.63977 0.69685

0.48906 0.54666 0.60476 0.66199 0.71730

54.882 60.242 64-575 68.151 71.166

0.52348 0.57766 0.63292 0.68781 0.74119

50.921 56.843 61.237 65-I57 68.503

0.56140 0.6x223 0.66462 0.71708 0.76844

47-471 53-288 58.200 62.393 66.018

o-S o-SS 0.6 0.65 0.7

0.75088 0.80127 0.84754 0.88927 0.92612

59-405 64-327 68.215 71-371 73-999 76.238 78.185 79.911 81.464 82.887

0.76989 0.81912 0.86443 0-90539 0.94161

73-759 76.028 78.049 79.878 81.557

0.79220 0.84012 0.88436 0.92444 0.95994

71.408 73-971 76.267 78.353 80.274

0.81775 0.86426 0.90732 0.94642 0.98113

69.196 72.021 74-568 76.893 79-043

0.84649 0.89149 0-93330

0.97136 1.00521

67.125 70.184 72.958 75-504 77.868

0-7S 0.8 0.8s 0.9 0-9S

0.95779 0.98403 1.00464 1.01949 1.02843

84.207 85-450 86.635 87.779 88.896

0.97277 0.99862 1.01894 1-03357 1.04239

83.119 84-593 85.999 87-358 88.687

0.99052 1.01592 1.03590 1.05029 1.05898

82.068 83.762 85-383 86.951 88.484

ι.01107 1-03597 I-°5557 1.06970 1.07822

81.056 82.962 84.788 86.556 88.287

1.03446 1.05880 1.07798 1.09182 1.10017

80.087 82.194 84.216 86.177 88.098

ι.o ι.05 1.X 1-I5 1.2

1-03141 1.02843 1.ο 1949 1.00464 0.98403

90.000 91.104 92.221 93-36S 94-550

I-04534 1.04239 I-03357 1.01894 0.99862

90.000 9I-3I3 92.642 94.001 95-407

1.06188 1.05898 1.05029 1.03590 1.01592

90.000 91.516 93-oSo 94.617 96.238

1.08107 1.07822 1.06970 I-05557 I-03597

90.000 9 r -7i3 93-444 95.212 97.038

1.10297 I.IOOI7 1.09182 1.07798 1.05880

90.000 91.902 93-823 95-784 97.806

1 25 1-3 I-3S 1-4 ι-45

0.95779 95-793 0.926x2 97-113 0.88927 98-536 0.84754 100.090 0.80127 101.815

0.97277 96.881 0.94161 98-443 0.90539 100.122 0.86443 101.951 0.81912 103.972

0.99052 97.932 0-95994 99.726 0.92444 101.647 0.88436 103.733 0.84012 106.029

1.01107 0.98113 0.94642 0.90732 0.86426

98.944 100.957 103.107 105.432 107.979

1.03446 1.005 21 O.97136 O.9333O O.89I49

99-9I3 102.132 104.496 107.042 109.816

1-5 1-55 1.6 1.6S 1-7

0.75088 0.69685 0.63977 0.58036

0.76989 0.7x730 0.66199 0.60476 0.54666

106.241 108.834 111.849 II5-425 119.758

0.79220 0.74x19 0.68781 0.63292 0.57766

108.592 l i 1.497 114.843 118.763 123.157

0.81775 0.76844 0.71708 0.66462 0.61223

110.804 O.84649 1x2.875 113.982 . O.79895 116.288 117.607 O.74969 120.144 121.800 O.69968 124.546 126.712 O.65OI2 129.626

0.48906 0.43385 0.38370 0.3423S 0.31447

125.118 131.878 140.507 151.467 164.882

0.52348 0.47231 0.42671 0.38994 0.36571

129.079 135-992 144.484 154.786 166.831

0.56140 0.51401 0.47246 0.43953 0.41818

132.529 139.464 147.712 157-371 168.297

I-7S 1.8 1.8s 1.9 ι-95 2.0

103.762 106.001 108.629 111.785 0.51954 "5-673 120.595 0.45854 127.008 Ι ο·399 3 135-057 0-3439Ó 147.no 0.29782 162.186 0.26452 0.25261 180.000 Example,

0.30452 180.000

0.35719 180.000

0.41075 180.000

0-46534 0.47191 0.49093 0.52062 0.55860 0.60249 0.65012 0.69968 0.74969 0.79895

0.60249 0.55860 0.52062 O.49O93 O.47I9I

44-473 50.374 55-454 59-856 63.712

I3S-S27 142.399 150.358 159.424 169.433

0.46534 180.000

sinh (0.40 + i 0.25) = 0.56140 /47°.47i = 0.56140 /47°.28'.I6'.

[91]

O.OOO

IO.567 20.576 29.642 37.601

TABLE X .

χ --- 0.5

o 0.0s O.I

0.32110 0.52697 0.54407 0.57100 0.60583

0.000 9.665 18.918 27-453 3S-III

0.64652 0.69112 o-73793 0.78552 0.83266

41.871 47-793 52.980 57-542 61.584

o-5 °-55 0.6 0.65 0.7

0.87837 0.92182 0.96232 0.99927 1.03220

0-75 0.8 0.85 0.9 °-95

HYPERBOLIC

χ = 0.55

SINES,

s i n h (x + iq)

χ - 0.6

8.936 17-559 25.625 32-990

0.63665 0.64147 0.65559 0.67810 0.70769 0.74282 0.78194 0.82361 0.86650 0.90946

65.198 68.462 71.441 74.189 76-751

0.69333 0.73510 0.77927 0.82447 0.86951 0.91338 0-95524 0.99437 1.03004 1.06214

39.610 45-5» 50-759 55-437 59.628 63.411 66.854 70.016 72.948 75-693

1.06070 1.08446 1.10320 1.11672 1.12489

79.164 81.461 83.669 85.814 87.917

1.08987 1.11300 1.13127 1.14446 1.15244

1.0 ι.05 1.1 1-iS 1.2

1.12763 1.12489 1.11672 1.10320 1.08446

90.000 92.083 94.186 96-33i 98-539

1.15510 1-15244 1.14446 1.13127 1.11300

1-25 1-3 i-35 1.4 ι -45

1.06070 1.03220 0.99927 0.96232 0.92182

1-5 1-55 1.6 1.6s 1-7

CONTINUED

X = 0.65

X = 0.7

31-174

0.69675 0.70115 0.71409 0.73482 0.76220

0.000 7-839 15.486 22.781 29.613

0.75858 0.76263 0.77455 0.79369 0.81911

37.642 43-494 48.769 53-529 57-838

0.79492 0.83160 0.87090 0.91156 0.95249

35-926 41.710 46.989 51-803 56.204

0.84965 0.88406 0.92112 0.95966 0.99861

0.95149 0.99171 1.02948 1.06411 1.09509

61.762 65.360 68.685 71.784 74.696

0.99270 1-03134 1.06769 I.IOIII 1.13108

60.245 63.976 •67-445

70.694 73.760

1.03704 1.07409 1.10904 I.I4I25 I.I70I9

58-853 62.698 66.294 69.677 72.884

78.287 80.763 83.148 85.467 87.744

1.12200 1.14448 1.16226 I.I75IO 1.18287

77-459 80.102 82.653 85-121 87.580

1-15715 1.17892 1.19623 1.20871 1.21626

76.678 79-477 82.185 84.826 87.424

I.I954I 1-21653 1.23327 1.24538 1.25271

75-946 78.890 81.744

90.000 92.256 94-533 96.852 99.237

1.18547 1.18287 I.I75IO 1.16226 1.14448

90.000 92.420 94.879 97-347 99.898

1.21879 90.000 1.21626 92.576 1.20871 95-174 1.19623 97-8IS 1.17892 100.523

1-25517 1.25271 1.24538 1.23327 1.21653

90.000 92.723 95-468 98.256 IOI.IIO

100.836 103.249 105.811 108-559 111.538

1.08987 101.713 1.06214 104.307 ι .03004 107.052 0.99437 109.984 0.95524 113.146

1.12200 1.09509 1.06411 1.02948 0.99174

102.541 105.304 108.216 111.315 114.640

1-15715 103.322 1.13108 106.240 I.IOIII 109.306 1.06769 112.555 1-03134 116.024

I.I954I I.I70I9 I.I4I25 1.10904 1.07409

104.054 107.116 110.323 113.706 117.302

0.87837 0.83266 0-78552 0.73793 0.69112

114.803 118.416 122.458 127.020 132.207

0.91338 0.86951 0.82447 0.77927 0.73510

116.589 120.372 124-563 129.241 134.489

0.95149 0.90946 0.86650 0.82361 0.78194

118.238 122.162 126.471 131.231 136.506

0.99270

0.95249 0.91156 0.87090 0.83160

123.796 128.197 133.011 138.290

119.755

1.03704 0.99861 0.95966 0.92112 0.88406

121.148 125.284 I2 9-755 134-603 139-867

1-75 1.8 1.8s 1.9 ι-95

0.64652 0.60583 0.57100 0.54407 0.52697

138.129 144.889 152-547 161.082 170.335

0-69333 0-65555 0.62350 0.59894 0.5834s

140.390 147.010 154-375 162.441 171.064

0.74282 0.70769 0.67810 0.65559 0.64147

142.358 148.826 I55-9I4 163.568 171.663

0.79492 0.76220 0.73482 0.71409 0.70115

144.074 150.387 157.219 164.514 172.161

0.84965 0.81911 0.79369 O-77455 0.76263

145-574 151-737 158.335 165.315 172.581

2.0

0.52110 180.000

O-IS

0.2

0.2s °-3 0-35 0.4 0.4s

Example.

0.57815 0.58345 0.59894 0.62350 0.65555

0.57815 180.000

0.000

= r ¿y.

8.337

16.432 24.086

0.63665 180.000

0.69675 180.000

34.426 40.133

45-397 50.245 54-716

84-532

87.277

0.75858 180.000

sinh (0.70 + i 1.70) = 0.88406 /i39°.867 = 0.88406 /i30°.52,.0i''.

[92]

0.000 7.419 14-685 21.665 28.263

TABLE X .

Χ - 0.75

?

r

y 0

HYPERBOLIC Χ -

r

SINES,

0.8

s i n h (x + iq)

= r [y^

X = 0.85

y

0.82232 0.82605 0.83706 0.85481 0.87846

0.000 7.064 14.002 20.706 27.093

0.88811 0.89157 0.90178 0.91828 0.94033

O 0.000 6.759 13.415 19.877 26.073

0.25

0.90700

0-3 0-35

0-93932

33-III 38.737 43-974

0.96705 0.99742 1.03041 1.06500 1.10024

r

y o

CONTINUED

Χ = 0.9

Χ = 0.9S

r

y o

r

y o

0.95612

0.000

0-95933 0.96883 0.98420 1.00481

6.497 12.909 19.158 25.181

1.02652 I.0295I 1.03837 1.05273 1.07202

0.000 6.270 12.468 18.529 24.399

1.09948 1.10227 I.IIO56 1.12400 1.14209

0.000 6.073 12.085 17.980 23.712

31.955 37.500 42.702 47.574 52.135

1.02985 1.05843 1.08957 Ι.Ι 2234 1.15583

30.938 36.401 41.565 46.433 51.022

1-09553 1.12243 I.15184 I.18289 I.2147I

30.039 35-426 40.547 45-407 50.014

1.16418 1.19227 Ι.21732 1.24674 1.27697

29.245 34-557 39.637 44-483 49.102

1.13522 1.16917 1.20135 I.23II5 1.25802

56.414 60.441 64.245 67.857 71.307

1.18918 1.22163 1.25246 1.28107 1.30693

55.353 59-450 63.399 67.048 70.602

I.24649 1.27748 I.30700 I.33444 I.35927

54.386 58.543 62.507 66.301 69.949

1-30723 1.33682 1.36505 1.39136 1.41519

53-507 57.714 61.743 65.613 69.347

78.339 81.330 84-255

1.28152 I.30I24 X. 31691 1.32825

74.621 77.825 80.942 83.996 87.008

Χ ·3 2 955

1.34858 1.363Ò9 1.37466 1.38130

74-020 77-344 80.580 83.754 86.887

I-38I05 1-39937 1.41395 1.42452 1.43094

73-474 76.898 80/242 83.527 86.773

1.43611 1.45371 1.46778

72-963 76.484 79.929

1-47797 1.48415

83.316 86.668

1-33743 I-335I3 1.32825 1.31691 1.30124

90-000 92-992 96.004 99.058 102.175

1.38130 1.37466 1.36369 1.34858

90.000 93.113 96.246 99.420 102.656

1.43309 1.43094 1.42452

90.000 93.227 96.473

I-4I395 1-39937

99-758 103.102

1.48623 1.48415 1.47797 1.46778

90.000 93.332 96.684 100.071

1.2

90.000 1.29468 92.862 1.29230 1.28520 95-745 98.670 1.27346 1.25726 101.661

1-25 1-3 1-35 1-4 1-45

1.23689 104.740 1.21248 107.933 1-18457 111.267 I . I 5 3 5 6 114.772 1.12001 118.478

1.28152 1.25802 1.23115 Ι.20135 1.16917

105.379 108.693 112.143 115.755 119.559

1.32955 1.30693 1.28107 1.25246 1.22163

105.974 109.398 112.952 116.601 120.550

1.38105

106.526

!·359 2 7 1.33444 1.30700 1.27748

1-5 1-55 1.6 1.65

1-08453 1.04785 1.01079 0.97427

1-7

0.93932

122.422 126.637 131.160 136.026 141.263

1.13522 1.10024 1.06500 I.0304I 0.99742

123.586 127.865 132.426 137.298 142.500

1.18918 1.15583 1-12234 1.08957 1.05843

124.647 128.978 133.567 138.435 143-599

1-75

0.96705 0.94033 0.91828 0.90178 0.89157

148.045 153.927 160.123 166.585 173.241

1.02985 1.00481 0.98420 0.96883

149.062 154.819 160.842 167.091

1-95

0.90700 146.889 0.87846 152.907 0.85481 159.294 0.83706 165.998 0.82605 172.936

0.95933

2.0

0.82232 .180.000

0.88811

180.000

0.95612

0.00 0.05 O.IO 0.15 0.2

0.45

0.97427 1.01079 1.04785

0.5 0-55

1-08453 1.12001

0.4

0.6 0.65 0.7 0.75 0.8 0.85 0.9

0-95 I.O

i.os

I.I I

-I5

1.8 1.85 1.9

48.840

53-363 57-578

I-I5356

61.522 65.228

1-18457 1.21248

68.733 72.067

1.23689 •1.25726 1.27346 1.28520 1.29230

75.260

87.138

Example.

I-4537I

103.516

110.051 113-699 117.493 121.457

1.43611 1.41519 1.39136 1.36505 1.33682

107.037 110.653 114.387 118.257 122.286

1.24649 1.21471 1.18289 1.15184 1.12243

125.614 129.986 134-593 139.453 144-574

1.30723 1.27697 1.24674 Ι.21732 Ι 19227

126.493 130.898 135.517 140.363 145.443

I73-503

I-09553 1.07202 1.05273 1.03837 1.02951

149-961 155.601 161.471 167.532 I73-730

1.16418 1.14209 1.12400 1.11056 1.10227

150.755 156.288 162.020 167.915 173.927

180.000

1.02652

180.000

1.09948

180.000

sinh (0.90 + ¿LO) = 1.43309 /90O.

[93]

TABLE X .

HYPERBOLIC

sinh {x + iq) = r ¡y.

SINES,

χ = 1.05

o 0.05 O.I 0-I5 0.2

1.17520 1.17782 1.18552 1.19816 1-21515

0.2s °-3 0-3S 0.4 0-45

1-23594 1.25984 1.28612 1.31400 1.34272

43-651 48.276

°-5 o-55 0.6 0.65 0.7

I-37IS3 1.39976 1.42675 I-45I93 1.47479

0-7S 0.8 0.85 0.9 0-95

1.49487 1.51182 I-52532 I -535 I 3 1.54108

1.0 1.05 1.1 1-IS 1.2

1.54308 90.000 1.54108 93.276 I-535I3 9 6 . 8 7 8 I-52532 1 0 0 . 3 6 2 1.51182 103.899

I-2S 1-3 Ι·35 1.4 I-4S

1.49487 1.47479 I -45 I 93 1.42675 1.39976

107.509 III.209 II5.0I9 118.957 123.043

i-S I-5S 1.6 1.6S 1-7

I-37I53 1.34272 1.31400 1.28612 1.25984

I-7S 1.8 1.8s 1.9 ι-95

1.23594 I.2I5I5 1.19816 1.18552 1.17782

2.00

1.17520 1 8 0 . 0 0 0

0.000 5.900 11.748 17.496 23-io5 28.541 33-784 38.821

CONTINUED

» = 1.15

1.25386 1.25631 1.26358 1.27540 1.29137

0.000 5-748 11-453 17.071 22.568

I-33565 1-33795 1.34478 1-3559° i-37°93

0.000 5-6i5 il.192 16.694 22.092

1.42078 1.42294 1.42936 I-43983 1-45399

0.000 5-497 10.961 16.361 21.670

1.50946 1.5115° I-5I755 I-5274I i-54°77

0.000 5-393 IO-757 16.065 21.294

I -3 I °95

I-35837 1.38479 1.41207

27-9I5 33-°93 38.090 42.902 47-53°

1-38939 1.41070 1.43421 1.45926 1.48517

27-359 32-477 37-435 42.227 46.855

1.47141 I-49I55 1.51381 1-53756 1.56217

26.864 31.926 36.847 41.620 46.245

I-5572I 1.57626 1-59733 1.61987 1.64325

26.421 31-433 36.319 4I-073 45-693

52.707 56.957 61.043 64.981 68.791

1-4395° 1.46641 1.49220 1.51630 1.53820

51.981 56.268 60.403 64.401 68.280

1.51128 1-53694 1.56156 1.58460 1.60558

5I-323 55-640 59.818 63.870 67.811

1.58701 1.61147 1.63497 1.65699 1.67705

50.725 55-069 59.284 63-384 67.380

1.66688 1.69018 1.71260 I-73363 1.75282

50.184 54-549 58.797 62.939 66.986

72.491 76.101 79.638 83.122 86.724

1-55747 1-57374 1.58671 1.59614 1.60187

72.056 75-747 79.369 82.941 86.479

1.62404 1.63965 1.65211 1.66117 .66667

71.656 75-421 79.121 82.774 86.395

1.69472 ι.70971 1.72166 1.73036 I-73564

71.287 75.120 78.893 82.620 86.317

1.76975 70.950 1.78409 • 74.844 1-79555 7 8 . 6 8 2 1.80389 82.478 1.80895 86.246

1.60379 90.000 1.60187 93-521 1.59614 97-059 1.58671 100.631 1-57374 1 0 4 . 2 5 3

1.66852 1.66667 1.66117 1.65211 1.63965

90.000 93-6o5 97.226 100.879 104.579

I-7374I 9 0 . 0 0 0 1-73564 9 3 - 6 8 3 97.380 1.73036 1.72166 101.107 ι . 7 0 9 7 1 104.880

1.81066 90.000 1.80895 93-754 1.80389 97.522 1-79555 1 0 1 . 3 1 8 1.78409 105.156

1-55747 1.53820 1.51630 1.49220 1.46641

107.944 III.720 "5-599 119-597 123.732

1.62404 1.60558 1.58460 1.56156 I-53694

108.344 112.189 116.130 120.182 124.360

1.69472 108.713 1.67705 112.620 1.65699 116.616 1.63497 120.716 1.61147 124.931

1.76975 1.75282 1-73363 1.71260 1.69018

109.050 113.014 117.061 121.203 125.451

127.293 131.724 136.349 141-179 146.216

1-4395° 1.41207 1.38479 I-35837 I-3335I

128.019 132.470 137.098 141.910 146.907

1.51128 1.48517 1.45926 1.43421 I.4I070

128.677 133-145 137-773 142.565 I47-523

129.275 133-755 138.38° 143-153 148.074

1.66688 1-64325 1.61987 1-59733 1.57626

129.816 i34.3°7 138.927 143.681 148.567

151.459 156.895 162.504 168.252 174.100

I-3I095 ι.29137 1.27540 1.26358 1.25631

152.085 157-432 162.929 168.547 174.252

1-38939 1-37093 1-35590 1.34478 1-33795

152.641 157.908 163.306 168.808 I74-38S

1.58701 1.56217 1-53756 1.51381 I -49 I 55 1.47141 1-45399 1-43983 1.42936 1.42294

I53-I36 158.33° 163.639 169.039 i74-5°3 1.42078 1 8 0 . 0 0 0

I-5572I i-54°7 7 1.52741 I-5I755 1.5115°

153-579 158.706 163.935 169.243 174.607

Example.

I -3335 I

1.25386 1 8 0 . 0 0 0

1.33565 1 8 0 . 0 0 0

1.50946 180.000

sinh (1.20 + i 1.25) = 1.76975 /I09°.050 = 1.76995 /ιοςΛο.ν.οο"-

L 94

]

TABLE X .

Χ - 1.25

HYPERBOLIC

χ = ι·3

SINES,

s i n h (χ + ig)

* = Ι ·35

= r ¿y. χ

0.2

1.60192 1.60384 I.Ó0954 1.61884 1-63145

ο.οοο 5·3°ι 10.576 ΐ5·8θ3 20.958

1.69838 1.70019 ΐ·7°557 Ι ·7 Ι 435 1.72627

ο.οοο 5·2ΐ8 10-415 15-568 20.659

ΐ·799°9 ι.8οο8ο 1.80588 1.81417 1.82544

ο.οοο 5-Ϊ45 10.271 15-359 20.392

1.90430 ΐ·9°592 1.91072 1.91856 1.92921

0.2S °·3 °-35 °·4 °·45

1.64699 1.66501 1.68497 I-70Ó35 1.72856

26.026 30.991 35-845 40.580 45-195

1.74096 1.75802 1.77694 1-79722 1.81832

25-673 3°·595 35-418 40-135 44-745

1-83934 1-85549 1-87343 1.89268 1-91273

25-356 .30.240 35·°35 39-735 44-338

1-94237 1.95767 1.97468

°·5 °-55

Χ.75Ι04 1-77323 1.79462 ι.8ΐ47° 1.83304

49-693 54-077 58-354 62-533 66.625

1.83970 1.86084 1.88123 1.90040

49-248 53-648 57-95° 62.163 66.295

ΐ-933°6 Ι-953Ι9 I-972Ó2 1.99091 2.00764

1.84924 1.86297 1.87394 1.88193 1.88679

70.640 74-59° 78.489 82.348 86.ι8ο

Ι-9334Ι 1-94654

70-356 74-358 78.311 82.228 86.Ι20

2.02245 2.03500 2.04505 2.05238 2.05684

48.845 53-258 57-583 61.825 65-994 70.097 74-145

ο 0.05 ο.ι °·τ5

0.6

ο.ό5 °-7 °·75 0.8

0.8S 0.9 °·95

1.9179

2

1.95704 1.96470

Ι·9 6 935

ι.ο Ι.05 Ι.Ι I-IS

1.2

1.88842 90.000 1.88679 93-820 1.88193 97.652 1-87394 ι ο ί . 5 1 1 1.86297 Ι05-4ΙΟ

I-2S Ι·3 I-3S ΐ·4 I-4S

1.84924 1.83304 1.81470 1.79462 1-77323

109.360 "3-375 117.467 121.646 Ι25.923

ι-93341 1.91792 1.90040 1.88123 1.86084

109.644 ΙΙ3·7°5 117-837 122.050 126.352

ΐ·5 I-5S

Ι.75Ι04 1.72856 1.70635 1.68497 1.66501

130.308 134-805 Ι39-420 144-155 Ι49·°°9

1.8397° 1.81832 1.79722 1.77694 1.75802

Ι3°·752 135-255 139-865 144-582 Ι49·4°5

1.64699 1-63145 1.61884 Ι.60954 1.60384

153-974 159-042 164-197 169.424 174-699

1.74096 1.72627 Ι-7Ι435 ΐ·7°557 ι.70019

154-327 159-341 164.432 169.585 174-782

1.6 1.6$

ΐ·7

Ι ·75

1.8 1.85 1.9 I-9S 2.0

1.60192 ι8ο.οοο

Example.

90.000 1-96935 93·88Ο 1.9647° 97-772 1.95704 101.689 1.94654 105.642 1.97091

1.69838 ιδο.οοο

78.149 82.118

86.065

CONTINUED

- 1-4

ο.οοο 5·°8° 10.143 15.172

20.Ι53

2.0Ι200

25·°73 29-921 34-689 39-373 43-97°

2.03Ι35 2.0505Ι 2.06903 2.08647 2.Ι0244

48.480 5 2 ·9°5 57-249 6ΐ·5ΐ8 65-719

2.11658 2.12858 2.13819

69.861 73-951 78.000 82.018 86.0Ι4

1.99295

2.Ι4520

* = Ι·45

2.01427 2.01580 2.02034 2.02775 2.03784 2.05030 2.06479 2.08093 2.09828 2.11637

ο.οοο 5-02Ι Ι0.028

Ι5-0Ο5 19-939 24-8ι8

29.634 34-379 39-047 43-638

2.13478 2.Ι5302 2.17067 2.18731 2.20254

48.150 52-584 56-945 61.238 65.469

2.2IÓ04 2.2275 1 2.23669 2-24334 2.24747

69.645 73-774 77-864 81.925 85.968

2.05833 90-000 2.05684 93-935 2.05238 97-882 2.04505 101.851 2.03500 105.855

2.14947 2.15090 90.000 2.14947 93-986 2.Ι4520 97.982 2.13819 Ι02.000 2.12858 106.049

2.24884 90.000 2.24747 94-032 2.24334 98.075 2.23669 102.136 2.22751 106.226

2.02 245 2.00764 Ι.9909Ι 1.97262 Ι ·953 Ι 9

2.11658 2.Ι0244 2.08647 2.06903 2.05051

ΙΙΟ.Ι39 ΙΙ4·28ι 118.482 122-751 127.095

2.21604 2.20254 2.1873Ι 2.17067 2.Ι5302

"0-355 "4-531 118.762 Ι 2 3·°55 127.416

2 ·°3 Ι 35

131.520

1-99295 140.627 1-97468 1 4 5 - 3 " 1.95767 150.079

2.13478 2.11637 2.09828 2.08093 2.06479

Ι36.362 140-953 145-621 150.366

1-94237 1.92921 1.91856 1.91072 1.90592

2.05030 2.03784 2.0277S 2.02034 2.01580

155-182 ι6ο.ο6ι 104-995 169.972 174-979

109.903 ιι4-οο6 118.175 122-417

126.742 ι·933°6 1.91273 Ι 3 Ι · Ι 5 5 1.89268 135-662 1-87343 140.265 1-85549 144-965 149-760 1-83934 154-644 1-82544 ΐ59·6ο8 1.81417 164-641 1.80588 169.729 ι.8οο8ο 174-855 ι·799°9 ι8ο.οοο

2.0Ι200 136.030

154-927 159-847 164.828 169.857 174-92°

1.90430 ι8ο.οοο

2.01427 ιβο.οοο

sinh (1.45 + i 1.7°) = 2.06479 Λ5ο°·366 = 2.06479 /ΐ5°°·22'.οι 1 '.

[95]

TABLE X . χ - 1.50

H Y P E R B O L I C SINES, » = i-SS

o 0.05 O.I 0.15 0.2

2.12928 2.13073 2.13502 2.14204 2-ISIS9

o.ooo 4.969 9-925 14.855 19.746

2.24961 2.25098 2.25504 2.26169 2.27074

0.25 °·3 °·35 0.4 °·45

2.16340 2.17714

24.590 29.376 34-°99 38.753 43-337

2.28193 2.29496 2.30949 2.32513

°·5 0.6 0.65 0.7

2.24362 2.26098 2.27779 2.29365 2.30819

56.670 60.984 65.241

2.35812 2.37465 2.39066 2.40577 2.41976

°·75 0.8 0.8s 0.9 0-9S

2.32107 2.33202 2.34080 2.34720 2.3511°

69.448 73.611 77.740 81.842 85-925

2.43193 2.44238 2.45076 2.45688 2.46061·

1.0

2.35241 90.000 2.3511° 94-075 2.34720 98.158 2.34080 102.260 2.33202 106.389

2.46186 90.000 2.46061 94-113 2.45688 98.235 2.45076 102.374 2.44238 106.536

i-45

2.32107 2.30819 2.29365 2.27779 2.26098

110.552 114-759 119.016 123.33° 127.706

2.43193 2.4197Ò 2.40577 2.39066 2.37465

1-5 i-55 1.6 1.6s 1-7

2.24362 2.22612 2.20892 2.19245 2.17714

132.150 136.663 141.247 145.901 150.624

1-75 1.8 1.85 1.9 ι-95 2.0

0-55

1.05

1.1 1-IS 1.2 I-2S 1-3 I-3S 1.4

2.19245

2.20892 2.22612

47.850 52-294

2.34148

4-923 9-833 14.721 19-574 24-385 29.144 33-846 38.488 43.066 47.580 52-030 56-420 60.752 65-°33 69.268 73464 77.626 81.765 85.887

sinh (x + ig) = r /y.

χ = 1.60

» = 1.65

2-37557 2.37686 2.38071 2.38701 2.39558

0.000 4.881 9-751 14.600 19.419

2.50746 2.50869

2.40619 2.41856

24.200 28.935 33-619 38.248 42.820

2.5365° 2.54824 2.56133 2-S7S44 2.59021 2.60527 2.62024 2.63476 2.64847 2.66108

2.43235

2.44721 2.46274 2-478S7

2.53717

47-334 51-791 56.192 60.542 64.845

2.54890 2.55887 2.56687 2.57272 2.57627

69.105 73-329 77-523 81.695 85.851

2.49430

2-5°955 2.52395

CONTINUED

2.51234

2.51831 2.52644

2.67226 2.68178 2.68941 2.69499

2.69839

χ = 1.70

0.000 4-843 9.677 14.492 19.280 24.034 28.747 33-415 38.032 42-599 47-113 51-574 55-986 60.351 64.673

2.64563 2.64679 2.65025 2.65591 2.66361

0.000 4.809 9.610 14-395 19-155

2.67316 2.68430 2.69673 2.7IOI4 2.72418

23.884 28.578 33.229 37-837 42.398

2.73850 2.75274 2.76657 2.77963 2.79164

46.911 51-378 55-799 60.178 64-5I7

68.956 73.206 77.429 81.631 85.819

2.80230 2.81138 2.81867 2.82399 2.82723

68.821 73-094 77-343 81.573 85.790

2.57746 90.000 2.57627 94.149 2.57272 98-305 2.56687 102.477 2.55887 106.671

2.69951 90.000 2.69839 94.181 2.69499 98.369 2.68941 102.571 2.68178 106.794

2.82832 90.000 2.82723 94.210 2.82399 98.427 2.81867 102.657 2.81138 106.906

110.732 114.967 119.248 123.580 127.970

2.54890 2-53717 2-52395 2-5°955 2.49430

110.895 "5-155 119.458 123.808 128.209

2.67226 2.66108 2.64847 2.63476 2.62024

111.044 115-327 119.649 124.014 128.426

2.80230 2.79164 2.77963 2.76657 2.75274

III.179 115-483 119.822 124.201 128.622

2.35812 2.34148 2.32513 2.30949 2.29496

132.421 136.934 141.512 146.154 150.856

2-47857 2.46274 2.44721 2.43235 2.41856

132.666 137.180 141.752 146.381 151.065

2.60527 2.59021 2-57544 2.56133 2.54824

132.887 137.401 141.968 146.585 151-253

2.73850 2.72418 2.7IOI4 2.69673 2.68430

133-089 137.602 142.163 146.771 151.422

2.16340 i55-4i° 2· Ι 5 Ι 59 160.254 2.14204 165.145 2 . I 3 5 0 2 i7°-°75 2.13073 i75-°3i

2.28193 2.27074 2.26169 2.25504 2.25098

155-615 160.426 165.279 170.167 175.077

2.40619 2.39558 2.38701 2.38071 2.37686

155.800 160.581 165.400 170.249 I 75- I I 9

2.5365° 2.52644 2.51831 2.51234 2.50869

155.966 160.720 165.508 170.323 175-157

2.67316 2.66361 2-65591 2.65025 2.64679

156.116 160.845 165.605 170.390 175-191

2.12928 180.000

2.24961 180.000

Example,

sinh (1.55 +

i0.60)

2.37557 180.000

2.50746 180.000

= 2.39066 /56°·42ο = 2.39066

[96]

2.64563 180.000

TABLE X .

HYPERBOLIC

SINES,

χ = 1.75

o 0.0s O.I O.I s

0.2 0.25 °·3 °·35 °·4 °·45 °-SS

0.6 0.63 0.7 o-75 0.8 0.85 0.9 0-9S

1.0 1.05 1.1 Ι

·Ι5 1.2 1-25

ΐ·3 1-35 1.4 1-45 ΐ·5 1-55 1.6

1.65 ΐ·7 1-75

1.8 1.8s

1.9 1-95

2.0

0.00

x

2.79041 2.79151 2.79479 2.80016 2.80747

14-307 19.042

2.94217 2.94322 2.94633 2.95142 2.95835

2.81653 2.82710 2.83891 2.85165 2.86499

23-750 28.425 33-063 37.661 42.216

2.96695 2.97699 2.98821 3.00031 3.01300

2.87861 2.89217 2.90532 2.91777 2.92921

46.730 51.200 55-630 60.020

46.565 51-039

64-375

3-02595 3-03885 3-05138 3-06323 3-07413

2·93938 2.94804 2.95498 2.96006 2-96315

68.698 72.992 77.265 81.520 85-763

3.08382 3.09206 3.09869 3-10353 3.10648

68.586 72.900

4-779 9-551

2.96419 90.000 2.96315 94-237 2.96006 98.480 2.95498 102.735 2.94804 107.008

0.00 4-752 9-497

14.228 18.941 23.629 28.287 32.912 37-50I

42-053

55-476 59-877

64.246

77-194

81.471 85-738

=

3.10129 3.10228 3-10523 3.11006 3.11665 3.12481 3-Ï3434

3.14500 3-15650 3.16856

111.414

2.87861 2.86499 2.85165 2.83891 2.82710

133.270 137-784 142-339 146.937 151-575

3-02595 3.01300 3.00031 2.98821 2.97699

133-435 137-947

2.81653 2.80747 2.80016 2.79479 2.79151

156.250 160.958 165.693 170.449 175.221

2.96695 2.95835 2.95142 2.94633 2.94322

2.79041

180.000

2.94217

II5-754

120.123 124.524 128.961

23-520 28.163 32.776 37-357

41.904

3-23598

68.484 72.816 77.129 81.428 85.716

3-24384 3-25015 3-25477 3-25759

3-23598 3.22675 3.21636 3.20507 3-I93IS

3-08382 3-07413 3-06323 3-05138 3-03885

4.727 9.448 14.158 18.850

46.416 50.893

3-25853

II 1.302 115-625 119.980 124.370 128.800

I.Ss

3.18088 3-I93I5 3.20507 3.21636 3.22675

3.10747 90.000 3.10648 94.262 3-10353 98.529 3.09869 102.806 3.09206 107.100

2.93938 2.92921 2.91777 2.90532 2.89217

Example.

sinh (x + ig) = r / γ .

55-337 59-748

64.129

90.000 94.284 98.572 3-25015 102.871 3-24384 107.184

3-25759 3-25477

111.516 115.871 120.252 124.663 129.107

χ -

3.26816 3.26911 3-27191 3.27649 3.28274 3.29049 3-29955

3-30967 3.32060 3-33207 3-34378 3-35546

3-36681 3-37756 3-38745

3-39624 3-40373

3-40975 3-4I4IS 3.41683

CONTINUED

1.9

9-405

3-44675 3-45112 3-45702

0.00 4.685 9365 14-036 18.693

23.421 28.051 32.654 37.227 41.770

3.46441

23-332

3-49302 3-50393

37.110 41.649

46.281 50.761 55-211 59-631 64.023

3-SI507 3.52617 3-53698

46.160 50.642 55-096

3-S5663

63.927

68.392 72.740 77.071 81.388 85.696

3-56500

68.308 72.670 77.017 81.352 85.678

3-44321

4-705 14.094 18.767

3-48263

3-54721

3-57213 3-57788

3.58207 3.58462

27.950

32-543

59-524

90.000 94.322 98.648 3 - 5 7 7 8 8 102.983 3-57213 107.330

3-39624 I I 1.608

3-56500 111.692 3-S5663 116.073 3-54721 120.476 3-53698 124.904 3.52617 129-358

3-38745 3-37756

115-977

120.369 3-36681 124.789 129.239 3-35546 3-34378 3-33207

3-29955

iSi-949

156-371 161.059 165.772 170.503 175.248

3.12481 156.480 3.11665 161.150 3.11006 165.842 3 - i o 5 2 3 170.552 3.10228 175-273

3.29049 3.28274 3.27649 3.27191 3.26911

156.579 161.233 165.906 170.595

180.000

3.1012g

3.26816 180.000

I33-7I9 138.230 3.32060 142.773 3-30967 147-346

175-295

sinh (1.90 + i 2.0) = 3.26816 /i8o°.o = 3.26816 \i8o°.o.

[97]

3-4730I

3-58S48 3.58462 3-58207

133-584 138.096 142.643 147.224 3-13434 151-837

180.000

3.44410

90.000 3-41683 94·3θ4 3-4I4I5 98.612 340975 102.929 3-40373 107.260 3-41773

3.18088 3-16856 3-15650 3.14500

142.499 147.088 I5I-7I3

ζ = ι.95

3-5I507 133.840 3-50393 138.351 3-49302 142.890 3-48263 147-457 3-47301 152.050 3.46441

156.668 161.307 165.964 170.635 3.44410 I75-3I5 3-45702 3-45112 3-44675

3.44321

180.000

TABLE X .

HYPERBOLIC χ -

SINES,

sinh ( * + ig) = r / γ .

CONTINUED

χ - 2.15

2.05

0 0.05 O.I 0.15 0.2

3.62686 3.62771 3.63023 3-63437 3.64000

0.000 4.667 9-330 13.984 18.626

2.81950 3.82039 3.82279 3.82671 3.83206

0.000 4.651 9.298 13938 18.566

4.02186 4.02263 4.02490 4.02863 4-0337I

0.000 4.636 9.269 I3-89S 18.511

4.23419 4.23491 4.23707 4.24061 4-24544

0.000 4.623 9-243 I3-857 18.462

4-457" 4-45779 4·45985 4.46322 4.46780

0.000 4.611 9.220 13-823 18.418

0.2S 0-3 0-3S 0.4 0.4S

3.64699 3-655I7 3.66430 3.67418 3-68455

23.252 27.858 32-443 37.004 41-539

3-83870 3-84648 3-855I5 3-86454 3.87440

23.180 27.776 32-353 36.908 41.440

4.04002 4.04740 4.05566 4.06459 4.07396

23.114 27.701 32.271 36.821 41-351

4-25I45 4.25846 4.26630 4.27479 4.28371

23-055 27.634 32.197 36.743 41.270

4-47350 4.48017 4.48763 4-49570 4.50417

23.002 27-573 32-130 36.671 41.196

0.5 0.5S 0.6 0.65 0.7

3-695I5 3-70572 3.71600 3-72574 3-73470

46.049 50.533 54-992 59-427 63.840

3-88448 3-89453 3-90432 3-9*359 3.92213

45-950 50.435 54-898 59-340 63.761

4-08355 4.09311 4.10243 4.11125 4.11938

45-859 SO-347 54-813 59.260 63.689

4.29282 4-30I93 4.31079 4.31918 4-32693

45-778 50.266 S4-736 59-i88 63.624

4-51285 4-52150 4-52993 4-53793 4-54529

45-703 50.193 54.666 59-123 63-565

0-7S 0.8 0.8s 0.9 0-95

3.74268 3-74948 3-75495 3-75894 3-76137

68.232 72.608 76.969 81.319 85.661

3.92972 3.93620 3.94142 3-94521 3-94753

68.164 72.551 76.925 81.289 85.646

4.12661 68.101 72.499 4.13278 4-13774 7 6 . 8 8 5 4.14136 ,81.262 4-14357 8 5 . 6 3 2

4-3338I 4-33968 4.34440 4-34785 4.34996

68.044 72.452 76.849 81.237 85.620

4-55184 4-55744 4.56167 4-56523 4-56723

67-993 72.409 76.816 81.215 85.609

1.0 1 ·°5 1.1 1-iS 1.2

3.76220 90.000 3-76137 94-339 3-75894 9 8 . 6 8 1 3-75495 1 0 3 . 0 3 1 3-74948 1 0 7 . 3 9 2

3-94832 9 0 . 0 0 0 3-94753 94-354 3-94521 9 8 . 7 1 1 3.94142 103.075 3.93620 107.449

90.000 4-I443I 4-14357 9 4 . 3 6 8 4.14136 98.738 4-13774 103-115 4.13278 107.501

4-35067 9 0 . 0 0 0 94.380 4.34996 4-34785 9 8 . 7 6 3 4-34440 1 0 3 . 1 5 1 4-33968 1 0 7 . 5 4 8

I.2S 1-3 I-3S 1.4 i-45

3.74268 3-73470 3-72574 3.71600 3-70572

111.768 116.160 120.573 125.008 129.467

3.92972 3.92213 3-91359 3-90432 3-89453

4.12661 4.11938 4.11125 4.10243 4.09311

111.899 116.311 120.740 125.187 129.653

4-3338I 4-32693 4.31918 4.31079 4-30I93

111.956 116.376 120.812 125.264 129.734

4.56791 90.000 4-56723 94-391 4-56523 98-785 4.56167 103.184 4-55744 1 0 7 . 5 9 1 4-55I84 1 1 2 . 0 0 7 4-54529 " 6 . 4 3 5 4-53793 1 2 0 . 8 7 7 4-52993 125.334 4-5215° 1 2 9 . 8 0 7

ϊ·5 i-SS

3-695I5 3-68455 3.67418 3.66430 3-65SI7

133-951 138.461 142.996 147-557 152.142

3.88448 3.87440 3-86454 3-8551S 3-84648

1-75 1.8 i.8S 1.9 ι-95

3.64699 3.64000 3-63437 3-63023 3.62771

2.0

3.62686

1.6

1.6S 1-7

111.836 116.239 120.660 125.102 129-565

4-08355 4.07396 4.06459 4.05566 4.04740

134.141 138.649 143-179 147.729 152.299

4.29282 4.28371 4.27479 4.26630 4.25846

134.223 138.730 143-257 147.803 152.366

4-51285 4.50417 4-4957° 4.48763 4.48017

I34-297 138.804 I43-329 147.870 152.427

156.748 161.374 166.016 170.670 175-333

134-051 138.560 143.092 147.647 152.224 3-83870 1 5 6 . 8 2 0 3.83206 161.434 3.82671 166.062 3.82279 170.702 3 - 8 2 0 3 9 175-349

4.04002 4-03371 4.02863 4.02490 4.02263

156.886 161.489 166.105 170.731 I75-364

4-25I4S 4-24544 4.24061 4.23707 4.23491

156.945 161.538 166.143 170.757 175-377

4-47350 4.46780 4.46322 445985 4-45779

156.998 161.582 166.177 170.780 I75-389

180.000

3.81958

4.02186

180.000

4 . 2 3 4 1 9 .180.000

4.45711

180.000

180.000

Example.

sinh ( 2 . 0 + i 1.0) = 3 . 7 6 2 2 0 / 9 0 o .

Γ98]

TABLE X . X = 2.25

0 0.05 O.I 0.15 0.2

4.69117 4.69182 4-69377 4.69697 4.70134

0.000 4.601 9.199 13-792

0.25 0-3 0.35 0.4 0.45

4.70675 4.71308 4.72017 4.72784 4-73592

05 0.55 0.6 0.65 0.7

4.744I5 4.75240 4.76042 4.76802 4-77503

22.954 27.518 32.070 36.608 41.130 4S-636 50.127 54-603 59.064 63.512

0.75 0.8 0.85 0.9 °-95

4.78127 4.78661 4.79088 4.79402 4-79593

67.947 72-371 76.786 81.195 85-599

1.0 1-05 I.I I-I5 1.2

4-79657 4-79593 4.79402 4.79088 4.78661

1.25 1-3 1-35 1-4 1-45

HYPERBOLIC X - 2.3

22.910 27.469 32.016 36-549 41.070

4-98734 4.99518 5.00281 5.01005 5.01672

45-576 50.068 54-546 59.011

5.02265 5-02773 5.03181 5-03479 5.03661

X = 2.4s

X - 2.4

5-75103 5-75IS6 5-753i6 5-75576 5-75932

0.000 4-567 9-133 I3-695 18.252

5-49775 5.50468

22.835 27-383 31.922 36.449 40.966

5-7637S 5.76892 5-77472 5.78099 5-78758

22.803 27-347 31.881 36.407 40.922

45-521 50.014 54-494 58.962 63-419

5-5II77 5-5I887 5-52578 5-53233 5-53837

45-471 49.965 54-447 58.919 63.380

5-79434 5.80108 5-80765 5-81389 5.81964

45-42 7 49.921 54-405 58.879 63·344

5.27662 5-28143 5-28531 5.28816 5-28989

67.866 72.304 76.735 81.160 85.581

5-54375 5-54834 5-55204 5-55474 5-5S639

67.831 72.275 76.712 81.144 85-573

5.82476 5-82913 5-83265 5-83522 5-83680

67.800 72.249 76.692 81.131 85.566

5.29047 5.28989 5.28816 5-28531 5-28143

90.000 94.419 98.840 103.265 107.696

5-55695 5-55639 5-55474 5-55204 5-54834

90.000 94.427 98.856 103.288 107.725

22.871 27.424 31.966 36-497 41.016

63-463 67.904 72.336 76.759 81.176 85-589

90.000 94.401 98.805 103.214 107.629

5.03722 90.000 94.411 5.03661 5-03479 98.824 5-03181 103.241 5-02773 107.664

4.78127 4-775°3 4.76802 4.76042 4-75240

112.053 116.488 120.936 125.397 129.873

5-02265 5.01672 5.01005 5.00281 4.99518

112.096 116.537 120.989 125-454 129.932

5.27662 112.134 5.27096 116.581 5.26461 121.038 5-25772 125.506 5.25046 129.986

1-5 1-55 1.6 1-65 1-7

4-74415 4-73592 4.72784 4.72017 4.71308

134-364 138.870 143-392 147-93° 152.482

4-98734 4-97950 4.97183 4-96454 4.95780

134.424 138.930 143-451 147.984 152.531

5-23728 5-23553 5-22825 5-22131 5.21490

134-479 138.984 143-503 148.034 152.576

1-75 1.8 1-85 1.9 1-95

4.70675 157-046 4-70I34 161.622 4.69697 166.208 4-69377 170.801 4.69182 175-399

4-95177 157.090 4.94662 161.659 4.94249 166.236 4-93944 170.820 4-93759 175.409

5.20917 5.20428 5-20035 5-19745 5-19569

2.0

4.69117

4.93696

5.195x0

180.000

CONTINUED

0.000 4-574 9-I47 13.716 18.279

5.20917 5.21490 5-22131 5.22825 5-23553 5.23728 5.25046 5-25772 5.26461 5.27096

Example.

4-95177 4.95780 4·96454 4.97183 4-97950

* = 2-35

0.000 4.582 9.163 13-739 18.309

180.000

4-591 9.180 I3-764 18.341

sinh (x + iq) = r /y.

5.19510 5-I9569 5-19745 5-20035 5.20428

18.378

4.93696 4-93759 4-93944 4.94249 4.94662

SINES,

5.46623 5.46679 5.46847 5.47122 5-47495 5.47961 5-48505 5-49115

90.000 5-83732 5-83680 94-434 5-83522 98.869 5-83265 103.308 5-82913 107.751 5.82476 5.81964 5-81389 5.80765 5.80108

112.200 116.656 121.121 125.595 130.079

5-79434 5-78758 5.78099 5-77472 5.76982

134-573 139.078 143-593 148.119 152.653

157.129 161.691 166.261 170.837 175.418

5-54375 112.169 5-53837 116.620 5-53233 121.081 5-52S78 125-553 130.035 5-51887 I34-529 5-51177 I39-034 5.50468 143-551 5-49775 148.078 5-49115 152.617 5-48505 5.47961 157-165 5-47495 161.721 5.47122 166.284 5.46847 170.853 5.46679 175.426

5-76375 5-75932 5-75576 5-753I6 5-75I56

I57.I97 161.748 166.305 170.867 175-433

180.000

5.46623

5.75103

180.000

180.000

s i n h ( 2 . 4 0 + i 0^4) = 5 - 4 9 7 7 5 / 3 6 ° · 4 4 9 = 5 - 4 9 7 7 5 /z6".26'.s6".

[99]

TABLE X. χ = 2.5

HYPERBOLIC SINES, « = 2.55

sinh (x + ig) = r /γ.

χ - 2.6

O.I 0.15 0.2

6.05020 6.05071 6.05223 6.05471 6.05808

0.000 4.561 9.120 13.676 18.228

6.36451 6.36499 '6.36644 6.36879 6.37201

0.000 4-555 9.109 13.660 18.206

6.69473 6.69518 6.69656 6.69880 6.70185

0.000 4-550 9.098 13.644 18.187

0.25 0-3 0-3S 0.4 o-45

6.06229 6.06722 6.07273 6.07869 6.08496

22.774 27-3I4 31-845 36.368 40.882

6.37601 6.38068 6.38592 6.39160 6-39755

22.748 27.284 31.812 36.333 40.845

6.70565 6.71011 6.71509 6.72048 6.72616

o-S 0-55 0.6 0.6s 0.7

6.09139 6.09781 6.10406 6.10999 6.11547

45-386 49.881 54-366 58·843 63-311

6.40367 6.40977 6.41572 6.42658

45-350 49.845 54·33 2 58.810 63.282

0-7S 0.8 0.85 0.9 0-95

6.12034 6.12450 6.12785 6.13030 6.13179

67.772 72.226 76.674 81.119 85.560

6.43121 6.43518 6.43836 6.44069 6.44212

ι.00 1.05 1.1 1-iS 1.2

CONTINUED

χ - 2.6s

2.7

7-04343 7-04556 7.04847

4-545 9.089 13-631 18.169

7.40626 7.40668 7.40791 7.40994 7-41271

0.000 4-540 9.080 13.618 18.153

22.725 27.257 31.782 36.301 40.813

7.05208 7-05631 7.06105 7.06618 7.07158

22.703 27.232 3I.756 36.273 40.783

7.41614 7.42017 7.42467 7·42955 7.43468

22.684 27.210 31-731 36.246 40.756

6.73197 6.73778 6-74343 6.74881 6.75376

45·3ΐ6 49.812 54-30O 58.781 63.255

7.07711 7.08264 7.08802 7-09313

45-286 49.782 54·272 58.754 63.230

7-43994 7-44519 7-45032 7·455ΐ8 7.45966

45-259 49-755 54.246 58.730 63.209

67.746 72.204 76.658 81.107 85-554

6.75818 6.76195 6.76499 6.76720 6.76855

67.701 72.167 76.629 81.088 85-544

7.46366 7.46708 7.46982 7-47183 7.47306

67.682 72.152 76.617 81.080 85-S40

6.13229 90.000 6.13179 94.440 6.13030 98.881 6.12785 103.326 6.12450 107.774

6.44259 90.000 6.44212 94.446 6.44069 98.893 6.43836 103.342 6.43518 107.796

6.76901 6.76855 6.76720 6.76499 6.76195

67.722 72.185 76.643 81.097 85-549 90.000 94-451 98.903 103.357 107.815

7-11234 90.000 7.11191 94.456 7.11062 98.912 7.10851 103-371 7-10563 107-833

7-47347 90.000 7.47306 94.460 7-47I83 98.920 7.46982 103.383 7.46708 107.848

1.25 1-3 I-3S 1.4 ι-45

6.12034 6-11547 6.10999 6.10406 6.09781

112.228 116.689 121.157 125-634 130.119

6.43121 6.42658 6.42137 6.41572 6.40977

112.254 116.718 121.190 125.668 130.155

6.75818 6.75376 6.74881 6-74343 6.73778

112.278 116.745 121.219 125.700 130.188

7.10204 7.09784 7-09313 7.08802 7.08264

112.299 116.770 121.246 125.728 130.218

7.46366 7-45966 7-45518 7-45032 7-44519

112.318 116.791 121.270 125.754 130.245

1-5 1-55 1.6 1.65 1-7

6.09139 6.08496 6.07869 6.07273 6.06722

134.614 139.118 143.632 148.155 152.686

6.40367 6-39755 6.39160 6.38592 6.38068

134-651 I39.I55 143.667 148.188 152.716

6.73197 6.72616 6.72048 6.71509 6.71011

134.684 139.187 143.699 148.218 152.743

7.07711 7.07158 7.06618 7.06105 7-05631

I34-7I3 139.217 ΐ43·727 148.244 152.768

7-43994 7.43468 7-42955 7.42467 7.42017

I34.74I 139-244 143-754 148.269 152.790

1-75 1.8 1.85 1.9 ι-95

6.06229 6.05808 6.05471 6.05223 6.05071

157.226 161.772 166.324 170.880 175-439

6.37601 6.37201 6.36879 6.36644 6.36499

157-252 161.794 166.340 170.891 175-445

6.70565 6.70185 6.69880 6.69656 6.69518

157-275 161.813 166.356 170.902 175450

7.05208 7.04847 7.04566 7-04343 7.04213

157.297 161.831 166.369 170.911 175-455

7.41614 7-41271 7.40994 7.40791 7.40668

I57-3I6 161.847 166.382 170.920 Ι75·46Ο

2.0

6.05020 180.000

o O.OS

Example.

6.42137

6.36451 180.000

6.69473 180.000

7.04169 7.04213

7.09784 7.10204 7-10563 7.10851 7.11062 7.11191

7.04169 180.000

7.40626 180.000

sinh (2.6 + i o6) = 6.74343 /54°·300 = 6.74343 /54°.i8,oqit.

[100]

TABLE X .

HYPERBOLIC

SINES,

χ = 2.75

sinh (x + iq) = r ¿y.

» = 2.85

CONTINUED

X = 2.9

* =

2 -95

o 0.05 O.I O-IS 0.2

7-78935 7-78975 7.79092 7.79285 7-79547

0.000 4-537 9-073 13.607 18.138

8.19192 8.19230 8.19341 8.19524 8.19774

4-533 9.066 I3-596 18.125

8.61497 8.61532 8.61639 8.61813 8.62051

0.000 4-530 9.060 I3-587 18.113

9-05956 9.05990 9.06091 9.06257 9.06483

0.000 4-527 9-054 13-579 18.102

9.52681 9-527I3 9.52809 9.52967 9.53182

0.000 4-525 9.049 13-571 18.093

0.25 °·3 0-35 °·4 0-45

7-79875 7.80257 7.80685 7.81149 7.81637

22.666 27.190 31.709 36.223 40-73I

8.20085 8.20449 8.20857 8.21298 8.21762

22.650 27.172 31-689 36.202 40.710

8.62347 8.62691 8.63080 8.63500 8.63941

22.636 27-155 31-671 36-183 40.690

9.06764 9.07093 9.074Ò1 9.07861 9.08281

22.623 27.141 3I-65S 36-165 40.671

9-53450 9-53762 9-54II3 9-54492 9.54892

22.611 27.127 31.640 36-150 40.655

°·5 0-55 0.6 0.65 ο·7

7.82138 7.82638 7-83125 7-83588 7.84015

45-234 49-731 58.708 63.189

8.22238 8.22713 8.23177 8.23617 8.24024

45.212 49.709 54.201 58.688 63.171

8.64395 8.64846 8.65287 8.65706 8.66092

45.192 49.689 54.182 58.670 63-I55

9.08711 9.09142 9.09561 9.09959 9.10327

45-173 49.671 54-165 58.654 63.140

9-5530I 9-557" 9.56109 9-56488 9-56838

45-157 49-655 54-149 58.640 63.127

0-75 0.8 0.85 °·9 0-95

7-84395 7.84720 7.84980 7.85172 7.85288

67.665 72.137 76.606 ¿I.072 85-536

8.24385 8.24694 8.24942 8.25124 8-25235

67.649 72.124 76.596 81.065 85-533

8.66436 8.66731 8.66967 8.67140 8.67246

67.636 72.112 76.587 81.059 85-530

9.10655 9.10934 9.11160 9.11324 9.11424

67.622 72.102 76.579 81.053 85-527

9-57I50 9.57416 9-57630 9-57787 9-57882

67.611 72.092 76-57I 81.048 85-524

Ι.Ο Ι.05 Ι.Ι Ι·Ι5 1.2

7.85328 9O.OOO 7.85288 94.464 7.85172 98.928 7.84980 IO3.394 7.84720 IO7.863

8.25273 90.000 8-25235 94.467 8.25124 98-935 8.24942 103.404 8.24694 107.876

1-25 1-3 1-35 1-4 1-45

7-84395 7-84015 7-83588 7-83125 7.82638

"2-335 116.811 121.292 125.778 130.269

8.24385 8.24024 8.23617 8.23177 8.22713

1-5 1-55 1.6 1.65 1-7

7-82138 7.81637 7-81149 7.80685 7-80257

134.766 139.269 143-777 148.291 152.810

1-75 1.8 1.85 1-9 1-95

7-79875 7-79547 7.79285 7-79092 7-78975

2.0

54-222

8.67281 90.000 8.67246 94.470 8.67140 98.941 8.66967 103.413 8.66731 107.888

9.11458 90.000 9.11424 94-473 9.11324 98.947 9.11160 103.421 9.10934 107.898

9-57915 90.000 9-57882 94-476 9-57787 98.952 9-57630 103.429 9-574I6 107.908

112.351 1x6.829 121.312 125.799 130.291

8.66436 8.66092 8.65706 8.65287 8.64846

112.364 116.845 121.330 125.818 i30-311

9.10655 9.10327 9-09959 9.09561 9.09142

112.378 116.860 121.346 125-835 130-329

9-57I50 9-56838 9.56488 9.56109 9-557"

112.389 116.873 121.360 125.851 130.345

8.22238 8.21762 8.21298 8.20857 8.20449

I34-788 139.290 143.798 148.311 152.828

8.64395 8.63941 8.63500 8.63080 8.62691

134.808 139.310 143.817 148.329 152.845

9.08711 9.08281 9.07861 9.07461 9.07093

134.827 r 39-329 I43-83S 148.345 152.859

9-5530I 9.54892 9.54492 9-54"3 9-537Ó2

I34-843 139-345 143-850 148.360 152.873

157-334 161.862 166.393 170.927 I7S-463

8.20085 8.19774 8.19524 8.19341 8.19230

I57-350 161.875 166.404 170.934 175-467

8.62347 8.62051 8.61813 8.61639 8.61532

157-364 161.887 166.413 170.940 I75-470

9.06764 9.06483 9.06257 9.06091 9.05990

157-377 161.898 166.421 170.946 175-473

9-53450 9.53182 9.52967 9.52809 9-527I3

I57-389 161.907 166.429 170.951 175-475

7-78935 180.000

8.19192

180.000

8.61497 180.000

Example.

9.05956 180.000

9.52681 180.000

sinh (2.95 + i 1.95) = 9.52713 /i75°-475 = 9-5 2 7i3 /i7.S°-28'·.^.

[ιοί]

TABLE X . » = 3.0

HYPERBOLIC χ = 3.05

SINES,

sinh (x + ig) = r /y. X = 3.

CONTINUED

* = 3-15

χ =

3.2

o 0.05 O.I o-iS 0.2

10.01787 10.01820 10.01910 10.02060 10.02260

0.000 4.522 9.044 I3-56S 18.084

10.53399 10.53430 10.53520 10.53660 10.53850

0.000 4.520 9.040 13-559 18.076

11.07645 11.07670 11.07750 11.07890 11.08080

0.000 4.518 9-036 13-553 18.068

1.64661 1.64690 1.64770 1.64890 1.65070

0.000 4.516 9-033 13-548 18.062

12.24588 12.24610 12.24690 12.24810 12.24980

4-515 9.030 13-543 18.056

0.2s °-3 0-3S 0.4 0-4S

10.02520 10.02820 10.03150 10.03510 10.03890

22.601 27.115 31.627 36-I35 40.640

10.54090 10.54380 10.54690 10.55040 10.55400

22.591 27.104 3!-6iS 36.122 40.627

11.08310 11.08580 11.08880 11.09200 11.09540

22.583 27.094 31.604 36.111 40.615

1.65290 1.65540 1.65830 1.66140 1.66470

22.575 27.085 3 I -594 36.100 40.604

12.25180 12.25430 12.25700 12.26000 12.26310

22.568 27.077 31-585 36.091 40.594

°-5 o-SS 0.6 0.65 0.7

10.04280 10.04670 10.05050 10.05410 10.05743

45.142 49.640 54-135 58.626 63.114

10.55770 10.56140 10.56500 10.56840 10.57160

45-129 49.627 54.122 58.614 63.104

11.09900 45.116 il.10250 49.615 II. 10600 54.110 II.10920 58.604 I I.I 1220 63.094

I.668IO 1.67140 1.67470 1.67780 1.68060

45- i o 5 49.604 54.100 58.594 63-085

12.26630 12.26950 12.27260 12.27550 12.27820

45-095 49-594 54.090 58.585 63.077

0-7S 0.8 0.85 0.9 °-95

10.06040 10.06292 10.06500 10.06645 10.06737

67.600 72.083 76.564 81.044 85.522

10.57440 10.57680 10.57880 10.58020 10.58110

67-59I 72.075 76.558 81.040 85.520

II.II490 II.II720 II.II900 II.12040

1.68320 1.68540 1.68710 1.68840 1.68920

67-574 72.062 76.548· 81.032 85.516

12.28070 12.28280 12.28440 12.28560 12.28640

67.567 72.056 76.543 81.029 8S-5I5

1.0 1.05 1.1 1-IS 1.2

10.06766 90.000 10.06737 94.478 10.06645 98.956 10.06500 103.436 10.06292 107.917

10.58135 90.000 10.58110 94.480 10.58020 98.960 10.57880 103.442 10.57680 107.925

II.I2I50

90.000 94.482 II.12040 98.964 II.II900 103.447 II.II720 107.932

1.68946 90.000 1.68920 94.484 1.68840 98.968 1.68710 103.452 1.68540 107.938

12.28665 90.000 12.28640 94-485 12.28560 98.971 12.28440 103.457 12.28280 107.944

1-25 1-3 I -3S 1.4 1-45

10.06040 10.05743 10.05410 10.05050 10.04670

112.400 116.886 121.374 125.856 130.360

10.57440 10.57160 10.56840 10.56500 10.56140

112.409 116.896 121.386 125.878 130.373

I I.I 1490 11.11220 II.10920 II.10600 II.10250

112.418 116.906 121.396 125.890 130.385

1.68320 1.68060 1.67780 1.67470 1.67140

112.426 116.915 121.406 125.900 130.396

12.28070 12.27820 12.27550 12.27260 12.26950

112.433 116.923 121.415 125.910 130.406

1-5 1-55 1.6 1.65 1-7

10.04280 10.03890 10.03510 10.03150 10.02820

134.858 139.360 143.865 148.373 152-885

10.55770 10.55400 10.55040 10.54690 10.54380

134.871 139-373 143.878 148.385 152.896

11.09900 11.09540 11.09200 11.08880 11.08580

134-884 139-385 143-889 148.396 152.906

I.668IO 1.66470 1.66140 1.65830 1.65540

I34-895 I39-396 143.900 148.406 152.915

12.26630 12.26310 12.26000 12.25700 12.25430

I34-905 139.406 143.909 148.415 152.923

I-7S 1.8 1.85 1.9 ι-95

10.02520 10.02260 10.02060 10.01910 10.01820

157-399 161.916 166.435 170.956 I75-478

10.54090 10.53850 10.53660 10.53520 10.53430

157.409 161.924 1 166.441 . * 170.960 175.480

11.08310 11.08080 11.07890 11.07750 11.07670

157-417 161.931 166.447 170.964 175.482

1.65290 1.65070 1.64890 1.64770 1.64690

I57-425 161.938 166.452 170.967 175.484

12.25180 12.24980 12.24810 12.24690 12.24610

157-432 161.944 166.457 170.970 I75-485

2.0

10.01787 180.000

Example.

10.53399 180.000

I I.I2120

67.582 72.068 76.553 81.036 85.518

I I.I 2 1 2 0

11.07645 180.000

11.64661 180.000

12.24588 180.000

sinh (3.2 + ì l i ) = 12.28560 /g8°.97i = 12.28560 /q8°..s8'.i6*.

[ 102]

Table X.

* = 3-25

2.0

HYPERBOLIC

SINES,

s i n h ( χ + iq) = r /y.

* = 3-3

3-35

Continued

» = 3-4

2.87578 2.87600 2.87670 2.87790 2.87949

0.000 4-514 9.027 13-539 18.051

13-53788 13-53810 13-53878 ΐ3·5399ΐ Ι3·54ΐ4ΐ

0.000 4.512 9.024 I3-536 18.046

14.23382 14.23403 14.23470 14-23573 14-23718

0.000 4-5II 9.022 13-532 18.041

14.96536 14.96556 14.96616 14.96716 14.96855

4.510 9.020 Ι3·529 18.038

2.88146 2.88380 2.88637 2.88910 2.89215

22.561 27.070 31-577 36.082

13.54322 13.5455° I3-54796 13.55062 13-55346

22-555 27.063 3!-569 36.074 40-577

14.23897 14.24106 14.24339 14-24595 14.24863

22.550 27.057 31-563 36.067 40-570

14.97023 14.97225 14.97448 14.97690 14-97945

22.545 27.052 31-557 36.061 40.563

2.89518 2.89820 2.90117 2.90398 2.90658

45.086 49-585 54.082 58.577 63.070

13-55633 Ι3·559ΐ8 13.56203 13-56470 13.56718

45.078 49-577 54-074 58.569 63.063

14-25137 14.25412 14.25614 14-25933 14.26167

45-071 49-570 54.067 58.563 63·°57

14.98205 14.98466 14.98721 14.98960 14.99186

45.064 49·563 54.061 58.557 63.052

2.Ç 2.91085 2.91240 2.91360 2.9143Ο

67-561 72.051 76.539 81.027 85-514

13-56936 I3-57I23 I3-57275 I3-57387 13-57455

67-555 72.046 76-535 81.024 85-512

14.26377 14.26556 14.26700 14.26805 14.26870

67-550 72.041 76-532 81.022

2.91456 90.000 94.486 2.9143Ο 98·973 2.91360 2.91240 103.461 2.91085 107.949

I3-S7476 13-57455 I3-57387 I3-57275 I3-57I23

90.000 94.488 98.976 103.465

14.26891 14.26870 14.26805 14.26700 14.26556

90.000 94.489 98.978 103.468 107.959 112.450 116.943 121.437 125-933 130.430

40-585

107.954

85-511

22.541 27.047 3I.55I 36-055

I4-99385 67-545 72.038 14-99555 76.529 14.99692 81.020 14.99790 85.510 I4-99853 14.99874 90.000 I4-99853 94.490 14.99790 98.980 14.99692 103.471 14-99555 107.963

15.76144 15-76303 I5-76433 15-76530 15-76587

67-54I 72.034 76.526 81.018 85-509

I4-99385 1 1 2 . 4 5 5 14.99186 1 1 6 . 9 4 8 14.98960 1 2 1 . 4 4 3 14.98721 125-939 14.98466 I30-437 14.98205 134-936 14-97945 139-437 14.97690 143-939 14.97448 148.443 14.97225 152.948

15.76144 112.459 I5-75950 1 1 6 . 9 5 3 I5-75740 121.449 I 5 - 7 5 5 I O 125.945 I 5 - 7 5 2 7 0 130.443

157.455 161.962 166.471 170.980 175.490

I5-73898 157-459 I5-73736 161.96Ò 15-73603 166.474 1 5 - 7 3 5 1 0 170.982 I5-73450 175-491

180.000

15.73432

13-56936 13.56718 13.56470 .13.56203 I3-559I8

112.445 116.937 121.431 125.926 130-423

14.26377 14.26167 I4.25933 14.25614 14.25412

2.89518 2.89215 2.889IO 2.88637 2.88380

134.914 139-415 143-918 148.423 I52-93°

13-55633 ΐ3·55346 13.55062 I3.54796 13-5455°

134.922 I39-423 143.926 148.431 I52.937

14.25137 14.24863 14.24595 14.24339 14.24106

134.929 139.430 143-933 148.437 152.943

2.88146 2.87949 2.87790 2.87670 2.87600

157-439 161.949 166.461 170.973 I75-486

Ι3·54322 157-445 I3-54I4I i6i-9S4 I 3 . 5 3 9 9 I 166.464 I3-53878 170.976 1 3 . 5 3 8 1 0 175-488

14.23897 14.23718

157.450 161.959

14.23573 14.23470 14.23403

166.468 170.978 175.489

14.97023 14.96855 14.96716 14.96616 14.96556

12.87578

180.000

13.53788

14.23382

180.000

14.96536

40.557

15.76607 90.000 94.491 15-76587 98.982 15-76530 I5-76433 103.474 15-76303 107.966

15.75020 15.74772 15.74529 15.74300 15.74088

s i n h (3.3 + i 1 ^ ) = 1 3 . 5 6 7 1 8 / ι ι 6 ° · 9 3 7 = 1 3 . 5 6 7 1 8 / ι ι 6 ° . ; ; 6 ' · ΐ 3 ' ' .

tío 3]

4-509 9.018 13-526 18.034

45-058 49-557 54-055 58.551 63.047

112.439 116.930 121.423 125.918 130.415

Example.

ΐ5·73432 15-73450 I5-735IO 15-73603 I5-73736 15-73898 15.74088 15.74300 15-74529 15-74772 15.75020 I5-75270 I5-755IO I5-75740 I5-75950

2.90008 2.90658 2.90398 2.90117 2.89820

180.000

* = 3-45

134-942 139-443 143.945 148.449 152.953

180.000

TABLE X . x =

o 0.05 O.I o.iS 0.2

* = 3-55

3-5

SINES,

s i n h ( χ + iq)

x = 3-6

= r [_7.

CONTINUED

« = 3.7

X = 3-65

16.54428 16.5455°

16.54337

0.000 4.508 9.016 13-524 18.031

17.39230 17.39248 17.39300 I7-39386 I7-39504

0.000 4-507 9.015 I3-522 18.028

18.28546 18.28560 18.28611 18.28694 18.28805

0.000 4-507 9.013 I 3-5 I 9 18.025

19.22434 19.22448 19.22496 19.22577 19.22681

0.000 4.506 9.012 I3-5I8 18.023

20.21129 20.21140 20.21189 20.21268 20.21365

0.000 4.506 9.011 I3-5I6 18.021

0.25 °-3 0-3S 04

16.54702 16.54885 16.55087 16.55307

0-4S

22-537 27.042 31-547 36.050

16.55538

40.552

17.39650 17.39822 17.40015 17.40222 17.40441

22.534 27.038 31-542 36-045 40-547

18.28942 18.29109 18.29292 18.29490 18.29699

22.530 27-035 31-538 36.041 40.542

19.22813 19.22968 19.23142 I9-2333I I9-23530

22.527 27.031 31-535 36.037 40-538

20.21490 20.21639 20.21804 20.21981 20.22170

22.525 27.028 31-531 36-033 40.535

°-5 o-55 0.6 0.65

16.55774 16.56010 16.56240 16.56460 16.56660 16.56840 16.56996 16.57120 16.57210 16.57260

45-052 49-552 54-050 58.547 63.042 67-537 72.031 76.524 81.016 85.508

17.40665 17.40900 17.41110 17.41310 17.41510

45-047 49-547 54-045 58.542 63.038

18.29912 18.30126

45-043 49-542 54-04I 58.538 63-035

19.23732 19.23940 19.24134 19.24323 19.24496

45-039 49-538 54-037 58.535 63-031

20.22365 20.22560 20.22750 20.22930 20.23090

45-035 49-535 54-033 58.531 63.028

17.41680 17.41830 I7-4I945 17.42030 17.42083

67-533 72.028 76.521 81.015 85-507

18.30880 18.31020 18.31130 18.31210 18.31260

19.24650 19.24790 19.24890 19.24970 19.25015

67.527 72.023 76.518 81.012 85.506

20.23240 20.23365 20.23460 20.23540 20.23585

67-525 72.021 76.516 81.011 85.506

16.57282 90.000 16.57260 94.492 16.57210 98.984 16.57120 103.476 16.56996 107.969

17.42102 90.000 17.42083 94-493 17.42030 98.985 I7-4I945 103.479 17.41830 107.972

18.31278 18.31260 18.31210 18.31130 18.31020

67-530 72.025 76.519 81.013 85-507 90.000 94-493 98.987 103.480 107.975

19-25033 90.000 19.25015 94-494 19.24970 98.988 19.24890 103.482 19.24790 107.977

20.23601 90.000 20.23585 94.494 20.23540 98.989 20.23460 103.484 20.23365 107.979

16.56840 16.56660 16.56460 16.56240 16.56010

112.463 116.958 121.453 I25-950 130.448

17.41680 17.41510 17.41310 17.41110 17.40900

112.467 116.962 121.458 125.955 130.453

18.30880 18.30715 18.30530 18.30335 18.30126

112.470 116.965 121.462 125-959 130.458

19.24650 19.24496 19.24323 19.24134 19.23940

112.473 116.969 121.465 125.963 130.462

20.23240 20.23090 20.22930 20.22750 20.22560

16.55774 16.55538 16.55307 16.55087 16.54885

134.948 139.448 143-95° 148.453 152.958

17.40665 17.40441 17.40222 17.40015 17.39822

134-953 139-453 143-955 148.458 152.962

18.29912 18.29699 18.29490 18.29292 18.29109

134-957 139-458 143-959 148.462 152-965

134.961 139.462 I43-963 148.465 152.969

20.22365 20.22170 20.21981 20.21804 20.21639

16.54702 16.54550 10.54550 16.54428 16.54337 16.54281

I57-463 101.909 161.969 166.476 170.984 175.492

17.39650 17-38504 I7-39386 17.39300 17.39248

157.466 161.972 166.478 170.985 175-493

18.28942 18.28805 18.28694 18.28611 18.28560

I57-470 161.975 166.481 170.987 175-493

19.23732 19.23530 19-23331 19.23142 19.22968 19.22813 19.22681 19.22577 19.22496 19.22448

157-473 161.977 166.482 170.988 175-494

20.21490 20.21365 20.21268 20.21189 20.21140

0.7

0.75

0.8 0.8s 0.9 0-95 1.0 1.05 1.1 Ι Ι · 5 1.2

1-2S 1-3 I-3S 1.4

i-45 1-5 1-55 1.6 1.6s 1-7 I -7S 1.8 1.8S 1.9 ι-95 2.0

16.54263 16.54281

HYPERBOLIC

16.54263 180.000 Example.

T

7-39 2 3o 180.000

18.30335

18.30530 18.30715

18.28546 180.000

19.22434 180.000

I34-965 139-465 143.967 148.469 152-972 157.475 161.979 166.484 170.989 175.494

20.21129 180.000

sinh (3.65 + i 0.25) = 19.22813 /22°.527 = 19.22813 /22°.31'.37"'.

1104]

112.475

116.972 121.469 125.967 130.465

TABLE X . 3-75

HYPERBOLIC SINES, » = 3-8

o 0.05 O.I 0.15 0.2

21.24878 21.24891 21-24935 21.25006 21.25102

0.000 4-5°5 9.010 I3-SI4 18.019

22-33952 22-33995 22.34061 22.34153

0.25 0-3 o-35 0.4 0.4s

21.25221 21.25362 21.25520 21.25685 21.25869

22.522 27.026 3 i -5 2 8 36.030 40.531

o-5 0.55 0.6 0.65 0.7

21.26052 21.26236 21.26415 21.26586 21.26745

45-032 49-531 54-030 58-528 63.026

0.7s 0.8 0.8s 0.9 0-9S

21.26885 21.27000 21.27100 21.27170 21.27212

1.0 1-OS 1.1 i-iS 1.2

21.27230 21.27212 21.27170 21.27100 21.27000

67.522 72.019 76.514 81.010 85-S05 90.000 94-495 98.990 103.486 107.981

I-2S 1-3 1-35 1.4 1-45

21.26885 21.26745 21.26586 21.26415 21.26236

112.478 116.974 121.472 125.970 130.469

22.35850 22.35716 22.35565 22.35403 22.35230

1-5 1-55 1.6 1.6s 1-7

21.26052 21.25869 21.25685 21.25520 21.25362

1-75 1.8 1.8s 1.9 1-95 2.0

21.25221 21.25102 21.25006 21-24935 21.24891

134.968 139.469 143-970 148.472 i5 2 -974 I57-478 161.981 166.486 170.990 175-495

21.24878 180.000 Example,

* = 3-85

CONTINUED * = 3-95

* = 3-9

9.009 13-513 18.017

23-48589 23.48601 23.48640 23.48704 23.48791

0.000 4-504 9.008 13-512 18.015

24.69110 24.69120 24.69159 24.69223 24.69302

22.34270 22.34401 22.34550 22.34712 22.34883

22.520 27.023 31-526 36.027 40.528

23.48900 23.49028 23-49"5 23.49322 23.49486

22.518 27.021 3I-523 36.025 40.526

24.69406 24.69528 24.69662 24.69809 24.69964

40.523

22.35060 22.35230 22.35565 22.35716

22.35403

45.029 49.528 54-027 58.526 63.023

23.49652 23.49820 23.49980 23-50136 23.50277

45-026 49.526 54-025 58.523 63.021

24.70121 24.70280 24.70440 24.70580 24.70720

45-023 49-523 54.022 58.521 63.019

22.35850 22.35962 22.36055 22.36122 22.36163

67.520 72.017 76-513 81.009 85-504

23.50404 23-50512 23.50600 23.50664 23.50702

67.518 72.015 76.512 81.008 85-504

24.70840 24.70940 24.71024 24.71090 24.71120

67-517 72.014 76.511 81.007 85-504

22.36178 90.000 22.36163 94.496 22.36122 98.991 22.36055 103.487 22.35962 107.983

90.000 94.496 98.992 103.488 107.985

25-97731 90.000 25.97720 94.497 25.97680 98.993 25-97630 103.490 25-97550 107.987

112.482 116.979 121.477 125-975 130.474

24-71135 24.71120 24.71090 24.71024 24.70940 24.70840 24.70720 24.70580 24.70440 24.70280

90.000 94.496 98-993 103.489 107.986

112.480 116.977 121.474 125-973 130.472

23-50717 23.50702 23.50664 23.50600 23.50512 23.50404 23.50277 23.50136 23.49980 23.49820

112.483 116.981 121.479 125.978 130.477

25-974SO 25-97337 25.97206 25.97066 25.96920

112.485 116.983 121.481 125.980 130.479

22.35060 22.34883 22.34712 22.34550 22.34401

134-971 139-472 143-973 148.474 152.977

23.49652 23.49486 23.49322 23-49115 23.49028

134-974 139-474 143-975 148.477 152.979

24.70121 24.69964 24.69809 24.69662 24.69528

134.976 139-477 143.978 148.479 1.52.981

25.96770 25.96618 25.96471 25-96333 25.96205

134-979 139-479 143.980 148.481

22.34270 22.34153 22.34061 22.33995 22.33952

157.480 161.983 166.487 170.991 I7S-49Ó

23.48900 23.48791 23.48704 23.48640 23.48601

157.482 161.985 166.488 170.992 175.496

24.69406 24.69302 24.69223 24.69159 24.69120

157-483 161.986 166.489 170.993 175.496

25.96090 25-95991 25-959" 25-95854 25.95820

22.33941

0.000

sinh (x + ig) = r ¿y.

4-504

22.33941 180.000

23.48589 180.000

0.000 4-504

9.007 13-5« 18.014 22.517 27.019 31-521 36.022

24.69110 180.000

25.95806 25.95820 25-95854 25-95911 25-95991 25.96090 25.96205 25-96333 25.96471 25.96618 25.96770 25.96920 25.97066 25.97206 25-97337 25.97450 25-97550 25.97630 25.97680 25.97720

9.007 13-510 18.013 22.515 27.017 3I-5I9 36.020 40.521 45.021 49-521 54.020 58.519 63.017 67-5IS 72.013 76.510 81.007 85-503

152-983

157-485 161.987 166.490 170.993 175-497 25.95806 180.000

sinh (3.90 + i 1.90) = 24.69159 /ΐ7ο°·993 = 24.69159 /i7o".s737 12.526

0.00 3·043 5*995 8.771 11.297

Ι·35Ι75

1-38559 Ι-342Ι2

1.39606 1.38992 1.37189 1-34313 1-30536

·325Ι9 1.26128 1.19482 1.12803 1.06265

16.576 18.807 20.559 21.819 22.576

1.29101 1-23478 1.17576 1.11594 1.05693

14964 17.004 Ι8.6ΐ3 19-773 20-472

1.26062 1.21 ΙΟΙ

13-513 15-375 16.848 17-914 18.557

1-23358 1.18971 1.14297 1.09490 1.04687

Ι2.2θ8 13-904 15-250 16.220

Ι.ΟΟΟΟΟ 22.828 o . 9 4 r ° 4 22.576 0.88650 2 1 . 8 1 9 0.83695 20.559 0.79285 18.807

ι .οοοοο

20.70Ö 20-472 19-773 ι8.6ΐ3 17-004

18.772 Ι8·557 Ι7-9Μ 16.848 15-375

Ι.ΟΟΟΟΟ

0.94614 0.89611 0.85051 0.80986

Ι.ΟΟΟΟΟ 0.95086 0.90504 0.86316 0.82576

0.95523 °·9Ι333 0.87492 0.84054

17.013 ι6.8ι6 16.226 15-250 13-904

0.75461 0.72264 0.69730 0.67892

0-77459 0.74509 0.72172 0.70478

14-964 12.526 9-737

0.66777

16.576 13.898 10.819 7.411 3.767

0.6945 1

3-384

0.79326 0.76607 ο-74453 0.72892 0.71947

13-513 11.297 8.771 5-995 3-043

0.81065 0.78561 0.76578 0.75UI 0.74269

Ι2.2θ8 10.196 7-909 5.401 2-741

0.66403

0.00

0.69107

0.00

0.71629

Ο.οο

0.73978

0.00

Ι

1.41889

9-737

6.662 3-384

6.662

6.662

ΙΙ5853 Ι.Ι0492 Ι.Ο5168

17.914

17.914 15-375

1-34645

1.34645 1.33083 1.30586 1.27289

N e g a t i v e quantities are in h e a v y t y p e . t a n h (0.9 + i 1.0) = 1.39606 /o°. t a n h (0.95 + i I.5.P = 0.95523 \ i 6 ° . 8 i 6 = 0.95523 \ ι 6 ° . 4 8 ' · 5 8 ' .

[125]

ι6.8ι6

16.226 Ι5-250 Ι3·9°4

5-40Ι 2.741 0.00 2.741 5-401 7-909 10.196

ι6.8ι6

TABLE X I I .

HYPERBOLIC χ -

o 0.0s O.I o.is 0.2 0.23 0-3 0-35 0.4 0-4S

0.76159 0.76428 0.77229 0-78552

ο.οο 2-470 4.870 7-134

0.80376

Ç.200

0.82678 0.85425 0.88580 0.92098 °·959 2 5

ΙΙ.032

0.78181 0.78427 0.79164 0.80380 0.82058

TANGENTS,

ο.οο 2.228 4-394 6.440

1.15

8-315

ο.οο 2.0 ΙΟ 3.966 5·8ι5 7-513

0.84172 9-971 0.86693 11-373 0.89585 12.492 0.92802 Ι 3 · 3 ° 5 0.96294 13-798

0.85551 0.87862 0.90509 0-93449 0.96632

9-ΟΙ5 Ι0.288 "·3°5 Ι2.045 12-495

I3-9Ó3 1.03849 13-798 1.07756 Ι 3 · 3 ° 5 1.11626 12.492 Ι·Ι5349 »•373

Ι.ΟΟΟΟΟ 12.646 1.03486 12.495 ι.07010 12.045 1.10486 Ι Ι · 3 ° 5 1-13815 Ι0.288

9-ΟΙ5 7-513 5·8ι5 3.966 0.00

°-75 0.8 0.85 0.9 0-95

1-20951 Ι Ι . 0 3 2 9·2ο6 1.24415 Ι·27305 7-134 4.870 1.29485 2-47° Ι·3°843

1.18804 1.21866 1.24409 1.26320 1.27506

9-971 8-315 6-440 4-394 2.228

1.16889 1.19602 1.21846 1-23527 1.24569

1.0 1-OS 1.1 1 Ι · 5

Ι·3Ι3°4

1.27908 1.27506 1.26320 ι.244°9

1.2

1.24415

0.00 2-470 4.870 7-134 9·2Ο6

0.00 2.228 4-394 6.440 8.315

1.24922 Ι.24569 1-23527 1.21846 1.19602

1-25 ΐ·3 Ι·35 ΐ·4 1-45

Ι.2095Ι 1.17061 1.12892 1.08581 1.04248

11.032 12-575 13.802 14.694 15-233

Ι.Ι88Ο4

9-971 11-373 12.492 13-305 13-798

1.16889 1.13815 1.10486 Ι.070ΙΟ 1.03486

Ι.ΟΟΟΟΟ

ΐ·7

1.6s

0-9S925 0.92098 0.88580 0.85425

15-414 15-233 14-694 Ι3·8θ2 12.575

0.96294 0.92802 0.89585 0.86693

I3-9Ô3 13-798 13-305 12.492 11-373

ΐ·75 1.8 1.85 1.g ΐ·95

0.82678 11.032 0.80376 9-206 0-78552 7-134 0.77229 4.870 0.76428 2.470

0.84172 0.82058 0.80380 0.79164 0.78427

2.0

0.76159

0.78181

°·°ο

χ -

0.80050 0.80277 0.80954 0.82071 0.83611

0.5 °-SS 0.6 0.65 0.7

ΐ·5 i-SS 1.6

= r

1.05

I2.57S 13.802 14.694 15-233 ι.οοοοο 15-414 1.04248 15-233 1.08581 14.694 1.12892 13.802 1.17061 12-575

Ι·3°843 1.29485 1.27305

t a n h ( * + iq)

Ι.ΟΟΟΟΟ

1.21866

Ι-Ι5349 1.11626 1.07756 1.03849 Ι.ΟΟΟΟΟ

2.ΟΙΟ

0.8Ι775 0.81984 0.82605 0.83630 0.85043

¿y.

ο.οο ι.8ΐ5 3-582 5-254 6.790

CONTINUED

0.83365 0.83557 0.84127 0.85067 0.86361

ο.οο 1.639 3-235 4-747 6.138

0.86822 8.151 0.88939 9·3°6 ° · 9 Ι 3 5 9 10.231 0.94042 10.904 0.9694 1 Ι Ι · 3 Ι 3

7-371 0.8799 1 0.89927 8.419 0.92138 9-2 5 8 0.94585 9.870 0.97223 10.242

Ι.ΟΟΟΟΟ

»•451 »•313 1.06335 10.904 1.09458 10.231 1-12437 9·3°6

Ι.ΟΟΟΟΟ

10.368

Ι·°3Ι55

1.02854 Ι-05725 1 ·ο8533

ΙΟ. 242

8.151 6.790 5-254 3-582 Ι·8Ι5

1.13649 Ι·Ι5793 Ι-Ι7555 1.18868 1.19680

7-371 6.138 4-747 3-235 1.639

ο.οο

Ι-Ι9954 1.19680

0.00 1.639 3-235 4-747 6.138

Ι·Ι5Ι79

1-17587 Ι-Ι9574 1.21058 1.21976

9.870 9-258 8.419

3-966 5-815 7-513

1.22286 Ι.2Ι976 Ι.21058 1.19574 1-17587

9.015

Ι·Ι5Ι79

11.305 12.045 12.495

8.151 9·3θ6 1.12437 1.09458 10.231 1 ·°6335 10.904 Ι·°3Ι55 ΙΙ-3Ι3

7-371 1.13649 Ι.ΙΙ20Ι 8.419 Ι·Ο8533 9.258 Ι-05725 9.870 1.02854 10.242

0.96632 0-93449 0.90509 0.87862

12.646 12-495 12.045 11.305 10.288

Ι.ΟΟΟΟΟ 11-451 0.9694 1 Ι Ι . 3 Ι 3 0.94042 10.904 0-9Ι359 10.231 0.88939 9-306

Ι.ΟΟΟΟΟ 10.368 0.97223 10.242 0.94585 9.870 0.92138 9-258 8.419 0.89927

9-971 8-315 6.440 4-394 2.228

0.85551 0.83611 0.82071 0.80954 0.80277

9-015 7-513 5-815 3-966 2.010

0.86822 0.85043 0.83630 0.82605 0.81984

8.151 6.790 5-254 3-582 ι.8ΐ5

0.8799 1 0.86361 0.85067 0.84127 0-83557

7-371 6.138 4-747 3-235 1.639

ο.οο

0.80050

ο.οο

ο.8ΐ775

°·00

0.83365

ο.οο

Ι.ΟΟΟΟΟ

2.010

10.288

Note.

Negative quantities are in h e a v y type.

Examples,

tanh (1.1 + i ojr) = 1.13815 /IO°.288 = 1.13815

I.8IS 3-582 5-254 6.790

Ι-Ι7555 Ι·Ι5793

/το°.τη'.ιη".

tanh (1.2 + ¿ 1 7 ) = 0.89927 \8°.4i9 = 0.89927 \8°.25'.o8".

[126]

Ι.Ι8868

TABLE X I I .

HYPERBOLIC TANGENTS,

* = 1.25

o 0.0s O.I 0.15

χ = 1.3

t a n h (x + iq) = r / γ .

χ = 1-35

* = 1.4

0.84828 0.85004 0.85526 0.86387 0.87573

0.00

1.481 2.924 4.291 5-549

0.86172 0.00 0.86333 1 . 3 3 8 0.86811 2.643 0.87599 3-879 0.88684 5 . 0 1 7

0.87405 0.87552 0.87990 0.88711 0.89702

0.00 1.210 2.389 3-508 4.538

0.88535 0.88669 0.89069 0.89728 0.90634

0.00 1.094

0.89063 0.90833 0.92852 0.95082 0.97481

6.666 7.616 8.378 8-933 9.272

0.90047 6.029 0.91663 6.890 0.93504 7 . 5 8 1 0 - 9 5 5 3 4 8.085 0.97715 8.393

0.90947 0.92421 0.94099 0.95947 0.97928

5-454 6.234 6.860 7-317 7-596

0.91769 0.93114 0.94642 0.96323 0.98122

0.5 o-SS 0.6 0.65 0.7

ι.00000 1.02584 I-OS 173 1.07699 1.10092

9-38S 9.272 8-933 8.378 7.616

ι.00000 1.02338 1.04674 1.06948 1.09096

8.496 8-393 8.085 7-58I 6.890

1.00000 1.02116 1.04224 1.06271 1.08200

7.689 7-596 7-317 6.860 6.234

0.75 0.8 0.8s 0.9 0-95

1.12280 1.14192 1.15758 1.16924 1.17642

6.666 5-549 4.291 2.924 1.481

1.11054 1.12760 1.14156 1-15193 1.15831

6.029 5-OI7 3-879 2.643 1-338

I-09955 1.11483 1.12726 1.13650 1.14218

1.0 1.1 I-I5 1.2

1.17885 1.17642 1.16924 I.I5758 1.14192

0.00 1.481 2.924 4.291

1.16047 1.15831 1-15193 1.14156 1.12760

0.00 1.338 2.643

1-25 1-3 1-35 1.4 1-45

1.12280 1.10092 1.07699 1.05173 1.02584

6.666

1.5 1-55 1.6

1.00000 0.97481 0.95082 0.92852 0.90833

9-385

6.666 5-549

1.9 ι-95

0.89063 0.87573 0.86387 0.85526 0.85004

2.0

0.84828 0.00

0.2

0.2 s 0.3 0.3s 0.4 0.45

1-OS

1.6s 1-7

I-7S 1.8

1.8s

5-549 7.616

8.378 8.933 9.272

9.272

8-933

8.378 7.616

4.291

2.924 1.481

CONTINUED * = 1.4s

0.89569 0.89692 0.90057 0.90659 0.91485

0.00 0.989 1-953 2.869 3-712

4-934 5.640 6.207 6.622

0.92521 0.93746 0.95137 0.96665 0.98298

4-463 5-103 5-617 5-993 6.222

6.960

1 . 0 1 9 1 4 6.875 1.03817 6.622 1.05661 6.207 I-07395 5 . 6 4 0

1.00000 ι.01733 1-03450 1.05112 1.06671

6.299 6.222 5-993 5-617 5-103

5-454 4-538 3-508 2.389 ι.210

1.08963 I-I0335 1.11448 1.12272 1.12779

4-934 4.104 3.172 2.161 1.094

1.08084 1.09308 1.10304 I.II04I 1.1 1493

4-463 3-712 2.869 1-953 0.989

1.14410 1.14218 1.13650 1.12726 1.11483

0.00 1.210 2.389 3.508

1.12950 1.12779 1.12272 1.11448 I-I0335

0.00 1.094 2.161

1.11646 I.II493 I.II04I 1.10304 1.09308

0.00 0.989

I-09955 1.08200 1.06271 1.04224 1.02116

5-454

1.08084 1.06671 I.05II2 1.03450 I-OI733

4-4Ö3 5103 5-6i7 5-993

7.689 7.596

1.00000 0.98298 0.96665 0-95137

6.299 6.222

6.890

ι.00000 0.97928 0.95947 0.94099 0.92421

0 . 9 0 0 4 7 6.029 0.88684 5.017 0.87599 3-879 0 . 8 6 8 1 1 2.643 0.86333 1-338

0.90947 0.89702 0.88711 0.87990 0.87552

0.86172

0.87405 0.00

1.11054 1.09096 1.06948 1.04674 1.02338 1.00000 0.97715 0.95534 0.93504 0.91663

3-879 5017

6.029 6.890

7.581 8.083 8.393 8.496

8-393

8.085

7-58I

0.00

4-538

6.234 6.860

7-317

7.596

7-317

6.860 6.234

5-454

4.538 3.508 2.389 i.210

2.IÒI

3-172 4.104

6.875

3.172

4.104

1.08963 I-07395 1.05661 1.03817 I.OI9I4

4-934

1.00000 0.98122 0.96323 0.94642 0.93114

6.960

0.91769 0.90634 0.89728 0.89069 0.88669

5.640 6.207 6.622 6.875 6.875 6.622 6.207 5-640

4-934

4.104

3.172

2.161 1.094

0.88535 0.00

0.93746 0.92521 0.91485 0.90659 0.90057 0.89692 0.89569

Note.

Negative quantities are in heavy type.

Examples,

tanh (1.25 + i0.25) = 0 . 8 9 0 6 3 / 6 ° . 6 6 6 = 0 . 8 9 0 6 3 /6°.m'. expressed in polar coordinates.

nished by an ordinary 25 cm. slide rule. When a higher degree of precision than this is needed, arithmetical interpolation in the Tables must ordinarily be resorted to; but even then it is desirable to obtain a preliminary approximate value from the Atlas, in order to furnish a check against gross error.

[173]

EXPLANATORY

GRAPHIC

TEXT

REPRESENTATIONS

Figs. 25 and 26 present the results obtained from Table I to true polar coordinates. Each intersection of the curves corresponds to an entry in the table. Fig. 25 relates to pages 2, 3, and Fig. 26 to the rest of the table. T h e curves of constant ρ intersect those of constant δ perpendicularly. T h a t is, each intersection occurs theoretically at right angles. If, however, an attempt is made to prepare plates corresponding to Figs. 25 and 26 on a large scale, for a reasonable degree of precision, in rapid interpolation b y graphical inspection, difficulties present themselves. Firstly, it has been found impracticable to procure polar coordinate ruled sheets large enough. Secondly, regular polar coordinate charts of the type presented in Figs. 25 and 26'necessarily offer very little graphical interpolation precision at small radial distances from the origin of co1 2

3 4m 1 h ί f à / /

\

b Γ χ

\

ν

Ρ η 4 FIG. 27. — Polar Coordinate Diagram Regular Presentation on Circular Sheet.

FIG. 28. — Polar Coordinate Diagram Squared Presentation on Rectangular Sheet.

ordinates, where the radial lines, sharply converging, crowd the diagram. On the other hand, they offer relatively great apparent interpolation precision at large distances from the origin, as the radii diverge. In preparing interpolation charts, therefore, the author has devised the scheme of using squared paper sheets for presenting such polar coordinate quantities as appear in Figs. 25.and 26. Fig. 27 represents the regular polar coordinate r/y diagram, in which the lines of constant r are circles concentric at 0, and the lines of constant y are radii diverging from 0. Fig. 28 represents the corresponding squared polar coordinate diagram, in which the lines of constant r are the vertical straight lines 1-1, 2-2, 3-3, and 4-4. The lines of constant y are the parallel horizontal lines, while the origin point 0 in Fig. 27 becomes expanded into the original straight line 0-0 in Fig. 28. T h e straight, broken line DU is transformed into the curved line du, and, in general, orthogonally intersecting curves of one diagram do not transform into orthogonally intersecting curves on the other. [174]

EXPLANATORY TEXT INTERPOLATION

CHARTS

Plates IA, IB, and Ic of the Atlas correspond to Table I and present, to squared polar coordinates, the results in that table. Each intersection of curves in the plates corresponds to one entry in the table. Plate IA includes the entries on pages 2 and 3 of the table; while Plates IB and Ic include the entries in the remainder of the table. The curves of constant ρ and constant δ intersect one another at various angles, but the method of interpolation requires little explanation. The entering quantity will fall within some particular curvilinear parallelogram. The respective opposite sides may be subdivided into tenths in any of the three following ways: (1) by direct inspectional estimate, (2) by graphical subdivision on a sheet of tracing paper laid over the chart, (3) by means of a radiating decimal scale of lines, prepared in advance, on tracing paper or thin celluloid. It is not, in general, worth the effort of attempting a closer subdivision than tenths of the sides of any parallelogram. The point of intersection of lines parall'el to the sides, through the correct decimal points, is then to be marked on the covering tracing paper, or held with a blunt pointer, such as a knitting needle, on the chart itself, and the rectangular coordinates of this point read off from the parallel ruling or background of the plate. That is, the charts are always used with the entering variable on the curvilinear coordinates, and with the result found on the rectilinear framework in the background; except when inverse functions are sought, and the procedure consequently reversed. TABLE

I

sinh (ρ/δ) = r / y POLAR

HYPERBOLIC

S I N E S OF A P O L A R

VARIABLE

Table I, pages 2 to 7, gives the hyperbolic sine of vectors up to 3.0 in. modulus, by steps of ο. ι, for each degree of argument from 45o to 90o. The results are also expressed in polar coordinates, as plane vectors, corresponding to the relation: — sinh (ρ/δ) = r/y or

ρ [δ_ = s i n h - 1

(30)

(r/y).

(31)

The graphs of the results to true polar coordinates appear in Figs. 25 and 26, where the curves of constant ρ always intersect orthogonally the curves of constant δ ; so that at any point of intersection the angles of intersection are right angles. In Plates IA, IB, and Ic of the Atlas, the same graphs are given to squared polar coordinates, the disadvantages of the distortion being more than outweighed by the advantages in facility of graphic interpolation. In these charts the curves of constant ρ do not intersect the curves of constant δ orthogonally.

[i7S]

EXPLANATORY TEXT INTERPOLATION.

FIRST CASE.

I N MODULUS ONLY

If Table I is entered with a vector quantity of more than one decimal in modulus and of some exact degree of argument, such as 2.76 /70°; then the result will lie nearly on the line between the results for 2.7/7ο 9 and 2.8/70°; namely, between 1.2031 /136.48g and i.2i36/i43°.oo5. A first approximation may be obtained by proportional parts between them, thus : — Required sinh 2.76 /70 o by Table, sinh 2.80/70° = 1.2136/i43°.oog. by Table, sinh 2.70/70 o = 1.2031 /136°.48g. Difference 0.10/70° = 0.0105

/6°·5ΐ6.

Proportion for 0.06/70° = 0.0063

/3°.gio.

sinh 2.70/70° = ι.2031 /i36°-48q. Result sinh 2.76/70° = 1.2094/140°.399. The true value is

1.2086/140°.366.

INTERPOLATION BY THE U S E OF TAYLOR'S THEOREM

When more precise interpolation is required than that by simple intermediate proportion, we may use Taylor's theorem in the following form; since ¿(MnM

=

cosh0)

( sinhθ "> = sinhΘ,

sinh (θ + Δ0) = sinh θ + ΑΘ cosh θ + ^^ 21 Let θ = ρ [δ and ΑΘ = Αρ ¿δ. Then sinh \(ρ + Αρ)/δ\

= sinh (ρ/δ) + Αρ/δ.

etc.

de2

άθ

sinh θ + ^ ^ cosh θ + ... (32) 3'

cosh (ρ¿δ)+^-¿28.

sinh (ρ¿δ) + . . .

(33)

The number of correction terms to be retained depends on the interval, and on the degree of precision desired. It is seldom that more than two correction terms have to be retained. Thus in the case already considered: — sinh(2.76/7o°) = sinh (2.7/70 o ) + 0.06/70°. cosh (2.7/70 o ) +0-0018/140°.sinh(2.7 /70 o ). By Table II, page 13, cosh (2.7/70 o ) = 1.3422/153 o .322. Consequently dealing first with the first correction term only: sinh (2.76/70°) = i.2Q3i/i36°.48q + 0.06/70° X 1.3422/153 o .322 = 1.2031 /I36°.48Q + 0.08053 /

22

3°·322

= I.203I/136°.489 (χ + 0.066937/86°.833) = Ι.203Ι /136°.489 (ι + 0.003698 + i 0.06684) = χ 2 θ 3 ΐ / ΐ 3 6 ° · 4 8 9 (Ι.ΟΟ59 2 /3°·8ΙΟ)

= ι.2ΐθ2/ΐ4ο°·299· [ΐ7ό]

EXPLANATORY

TEXT

Taking next the second correction term into account. sinh ( 2 . 7 6 / 7 0 o ) = ι . 2 1 0 2 / 1 4 0 ° . 2 9 9 + 0 . 0 0 1 8 / 1 4 0 ° X = 1.2102 /I4Q°.29Q +

1.2031/136^482

0.002166/226^482

= 1.2 I 0 2 / 1 4 0 o . 2 9 9 (1 + 0.001789 /1,36°. 190) = 1.2102 /I4Q°.299 (1 — 0.00129 + í 0.00124) = I . 2 I Q 2 / l 4 0 ° . 2 9 9 (0.99871 + Í 0.00124) = I.2IQ2/l4Q°.299 (0.99871/O°.067) = 1 . 2 0 8 6 / 1 4 0 ° .366. T h e correct result is

(34)

1.2086/140^366.

SECOND AND G E N E R A L C A S E .

INTERPOLATION BOTH IN MODULUS AND ARGUMENT

L e t the entered quantity be sinh ( 1 . 0 2 5 / 8 0 . 7 5 ° ) . W e have from T a b l e I the four nearest results as follows: — sinh ι . 0 / 8 0 ° =

0.85125/83^489..

sinh 1 . 1 / 8 0 ^ =

0.90416/84Λ286.

sinh 1.0 / 8 i ° =

0.84940/84^56.

sinh 1 . 1 / 8 1 ° =

0.90172 / 8 4 ° . 8 7 7 .

Difference for i °

= — 0.00185 + ο°.667·

Proportion for 0 . 7 5

0

Diff. for i °

= — 0.00244 +

o°.59i.

— 0.00183 +

°·443·

= — 0.001388 + 0.500.

sinh Ι.Ο/8Ο°·75 =

0.84986/83^282..

sinh 1.1 / 8 o . 7 5 =

0.90233/84°·729·

Difference for 0.1 = +

0.05247

/ο°·74ο.

Proportion for 0.025 = +

0.01312

/O°.I85·

sinh 1.025/8o°-75 =

0.86298 / 8 4 ° . 1 7 4 .

T h e true value =

0 . 8 6 3 7 2 / 8 4 ° · 166.

sinh I.I/8Q°.75

=

0.90233

/84°·729·

(35)

D U A L INTERPOLATION BY THE U S E OF TAYLOR'S

THEOREM

L e t the nearest tabular function be sinh θ = sinh (ρ/δ) and the required function sinh (θ + ΑΘ) = sinh \(p + Δρ) /δ + A 5 j . T h e n (ρ + Ap) /δ + A3 = ρ / δ +

(Aρ + iρΑδ).

(36)

where the increment Ap + i ρΑδ is taken with reference to the vector axis ρ / δ .

Refer-

ring this increment to the initial axis of reference, θ + ΑΘ = (ρ + Αρ)/δ + Α δ = ρ / δ + So t h a t ΑΘ = V (Ap)2 +

V(Ap 2 +

(ρΑδ)2^ + tan"1

(ρΑδ)2 / δ + t a n ~ ' (ρΑδ/Αρ).

(37) (38)

When, however, Αδ is not very small, the last formula may contain an appreciable error, and the following method of deducing ΑΘ, using rectangular complex quantities, is to be preferred. [ 177]

EXPLANATORY TEXT Let , θ = p¿$ = χ + iy. and

(39)

θ + AO = (ρ + Δρ) /δ + Αδ = χ + Αχ + i (y + Ay). Then Δ0 = Αχ + i Ay

(40)

= V(Ax)2 + (Ay) 2 /tan - 1 (Ay/Αχ).

(4ι)

We then h'ave by Taylor's theorem, as before, (Αθ)2 (Αθ)ζ sinh (θ + ΑΘ) = sinh θ + Αθ. cosh θ + . sinh θ + . cosh θ + . . . (42) 2 3.2 a series in which two correcting terms only need ordinarily be retained. Thus, in the example last considered, θ = i . o / 8 o ° and θ + Αθ = ι.ο25/8ο°·75· If we form A0 by the use of (37), we have A ρ = 0.025, Αδ = ο°·75 = 0.01309 radian, ρΔδ = 0.01309. Αθ = \/(°·θ25) 2 + (0.01309)2 /8ο° + t a n - 1 (0.01309/0.025) = 0.02822/80° + 27°.637 = 0.02822/107^637. If we form Αθ by the use of the rigid formula (41) θ + Αθ = 1 . 0 2 5 / 8 0 = 0.164761 + i 1.0116715. 0 = 1.0/80 0 = 0.173648 + ¿0.9848078. Αθ = - 0.008887 + i 0.0268637 = 0.028295/108^306. Entering now the correction formula (42), we find in the tables: sinh 1.0/80° = 0.85125/83 o .489, cosh 1.0/80° = 0.57991/14Λ521. so that sinh ι.025/80°.75 = sinh 1.0/80° + 0.028295

/ιο8°·3ο6 X cosh 1.0/8o°

+ (°-°^95)2/2i6q

6i2

+ (°·°^95)3Λ24Ο

9ι8 x

x

sinh x

cosh

o¿3j

I

= 0.85125/83°.48Q + 0.028295 /108°.306 X 0.57991/i 4 °.52i + o.ooo4oo3/2i6°.6i2 X 0.85125/8^^483 + 0.000001 /324 o .918 X 0.57991/14^521. I t is evident that in conformity with the precision of the tables, only the first two correction terms need be included. Taking the first into account, we have: sinh 1.025/8ο°·75 = 0.85125/82^483 + 0.028295/ιο8°·3ο6 Χ 0.57991/14^521 = 0.85125/82^483 + O.oi6 4 O8/I22°.827 = ο.85Ι25/83°· 4 89 (Ι + 0.019276/39°·3.38) = 0.85125/83°.4^9 (ι + ο.οΐ49°9 + ¿0.012219) = 0.85125/82^483 ( Ι · ο ΐ 4 9 ° 9 + ίο.012219) = 0.85125/82^483 Χ 1.01498/0^630 = 0.86400/84°.! 79· [178]

EXPLANATORY TEXT Taking up the second sinh 1.025 /8o°-75 = = = = = = = The true value is

correction term: — 0.86400/84°:i79 + 0.0004003/216^612 X 0.85125/83^483 0.86400/84°.! 79 + 0.0003405 /300 o .101 Ο.864ΟΟ/84o.I 79 (i + O.OOO395 /2I5°-922) 0.86400/84 o .179 (x — 0.00032 — i 0.000232) 0.86400/84 o .179 (0.99968 — i 0.000232) 0.86400/84 o .!79 Χ ο·99968\ο°.οΐ3 0.863727 /84°.166. 0.86372 /84°.166. CONCLUSIONS

In general, dual interpolation by simple proportion, as in (35), will give a result of the third order of precision. In order to secure precision of the fourth order, interpolation by the use of Taylor's theorem as in (42) may be required. EXTENSION o r

T A B L E BY U S E OF F O R M U L A FOR 2Θ

Although Table I is only carried as far as 3.0 in modulus (p = 3); yet it may be used with a little additional calculation in conjunction with Table II, for obtaining the hyp. sines of plane vector quantities of moduli up to 6.0, by means of the formula : sinh 20 = 2 sinh θ cosh θ (43) Example: Required sinh 5.0/77 o , a quantity outside of Table I. Here θ = 2.5/77° is within the limits of the Table; so that sinh 5.0/77° = 2 X sinh 2.5/77° X cosh 2.5/77 0 (44) = 2 X 0.87843/i2o°.89i X 0.96459/156^524 = 2 X 0.87843 X 0.96459/277°·4ΐ5 = 1.75686 X 0.96459/277°·4ΐ5 = 1.69465/277^415. This method ordinarily calls for interpolation both in sinh θ and cosh Θ. For this reason, it may be preferable to obtain the required result by the use of either Table V I I or Table X, the limits of which are less restricted.

TABLE I I cosh (ρ/δ) = r / y P O L A R H Y P E R B O L I C C O S I N E S OF A P O L A R

VARIABLE

Table I I gives the value of cosh ρ/δ between the limits of ρ = o and ρ = 3.0 by steps of 0.1, and the limits δ = 45 o and δ = 90°, by steps of i°. The graphs of these quantities, to squared polar coordinates, appear in Plates IIA and IIB of the Atlas. [ 179]

EXPLANATORY TEXT INTERPOLATION BY SIMPLP

PROPORTION

In general, as in the case of Table I, a very fair degree of precision in interpolation can be obtained by taking first simple proportional parts in argument, and then simple proportional parts in modulus. Example: Required cosh (0.93105 We have from Table I I : — cosh 0.9/57° —

0.88922/23°.i4o.

cosh 1.0/57 0 =

0.87976/28^212·

cosh 0.9/58° =

0.87602/23^003.

cosh 1.0/58° =

0.86350/28^823.

Difference for 1° = — 0.01320 /o°.i37· Diff. for

0.518 o

Difference for 1° = — 0.01626 /o°.o94.

= - 0.00685 /ο°·ο7ΐ·

cosho.Q/57 0 .5i8 =

Diff· for

0.88237 ^".OÓQ.

0.518° = - 0.00844 /o°.Q49·

cosh i.o/57 o .518 =

0.87132/28^868

cosh 0.9 /57.518 =

0.88237 /23.069

Difference for ο.ι = — 0.01105 /5°-799 Diff. for 0.3105 = — 0.00343 /i°.8oo cosh 0.93105/57°·5ι8 = 0.87894/24°.869 The correct value is INTERPOLATION OF TAYLOR'S

0.87837 /24.803

THEOREM

When a higher degree of precision is required than can be expected from simple proportional parts, we may use Taylor's Theorem in the following form: — cosh (0 + Δ0) = cosh θ + ΑΘ sinh θ +

cosh θ + ^ f - sinh θ + . . . (45) 2.

3.

Example: Required cosh 0.93105/57°-5i8 having given in Table I I cosh 0.9 /57° = 0.88922/23°. 140 and in Table I sinh 0.9/57° = 0.85414/64^218. 0.93105/57.518 = 0.500 0.900

/57°

+¿0.785398.

= 0.49018 + 20.754804. ΔΘ = 0.00982 + ζ 0.030594 = O.q32I4/72°.I96.

cosh 0.93105/57°·5ΐ8 = cosh0.9/57° + 0-03214/η2°.ιφ X 0.85414/64^218 + 0.00052/144°·392 Χ 0.88922/23°.i40. [180]

EXPLANATORY TEXT I t is evident that for the Tables here considered only two correction terms need be included. Taking up the first correction term, cosh o.Q3io5/57°.5i8 = = = = = =

0.88922/23°. 140 0.88922 / 23°.i4Q 0.88922/23^140 0.88922 /23 o .140 0.88922/23°.i4Q 0.88922/23^14?

+ 0.03214/22^126 X o.8s4i4./64°.2i8 + 0.02745 / I30°.4i4 (1 + 0.03087/113°·274) (1 — 0.01220 + i 0.02835) (0.98780 + i0.02835) X 0.98780 (1 + i 0.02870)

= 0.88922/23°.I4Q X 0.98780 X I.OQQ4I/I°.645

= 0.87873/24^285. Taking up the second correction term: — cosh 0.93105/57°.518 = 0.87873/24°·785 + 0.00052/144°.392 X 0.88922/23^140 = 0.87873/24°·785 + Ο.ΟΟΟ46/Ι67°·532

= = = = = = The correct value is

0.87873/24^285. (ι + 0.000524/142^247 0.87873/24^285 (ι — 0.000416 + i 0.000317) 0.87873/24 o .785 (0.999584 + í 0.000317) 0.87873 X O.999.S84/24°-78.S (I + i 0.00032) 0.87837M"·?«* 0 X i/0.018 o 0.87837/24^803. 0.87837/24°.803. GRAPHICAL INTERPOLATION

For rapid but less precise work, interpolation may be made by proportional parts on Plate IIA or Plate IIB, without arithmetical computation. TABLE I I I tanh (ρ/δ) = r / γ POLAR HYPERBOLIC TANGENTS OF A POLAR VARIABLE

Table I I I gives in polar coordinates the value of tanh ρ j_δ between the limits ρ = o and ρ = 3·ο by steps of 0.1, and the limits δ = 45° and δ — 90o, by steps of 1°. The graphs of these quantities, to squared polar coordinates, appear in Plates IIIA and IIIB of the Atlas. INTERPOLATION BY SIMPLE PROPORTION

In general, as in the cases of Tables I and II, a very fair degree of precision can be obtained by taking first simple proportional parts in argument and then simple proportional parts in modulus. Example: Required tanh (0.93105/57°.5i8). [181]

EXPLANATORY

TEXT

We have from Table I I I : — tanh 0.9/57° = 0.96056/4i°.o78.

tanh 1.0/57 0 = 1.06648 /37°·θ35·

tanh 0.9/58° = 0.97069/42^x11.·

tanh 1.0/58° = 1.08054 /38 o .004.

Difference for 1° = 0.01013 /i°-033. Diff. for 0.518

o

= 0.00525 /ο°.535·

Difference for 1° = 0.01406

/o°.q6q.

Diff. for 0.518° = 0.00728

/ο°·5θ2.

tanh 1.0/57°.518 = 1.07376 /37°·537· tanh 0.9/57.5x8 = 0.96581 / 4 i ° . 6 i 3 .

tanh 0.9/57°.5i8 = 0.96581/41^613.

Difference for 0.1 = 0.10795/— 4°-Q76. "

for 0.3105 = 0.03352/— i°.266.

Inferred value of tanh 0.93105/57°·5ΐ8

= 0.99933

/40°.347.

Correct value of tanh 0.93105/57°-5i8

=

/40

1.0000

.395.

INTERPOLATION BY TAYLOR'S THEOREM

For a higher degree of interpolation precision than by simple proportion, we may use Taylor's theorem in the following form: — tanh (θ + ΑΘ) = tanh θ + ΑΘ sech 2 0 - — p . 2 sech20 tanhô 2 sech20 (sech2 0 - 2 tanh 2 0) + . . . (46) 3! Example: Required tanh 0.93105/57 o .518. having given in Table I sinh 0.9/57 0 = 0.85414/64.218. I I cosh 0.9/57° = 0.88922/23°.i4Q. I I I tanh 0 . 9 = 0.96056 /4i°.o78. Here Δ0 = o.o32i4/72°.i96, as given by (41). Hence by Taylor's theorem as far as the second correction term inclusive, 0.03214 /72.I9Ó tanh 0.93105ZS2IÜ8 = tanh 0 . 9 / 5 2 . + ( o - 8 8 q 2 2 ) V 4 Ó 2 g o -

(0.03214) 2 /i44°-392 - (0.88922) 2 /46 o .280

X

°-96O S 6Z4^O28.

Taking up the first correction term: — tanh 0.93105/57°·5ΐ8 =

O.Q6O.S6/4I°.O78 +

/A.S°-QI6

= 0.96056/41°.078 + 0.04065/25 e .916 = 0.96056/^1^028 (i + 0.04232\l5°.l62) = 0.96056/41^028 (1 + 0.04084 — ¿0.01107) = 0.96056/41°.078 (1.04084 — iΟΛΙΙΟ7) = 0.96056 /4I.078 (1.04090X0°.609) = 0.99985/40^462· [182]

EXPLANATORY

TEXT

Taking up next the second correction term: — tanh 0.93105/57°.5i8 = 0.99985/40^469 — 0.00103Λ44°·392 Χ 0.96056/41^028 (O.88922)V46°.28O

~~

o

= 0.99985/40 .469 + 0.00103 Χ 0·96056\40°.8ΐ0 0.79071 = 0.99985/4°°·469 + 0.00126X40°.810 = 0.99985/4°°·469 C1 + 0.00126X81.279) = 0.99985/40^469

(ι + Ο.ΟΟΟΙ9 — i 0.00125)

= 0.99985 /4°°·469 (1.00019 — i 0.00125) = 0.99985/42^469 Χ ι.οοοΐ9\ο°.ο72 = ι.οοοο /4ο°·397· Correct value = 1.0000 /40°.395. When more than two correction terms have to be retained, it is often easier to determine sinh (0 + Δ0) and cosh (θ + ΑΘ) by Taylor's theorem, as already described, and then to take their ratio for tanh (0 + ΔΘ). T A B L E IV Θ

POLAR RATIO

0

·

FO

R

P O L A R V A L U E S OF 0

Table IV has been prepared by dividing the values of sinh0 found successively in Table I by their respective values of 0. T h e object of the table is to facilitate the computation of the equivalent Τ or Π of any uniform alternating-current line of known electrical constants.* T h a t is, the table pertains more particularly to the applications of hyperbolic functions than to the fundamental properties of those functions. T h e sinh (ρ/δ) table gives the vector value of rr for the range ρ = o to ρ = 3 by steps of 0.1, P¿±

and for δ = 45 o to δ = go0 by steps of i°. The graphs of the values contained in the tables are plotted to squared polar coordinates in Charts IVA and IVB of the Atlas, for rapid graphic interpolation. I N T E R P O L A T I O N BY S I M P L E

PROPORTION

A fair degree of precision in interpolation can ordinarily be obtained by first taking simple proportional parts in argument and then simple proportional parts in modulus. sinh (1.025/80^25) Example: Required 7^-5 . 1.025 /80 -75 * " The Application of Hyperbolic Functions to Electrical Engineering Problems, " by A. E . Kennelly, University of London Press, 1914, Chap. III. [183]

EXPLANATORY TEXT We have from Table IV the following values of For 1.0/80 0 =

0.85125

1.0/81° =

0.84940

Difference for 1° = — 0.00185/-- Ο°.33.3· o-7S° = - 0.00139/ - o°.25o. For ι .0/80°. 7 5 =

For 1.1/80 0 =

Λ β · 48g. /3°.I.S6.

0.84986

1.1/81 0 =

0.82196 0.81975

/4°-286. /.f.877.

Difference for i° = — 0.00221 / — o°.4og. °°-7S = — 0.00166 / -

0

°·3°7·

For 1.1 /8o°.75 =

0.82030

/3°·979·

For i.o/8o°.75 =

0.84986

/3°·239·

Difference for 0.1 = — 0.02956 " for 0.025 = - 0.00739

/ ο°·74ο.

For 1.025/80°.75 =

0.84247

f f . 424.

Correct value

0.84265

/3.°416.

2

/ 3°· 39·

=

/o°.i8 S .

When a higher degree of precision is required than can be expected from proportional parts, the proper value of sinh (0 + Δ0) should be obtained by Taylor's theorem as already explainéd in connection with Table I, and this value divided by (θ + Δ0) ; because the expansion of

^ (0 + Δ0) conveniently for computation. EXTENSION

FOR THE R A N G E

directly, by Taylor's theorem, does not lend itself

OF T H E T A B L E

BY T H E U S E

or

FORMULA

FOR

20

Although Table IV is only carried as far as 3.0 in modulus (p = 3); yet it may be used with a little additional calculation, in conjunction with Table I I , for obtaining for vector values of 0 with moduli up to 6.0, by means of the formula: — 0 sinh 20 = 2 sinh 0. cosh 0 sinh — 20 = —-— sinh θ . cosh, 0. .

, whence

20

Consequently, to

find

0

0

(47) , ολ (48)

^ for the double of any quantity within the range of Table

IV, find the value of

^ for the quantity, by interpolation directly in Table IV, and 0 multiply the result by the hyperbolic cosine of the quantity as obtained from Table I I . Corresponding steps may be taken with Charts I I and IV. sinh (5.0/77°) . . Example: Required -,—5— , this being outside of the limits of Table I V ; 5-0/77 but not outside twice the value therein obtainable. Here and cosh 0

for 0 = 2 . 5 i s by Table I V " "

"

"

" "

"

II [184]

0.35137 / 4 3 ° - 8 9 i . 0.96459/156^524.

EXPLANATORY TEXT Hence

sinh (5.0/77°) = 0.35137/43^821 X 0.96459/156^524. g q ^ ^ = 0.33893 /200°.415.

This procedure calls for interpolation both in

^ and in cosh Θ.

For this reason

θ

it may be preferable to obtain the required result by the use of either Table VII or Table X, the limits of which are less restricted. TABLE V POLAR RATIO

TAI

^

FOR POLAR VALUES OF θ

Θ

Ό

Table V, like Table IV, has been prepared for electrical engineering applications of hyperbolic functions, rather than for developing these functions alone. I t gives the tanh (ρ/δ) vector value of rr for the range ρ = o to ρ = 3.0 in modulus, by steps of 0.1, Ρ ¿2

and for the range δ = 45 o to δ = 90o in argument, by steps of i°. I t was computed directly from Table I I I by dividing the resulting values successively by their respective values of Θ. The graphs of the values in Table V are presented to squared polar coordinates in Chart V, for rapid graphic interpolation. INTERPOLATION BY SIMPLE PROPORTION

Except where a high degree of precision in interpolation is required, it is preferable to interpolate first by simple proportion in argument, and then by simple proportion in modulus; although this order of operations may be inverted. Example: Required

a

^ for θ = 0.93105/57^518.

We have from Table V: — For θ = 0.9/57°

=

1-06729 \ΐ5°·922.

F o r ò = 1.0/57° = 1.06648X19°.965.

θ = 0.9/58^ = 1.07854 \ι5°.889·

θ = 1.0/58! = 1.08054

Difference for 1° = ο . ο ι ΐ 2 5 \ — 0.033. "

Difference for 1° = 0.01406 \o°.o3i.

for 0.518° - 0.00583\— 0.017.

"

for o°-5i8 = 0.00728 \o°.oi6.

F o r θ = Ι.Ο/57°·5Ι8 = 1.07376X19°.981.

For g - ο·9/57°·5ΐ8 = 1.07312 \ΐ5°·9θ5·

θ = ο · 9 / 5 7 ° · 5 ΐ 8 = I-Q73" V5-9Q5· Difference for ο.ι = 0.00064 \4°.o7Ó. " for 0.03105 = 0.00020 \i°.2Ó6.

For θ = ο.93ΐο5/57°·5ΐ8 = ΐ · 0 7 3 3 2 \ ΐ 7 ° · Ι 7 ΐ · Correct value, [ ι85 ]

1.07406\17°.123.

EXPLANATORY TEXT When a higher degree of precision is needed than simple proportion can give, it is preferable to find the proper interpolated value for tanh θ from preceding tables and then to divide by 0; since the function ^ ^ g ^

d ° e s n ° t lend itself to expansion

by Taylor's theorem in a simple form. Tables IV and V jointly, with their respective graphs in the Atlas, enable the equivalent Τ or Π of any uniform alternating-current line in the steady state, at a single frequency, to be completely determined, provided θ does not exceed six radians in modulus (δ lying between 45o and 90°) ; because although in both tables, θ is not carried beyond three radians; yet

li

^ can be found by extension up to six radians, and in the formu-

las for deducing the equivalent Τ or Π,

0

^ has only to be carried to half the modulus

. sinh θ θί-θ~·

The following example may illustrate the use of Tables IV and V either with or without the aid of the graphic interpolation Charts IV and V of the Atlas. An alternating-current line of uniform electrical constants is 250 km. long and has, at a certain frequency, a total conductor impedance of 565.711 /84o.777 ohms, associated with a total distributed insulation admittance of 4.3707 X io~3/qo° mhos. Its hyperbolic angle is therefore V.S.6.S7H X 4,3707 X /174°·777 = 1.5724 Z ^ M hyperbolic radians. Interpolating either from the tables or the Charts IV and V, we obtain sinh θ , Q / a , -^-=0.638/^

1 tanh (§)

t a n h o . 786 2 / 8 7 ^ 8 8 0.7862/8 7 °.388

If we multiply the conductor impedance by

θ

- " 7 ^ 5 ·

^, we have

565.711 /84°·777 Χ 0.638/2°.6 = 36ο.6ς)/87°·377 ohms, and if we multiply half the insulation admittance by 2.1854 X

(I)

, we have

I.27\i^5 = 2.78 Χ ιο~3/88°·5 mhos = 359. 7 7 \88 0 .5 ohms.

If now we apply an artificial condenser leak of 2.78 Χ ιο~3/88°·5 mhos to each end of a localised impedance coil of 360.69 /87o.377 ohms, we obtain the "equivalent Π " of the line at the frequency considered, and such a combination of localised impedance and admittances would behave exactly like the line, at its terminals, or outside them, so as to be capable of replacing the line in any electrical system, at that frequency.

[186]

EXPLANATORY TEXT TABLE VI POLAR FUNCTIONS OF POLAR SEMI-IMAGINARY

QUANTITIES

A semi-imaginary quantity is a complex numerical quantity which, when expressed in rectangular coordinates, has equal real and imaginary components; or, when expressed in polar coordinates, has an argument of 45°. That is x/± 45o = a ± i a. The interest of the table pertains primarily to the application of hyperbolic functions to uniform alternating-current lines of negligibly small linear inductance and leakance, a case approximated to by cabled lines at low frequencies. The table wàs first published by the author in the transactions of the International Electrical Congress of St. Louis (1904). The arguments of the results are given in degrees and minutes, and not in degrees and decimals like the rest of the tables. The table gives the hyperbolic sine, cosine, tangent, cosecant, secant, and cotangent of the vector X / f o r the range Χ = o to Χ = 20.5, by steps in Χ of 0.1 up to Χ = 6, and of 0.05 beyond that point. At Χ = 6, the values of the hyp. sine and cosine so nearly coincide, that they are taken as equal in the table, thus bringing sech χ and cosecha; into equality as well as tanh χ = coth χ = ι. Graphs of the functions are given in Chart VI as far as χ = 4, approximately. INTERPOLATION BY SIMPLE PROPORTION

In general, interpolation may be quickly effected by simple proportional parts of modulus since the argument is constant at 45o. This procedure is sufficiently evident to require no exemplification. INTERPOLATION OF TAYLOR'S THEOREM

When precise interpolation is necessary, we have the following expansions for f

(Θ +

ΔΘ)

sinh [ (s +

AX)¿4¿

} = sinh (x¿4¿) sinh

(X¿4¿)

cosh { (* + Λ*)/45_°} = cosh (x¿4¿

+

cosh

(AX)

+ ^Zí£L°· +

(AX)/A¿-

cosh ( x U ¿ ) + tanh ( (* +

AX)¿4¿

} = tanh (*/ 4 ,S° +

{X¿A¿)

cosh

sinh

+

(*/4S!) +

(XU¿)

+

sinh (*/4S!) + /4,S°. sech2(*/4,S°) -

{AX)

(49)

(So) ~F/QQ°.

3

2 sech2

(X¿4¿).

tanh (x/45!) "

(Ax)

2. sech2 (x/ 4 ,S°)·

2 i sech2 (X¿I¿) - 2 tanh (XU¿) } + . . . . Example: Required cosh (3.1/45°), having given in Table VI sinh (3.0/45°) = 4.1986/120^48^. cosh (3-o/4.S°) = 4-1443/i22°.i6'. [187]

(51)

EXPLANATORY T E X T Here cosh ( 3 . 1 / 4 5 ! ) = 4-1443 ¡ 1 2 2 ° . 1 6 ' + 0 . 1 / 4 5 ° X 4- 1 9 8 6 / 1 2 0 % ^ O.OI /90 o o 001 + Χ 4-1443 + - ^ - / l ¿ 5 ! Χ 4 . 1 9 8 6 / 1 20°.48' = 4 - 1 4 4 3 / i 2 2 ° . i 6 ' + 4·ΐ986/ΐ2ο°·48 ( 0 . 1 / 4 5 ° + 0-000x7/135°) + 4 - 1 4 4 3 / I 2 2 ° . I 6 ' (0.005/90°) = 4 . 1 4 4 3 / I 2 2 ° . I 6 / ( I + ¿0.005) + 4-1986/165^48^(0.1 + i 0.00017) = 4.1443 (1.0000/0°. 17') + 4.1986 /165°.48' (q.I/0°.0IQ = 4-1443/ι + 0.4199/I65°-49 / r = 4 . 1 4 4 3 / i 2 2 ° . 3 3 (1 + 0 . 1 0 1 3 2 / 4 3 ^ 1 6 0 = 4 - I 4 4 3 / I 2 2 ° - 3 3 ' (1 + 0.07378 + i0.06944) = 4 . 1 4 4 3 / 1 2 2 ! ^ (1.07378 + ¿0.06944) = 4· Ι 443 / I 2 2 ° - 3 3 / X 1 . 0 7 6 0 / 3 % ^ = 4.4590/126!^. which is in substantial agreement with the tabulated value of cosh 3 . 1 . Beyond χ = 6, the value of either sinh (s/45 o ) or cosh (x /45 o ) was computed from the formula : — X va Ι χ sinh (x/45°)

= cosh (x¿4¿)

where e = 2.71828 . . . Thus, with χ = 7,

= -L_

radians.

(52)

Va

= 4.9498,

— =

= 70.57 at the argument of 4.9498 cir-

cular radians = 283°.36'; so that: — sinh (7/45!) = cosh ( 7 / 4 5 ! ) = which coincides with the tabulated value in Table V I . T A B L E

VII

sinh (x + iy) — u + iv RECTANGULAR

HYPERBOLIC

SINES

OF A R E C T A N G U L A R

VARIABLE

Tables I to V I contain certain restrictions in range which limit their general application. They are primarily designed to cover particular applications of hyperbolic functions to electrical engineering. Tables V I I to X I V , however, are free from such restrictions, and are intended to furnish the circular as well as the hyperbolic sine, cosine, and tangent of a complex angle, and to furnish this result either in the rectangular or polar form. That is, they furnish: — sinh (χ + iy) ì or sin (x + ty) J

form

cosh or cos tanh or tan

form u + iv; also in the form r ¿ /y — f o r m u _j_ iv. a i s o ¡ n the form r ¡7 —

(χ (χ (χ (x

+ + + +

iy) ty) iy) ty)

Ì J Ì J

u

-f. ¿ v ·

[188]

ais0

j n the form r l·y 1 —

EXPLANATORY

TEXT

between the limits, for the hyperbolic functions, χ = o and χ = ± i o , b y steps of 0.05 and between the limits y = o and y = ± oc , b y steps of 0.07854 = π/40. P E R I O D I C P R O P E R T I E S OF THE R E C T A N G U L A R SINES AND

COMPLEX

HYPERBOLIC

COSINES

I t is well known that sinh \x i (y + τηπ) j = sinh \x + iy\ and cosh ¡x + i (y + 2 η τ ) \ = cosh \x + iy\

(53) (54)

where η is any integer. T h i s means that, keeping χ constant, the values of the h y p . sine and h y p . cosine repeat themselves as iy passes through increments of i. 2w, or they are periodic functions of iy, having the period 2 π i. T h e matter m a y be visualised more clearly from geometrical reasoning. ing the exponential form of the hyperbolic cosine, I + is

e

—(* + iy)

e

cosh (χ + iy) =

.

ex T h i s m a y written in the f o r m : — . 6

H

Consider-

t~x — e~ l ".

(SS) If χ be kept constant,

we

require to study the changes produced in this form of the h y p . cosine b y v a r y i n g y. q'

o

- A

o ^ C * Ρ

FIG. 2g. — Geometrical constructions for cosh (x + iy) and sinh (x + iy)·

In Fig. 29, OA is an initial line and OP a radius vector of length or modulus e 1 / 2 ; multiplied b y e *»; that is rotated positively about O, from OA through a circular angle of y radians. Similarly, op is a vector of length or modulus € _ I /2 rotated negatively through a circular angle of y radians from the initial line Oa. T h e equation (55) states that the hyperbolic cosine is the plane vector sum of OP and op, or O'p' in the Figure. If now we steadily increase the value of y, leaving χ constant, we cause OP to rotate steadily counterclockwise, and also op to rotate steadily clockwise, through Ay circular radians. W h e n Ay = 2x, both OP and op will have made one complete revolution and will h a v e returned to their initial positions indicated. Consequently, the value of cosh \x i (y + 2τ)\ repeats that of cosh \x + iy\. Since

sinh (χ + iy) =

(

χ + iy —

t

— (* +

iy)

(56)

the same reasoning applies; b u t the vector op is added in the negative or reversed direction; so t h a t O'q' is the hyperbolic sine of χ + iy. T h e above mentioned periodic property of the h y p . sine and cosine has been utilized for shortening the tables of those functions b y reducing the circular angle y of Fig. 29 [189]

EXPLANATORY TEXT from radians to quadrants. That is, any complex angle χ + iy represented by a point P, and radius vector OP, in the complex plane I F , Fig. 30, is first transferred to a new complex plane XQ, Fig. 31, at the point p = x, q, by keeping χ the same in both planes, but making the points ξ , ~ , ^ , ^ , . . . etc., on the Y axis of the X Y diagram, become the points 1, 2, 3, 4 . . . etc., on the Q axis of the XQ diagram.

Thus if

χ + iy = 2.5 + i 6.2832, χ + iq = 2.5 + i 4.00 where 4.00 is underscored to indicate quadrant measure, instead of the ordinary radian measure.

F I G . 30.

FIG 31.

Transference of a Complex Quantity from the X Y to the X Q Plane.

In the case indicated by Fig. 30, χ + iy = 3 + i 9 and χ + iq — 3 + i 5.74 in Fig. 31. Consequently, after a complex angle has been transferred from the complex plane XY to the complex plane XQ, the values of either sinh (x + iq) or cosh (χ + iq) exactly repeat themselves for each 4 units of increase in q; or with reference to Fig. 31 for each 4 quadrants of increase in the circular angle instead of 6.2832 . . . radians. The operation of transferring the complex angle from the XY to the XQ plane may therefore be described as quadranting y, i.e., changing the expression of y from circular radián units to circular quadrant units. All of the Tables V I I to X I I I inclusive require to be entered in terms of χ + iq; so that the complex entering value has to be quadranted by dividing its imaginary or ^-component by the numeric τ/2 = 1.57079 This preliminary step occupies a certain extra time and effort; but it actually economises the total time and effort involved. If the tables were computed for χ + iy, they would have to be repeated in bulk for each χ radians, or 2 quadrants, increase in y. In electrical engineering appli[ 190]

EXPLANATORY

TEXT

cations, y frequently rises to 100 radians, and might easily be much greater than 100. In order to go up to 100 radians, the bulk of the Tables VII to X I I I would have to be increased about thirty fold. Altogether, aside from the greatly increased bulk and expense of such tables, the extra time and effort consumed in turning over the numerous pages would be comparable with that saved by eliminating the preliminary step of quadranting the imaginary component or dividing it by -π¡2. R U L E S FOR THÉ U S E OF T A B L E

VII

Express the " angle " whose hyperbolic sine is required in the form of an ordinary rectangular complex quantity χ + iy. Quadrant the imaginary component y through the process of dividing it by π/2; i.e., transfer the quantity from the XY to XQ plane; so that the new expression of the complex quantity is χ + iq ; where q = y 11.57079. . . . If q is greater than 4.0, divide by 4 and retain only the remainder. If the remainder exceeds 2, subtract 2 therefrom, and apply a negative sign to the result found in the table. A change of 2 quadrants simply reverses the sign of the result. If the remainder on the other hand does not exceed 2, enter Table VII with it, and take out the result with unchanged sign. Example: Required the hyperbolic sine of 0.65 + i 25.75. Here χ = 0.65 and y — 25.75. That is y is 25.75 circular radians. Reduce this to quadrants through dividing by 1.57079. . . . 25-75 * log 25.75 = 1-4107772. I -57°79 · · · l°ë τ/2 = 0.1961199. log 16.393 = 1.2146573. The quadranted value χ + iq = 0.65 + i 16.393 NOTE. — It is found convenient to underscore quadrantal quantities to distinguish them from radianal quantities. Rejecting quadrant multiples of 4, i.e., 16 in this case, we enter Table VII with χ + iq = 0.65 + i 0.393. The nearest entry to this is χ = 0.65, q = 0.4, for which the hyperbolic sine is 0.56368 + i 0.71639, an ordinary rectangular complex quantity on the U V plane. Interpolation should be made in this result to meet the change from q = 0.40 to q = 0.393, aS will be explained later. Second Example: Required sinh (x + iy) = sinh (1.15 + i 10.10). Quadranting the imaginary, sinh (x + iq) = sinh ( 1.15 + i IQ'I0o \ i· 57 o8 = sinh (1.15 + ¿6.430). Rejecting 4's from the imaginary = sinh (1.15 + i 2.430). Deducting 2 from the residual and changing the sign = — sinh (1.15 + ¿0.430). * This operation would ordinarily be effected with the slide-rule, when a high degree of precision is not aimed at. [191]

EXPLANATORY

TEXT

We now enter Table V I I with χ = 1.15 and q = 0.43. T h e nearest entry is # = 1.15 q = 0.45, the result for which is 1.08037 + * 1.12836. B u t we must apply a negative sign to the whole of this result because of the 2 rejected in the quadrantal residuum. Hence, sinh (1.15 + i 10.10) = — (1.08037 + i 1.12836) = — I.08037 — * 1-12836 = « + »!) except for the interpolation from q = 0.4g to q = 0.43. will be discussed later on.

T h e operation of interpolation

Third Example: Required sinh (x + iy) = sinh (3.60 + i 18.1). Quadranting the imaginary, sinh (x + iq) = sinh ^3.60 + i = sinh (3.60 + i 11.523). Rejecting 4's from the quadrants = sinh (3.60 + ¿3.523). Deducting 2 from the residual imaginary and changing the sign = — sinh (3.60 + i 1.523). Entering Table V I I with q — 1.5, for which the result sign to this result because of finally: — sinh (3.60 +

χ = 3.6 and q = 1.523, the nearest entry is χ = 3.6 and is — 12.92978 + i 12.94910. B u t applying the negative 2 deducted from the quadrantal imaginary, and we have

¿18.1) = — (— 12.92978 + ¿12.94910) = 12.92978 — i 12.94910 = u + iv except for the interpolated correction from q = 1.500 to q = 1.523, to be considered later. RANGE o r

THE

TABLE

Table V I I extends b y steps of 0.05 in χ up to χ = 3.95, and in Table X I I I up to χ = 4.0. In y, the range is indefinitely great; because after dividing y b y τ / 2 so as to reduce it to quadrant measure, all multiples of 4 are rejected. From o to 2, in the remainder, the table gives the result directly and from 2 to 4, b y change of sign in the total. Cases of χ greater than 4.0 are dealt with in connection with Table X I V . R E P E T I T I O N S I N THE

If sinh I χ + i

— aj j =

u + iv

TABLE

(57)

it is easy to show that: sinh I χ + í' ^

+ a^j j = — u + iv.

(58)

I t follows that in any column of Table V I I , the entry for q = (1 — a) is the same as that for q = 1 + a except for a change in the sign of u. Consequently, the table might have been reduced to half its present size, if the responsibility for making this change of sign had been left to the reader. I t was considered, however, that since the reader is already charged with the duty of applying a negative sign to the total result when the q [ 192]

EXPLANATORY

TEXT

residuum lies between 2 and 4, the retention of the full size of the present table was warranted, especially as the duplication of the text in each column provides a certain check upon the numerical work of tabulation.

INTERPOLATION BY SIMPLE PROPORTION

As a first approximation, interpolation may be effected by simple proportion, first in regard to χ and second in regard to q. Example: Required sinh (0.15 + ¿0.25), having given: s i n h (0.2 + i 0.2) = 0 . 1 9 1 4 8 + ¿ 0 . 3 1 5 2 2 .

s i n h (0.2 + i o ^ ) = 0.17939 +

¿0.46310.

s i n h (0.1 + i 0^2) = 0.09526 + ¿ 0.31056.

s i n h (0.1 + ¿ o j ) = 0.08925 +

¿0.45626.

D i f f . f o r 0.1 χ = 0.09622 + i 0.00466. Diff. f o r 0.05a; = 0.04811 + ¿0.00233.

D i f f . f o r o.ix = 0.09014 + ¿0.00684. D i f f . f o r 0.05a; = 0.04507 + i 0.00342. s i n h (0.15 + ¿ o j ) = 0 . 1 3 4 3 2 + ¿0.45968.

s i n h (0.15 + ¿ 0 . 2 ) = 0 . 1 4 3 3 7 + ¿ 0 . 3 1 2 8 9 . s i n h (0.15 + i o S ) = 0 . 1 4 3 3 7 +

¿0.31289.

Diff. for q o . i o = — 0.00905 + ¿ 0 . 1 4 6 7 9 . D i f f . f o r q 0.05 = — 0.00453

¿0.07340.

s i n h (0.15 + ¿ 0 . 2 5 ) = 0.13885 +

¿0.38629.

Correct value

= 0.13910 + ¿0.38700.

INTERPOLATION BY TAYLOR'S THEOREM

When a higher degree of precision is desired than t h a t which can be expected by simple proportion, we may use Taylor's theorem in the following form: — sinh (θ + Δ0) = sinh θ + ΑΘ cosh θ + ^ ^ 2i

sinh θ +

2!

cosh θ +

. . . (59)

sinh \ (x + iy) + (Λ* + i Ay) £ = sinh (x + iy) + (Ax + i Ay) cosh (x + iy) Η

3 (Ax + i Ay)2 . . (ΔΛ; + i Ay) . ¡ — — sinh (χ + iy) + ¡ — c o s h (x + iy) + 3*

...

(60)

Quadranting imaginarles on both sides ; or transferring to the XQ plane, sinh £ (Λ; + iq) + (ΔΛ; + iAq) | = sinh (x + iq) + \ Ax + i (IR/ 2 ) , 1\Ax + i(Tr/2)Ay\2

+

. . .

.

\Ax + ¿ (ΤΓ/2) Aq\3

sinh (x + iq) + i

^

= sinh (χ + iq) + A'θ cosh (χ + iq) +

cosh (x + iq) +

. . .

(61)

(Α^θ)2 ^ sinh (x + iq)

3 + (Δ^) cosh (x + iq) + 3·

2!

...

where Α'θ = Αχ + iAy = Αχ + ¿ (τ/2) Aq. [ 193]

\ c ° s h (A; + iq)

(62)

EXPLANATORY TEXT Example ( i ) : With Aq = o. Required sinh (0.15 + i 0.2), having given in Table V I I and in Table V I I I : sinh (0.1 cosh (0.1

+ t o . 2 ) = 0.09526 + ¿0.31056. + ¿ 0^2) = 0.95582 + ¿ 0.03095.

Then by (60);

sinh (0.15 + i 0 2 ) = sinh (0.1 + ¿ 0^2) + 0.05 cosh (0.1 + ¿ 0.2) + sinh (0.1 + i 0^2) + ° " ° ° O I 3 j-Qgk

j

¿0

2

)

= 0.09526 + ¿0.31056 + 0.05 (0.95582 + ¿0.03095) + 0.00125 (0.09526 -f- ¿0.31056) + 0.00002 (0.95582 + ¿0.03095) = ι.00125 (0.09526 + ¿0.31056) + 0.05002 (0.95582 + ¿0.03095) = (0.09538 + ¿0.31095) + (0.04781 + ¿0.00155) = 0.14319 + ¿0.31250 which is the correct tabular value of sinh (0.15 + ¿ 0.2) in Table V I I . Example (2) : With Ax = o. Required sinh (0.1 + ¿0.25), having given in Table V I I and in Table V I I I : sinh (0.1 + ¿0^2) = 0.09526 + ¿0.31056. cosh (o.x + ¿0^2) = 0.95582 + ¿0.03095. Then by (62); sinh (0.1 + ¿0.25) = sinh (0.1 + ioS) + ¿0.05 X 1.5708 X cosh (0.1 + i 0.2) (0.05 X 1.5708) 2 , . + ι — — — sinh (1.0 + 10.2) „ (0.05 X 1.5708) 3 + i3 — ^ ^ cosh (1.0 + i 0.2) 3! — = 0.09526 + ¿0.31056 + ¿ X 0.07854 (0.95582 + ¿0.03095) 0.00617 , . — - (0.09526 + 0.31056) — ¿ ° · ° ° ° 4 8 (0.95582 + ¿0.03095) = (0.09526 + ¿0.31056) (ι — 0.00309) + i (0.95582 + ¿ 0.03095) (0.07854 — 0.00006) = 0.99691 (0.09526 + ¿0.31056) + 0.07848 (— 0.03095 + ¿0.95582) = 0.09497 + ¿0.30960 — 0.00243 + ¿0.07501 = 0.09254 + ¿0.38461. The tabular value is 0.09254 + ¿ 0.38460. Example (3) : Interpolation for both Ax and Aq. Required sinh (0.15 + ¿ 0.25), having given sinh (0.1 + ¿0^2) = 0.09526 + ¿0.31056 by Table V I I and cosh (0.1 + ¿ 0 ^ ) = 0.95582 + ¿0.03095 by Table V I I I . Here Δ0 in formula (59) = (0.05 + ¿ 0.05) and Α'θ in formula (62) = (0.05 + ¿(π/2) X 0.05) = (0.05 + ¿0.07854). [ 194]

EXPLANATORY TEXT Thus: — sinh (0.15 + ¿ 0-25) = sinh (0.1 + i 0.2) + Δ'0 cosh (0.1 + i 0.2) ,2 + ^ ^ sinh (0.1 + ¿ 0 2 ) + . . . 2! Δ'0 = o.os + ¿0.07854. (Δ'0)2 = + 0.0025 — 0.00617 + ¿ 0.00785 = — 0.00367 + ¿0.00785. (Δ'0)2

Η

,

·

•—-— = — 0.00184 + ί°·οο393· (Δ'0)3 = (0.05 + ¿0.07854) 3 = — ο.οοοδο + ¿ο.οοοιο. (Δ'0)3 , . —-— = — O.OOOI3 + Í 0.00002. ( (A'ffS^ } sinh (0.15 + ¿ 0-25) = sinh (ο.ι + ¿ 02) | ι Η ^ 1- . . . j + cosh (0.1 + i 02) I Δ'0 + ^ y ^

+ . . .J

= (0.09526 + i 0.31056) (0.99816 + i 0.00393) = 0.09386 + i 0.31036 + (0.95582 + i 0.03095) (0.04987 + ¿0.07856) = 0.04524 + ¿0.07664 = O.I39IO + ¿0.38700. The correct tabulated value is = 0.13910 + ¿0.38700. EFFECTS OF CHANGES OF SIGN IN THE ENTERING QUANTITY

Table VII expresses the relation sinh (χ + iq) = u + iv.

(63)

(a) If χ be taken with negative sign, we have sinh (— χ + iq) = — u + iv

(64)

so that changing the sign of the real component entering the table changes the sign of the real component in the result; but leaves the sign of the imaginary component unchanged. (£>) If q be taken with negative sign, we have sinh (x — iq) = u — iv

(65)

so that changing the sign of the imaginary component in the entering quantity changes the sign of the imaginary component in the result, leaving the sign of the real component unchanged. (c) If both χ and q be taken with negative sign, we have sinh (— χ — iq) = sinh \ — (x + iq) \ = — u — iv = — (u

iv)

(66)

so that changing the sign of the total entering quantity changes the sign of the total result. The facts may be summed up by saying that changes in the sign of the entering quantity produce corresponding changes of sign in the result. [19s]

EXPLANATORY TEXT CIRCULAR SINES OF COMPLEX " ANGLES "

Since, as is well known: — sin θ = — i sinh (ίθ) we have sin (χ + iy) — — i sinh (ix — y) = i sinh (y — i x ) .

(67) (68) (69)

Consequently, in order to find the circular sine of the complex quantity (χ + i y ) , enter Table VII for sinh (y + i x ) , which on being quadranted, becomes sinh \ y + ix/(tv/2) \ and let the result be (u + iv). Then sinh (y — ix) = 11 — iv and sin (x + iy) = ν + iu. In other words, invert the entering components, and then invert the components of the result. Example: Required sin (1 + i 2) from Table VII. Here θ = (i + i 2). Enter the Table with — sinh (id) = sinh (— iff) = sinh (2 — i χ). Quadranting the imaginary, we enter the table with (χ — iq) = (2 — i 0.6366). The nearest entry is (2 — i 0.65), for which the hyp. sine is given as 1.89503 — i 3.20780. Consequently, sin (1 + i 2) = 3.2078 + i 1-89503, except in so far as interpolation is needed to reduce sinh (2 — i 0.6366) from sinh (2 — i 0.65). In this way any circular sine of a complex quantity can always be obtained from the table of hyperbolic sines, between the limits of o and ± 4 in y, and of o and ± in x. GRAPHIC INTERPOLATION BY M E A N S o r

CHARTS V I I A , V I I B ,

VIIC

Charts V I I - V I I I A, B, and c, serve for the evaluation of either sinh (x + iq) or cosh (x + iq), according to the axis of reference selected. Thus, taking Chart V I I VIIIB, if this is held with the line as the axis of reference or initial line; then by comparison with the entries in Table VII, it will be found that sinh (Λ; + iq) can be read from it directly over the range q = o to q = 4, beyond which the values repeat themselves indefinitely. On the other hand, if the chart be turned through 900, so as to bring the line CC as the axis of reference, it will be found by comparison with the entries of Table VIII, that cosh (χ + iq) can be read from it directly over the range q = o to q = 4. Chart VII-VIIIA gives sinh (x + iq) and cosh (x + iq) for values of χ up to about 0.9. Chart VII-VIIIB gives the corresponding results for values of χ up to about χ = 2. Finally, Chart VII-VIIIc provides for values of χ up to χ = 4. In all of these charts, interpolation can be made for both χ and q to 0.01, by direct inspection. The graphs on these charts are undistorted, since they give complex functions as results, in rectangular coordinates. The curves therefore always intersect orthogonally, and they represent a confocal system of ellipses and hyperbolas, the common foci being at two points at unit distances from the center, along one of the reference axes. The curvilinear rectangles into which the charts are divided have pairs of sides the ratio of whose lengths tends to the value 71-/2. If the preliminary process of quadranting the imaginary of the entering quantity were not adopted; that is, if the graphs were entered in terms of (χ + i y ) , instead of [196]

EXPLANATORY TEXT ( χ + iq) ; then it would be necessary* to have a new chart for each range of 2π units in y; or some 16 sets of Charts A, B, and C, in order to reach y = ioo. That is, 48 charts would have to be computed, prepared, drawn, lithographed, bound, sold and operated instead of the 3 charts actually presented. Moreover, if y were needed greater than 100, the set of 48 would fail; whereas, working with quadrant imaginaries, the three charts serve up to indefinitely great values of q and y. GRAPHIC CHART

VII-VIIIA

This chart corresponds to Tables V I I and V I I I at least as far as χ = o.g, or for pages 42 to 45, and 58 to 61 of this book. T o find hyperbolic sines from the chart, place it facing the observer with the axis 0 0 vertical. This is the major axis of all the ellipses shown. Starting from this central axis towards the right hand, the successive ellipses marked 0.1, 0.2, 0.3, etc., represent values of x\ while the successively rising hyperbolas 0.1, 0.2, represent values of q. These values of q will be found to extend over two quadrants. Enter the chart on the curvilinear coordinates for χ and q. A t the proper intersection read off the u and ν coordinates of the rectilinear ruling, u being the abscissas and ν the ordinates. Conversely, to find sinh - 1 (w + iv) within the limits u = o and « = ± 1 , 1 1 = 0 to ν = 2.0, enter the chart with the same aspect on the rectilinearly ruled coordinates and read off at the proper intersection the curvilinear values taking χ on the ellipses and q on the hyperbolas. T o find hyperbolic cosines from the chart, rotate it clockwise 90o; so as to have the axis 0 0 horizontal. Then enter on the curvilinear coordinates with χ on the ellipses and q on the hyperbolas. The first and fourth quadrants only will be presented to the observer; but from the symmetry of the diagram, it will be easy to reverse the chart, so as to present the second and third quadrants. Read off the result on the rectilinear background using u for abscissas and ν for ordinates. Conversely, to find cosh - 1 (u + iv) from the chart with the axis 0 0 horizontal, enter on the rectilinear background and read off at the proper intersection from the curvilinear coordinates in χ and q, taking the ellipses as parts of the ^-system and the hyperbolas as part of the ç-system. GRAPHIC CHART

VII-VIIIB

This chart gives the graph of the functions sinh (χ + iq) and cosh (x + iq) from χ = 0.8 at least as far as χ = 2.05 along the ellipses and from q = o to q = 00 by virtue of successive rotations. In this and the following charts, the numerical values of q are all underscored, an indication which may serve readily to distinguish the imaginaries q, from the reals, x. * A single set of charts entered in terms of (x + iy) could be used up to y = 6.2832 in one revolution, and could be used for all larger values b y throwing out multiples of 2τ. This operation of dividing y b y 2ir would, however, take as much time as the operation of quadranting, and would also lead to a dissymmetrical chart in the hyperbolas.

[197]

EXPLANATORY

TEXT

In all of the Charts V I I to I X inclusive, the curvilinear rectangles all tend to have sides in the ratio τ : 2; that is the long side approximates to being 1.57 times the short side. In IXA exceptions are found; because extra curvilinear coordinates are supplied. To find sinh (χ -F iq) from Chart V I I - V I I I B , hold the minor axis 5 5 horizontal. Enter on the curvilinear coordinates with χ on ellipses and q on hyperbolas. At the proper intersection read off on the rectilinear background in u and v. Proceed inversely to find the inverse function sinh - 1 (u + iv). To find cosh (x + iq) from the same chart, hold the major axis CC horizontal. Enter on the curvilineár coordinates with χ on ellipses and q on hyperbolas. Read off on the rectilinear background. All four quadrants appear in this and the following chart, so that it is not necessary to limit the value of q to less than 2 quadrants. GRAPHIC CHART

VII-VIIIC

This Chart gives the graph of sinh (x + iq) and cosh (x + iq) from χ — 2.o, at least as far as χ = 3.90. The procedure is precisely the same as that for V I I - V I I I B already described. TABLE VIII cosh (χ + iq) = u + in RECTANGULAR HYPERBOLIC COSINES OF A RECTANGULAR

VARIABLE

Table VIII may be regarded as an inversion of Table VII; because: cosh θ = — i sinh (0 + ¿71-/2) (70) or in quadrant imaginarles, cosh θ — — i sinh (0 + i ι). (71) That is the hyp. cosine of any complex quantity (x + iq) is — i times the hyp. sine of that quantity with an additional quadrant in the imaginary. Thus cosh (0.5 + i 0.6) = — ¿sinh (0.5 + i 1.6) =

— ¿ ( — 0.42158 +

¿0.66280)

= + 0.66280 + ¿0.42158. All of the entries in Table VII thus reproduce themselves by inversion in corresponding parts of Table VIII, a fact which serves as a numerical check upon both. In order to find the value of cosh (χ + iy), quadrant the imaginary quantity y, by dividing it with ir/2, as in entering Table VII. The complex quantity + iy) "will now be expressed as (x + iq); or will in effect have been transferred from the XY to the XQ plane. Next throw out multiples of 4 from q, so as to leave a remainder less than 4. If this remainder exceeds 2, deduct 2 from it, but change the sign of the total result thereupon deduced. If the remainder, however, is not greater than 2, then, the result is taken directly from the table. Example: To find cosh (1 + ¿ 5 ) = cosh (χ + iy). Quadranting, we have cosh (1 + ¿ 3 . 1 8 3 ) = cosh (x + iq). Deducting 2 from q, cosh (x + i 1-183). [198]

EXPLANATORY

TEXT

With this we enter Table VIII. The nearest entry is χ + iq = ι + i 1^2, the result for which is — 0.47684 + ¿ 1.11768. This has to be corrected by interpolation from q = 1^2 t o q = 1 . 1 8 3 .

R e v e r s e t h e sign of t h e r e s u l t t o 0.47684 — i 1 . 1 1 7 6 8 f o r t h e

deduction of 2 quadrants. Example 2: Required cosh (0.25 + i 30 ) = cosh (χ + Quadranting, this becomes cosh (0.25 + i 19.099) = cosh (χ + Rejecting imaginary quadruples = cosh (0.25 = cosh (x + Deducting 2 quadrants = cosh (0.25 = — cosh (x

iy). iq). + ¿3.099) iq). + i 1.099) + iq).

The nearest entry is 0.25 + Ì L I for which the result is (— 0.16135 + I 0.24950). Applying the negative sign on account of the two deducted quadrants, the final result is, neglecting interpolation, cosh (0.25 + i 30) = 0 . 1 6 1 3 5 — i 0.24950 — u — iv. INTERPOLATION BY SIMPLE PROPORTION

A first approximation can be obtained by interpolating according to simple proportion. Example: Required cosh (0.55 + ¿0.55) = cosh (χ + iq\ having given c o s h (0.6 + Í O J ) = 0.83825 + i 0.45018. c o s h (0.5 + ¿ o j ) = 0.79735 + ¿ 0 . 3 6 8 4 7 .

c o s h (0.6 + ¿ 0.6) = 0.69680 + cosh (0.5 + ¿O^J) = 0.66280 +

¿0.51506. ¿0.42158.

D i f f . f o r Λ;Ο.Ι = 0.04090 + i 0 . 0 8 1 7 1 . D i f f . f o r χ 0.05 = 0.02045 ¿0.04086.

D i f f . f o r Λ; 0 . 1 = 0.03400 + D i f f . f o r Λ; 0.05 = 0 . 0 1 7 0 0 +

¿0.09348. ¿0.04674.

cosh (0.55 + Í 0 J 3 ) = 0.67980 + cosh (0.55 + ¿ 0 ^ ) = 0 . 8 1 7 8 0 +

¿0.46832. ¿0.40933.

Diff. for j o j = —0.13800 + D i f f . f o r q 0.05 = —0.06900 +

¿0.05899. ¿0.02950.

cosh ( 0 . 5 5 + ¿o-5) = 0 . 8 1 7 8 0 +

¿0.40933.

sh (0.55 + ¿0-55) =

Correct value =

0.74880 +

¿0.43883.

0.75018 + ¿0.43963.

INTERPOLATION BY TAYLOR'S THEOREM

For a higher degree of precision than simple proportion affords, reference may be had to Taylor's theorem in the following form : — cosh (θ + ΑΘ) = cosh β + ΑΘ sinh θ + ^ r ^ cosh θ + ^ ^ 2l 3!

sinh θ + . . .

cosh \ (χ + iy) + (Ax + i Ay) £ = cosh (x + iy) + {Ax + i Ay) sinh (x + iy) +

(A

* + / A y ) 2 cosh (* + ¿,) +

(A [

* + / A y ) 3 sinh (* + ¿y) + . . .

199]

(7,.)

EXPLANATORY

TEXT

Quadranting imaginarles on both sides, or transferring to the XQ plane, cosh ¡ (χ + iq) + (Ax + iAq) £ = cosh (x + iq) + (Δ* + iAq x/2) sinh (χ + iq) (Ax + iAq τ/2)2 + cosh (x + iq) 2Ì (Ax + iAq τ π/2) τ/ν 3 s i n h ( x + i q ) + _ + 3Î = cosh (x + iq) + Α'θ sinh (x + iq) +

cosh

( * + *?) +

· · ·

where Α'θ = (Ax + i Ay) = (Ax + iAq x/2).

(71b)

Example: Required cosh (0.5 + i 0.55) = cosh (x + iq) having given and Here Δ * = ο, cosh (0.5 +

cosh (0.5 + ¿ o j j ) = 0.79735 + ¿0.36847 in Table V I I I sinh (0.5 + ¿ o j j ) = 0.36847 + 0.79735 Table V I I . Δ^ = i 0.05, Α'θ = ¿0.05 Χ 1.5708 = ¿0.07854· i o^j) = cosh (0.5 + i o j ) I i + + ^y^- + + sinh (0.5 + i 0^) { Α'θ + ^

Α'θ

=

i 0.07854.

(Δ'0)2

=

— 0.00617.

(Α'θ)3 = — i 0.00048. (Α'θ)* =

+ 0.00004.

/ Λ 'Ñ'\ 2

2¡

=

—

+

°·003°9·

(Α'θ) 3 -—ρ- = — i 0.00008. 3! (Α'θ) 4 - — = o.ooooo.

cosh (0.5 + ¿0.55) = (0.79735 + i 0.36847) (χ - 0.00309) + (0.36847 + ¿0.79735) (i0.07854 - i0.00008) = (°·79735 + í 0.36847) 0.99691 + (0.36847 + i0.79735) i0.07846 = 0.79489 + i 0.36733 + i 0.02891 — 0.06256 = ° - 7 3 2 3 3 + * 0.39624. T h e tabulated value = 0.73233 + ¿0.39624. In view of the similarity of the interpolation operations by Taylor's theorem to those already discussed in relation to Table V I I , further examples are probably not needed. EFFECTS

OF CHANGES OF SIGN IN THE E N T E R I N G

Since if

QUANTITY

cosh ( χ + iy) = u + iv (72) cosh (— χ + iy) — u — iv (73) cosh ( χ — iy) = u — iv (74) cosh (— χ — iy) = u + iv (75) changing the sign of either the real or imaginary entering component only changes the sign of the imaginary component in the result; while changing the sign of the entering quantity as a whole, has no effect on the sign of the result. [ 200]

EXPLANATORY TEXT CIRCULAR COSINES OF COMPLEX " ANGLES "

It is well known that if θ be any angle, real or complex, cos θ = cosh (id).

(76)

cos (χ + iy) = cosh (— y + ix)

(77)

Consequently,

or, quadranting the imaginary component, cos (Χ + iy) = cosh (— y + i 2Χ/Έ) = U + iv.

(78)

T o find the circular cosine of any complex quantity χ + iy, we enter Table V I I I with (— y + ix/1.5708). The result is the desired cosine. Example: Required cos (0.4 + i 1.2). Thus we require cosh (— 1.2 + i 0.2546). We now enter Table V I I I with χ = — i.a and q = 0.2546 the nearest entry being χ = — and q = 0.25, for which the result is 1.67283 — ¿0.57765. Hence cos (0.4 + i 1.2) = 1.67283 — i 0.57765 neglecting interpolation from q = 0.25 to q = 0.2546. GRAPHIC CHART INTERPOLATIONS

The use of the Graphic Charts VII-VIIIA, B, C, for hyperbolic cosines has already been described in connection with sines, on pages 197-198.

TABLE

IX

tanh (x + iq) = u + iv RECTANGULAR HYPERBOLIC T A N G E N T S OF A RECTANGULAR

VARIABLE

Entering Process Let tanh (x + iy) be the required function. Quadrant the imaginary component, as described under Tables V I I and V I I I ; that is, divide y by τ / 2 ; so that y/iir/2) = q. The required function is now expressed in the form tanh (x + iq). Throw out multiples of 2 from q and retain only the remainder as q. Enter Table I X with (χ + iq), and find the result directly as « ± iv. It is a well-known property of tanh (Λ; + iy), that it is periodic in iy, and that the period is i.w circular radians; or, in quadrants, i. 2. That is tanh I χ + i (y + ηπ) } = tanh (χ + iy).

(79)

where « is any integer; or, in quadrant measure of the imaginary, tanh ^ χ + i (q + 2η) i = tanh (χ + iq)·

(80)

Example: Required tanh (0.25 + i30) = tanh (x + iy). Quadranting, tanh (0.25 + i 19.099) = tanh (x + iq). Rejecting multiples of 2, tanh (0.25 + i 1.099) = tanh (x + iq). We now enter Table I X with χ = 0.25 and q = 1.099, the nearest entry to which χ = 0.25, q = The result is 2.95122 — i 1.75011. [aoi]

EXPLANATORY

TEXT

INTERPOLATION

Interpolation may be approximately effected by simple proportion, first in χ and then in q, as indicated in connection with Tables V I I and V I I I ; or, when a higher degree of precision is required, recourse may be had to Taylor's theorem in the following form: — tanh (0 + ΔΘ) = tanh θ + Αθ sech2 θ - —f- 2 sech2 θ tanh θ +

2 sech2 θ (2 tanh2 θ - sech2 Θ) + . . .

(81)

or tanh ! ( ι + iy) + (Ax + ¡Ay) ( - tanh (r. + iy) + -

I

R

O

F

)

«

Κ

»

+

« + • · •

and quadranting, tanh \ (χ + iq) + (Δ* + iAq) } = tanh (* + iq) + -

cosh2 ( , + iq)

Unh

+···

so that as far as the second correction term: —

A'9 cosh2 (x + iq) {Α'ΘΥ tanh (x + iq) cosh2 (* + iq) ^ · · ·

tanh { (x + iq) + (ΔΛ; + iAq) \ = tanh (x + iq) +

where Α'θ = (ΔΛ; + i Ay) = F Ax + iAq (τ/2) \

lö4;

(85)

Example: Required tanh (0.5 + ¿0.55) = tanh {x + iq) having given cosh (0.5 + i OJJ) = 0.79735 "t~ ¿0.36847 by Table V I I I = 0.87837 /24o.803 by Table X I and tanh (0.5 + i0.5) = 0.76159 + ¿0.64805 by Table I X = 1.0/40 .395 by Table X I I . Here Α'θ = (o + i0.05 Χ ι·57° 8 ) = (° + ¿0.07854)· * W _L · % * ι · a Q tanh (0.5 + . a s s ) = 0.76159 + *Ο.648Ο5 +

¿0.07854

(o,878,7)V4qQ,6o6

, 0-006iyXi/40°.395 + ( o . g y g 3 y ) , •

, · ¿ Ο , 0-07854 /4Q°-3Q4 , = 0.76x59 + 10.64805 Η 0.77153 = 0.76159 + » 0.64805 + 0.10180/40°.394 + = 0.76159 + ¿0.64805 + 0.07753 + ¿0.06597 = 0.84702 + ¿0.71274. The correct value is 0.84752 + ¿ 0.71229.

0.00617 \9°.2I ι °-71 l S3 0.00800 \9°.2II + 0.00790 — ¿0.00128

As the third correction term is inconvenient for computation, it is often preferable to obtain a precise interpolation of tanh {x + iy) in working out the correct interpolations of sinh {x + iy) and cosh (ζ + iy) by the methods already illustrated, and then to take their ratio. [ 202 ]

EXPLANATORY TEXT EFFECTS

OF C H A N G E S

If then

tanh tanh tanh tanh

and

OF S I G N IN THE

( ( (— (—

ENTERING

QUANTITY

χ iy) u + iv χ — iy) — u — iv χ + iy) = — u + iv χ — iy) = — (w + iv).

(86) (87) (88) (89)

Consequently, changes in the sign of the entering quantity produce corresponding changes of sign in the result.

FIG. 32. — Graphs of * + iy and tanh (x + iy) in the XY and UV planes respectively. CIRCULAR

Since It follows that

TANGENTS

OF C O M P L E X

" ANGLES

"

tan θ = — i tanh (id). tan (a; + iy) = — ¿tanh (— y + ix)

(90) (91)

or, quadranting the imaginary: — tan (Λ; + iy) = — i tanh (— y + i 2χ/τ) = — i tanh (— y + iq) = — i (— M + tiι) = ν + iu.

(92)

Consequently, to find tan (x — iy) from Table I X , enter it with y as a; and with χ/(τ/2) as q. Invert the components of the result and the required function is obtained. Thus [ 103]

EXPLANATORY TEXT required tan (i + i 2 ) . We enter with χ = 2 and q = 0 . 6 3 6 6 . χ = 2 . 0 and q = 0 . 6 5 for which u + iv = 1.01623 + ¿0.03318.

The nearest entry is Therefore, inverting,

tan ( 1 + 2 2 ) = 0 . 0 3 3 1 8 + i 1.01623 neglecting the interpolation from q = 0 . 6 5 to q = 0 . 6 3 6 6 . GRAPHIC INTERPOLATION BY M E A N S OF CHARTS I X A , I X B , AND

IXC

These charts contain all of the entries in Table IX, and also a certain number of additional results. They present circles intersecting circles orthotomically; i.e., by rectangular intersection. It is clear that for values of χ less than 0 . 1 0 , the curves run off Chart IXA. In fact the first curve shown of χ = o.oi extends as far as u = xoo. By taking χ small enough, the corresponding values of u and ν may become indefinitely great. The entire UV plane is covered to infinity once between χ = o and χ = oo, q = o and q = 2. It is covered once more for each 2 quadrants increase in q. When entering for tanh (— χ ± iq); or for the inverse operation tanh - 1 (— u d= v), it must be remembered that the confocal conic-section diagrams VII and VIII are complete for negative as well as for positive values of χ and q; but that only half of the UV plane is presented in Charts IX. The full graph is indicated in Fig. 3 2 , by the aid of which the functions corresponding to negative real values are readily apprehended. TABLE

Χ

sinh (x + iq) = r / γ POLAR HYPERBOLIC SINES OF A RECTANGULAR

VARIABLE

This table corresponds completely to Table VII, already considered, except that it offers results in polar instead of rectangular coordinates. To find sinh (x + iy) expressed in polar coordinates, quadrant the imaginary, and express the entering variable as (x + iq). Reject multiples of 4 in q, and if the remainder exceeds 2 , reject 2 but change the sign of the total result. INTERPOLATION BY SIMPLE PROPORTION

Required sinh ( 0 . 1 5 + i 0 . 2 5 ) having given sinh ( 0 . 2 + ¿ 0 2 ) = 0.36882/s8°.723· sinh ( 0 . 2 + io^) = sinh ( 0 . 1 + i 0^2) = 0.32485/72°.Ç47. sinh ( 0 . 1 + i o ^ ) = Diff. for 0 . 1 χ = + 0 . 0 4 3 9 7 / — 1 4 ° . 2 2 4 . Diff. for 0 . 0 5 χ = 0.02x99/— 7°.112. sinh ( 0 . 1 5 + 1-02) =

0-34684/65°.835.

0.49663/68^825. 0.464QI/78°.q32.

Diff. for 0 . 1 χ = + 0 - 0 3 1 7 2 / — io°.iQ7· Diff. for 0 . 0 5 χ = 0 . 0 1 5 8 6 / — 5°.o54. sinh (0.15 + ^0^) = 0 . 4 8 0 7 7 / 7 3 ° . 8 7 8 . sinh ( 0 . 1 5 + i o 1 2 ) = ο·34684/65°.835· Diff. for i o . ι = 0 . 1 3 3 9 3 / 8°·θ43· Diff. for i0.05 = 0 . 0 6 6 9 7 / 4 ° . Q 2 2 .

sinh ( 0 . 1 5 + i'o.25) = 0 . 4 1 3 8 1 /Ó9°.857· Correct Value 0.41124/70°.229. [ 204]

EXPLANATORY TEXT INTERPOLATION BY TAYLOR'S THEOREM

For a higher degree of precision than is obtainable by simple proportion, it is convenient to use rectangular coordinates and apply formula (62). Thus, required sinh (0.15 + ¿0.25). Here referring to Table VII and to the work on page 195, we find for the result u + iv = 0 . 1 3 9 1 1 + i 0 . 3 8 7 0 1 . Here

^ 0 . 3 8 7 0 1 = 1.5877110 log 0 . 1 3 9 1 1 = 1 . 1 4 3 3 5 8 4 0.4443526 log tan 7 0 0 . 1 3 ' = 0.4440674

°· 7 2 =

2853 3968

70O. 1 3 ' . · ^ = 7o°.229

^ 0 . 1 3 9 1 1 = 1.1433584

log sec. 7o°.229 = 0.4707400 log 0 . 4 1 1 2 4 = χ.6140984 0 . 4 1 1 2 4 /ΗΟ°.22^.

Result Correct Value 0.41124/70°.229.

INTERPOLATION BY CHARTS X - X I A AND X - X I B

These charts present the polar coordinate results on rectangular coordinate sheets, so that they are not true graphs, but are merely to be regarded as interpolation diagrams. To find sinh (χ + iy), proceed as in the use of the tables and quadrant the imaginary so as to obtain the entering quantity in the form (χ + iq). Enter with the curvilinear coordinates, taking the more nearly vertical wavy lines for χ and the more nearly horizontal lines for q, starting from the line ¿'5' as the zero of q. Read oS the result on the rectangular background to the left-hand scale of ordinates. When we leave X - X I A and enter X - X I B , it is noticeable that the curves of constant χ approach vertical straight lines and the curves of constant q approach horizontal straight lines. At and beyond χ = 3.0, we may approximate to the modulus r at any required q, by taking the value of r at q = 0.5 and simple proportional parts between this and r at q = o or r at q = 1.0. The change in modulus r between q = o j , and either of the above limits is very nearly e -I /2. Thus at χ = 3.5 and q = 0^5, r by the tables is 1 6 . 5 5 7 7 4 . At q = o, r = 1 6 . 5 4 2 6 3 , a change of — 0 . 0 1 5 1 1 , and at q = i^o r = 1 6 . 5 7 2 8 2 , a change of + 0.01508. The value of e _3 - 5 /2 will be found to be 0 . 0 1 5 1 0 , and over the entire range of q from o to 1.0, the change in r follows in nearly simple proportion. Beyond χ = 3.2, the limit of Chart X-XIB, the values of r can be obtained by the above rule applied to Table X and with the aid of Chart VII-VIIIc. The values of the amplitude y beyond χ = 3 closely approximate to q quadrants. That ex is sinh (x + iq) approximates to — /_£_, with q in quadrant measure. TABLE X I cosh (χ + iq) = r /y POLAR HYPERBOLIC COSINES OF A RECTANGULAR VARIABLE

Table X I corresponds completely with Table VIII, except that it gives results expressed in polar instead of rectangular coordinates. It is entered with + iq) just as in Table VIII. [205]

EXPLANATORY TEXT Interpolation may be effected by simple proportion, as in the case of Table X , or when a higher degree of precision is required, it may be carried on by Taylor's theorem. In the latter case, it is more convenient to refer to the corresponding entries in Table VIII, interpolating according to formula (71 b). The rectangular coordinates duly interpolated are then transformed into polar coordinates, as in the last example on page 225. INTERPOLATION BY CHARTS X - X I A

AND

X-XIB

When Charts X - X I are used to find cosh (χ -f- iy), the imaginary is first quadranted by dividing with x/2, so as to obtain the entering variable in the form cosh (χ + iq). Starting then from q = o at the horizontal line CC, near the middle of the chart, the underscored figures correspond to q for a little more than the first quadrant. The manifest repetition of the curves enables the lower half of the sheet, however, to be used for the second quadrant. The result is read off on the rectangular background to the right-hand scale of argument. Beyond χ = 3.2, the limit of Chart X - X I B reference may be had to Chart V I I V I I I B ; or the approximate formula may be used: ex cosh (x + iq) = - /g.

(93)

the argument q of the result being interpreted in quadrant measure and converted into degrees. TABLE

XII

tanh (x + iq) = r /y POLAR HYPERBOLIC

TANGENTS

OF A R E C T A N G U L A R

VARIABLE

Table X I I corresponds completely with Table I X , except that it gives results expressed in polar instead of rectangular coordinates. If we desire to find tanh (χ + iy), we must first divide y by π/2 so as to obtain the entering quantity in the form (χ + iq). Multiples of 2 are then rejected in q leaving a remainder less than 2. With this remainder the table is entered. Interpolation may be made by simple proportion to a moderate degree of precision. GRAPHIC INTERPOLATION BY

MEANS

OF C H A R T S X I I A ,

B , C, D

These charts cover between them the full range of Table X I I . To find tanh (χ + iq) from them with q less than 2, find the proper chart, and enter on the curvilinear coordinates keeping the underscored number for q. Read off the result on the rectilinear background. For tan (x + iy) and also for the effects of changes of sign, see directions in the discussion on Table I X . To find tanh - 1 (r ¿y), enter immediately on the rectangular background of r and 7 in the proper chart, and read off at the correct intersection the corresponding values on the curvilinear coordinates. The result will appear in terms of (x + iq). The imaginary q must be dequadranted, or multiplied by π/2, in order to be expressed in terms of (χ + iy). [ 206]

EXPLANATORY TEXT TABLE

XIII

/ (4 + iq) = u + iv or r /y RECTANGULAR AND POLAR FUNCTIONS o r THE RECTANGULAR V A R I A B L E (4 +

iq)

In this table the hyperbolic sine, cosine and tangent of (4 + iq) are collected from q = o to q = 2.O. The results are expressed both in rectangular coordinates (u + iv), and in polar coordinates r ¿y. I t will be seen that the moduli of the tangents vary between 0.99933 a n ( i 1-00067, or differ from unity by two thirds of one per mil, at most. The arguments also differ from o° by less than 0.04O, o r about 2'. 1 7 " of arc. Beyond χ = 4, it is evident that the hyp. sine and cosine differ by so small a percentage, that no tabulation of these differences would ordinarily be required. TABLE

XIV

exj 2 and logi0 (e1/2) SEMI-EXPONENTIALS

This table enables the hyp. sine or cosine of any rectangular variable (χ + iq) to be found for values of χ greater than 4 and less than 10. It is shown in the preceding tablé that when χ reaches 4, the ratio of the sine to the cosine never differs from unity by more than two-thirds of 1 per mil. This deviation from unity rapidly diminishes as χ is further increased. Consequently, the sine and cosine may each be computed from the formula. e1 sinh (x + iq) = cosh (x + iq) = — ¿q^ (94) Example: Required the value of sinh (8.51 + i 25.75). The first step is to quadrant the imaginary by dividing with 71-/2, as on page 191. This gives the required function in the form sinh (8.51 + i 16.393). Rejecting multiples of 4.0 in q, we may then write it sinh (8.51 + i 0.393). Turning to the top of page 143, we find i 1 /2 = 2482.082 for χ = 8.51; so that the result is 2482.082 70.393 quadrant. Expressing the argument in degrees by multiplying with 90 and we have 0.393 X 90 = 35.37 o . Thus sinh ( 8 . 5 1 + i 1 6 . 3 9 3 ) = cosh ( 8 . 5 1 + i 1 6 . 3 9 3 ) = 2 4 8 2 . 0 8 2 / 3 5 ° · 3 7 · INTERPOLATION IN Χ

C· Since

£ΐ+ΔΧ

= - 6 Δ4, Χ

=

_«* |f

ι

_L Λ + Δ

λ

3 ,; +( Δ ^*) 2 + I 1( Δ_*) ^ +l

ì

/ \

{gs)

it follows that when Ax is a small quantity, it suffices to multiply the tabular value of €R/ 2 by (1 + Δ*) in order to arrive at the interpolated result unless (Δ*) 2 /2! the second correction term, is of sufficient magnitude to need consideration. [207]

EXPLANATORY

TEXT

Example: To find sinh (8.51 + iq), having given that sinh (8.50 + iq) = 2457.383 /q. Here Δλ; = o.oi. 2457-383 X 1 = 1 2457.383 2457·3 8 3 Χ Δ* = 0.01 24.574 (ΔΛ;) 2

2457·383 X — — = 0.00005

.123

2482.080 Result 2482.080 /q. Tabulated value 2482.082 TABLE

XV

f (x + i o) REAL HYPERBOLIC

FUNCTIONS

This is a short table of real, as distinguished from complex hyperbolic functions for convenience of reference. I t was prepared and published by the author in 1903 in relation to continuous-current electric circuit applications, taking the sines, cosines, and tangents from Ligowski's tables, and adding the corresponding computed reciprocals for the cosecants, secants, and cotangents. Much more extensive tables of real hyperbolic functions are, however, available. See Bibliography, page 211. TABLE

XVI

SUBDIVISIONS OF A D E G R E E

This is a short table for convenience in changing the expression of a circular angle from decimals of a degree to minutes and seconds, or inversely. By its aid, threedecimal subdivisions of a degree may be converted into minutes and seconds of arc, by direct inspection; or minutes and seconds may be read off as decimals of a degree ta three-digit accuracy. METHODS

EMPLOYED IN

COMPUTATION

Tables I to V, inclusive, were computed as one group, and Tables V I I to X I I I , inclusive, as a separate group. Tables I to V were computed, at first, by using the formulas: — sinh (λ; + iy) = V sinh2 χ + sin2 y / t a n - 1 (tan y / tanh χ) = ri /γι.

(96)

cosh (χ + iy) — V cosh 2x — sin2 y / t a n - 1 (tan y · tanh x) = r2 /γ 2 .

(97)

tanh (χ + iy) = (n/r 2 ) /γι — γ2.

(98)

At a later stage of the work, the following formulas, kindly suggested by Professor Bouton, were substituted: — sinh (x + iy) = V cosh 2X · sin ζ / tan - 1 (tan y / tanh x) = rx /γι.

(99)

cosh (χ + iy) = V cosh 2X · cos ζ / tan - 1 (tan y · tanh x) = r2 /γ 2 .

(ioo)

tanh (χ + iy) = tan ζ /γι — γ2·

(ιοί) [208]

EXPLANATORY

TEXT

Where the auxiliary circular angle s is defined b y : COS 2J cosh 2X

=

COS

22.

(102)

T h e arithmetical work was conducted with the aid of five-place logarithms, and was checked b y tabulating successive first and second differences in the tabulated results. Tables V I I to X I I were computed b y means of the following formulas: — sinh cosh , , tanh

(χ {χ , (x

+ + . +

iy) = sinh χ cos y + i cosh χ sin iy) = cosh χ cos y + i sinh χ sin . . sinh 2X . . ty) = — r ¡ \- 1 r cosh 2x + cos 2y cosh

y. y. sin 2y ¡-^ 2x + cos 2y

(103) (104) , . (105)

A standard schedule was prepared and seven-place logarithms used in the computation. T h e value of tanh (x + iy) was arrived at in two ways, first by dividing (103) by (104), and second b y the independent formula (105). If these two methods did not give identical results for tanh (χ + iy) to five decimal places, when expressed both in rectangular and polar coordinates, the steps of the computation were gone over afresh.* Complete agreement being secured, leads to the inference that the values of sinh, cosh, and tanh (x + iy) are correct, at least as far as their logarithms. Finally, all of the tables have been reduced to graphic form in the Atlas, each entry of the tables being marked off on its proper chart with a sharp needle, and the ruling pen drawn through the successive punctures. In this process a certain number of errors were discovered and rectified. T h e tables were then set up in type from the M S S . used in making the charts, and were proofread three times. B y this procedure it is hoped that the outstanding errors are neither large nor numerous. BIBLIOGRAPHY AND APPLICATIONS OF HYPERBOLIC

FUNCTIONS

Hyperbolic functions of a real variable are employed extensively in mathematics generally. In particular, they are used in the solution of cubic equations. In navigation, real hyperbolic functions enter in connection with Mercator sailing. In cartography, real hyperbolic functions are used in preparing maps on certain projections, especially on Mercator's projection, which appears to have been the first application of hyperbolic functions. In statics, real hyperbolic functions naturally present themselves in relation to the properties of the catenary and of the funicular polygon; also in the discussion of the forms and stresses of elastic bodies. In dynamics, the same functions present themselves in the theory of vibrations, and in the motion of bodies through a resisting medium. * T h e author desires to express his acknowledgement of the care and painstaking effort of his assistants engaged in computation, namely, Miss Ethel Smith, A . B . Radcliffe, 1911. Miss A . F . Danieli, A . B . Radcliffe, 1911. Miss M a r y M . Devlin, A . B . Radcliffe, 1912. Miss Hope M . Hearn, A . B . Radcliffe, 1912. [ 209 ]

EXPLANATORY

TEXT

A good summary of the historical development of real hyperbolic functions is given in Becker and van Orstrand's "Hyperbolic Functions," Smithsonian Mathematical Tables, 1909, together with a fine compendium of formulas involving these functions.* In electrical engineering, the earliest published application of real hyperbolic functions is perhaps in T . H. Blakesley's "Alternating Currents of Electricity," London, 1889, which also appends a short table of these (real) functions. The real functions were also introduced by Sir J. J. Thomson, in " T h e Electrician," Vol. X X V I I I , page 599, 1891. " O n the Heat Produced by Eddy Currents in an Iron Plate Exposed to an Alternating Magnetic Field." The fundamental differential equation of the alternating potential-current, steadystate distribution along a uniform conductor, involving hyperbolic functions, nominally real, seems to have been first published b y O. Heaviside in 1893, "Electromagnetic Theory," Vol. I, page 450. The first published application of complex hyperbolic functions to the last-named problem was by the author, " O n the Fall of Pressure in Long-Distance AlternatingCurrent Conductors," Electrical World, N . Y . , Vol. X X I I I , page 17, January, 1894 and " T h e Electrician," London (abstract), Vol. X X X I I , page 239, January 5, 1894. Complex hyperbolic functions also present themselves in the discussion of Hertzianwave reflections, and in other branches of electrical engineering. They naturally enter the subject of confocal ellipses and hyperbolas, such as Captain Weir's Azimuth diagram of these confocals, for indicating the azimuth of a celestial object in terms of the hourangle, latitude and declination. (Godfray's "Astronomy," § 222.) The mathematical discussion of hyperbolic functions is found in Greenhill's "Differential and Integral Calculus," Macmillan and Co., 1896; Ligowski's "Tafeln der Hyperbelfunctionen und der Kreisfunctionen," Berlin, Ernst & Korn, 1890; McMahon's "Hyperbolic Functions," Wiley and Sons, N . Y . , 1896; Becker and van Orstrand's "Hyperbolic Functions," Smithsonian Institution, 1909; Vassall's "Nouvelles Tables des Logarithmes," Paris, Gauthier-Villars, 1872; as well as other text-books. Works dealing with the applications of hyperbolic functions to electrical engineering are: " T h e Application of Hyperbolic Functions to Electrical Engineering Problems," by the author, The University of London Press, 1911, and Fleming's " T h e Propagation of Electrical Currents in Telephone and Telegraph Conductors," Constable & Co., London, 1911. A three-dimensional complex-angle geometrical model,f from which the hyperbolic sines and cosines of complex angles can be presented projectively, has been constructed and described. BRIEF BIBLIOGRAPHY o r TABLES OF HYPERBOLIC FUNCTIONS

(1) "Tafeln der Hyperbelfunctionen ynd der Kreisfunctionen," b y Dr. W . Ligowski, Berlin, 1890, Ernst and Korn, 104 pages, giving five-figure logarithms of sinh Θ, cosh Θ, and tanh θ up to θ = ç, by steps of 0.001 up to θ = 2 and from 2.0 to 9.0 by steps of * This compendium has, by permission, been included in this book at its second edition, as Table Χ Χ Π Ι . t " A new geometrical model for the orthogonal projection of the cosines and sines of complex angles" by A. E. Kennelly, Proc. Am. Ac. of Arts and Sciences, Vol. 54, April, 1919, pp. 371-378. [210]

EXPLANATORY

TEXT

o.oi; also the Gudermannian angle to two or more decimals of a second of arc, and other tables. (2) Smithsonian Mathematical Tables, " Hyperbolic Functions," by George F. Becker, and C. E. van Orstrand, Smithsonian Institute, Washington, D.C., 1909, 321 pages, giving five-figure logarithms of sinh Θ, cosh Θ, and tanh Θ, by steps of 0.0001 up to 0.1, by steps of 0.001 from 0.1 to 3.0, and by steps of 0.01 from 3.0 to 6.0; also similar five-figure tables of natural real hyperbolic functions, and various other tables. (3) "Alternating-Current Phenomena in Parallel Conductors," Vol. I by F. E. Pernot, John Wiley, New York, 1918, containing a Table of six-decimal logarithms of hyperbolic functions, up to 2.0 by steps of 0.001. These present a higher order of precision by one unit, than have been previously available for real hyperbolic functions.

The following is a list of all the tables of Complex Hyperbolic Functions known to the present writer, in the order of date of publication: — (4) Chrystal's "Algebra," Edinburgh, 1889, briefly discusses the theory of sinh Θ, cosh Θ, and tanh θ where θ is complex; or of the form χ + iy. Graphs are given in outline for these functions, from which a few numerical values may be read. (5) The paper on "Resonance in Alternating-Current Lines," by E. J. Houston and A. E. Kennelly, Transactions Α. I. E. E., April, 1895, Vol. X I I , pages 133-169, contains a Plate for the graphical evaluation of sinh θ and of cosh θ, θ be'ìng a complex variable χ + iq, between the limits of χ = o and χ = 1.25; q = o and q = oc, by steps of 0.05 in χ and q. The Plate is 40 cm. X 34 cm. and corresponds to Plates V I I — VIIIA of the Atlas prepared from tables in this book, except that it gives the result in regular polar coordinates instead of regular rectangular coordinates. It was produced, by a graphical process, for a precision of the 2.5th order. (6) The first tables of complex hyperbolic functions were a short set published by Dr. James McMahon in his Chapter IV, entitled "Hyperbolic Functions," of a book by Merriam and Woodward on "Higher Mathematics," pages 107-168. The tables gave sinh (at + iy) and cosh (χ + iy) from χ = o to χ = 1.5, by steps of 0.1, and also from y = o to y = 1.5, by steps of 0.1, Wiley & Sons, New York, 1896. The chapter has since been issued as a separate volume by the same publishers. (7) A table of hyperbolic functions of semi-imaginaries or sinh, cosh, tanh, coth, sech and cosech of χ /45°, by steps of 0.1 in χ up to χ = 20.5, was published by the present writer ih a paper on "The Alternating-Current Theory of Transmission Speed over Submarine Telegraph Cables," in the Proceedings of The International Electrical Congress of St. Louis, Section A, Vol. I, pages 68-105, 1904. This table is reproduced in Table V I of this volume. (8) Some short tables of sinh, cosh, tanh, coth, sech, and cosech ρ ¿5_ by steps of 0.1 in p, up to ρ = ι.5, for five particular values of δ, published by the present writer in an article on "The Distribution of Pressure and Current over Alternating-Current Circuits," in the Harvard Engineering Journal, 1905-06. [211]

EXPLANATORY

TEXT

(9) Short three-digit tables of sinh and cosh (χ + iy) up to χ = ι, and y = 1, by W. E. Miller, in a paper "Formulae, Constants, and Hyperbolic Functions for Transmission-Line Problems" in the General Electrical Review, Schenectady, Ν. Y., May, 1910. Supplement. (10) "Tables of Hyperbolic Functions in Reference to Long Alternating-Current Transmission Lines," published by the present writer in the Transactions of the American Institute of Electrical Engineers, December 1 9 1 1 , pages 2 4 9 5 - 2 5 0 6 . These give sinh, cosh, and tanh ρ [δ_ from ρ = o to ρ = 0.5, by steps of 0.1, and from δ = 6o° to δ = go" by steps of i°. These tables are incorporated in Tables I, II, and III of this volume. (11) "Tables of Sines, Cosines, Tangents, Cosecants, Secants, and Cotangents of Real and Complex Hyperbolic Angles," published by the present writer in The Harvard Engineering Journal," 1912. These gave sinh, cosh, and tanh ρ [δ_ from ρ = o to ρ = ι by steps of 0.1, and from δ = 45o to δ = qo° by steps of also corresponding tables of (sinh 0)/0 and of (tanh θ)/Θ. These tables are published in separate form by the Harvard Engineering Journal. They are incorporated in tables I, II, III, TV, and V of this volume. N E W TABLES INTRODUCED IN THE SECOND EDITION

Tables I to V in this volume were computed for the range of 45o to 90o in the slope or argument δ of the entering vector quantity; because at that time it did not appear that there would be any need for the range from o° to 45o. Alternating-current lines used for the transmission or distribution of power have linear hyperbolic angles a, the slope of which is commonly between 8o° and 90o, rarely falling as low as 45o. It has been shown during recent years, however, that railway-signal engineers employ tracksignaling circuits, formed of the rails. These are metallic circuits of low frequency, small linear capacitance and large distributed linear leakance. The linear hyperbolic angles α of such circuits develop slopes lying within the range δ = o° to 45o. It has therefore become desirable to cover this range, at least as far as ρ = ι. For that purpose, Tables XVII to XXI have been inserted. They run by steps of 0.05 in p, from o to 1.0, and by steps of 50 in δ, from o° to 45o. This new tabulated material is available for use in track-signaling and similar computations. It is hoped to incorporate it graphically into the associated Chart" Atlas at the first opportunity. Table XVII presents sinh ρ[δ_ as a polar planevector. It corresponds to and may be regarded as an extension of Table I. Similarly Tables XVIII, XIX, XX and XXI correspond respectively to Tables II, III, IV and V. Whereas, however, Tables I to V are carried to five decimal places in the sizes and three decimal places in the slopes of the evaluated quantities, the new tables are carried to six decimal places in sizes and four decimal places in slopes. They thus aim at one higher order of precision.* Table XXII is similar to Table XI and expands the region covered by the first six entries of the latter into a correspondingly magnified field. It has been found that in * The author desires to express his acknowledgment of the painstaking assistance, on these tables, by Miss Lillian L. Hodgdon, of the Harvard Observatory computing staff.

[ 212 3

EXPLANATORY TEXT dealing with short lengths of alternating-current line, having negligible linear inductance as well as negligible linear leakance, and therefore having a semi-imaginary linear angle, there is frequent need for a magnified table of this kind. It may be noted that whereas Table V I expresses slopes in degrees and minutes, Table X X I I expresses them as degrees, and four-place decimals of a degree. Table X X I I I is a useful collection of 238 formulas, with a few insertions, taken from Becker and van Orstrand's book of Tables of real hyperbolic functions, referred to in the footnote on page 210.

[«3 3

TABLE X V I I .

HYPERBOLIC SINES,

0 0 0 S IO 15 20

25

30 35 40 45

sinh ( p ¿ ^ = r / y .

0.000000 0.000000 0.000000 0.000000 0.000000

0 0.0000 5.0000 10.0000 15.0000 20.0000

0.05 0 0.050021 0.0000 0.050020 5.0042 0.050020 10.0081 0.050019 15.0119 0.050016 20.0153

0.000000 0.000000 0.000000 0.000000 0.000000

25.0000 30.0000 35.0000 40.0000 45.0000

0.050014 0.050011 0.050006 0.050003 0.05000

0.40 0 0.410752 0.0000 5-2625 0.410589 10.5172 0.410105 0.409316 I5-7569 0.408245 20.9742

0.45 0 0.465342 0.0000 0.465109 5-3314 0.464420 10.6531 0.463294 15-9558 21.2303 0.461767

0.406925 0.405397 0.403709 0.401910 0.40006

0.459886 0.457709 0-455303 0.452741 0.45010

0.35

0 0 S IO 15 20

0-3S7I90 0.357081 0.356757 0,356228 0.355512

0 0.0000 5.2014 10.3969 Χ5·58ο8 20.7472

25 30 35 40 45

0.354628 0.353606 0.35247S 0.351270 0-35003

25.8908 31.0089 36.0961 411503 46.1694

0 0 5 IO 15 20

0.70 o 0.0000 0.758584 0.757700 5-7875 0.755078 «•5533 0.750800 17-2755 0.745004 22-9344

25 30 35 40 45

0.737873 0.729632 0.720538 0.710872 0.700934

28.5103 33.9861 39·3467 44.5800 49.6767

Examples,

25.0183 30.0208 35.0222 40.0236 45-0236

26.1625 31.3161 36-4305 41.5017 46.5278

O.IO 0 0.100167 0.0000 0.100164 5-0167 O.IOOI57 10.0325 O.IOOI44 15.0478 20.0614 0.100128 O.IOOI07 0.100083 0.100057 0.100029 0.10000

25.0730 30.0828 35-0897 40.0942 45-0952

26.4686 31.6630 36.8086 41.8997 46.9338

0.75 0 0.822317 0.0000 5.9000 0.821227 0.817993 «•7753 0.812719 17.6019 0.805576 23·3567

0.80 0 0.888106 0.0000 0.886780 6.0192 0.882843 12.0108 0.876425 17.9481 0.867734 23.8044

0.796787 0.786636 0-775439 0-763544 0.751317

0.857048 0.844710 0.831105 0.816660 0.801819

29.0175 34·5647 39.9811 45-2525 50.3678

sinh (0.3s /35 o ) = 0.352475 /%6°.οφχ. sinh (0.80¿£) = 0.886780/6°.OIQ2.

CAI* 3

29-5561 35.1800 40.6567 45-9697 51-1064

TABLE X V I I .

H Y P E R B O L I C SINES,

sinh (p /J) = r ¿y.

CONTINUED

0.20 0 0.201336 0.0000 5.0661 0.201316 0.201256 10.1303 0.201157 15.1906 0.201024 20.2450

0.25 0 0.252612 0.0000 0.252573 51033 0.252455 10.2033 0.2522Ò3 15-2972 0.252002 20.3825

0.30 0 0.304520 0.0000 0.304452 51483 0.304248 10.2922 030391S iS-4275 0.303465 20.5500

0.200859 0.200669 0.200458 0.200233 0.20000

0.251680 0.251308 0.250896 0-250457 0.25001

0.302909 0.302265 0.301553 0.300795 0.30001

0 0 5 10 IS 20

0.15 0 0.0000 0.150563 0-150554 5-0372 0.150529 10.0733 0.150488 15.0172 0.150432 20.1380

25 30 35 40 45

0.150362 0.150282 0.150193 0.150098 0.15000

0 0 5 10 15 20

0.50 9 0.0000 0.521095 0.520776 5-4078 0.519829 10.8039 0.518282 16.1767 0.516184 21.5153

0.578152 0.577726 0.576463 0.574400 0.571604

0.0000 5-4919 10.9697 16.4197 21.8286

0.636654 0.636100 0.634456 0.631774 0.628138

0.0000 5-5833 II.1500 16.6842 22.1700

0.65 o 0.0000 0.696748 0.696042 5-6819 0.693947 11-3477 0.690530 16.9694 0.685898 22.5386

25 30 35 40 45

0.513601 0.510612 0.507309 0-503794 0.50017

0.568162 0.564179 0-559779 0-555099 0.550279

27.1844 32.4764 37-6956 42.8344 47.8880

0.623662 0.6x8485 0.612769 0.606689 0.600432

27-5933 32-9414 38.2036 43-3705 48.4369

0.680199 0.673609 0.666335 0.658601 0.650644

25.1644 30.1858 35.2017 40.2114 45.2146

26.8094 32.0503 37-2305 42-3439 47-3872

25.2919 30.3303 35-3586 40.3761 45-38I7

0.60

0-55

0.90

0.85

25-4561 30.5158 35-5600 40.5872 45-S965

0-95

25-6558 30.7419 35-8058 40.8456 45-8592

28.0355 33-4450 38.7539 43-9525 490330

I. 00

0 0 5 10 IS 20

0.956116 0.954520 0.949784 0.942064 0.931612

0.0000 6.1447 12.2592 18.3131 24.2775

1.026517 1.024615 1.018975 1.009783 0.997344

0.0000 6.2767 12.5200 18.6967 24-7753

1.099484 1.097239 1.090583 1.079736 1.0650Ö2

0.0000 6.4147 12.7928 19.0986 25.2969

1.175201 I-I72573 1.164779 1.152083 i-i349!3

0.0000 6-5592 130775 19.5180 25.8417

25 30 35 40 45

0.918768 0.903942 0.887604 0.870267 0.852463

30-1255 358314 4I-3730 467314 51.8917

0.982060 0.964429 0.945011 0.924414 0.903276

30-7253 36.5183 42.1294 47-5372 52.7242

1.047043 1.026266 1.003398 0.979156 0.954292

31-3544 37.2400 42.9253 48.3864 53-6033

I-113841 1.089558 1.062847 1034550 ι-005545

32.0122 37-9955 437603 49.2789 545292

Examples,

sinh (0.90/20°) = 0.997344/24o-7753· sinh ( i . o / o ° ) = 1.175201 /o°.

C«S3

T a b l e XVIII.

H Y P E R B O L I C COSINES,

c o s h (ρ /δ) = r ¿y.

ι.000000 1.000000 1.000000 1.000000 I.ÓOOOOO

0 0.0000 0.0000 0.0000 0.0000 0.0000

0.05 0 I.00I250 0.0000 I.OOI23I 0.0125 0.0244 I.OOII75 1.001083 0.0358 1.000958 0.0461

O.IO 0 1.005004 0.0000 1.004929 0.0497 1.004703 0.0978 1.004335 0.1428 1.003836 0.1836

25 30 35 40 45

I.OOOOOO I. OOOOOO 1.000000 I.OOOOOO 1.000000

0.0000 0.0000 0.0000 0.0000 0.0000

1.000804 1.000626 1.000428 1.000218 1.00000

I.00322I 1.002507 1.001718 1.000876 I.OOOOI

0 0 5 IO 15 20

0.35 0 1.061878 0.0000 0.5861 1.060965 1.058252 1.1561 1.053819 1.6950 2.1872 I.04779I

1.081072 1.079886 1.076361 1.070598 1.062762

0.0000 0.7567 1-4933 2.1911 2.8303

1.102970 I.IOI477 1.097041 1.089786 1-079919

0.0000 0-9453 1.8669 2.7411 3-545°

25 30 35 40 45

1.040347 I.03I702 1.02.2112 1.011859 I.OOI25

1.053079 1.041829 1.029339 I.OI5974 1.00213

3-3936 3-8639 4.2264 4.4678 4-S784

1.067721 I-053539 1.037782 1.020908 1.00341

4-2567 4-8SSO S-32o8 5-6375 5.7906

0 0 0 s IO IS 20

2.6189 2.9772 3.2506 3-4295 3-5071

0.0547 0.0619 0.0672 0.0705 0.0716

0.40

0.70 0

0.45

0.80

0.75

0 0 5 IO IS 20

1.255169 1.251669 1.241264 1.224242 1.201069

0.0000 2.1083 4-1750 6.1581 8.0155

25 30 35 40 45

1.172378 1.138949 1.101690 1.061615 1.019823

9-7039 11.1803 12.4000 13-3186 13-8911

Examples,

0.2189 0.2478 0.2689 0.2820 0.2864

1.294683 1.290690 1.278820 1.259402 1.232970

0 0.0000 2-3744 4.7044 6.9456 9.0520

1-337435 1.332918 I-3I9493 1-297535 1.267648

0 0.0000 2.6483 5-2500 7-7578 10.1236

1.200241 1.162104 1.119583 1.073820 1.026048

10.9775 12.6747 14.0942 15.1850 15-8947

1.230648 1.187536 I-I394S9 1.087691 1.033602

12.2978 14.2294 15.8644 17-1458 18.0136

cosh (0.10/25°) = 1.003221/0^2183. cosh (0.75/40°) = 1.07.3820/15°.1850.

[216J

TABLE X V I I I .

HYPERBOLIC

COSINES,

c o s h (ρ/δ) = r ¿ y .

0.2S

0.-IS 0

β

0

0

1.011271

0.0000

S

I.OIIIOI

O.IIII

1.020067 1.019765

1.010595 1.009769 1.008648

0.2189 0.3203 0.4II9

1.007265 1.005662 1.003887 I.OOI995 1.00005

0.4914

IO IS

20

25 30 3S

40 4S

O.5S6I

0.6042 0.6339 0.6445

1.017404 1.015415

1.030943 1.029547 1.027265 1.024167

1.012961 1.010116 1.006966 1.003604 1.00013

0.8703 0.9858 1.0719 1.1258 1.1458

I.020343 1.015908 1.010992 1.005746 1.00033

ι.018868

S IO IS

20 25

30 35

40 45

0.0000 1.3711 2.7x08 3.9872

6.5297

1.103034 1.082045 1.058692

6-9350

1-033637

4.3264

1.084345 1.066914 1.047534 1.026760 1.00520

5.2028 5-9453

7.1424

1.007598

0.85

20

1-383531 1.378460 1-363390 1-338746 I.3052I2

0.0000 2.9289 5.8092 8.5914 11.2253

25

1.263709

13-6589

0 5 IO 15

3 ° 35

40 45

1-215358 1.161440 1.103367 1.042645

I5-8375

17.7030 19.1936 20.2411

Examples,

0.0000 16053

7.3222 8.4006 9.2700 9.8980 10.2544

3-1753

0

1.044665 1.042663 1.039390 1.034944 1-029455 1.023084 1.016021 1.008476 1.00068

1.486225

0.0000

ι-479951

3-5053

I5-0545

1-338233

6-3794

9.4428

17.4914 19.6019 21.3192 22.5686

1.278634 1.212219 1.140689 1.065840

c o s h (0.25 /30 o ) = 1 . 0 1 5 9 0 8 / ι " · 5 3 4 4 . c o s h (1.00/40°) = 1 . 1 6 2 6 1 1 /25°·7689·

C « 7 ]

I-937S

2.1994

2-3975 2-5253 2-5773

0.65 1.218793 1.215756 1.206728 1.191956 1.171848

0.0000 1.8514 3.6644 5.4000 7.0194 8.4836

I-I46955

1.117959 1.085656 1.050938 I-OI4773

«

0.0000 3-215°

0.0000 0.4350 0.8578 1.2567 1.6200

I-045339

9-7533

10.7894 11-5539

12.0094

1.00

0.95

12.3525

053338

1.7901

1.123873 1.099026 1.071363 1.041655 1.010745

1-299533

ι

I-7567

6.2267 7.1292 7.8481 8.3566 8.6311

I-34577I

1.245685 1-185653 1.120988

1-6703

5-I694

1.461313 1.430851 1.389438

1-383145

1-5344

4.6750 6.0689

0

1.433086 1.427430 1.410623

1-3533

1.185465 1.182861 1.175118 1.162452 1.145211

0.90

0

0

0-3047 0.6006 0.8794 1-1328

0.60

1.155101 1.152899 1.146351 1.135641 1.121066

3 - 3 4 "

0.0000

1-031413

0-55

0.0000 1.1506 2.2736

1.127626 1.125794 I.I20349 I.III445 1.099330

0.30

0

0.0000 0.1964 0.3872 0.5664 0.7292

0.50 0

CONTINUED

0

1.543081

6.9589 10.3086

1-536155 1.515588 1.481987

13-5005

1-436333

16.4789 19.1844 21.5531 23-5142

24.9872

1.379926 I.3I432I 1.241261 I.162611 1.080307

0.0000 3

7989

7-5475

11.1858 14.6653 17.9267 20.9097 2 3 5 4 8 3

25.7689 27.4870

TABLE

XIX.

HYPERBOLIC TANGENTS, tanh (/> ¿¿) = r ¿y.

o

0 0 s IO IS 20

25 30 35 40

45

0 0 5 IO 15 20 25

30 35 40

45

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.05

0

0.0000

5.0000 10.0000 15.0000 20.0000

0.049958 0.049959 0.049961 0.049965 0.049968

25.0000 30.0000 35.0000 40.0000 45.0000

0.049974 0.04998 0 0.049985 0.049992 0.05000

ο·35 O 0.0000 0.336375 0.336562 4·6ΐ53 9.2408 0-337II9 0.338036 0.339296

13-8858 18.5600

0.340875 0.342740

23.2719 28.0317

32·8455

0.344849

0.347153 0.34959

37.7208 42.6623

20

25 30 35 40

45

0

0.604367 0.605352 0.608313 0.613278 0.620284 0.629382 0.640619 0.654029 0.669614 0.687309

0.0000

4.9917

9-9837

14.9761 19.9692

24.9636 29.9589

34-9550 39-9531 44.9520

9-9347

14.9050 19.8778

24.8541 29.8350 34.8208 39.8122 44.8087

0.45

I3-5658

9.0239

0.423339 0.425124

18.1439

0-427594

17-6853

22.7689 27-4522 32.2041

0.430717 0.434449

22.2119 26.8080 31.4878 36.2622

0.381011 0.382325

0.384135 0.386414 0.389121 0.392202

0-395591 0.39921

370339 41.9494

0-75 0

0.0000 3-5256

0-653363

I4-3047

18.8064 22.8058 26.9467 31.2614

0.663856 0.676907 0.692614 0.711054 0.732244

0

0.438727 0.443469 0.44857

41-1432

18.0400

21.8900 25.8869 30.0675

34-4731

0 0.664037 0.665292 0.669078 0.675454 0.684523 0.696420 0.7II3I3 0.729386 0.750820 0.775752

tank (0.75/25°) = 0.663856/18^0400. tanh (0.40/20°) = 0-384135 /18°. 1439·

f 218]

0.0000

0.80

14.9189

Examples,

O.IOOOO

4.9670

4-3861 8.7862 13.2147

7.0709 10.6563

35·7856

0.099786 0.099833 0.099886 0.099941

0 0.0000

0.421899 0.422260

11.1174

7·3783

0.099668 0.099673 0.099688 0.099712 0.099745

0 0.0000 0-379949 0.380215 4-5058

0-635150 0.636270 0.639647 0.645321

3.6792

O.IO

0.40

0.70

0 0 5 IO 15

0 0.0000

0.0000 3-3709

6.7608 10.1903 13.6808

17-2583 20.9506 24.7922 28.8239

330928

TABLE X I X .

HYPERBOLIC TANGENTS,

O.IS

0.20

CONTINUED

o. 25

0.30

0

0

0

0

tanh (P /£) = r ¿y.

0

0

0.148885

0.0000

0.197375

0.0000

0.244919

0.0000

0.291312

0.0000

s IO

0.148901 0.148951

4.9261

4.8697

0.244992

4.7986

0.245210

9.6027

0.149032

0.197718

9-7431 14.6242

0.291435 0.291799

IS 20

9-8544 14.7869

0.197414 0.197529

14.4178

0.292398

0.149142

19.7261

0.197972

I9-5I58

0.245567 0.246055

4-7133 9-4344 14.1708

19.2497

0.293219

18.9300

25

0.149277

24.6730

0.198289

24.4216

0.246663

24.1028

0.294242

30

0.149436 0.149611

29.6297

0.198659

0.295445

23-7183 28.5425

O.I 99071

33-8897

0.296798

33-4083

0.149799

39-5775

O.I995I4

39-2503

0.247373 0.248168 0.249027

28.9814

34-5975

29-3445 34.2867

0.298267

383203

0.14999

44-570I

0.19997

44-2359

0.24993

38.8305 43.8064

0.29981

43-2819

35 40 45

0.50

0.60

0.5s

0.65

0

0.462117

0.0000

0.500521

0.0000

0.537049

0.0000

0.571669

0.0000

5

0.462586

0.501107

4.1208

0.572518

8.2589

0.537764 0.539908 0.543484

3-978O

0.502867

7-9747 12.0092

0.575065

3-8305 7-6803

0.579325

11.5694

0.548491

16.1011

0.585313

I5-5I92

0.593047

19-5519 23.6917

IO

0.463988

15 20

0.466314

4-2572 8.5303 12.8356

0.469545

17.1889

0-505793 0.509875

12.4325 16.6592

25

0473651 0.478587

21.6066

0.515090

20.9577

0.554922

20.2711

26.1050

0.521400

24.5408

0.602535

0.484289

30.7008

0.528746

25-3472 29.8475

0.562757

35 40

28.9336

0.613763

27.9645

0.490664

35-4089

34-4778

32.3986

0.49759

40.2448

39-2569

0.594049

33-4725 38.1825

0.626679

45

0.537034 0.546130

0571953 0.582428

0.641172

37-0236

30

0.90

0.85 0

0

0.95

0

I . 00

0

0

0.691070 0.692454

0.0000

0.716298

0.0000

0.739782

0.0000

0.761595

0.717804

3.0617

2.7603

6.1406

5-5300

0.703692

0.730063

0.754611

0-713763

13-0522

0.741095

9-2539 12.4228

5-8339 8.7900

0.768533

15 20

6.4500 9.7217

0.722358

0.763317

0.696634

0.741402 0.746304

2.9094

IO

3-2158

0.766541

11.7964

0.777391 0.790146

8.3322 11.1764

0.755702 0.774215

15.6708 19.0269

0.782407

14-8755 18.0556

0.807174

14.0855

0.802627

0.828989

17.0858

0.797038

22.5275

0.827737

21.3722

0.856264

20.2120

0.824642

26.2180

0.858389

24.8722

0.889851

23.5100

0-895343

28.6161

0-930795

27.0422

0 5

25

0.727041

16.4666

30

0.743766

35 40

0.764227

19-9939 23.6700

45

0.788737 0.817596

27·5378 31-6506

Examples,

0.857537

30.1556

t a n h (0.25/25°) = 0.246663 / 24°· 1028· t a n h (0.90/30o) = 0.774215 /iq°.o26q.

[

219]

0.0000

TABLE XX.

CORRECTING FACTOR.

sinh θ θ

=

r¿V.

o.os 0 0 s IO 15 20

25 30 35 40 45

ι.000000 ι.000000 1.000000 1.000000 1.000000

0 0.0000 0.0000 0.0000 0.0000 0.0000

1.000420 1.000404 1.000398 1.000373 ι. 000317

0 0.0000 0.0042 0.0081 O.OII9 0.0153

1.001670 1.001638 ι. 001570 1.001442 1.001279

0 0.0000 0.0167 0.0325 0.0478 0.0614

1.000000 1.000000 1.000000 1.000000 1.000000

0.0000 0.0000 0.0000 0.0000 0.0000

1.000279 1.000216 1.000134 ι.000061 1.00000

0.0183 0.0208 0.0222 0.0236 0.0236

1.001071 1.000834 1.000571 1.000290 1.00000

0.0730 0.0828 0.0897 0.0942 0.0952

0.40

0.35

0.45

1.020543 1.020230 I.0I930S 1.017796 1.015748

0.0000 0.2014 0.3969 0.5808 0.7472

1.026880 1.026473 1.025264 1.023290 1.020612

0.0000 0.2625 0.5172 0.7569 0.9742

1.034093

0.0000

Ï-033577 1.032044 1.029542 1.026148

0-33I4 0.6531

0.8908 1.0089 1.0961

45

1.021969 1.017131 1.011784 1.006092 1.00023

1.4686 1.6630 1.8086 1.8997

1.00008

1.017312 I.OI3493 1.009272 1.004775 I.OOOI4

1.1625 1.3161

35 40

I.OI3224 1.010302 1.007070 1.003630

0 5 IO 15 20

25 30

1-1503 1.1694

0.70 0 0 5 IO 15 20 25 30 35 40 45

1.0836g! 1.082429 1.078682 1.072572 1.064292 I.054I05 1.042331 1.029340 I.OI5532 ι.001334

1.096423 1.094969 1.090658 1.083625 I.074I02

0-7875 1-5533 2-2755 2.9344

1.062383 1.048848 I.0339I9 1.018059 1.001756

3-5103 3.9861 4-3467 4.5800 4.6767

sinh

(0.40/25°)

—

0-40/25

C2203

0 0.0000 0.9000 1-7753 2.6019 3-3567 4-OI75 4-5647 4.9811 5-2525 5-3678

1-9338

0.80

0.75

0 0.0000

Example.

1-4305 1-5017 1.5278

0-9558 1.2303

1.110132 1.108475 1.103554 1.095532 1.084668 1.071311 1.055881 1.038881 1.020825 1.002274

= 1 . 0 1 7 3 1 2 / 1 .1625.

0 0.0000 1.0192 2.0108 2.9481 3.8044 4·556ι 5.1800 5-6567 5-9697 6.1064

TABLE X X .

CORRECTING

0.20

O.IS 0 5 10 IS 20 25 3O 35 40 45

1-003753 1.003696 1.003527 1.003253 1.002878 Ι.002412 1.001878 1.001287 1.000653 1.00000

0 0.0000 0.0372 0-0733 0.0172 0.1380

1.006680 1.006579 1.006278 1.005784 I.005II9

0.1644 0.1858 0.2017 0.2II4 0.2146

1.004296 1.003343 1.002290 1.001167 1.00001

o. so

15 20

0.0000 0.4078 0.8039 1.1767

25 30 35 40 45

1.027202 1.021223 1.014618 1.007587 1.00035

1.8094 2.0503 2.2305

5 10

I-5I53

2-3439 2.3872

0.85 0 5 10 15 20 25 30 35 40 45

1.124842 1.122964 1117393 1.108311 1.096014 1.080904 1.063461 I.04424I 1.023843 1.002897

1.051185 1.050411 1.048114 1.044363 1.039280 1.033021 I.O25779 1.017780 I.OO927O I.000508

0.1303 0.1906 0.2450 0.2919

1.006722 1.005233 1.003584 1.001830 X.00003

0.3303 0.Λ86 0.3761 0.3817

h .

0.4561 0-5158 0.5600 0.5872 0.5965

I.061090 I.060167 I.057426 I.052956 I.O46896

0.0000

2.1844

I.O39436 I.030808 I.02I28I 1.011148 1.000720

2.5933 2.94x4 3.2036

2.8344 2.888ο

0-5833 I.I500 1.6842 2.1700

33705 3-4369

3-3131 4-2775 5-1255 5-8314 6.3730 6.7314 6.8917

1.091177 1.071587 I.0500I2 I.027I27 1.003640

1.009696 1.007550 1.005177 1.002651 1.00005

0 0.0000 0.1483 0.2922 0.4275 0.5500 0.6558 0.7419 0.8058 0.8456 0.8592

I.07I920 1.070833 1.067611 1.062354 1.055228

0.0000 0.6819

1.046459 1.036322 1.025x31 I.OI3232 1.000991

30355 3-445° 3-7539 3.9525 40330

1-3477 1.9694 2.5386

1.00

0-95 0.0000 1.2767 2.5200 3.6967

1.015067 1.014839 1.014161 1.013051 I.OII549

o.6¡

O.OOOO O.4919 O.9697 I.4I97 I.8286

2.4764 2.6956

CONTINUED

0.30 0 0.0000 0.1033 0.2033 0.2972 0.3825

ο.6ο

I-I40574 1.138461 1.132194 1.121981 1.108160

Example.

= r

θ

1.010448 I.0I029I 1.009820 Ι. 009051 1.008007

0.90 0.0000 1.1447 2.2592

sinn υ

0.25 0 0.0000 0.0661

°·55

1.042190 1.041552 1.039657 1.036564 1.032369

0

FACTOR.

0.0000

1.175201

0.0000

1.4147 2.7928 4.0986 5.2969

1-172573 1.164779 1.152083 1.134913

1-5592 3-0775 4.5180

4-7753

I-I57352 1.154989 1.147982 1.136564 1.121117

5-7253 6.5x83 7.1294 7-5372 7.7242

1.102151 1.080280 1.056208 1.030691 1.004518

6-3544 7.2400

1.113841 1.089558 1.062847 1034550 1.00554s

7.0122

sinh (ο·95/25°) /

0

7-9253 8.3864 8.6033

,,0 = 1.Χ02151 / 6 .3544·

[221]

5-8417 7-9955 8.7603 9.2789 9.5292

TABLE X X I .

CORRECTING FACTOR.

0

o 0 S IO IS 20 25 30 35 40 45

0.05

ι.000000 ι.000000 I.OOOOOO 1.000000 1.000000

0.0000 0.0000 0.0000 0.0000 0.0000

1.000000 1.000000 1.000000 1.000000 1.000000

0.0000 0.0000 0.0000 0.0000 0.0000

0.999160 0.999177 0.999223 0.999291

0.0000 0.0083 0.0163 0.0239

Q-999359

0.0308

0-999475 0.999591 0.999707 0.999844 1.00000

0.0364 0.0411 0.0450 0.0469 0.0480

15 20 25 30 35 40 45

0.961071 0.961606 0.963197 0.965817 0.969419

0.0000 0.3847

0.973929 0.979258 0.985284 0.991867 0.99883

1.7281 1.9683

0.7592 1.1142 1.4400

2-1545 2.2792 2-3377

0.950538 0.952528 0.955812 0.960338

5 IO 15 20 25 30 35 40 45

0.863381 0.864789 0.869019 0.876111 0.886121 0.899117 0.915169 0.934328 0.956592 0.981870

1.4342 1.8561 2.2311

0.966036 0.972801 0.980505 0.988977 0.99800

6.1936 7.1942 8.0533 8.7386 9.2144

0.885142 0.902542 0.923485 0.948072 0.976325

0.1459 0.1630 0.1792 0.1878 0.1913

0.998856 0.999414 0.99999

2.5478 2-7959 2.9661 3.0506

0 0.0000 0.6139 1.2138

0-937553 0-938355 0.940753 0.944719 0.950209

1-7853 2.3147 2.7881

0.957IS0 0.965442 0.974949 0.985487 0.99684

3.1920 3.5122 3 7378 3-8568

0.80 0.0000

0

1-4744 2.9291

0.830046 0.831615 0.836347

4-3437 5-6953

0.844318 0-855654

6.9600 8.1100 9.1131

0.870525 0.889141 0.911732

9-9325 10.5269

0-938525 0.969690

Note.

Negative quantities are in h e a v y type.

Example.

t a n h (0.75/25o) — - —

C «»3

0.1222

0.997858 0.998331

0 0.846867 0.848360 0.852862 0.860428 0.871150

0.0330 0.0633 0.0950

0-997II9 0.997448

0-75 0 0.0000 1.3208 2.6217 3.8826 5.0811

0.0000

0.45 0 0.0000 0.4942 0.9761

0.949872

0.70 0 0

0 0.996680 0.996732 0.996882

0.40 9

5 IO

0.10 0

0

0.35 0

= r ¿y.

= 0.885142

\6°.ç6oo.

0.0000 1.6291 32392 4.8097 6.3192 7-7417 9.0494 10.2078 11.1761 11.9072

TABLE X X I .

CORRECTING FACTOR.

5 IO

0.992567 0.992676 0.993006

IS

0-993547

20

0.994280

40

0.995182 0.996237 0.997410 0.998660

45

0-9999S

25 30 35

0 0.0000 0.0739 0.1436 0.2131 0.2739

0.986875 0.987069 0.987642 0.988588 0.989860

0.3270

0.991446

0.3703 0.4025 0.4225 0.4299

0.993295 0.995357 0.997571

0.99987

0.50 0 0 5 10 15

20 25 30 35

40

45

0.947302 0.968577 0.981327 0.99516

0.936527 0.948001 0.961356 0.976426 0.992963

45911 4-7552

5

10 15

20 25 30 35

40 45

0.813024 0.814651 0.819569 0.827873 0.839722 0-855342 0.875019 0.899091 0.927926 0.961877

0.986650 0.989493 0.992673 0.996106 0.99971

0 0.0000 1.7842 3-5500

0.795887 0.797560 0.802620 0.811181 0.823438

5-2783 6.9478 8.5334 10.0061 11.3300 12.4622 13-3494

0.839669 0.860239 0.885598 0.916269 0.952819

0 0.0000 0.2014 0.3973 0.5822 07503 0.8972 1.0186 ι.1x03 1.1695 1.1936

1.7411 2.5675 3.3408

0.895082 0.896274 0.899847 0.905806 0.914151

40423 4-6528

0.924870 0.937928

5.1525 5.5222 5-7431

0 0.0000 1.0220 2.0253 2.9908 38989 4.2289 5-4592 6.0664

0-953254

0.970713 0.990081

6.5275 6.8175

0.9s 0 0.0000 1-9383 3.8594 5.7461 7-5772

9.3292 10.9731 12.4725 13.7820 14.8444

0.778718 0.780424 0.785583 0.794327 0.806886 0.823586 0.844870 0.871302 0.903567 0.942466

0.980807 0.984817 0.989327 0.994224 0.99937

1.2817 1-4575 I-59I7 1.6797 1.718t

0 0.0000 1.1695 2.3197 3-4306 4.4808

0.879491 0.880796 0.884716 0.891269 0.900482 0.912380 0.926976 0.944250 0.964122 0.986419

54481 6.3083 7.0355 7.6014 7-9764

1.00 0 0.0000 2.0906 4.1661 6.2100 8.2036

10.1245 11-9444 136278 15.1278 16.3839

Note.

Negative quantities are in heavy type.

Example.

tanh (1.0/10°) -,—ζ = 0.768533X4^4700.

[223]

0 0.0000 0.2867 0.5656 0.8292 1.0700

0.971040 0.971449 0.97 2665 0.974660 0.977396

0.6s

0.60

0.90

0.8s 0 0

0.5784 0.6555 0.7133 0.7497 0.7641

0 0.0000 0.8792

0.910038 0.911104 0.914304 0.919624 0.927046

3 3934 3-8950 4.2992

0.957I74

0.1303 0.2569 0.3758 0.4842

0.979676 0.979968 0.980839 0.982269 0.984222

0.55 0 0.0000 0.7428 1.4697 2.1644 2.8111

0.924234 0.925172 0.927976 0.932627 0.939089

0 0.0000

CONTINUED 0.30

0.2S

O.I s o 0

= r ¿y.

0.790146

0 0.0000 2.2397 4.4700 6.6678 8.8236

0.807174 0.828989 0.856264 0.889851

10.9145 12.9142 14-7880 16.4900

0.761595 0-763317 0-768533 0-77739Ϊ

0.930795 1 7 9 5 7 8

TABLE X X I I .

FUNCTIONS OF

COMPLEX VARIABLE θ hyp. rads.

Sinh θ /45° Size Slope numeric degrees

0.00 0.01 0.02 0.03 0.04 0.05

0.00000 0.01000 0.02000 0.03000 0.04000 0.05000

45.0000 45.0009 45-0037 45.0084 45-0151 45.0236

0.06 0.07 0.08 0.09 0.10

0.06000 0.07000 0.08000 0.09000 0.10000 0.11000 0.12000 0.13000 0.14000 0.15000

45-0342 45-0465 45.0608 45.0770 45-0952 45-1152

O.II 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33

0-34 0-35 0.36 0.37 0.38

0-39

0.40 0.41 0.42

0-43

0.44 0.45 0.46 0.47 0.48 0.49 0.50

0.16000 0.17000 0.18000 0.19000 0.20000 0.21000 0.22000 0.23000 0.24000 0.25001 0.26001 0.27001 0.28001 0.29001 0.30001 0.31002 0.32002 0.33002 0.34002 0.35003 0.36003 0.37004 0.38004 0.39005 0.40006

45-I372

45.1611 45.1869 45.2146 45.2442

45-2757 45-3092 45-3445 45-38I7 45.4208 45.4619

45-5048 45-5497 45-5965 45-6453 45-6959 45-7485

45.8029

45-8592 45-9174 45-9775 46.0396 46.1036 46.1694 46-2372 46.3069 46.3786 46.4522 46.5278

0.41006 0.42007 0.43008 0.44009 0.45010

46.6053 46.6846 46.7658 46.8489

0.46012 0.47013 0.48014 0.49016 0.50017

47.0206 47.1093 47.2000 47.2926

46·9338

47·3872 Note.

/45O

Cosh 0/45° Size Slope numeric degrees ι.00000 0.0000 1.00000 0.0028 1.00000 0.0114 1.00000 0.0257 1.00000 0.0458 1.00000 0.0716 1.00000 0.1031 1.00000 0.1403 1.00000 0.1833 1.00001 0.2320 I.OOOOI 0.2864 1.00001 0.3466 1.00002 1.00003 1.00004 1.00005

0.4125 0.4841 0.5614 0.6445

1.00006 1.00008 1.00009 I.OOOII I.OOOI3 I.OOOI6 1.00020 1.00024 1.00028 1.00033 1.00038 1.00044 1.00051 1.00060 1.00068 1.00077 1.00087 1.00098

0.7333

1.00140 1.00156 I.OOI74 1.00193 I.002I3 1.00235 1.00259 1.00285 1.00312 I.0034I X.00372 1.00406 1.00442 1.00480 1.00520

3.7100

0.8278 0.9281 1-0341 1.1458 1.2632 1.3864

1-5152

1.6498 Ι.7901

1.9361

2.0878 2.2452 2.4084

2-5773 2.7520 2.9323

3·»83 I.OOIII 3-3099 1.00125 3-5071 3-9i8s

4.1328

4-3528 4-5784 4.8096 5.0464 5.2889

5-5369

5.7906 6.0498 6.3146 6.5850 6.8609 7.1424

SEMI-IMAGINARIES

(slope constant). tanh θ /45° Size Slope numeric degrees 0.00000

45.0000

O.OIOOO 44.9981 0.02000 44-9923

0.03000 44.9827 0.04000 44-9693 0.05000 44.9520 0.06000 44.9310 0.07000 44.9062 0.08000 44-8775 0.09000 44.8450 O. IOOOO 44.8087 O.I 1000 44.7686 0.12000 44-7247 0.13000 44.6770 0.13999 44-6255 0.14999 44-5701 0.15999 44.5109 0.16999 44-4479 0.17998 44.3811 0.18998 44.3104 0.19997 44-2359 0.20997 44-I576 0.21996 44-0755 0.22995 43.9896 0.23994 43.8999 0.24993 43.8064 0.25991 43.7092 0.26989 43.6081 0.27987 43-5032 0.28984 43-3945 0.29981 43.2819 0.30978 43-1654 0.31974 43-0452 0.32970 42.9213 0.33965 42.7937 0-34959 42.6623

0.35953 42.5272 0.36946

0-37939

0.38930 0.39921 0.40910 0.41899 0.42886 0.43872 0.44857 0.45841 0.46823 0.47803 0.48782

16.66667 14.28571 12.50000

II.IIIII

10.00000 9.09091

8-33333

7.69231 7.14286 6.66667 6.25000 5-88235

5-55556

5.26316 5.00000 4.76190

4-54545 4-34783 4.16667 3.99984 3.84601

3-70357 3-57130 3.44816

3-33322

3.22560 3.12481 3.03012 2.94100 2.85690

2-77755

42.3884 42.2458 42.0994 41.9494

2.70241 2.63130

41-7957

2.43867 2.38056

41.6382 41.4778 41.3120 41.1432 40.9708 40.7947 40.6150

40.4317 0.49759 40.2448

Negative quantities are in heavy type.

Examples, sinh 0.13 /45 O = ο. 13000 /45O. 1611. cosh o.26 /45 O = 1.00038 /I.°936I. [ 224]

Coseche/45o Size Slope numeric degrees oc 45.0000 100.00000 45.0009 50.00000 45-0037 33-33333 45.0084 25.00000 45.0151 20.00000 45.0236

2·56377 2.49963

2-32515

2.27226 2.22173

2-17335 2.12707 2.08273 2.04015 1.99933

45-0342

450465 45.0608 450770 450952

45.1152 45.1372 45.1611 45.1869 45.2146 45.2442 45-2757 453092 45-3445 45-3817 45.4208 45.4619 45.5048 45 5497 455965 45.6453 45.6959 45.7485 45.8029 45-8592 459174 45-9775 46.0396 46.1036 46.1694 46.2372 46.3069 46.3786 46.4522 46.5278 46.6053 46.6846 46.7658 46.8489 46.9338 47.0206 47.1093 47.2000 47.2926 47.387a

TABLE X X I I .

FUNCTIONS

OF

SEMI-IMAGINARIES.

CONTINUED

COMPLEX VARIABLE θ /45O (slope constant). θ

hyp. rads.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

O.IO O.II 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

031

0.32

°-33 0-34 0-35

0.36

0-37 0.38

0-39 0.40 0.41 0.42

0-43 0.44

°-45

0.46 0.47 0.48 0.49

aso

Sech ι» / 45° Size Slope numeric degrees ι.00000 0.0000 1.00000 0.0028 1.00000 0 . 0 1 1 4 1.00000 0.0257 ι. 00000 0.0458 1.00000 0.0716 ι. 00000 0.1031 1.00000 0.1403 1.00000 0.1833 0.99999 0.2320 0.99999 0.2864 0.99999 0.3466 0.99998 0.4125 0.99997 0.4841 0.99996 0.5614 0.9999s 0.6445 0.99994 0.99992 0-7333 0.8278 0.99991 0.9281 0.99989 1.0341 0.99987 1 . 1 4 5 8 0.99984 1.2632 0.99980 1.3864 0.99976 15152 0.99972 1.6498 0.99967 Ι . 7 9 0 1 0.99962 1.9361 0.99956 2.0878 0.99949 2.2452 0.99940 2.4084 0.99932 2.5773 0.99923 2.7520 0.99913 2.9323 0.99902 3 . 1 1 8 3 0.99889 3 3 0 9 9 0.99875 3.5071 0.99860 3.7100 0.99844 3.9185 0.99826 4.1328 0.99807 4 3 5 2 8 0.99788 4 5 7 8 4 0.99766 4.8096 0.99742 5.0464 0.99716 5.2889 0.99689 5-5369 0.99660 5.7906 0.99629 6.0498 0.99596 6.3146 0.99560 6.5850 0.99522 6.8609 0.99483

7.1424

Coth θ Size Slope numeric degrees oc 45.0000 100.00000 44.9981 50.00000 44 9923

33-33333

44.9827

II.IIIII

44.8450 44.8087

25.00000 44 9693 20.00000 449520 16.66667 4 4 9 3 1 0 14.28571 44.9062 12.50000 44-8775 10.00000 9.09091

44.7686

7.69231

44.6770

8-33333 44-7247 7-14337

6.66712 6.25039 5.88271

5-556x8

44-6255

44 5701

44.5109 44 4479 44.3811

5.26371 44-3104 5.00050 44-2359 4.76258 44.1576 4.54628 44-0755 4.34878 43-9896 4.16771 4 3 8 9 9 9 4.00112 43.8064 3.84748 4 3 7 0 9 2 3.70521 43.6081

3-57309 43-5032

3.45018

3-33544 3.22810

3-12754 3-03305

43-3945 432819 431654 430452 42.9213

2.94421 42-7937 2.86049 42.6623 2.78141 42 5272 2.70665 42.3884 2.63580 42.2458 2.56870 42.0994 2.50494 42.9494 2.44438 41-7957 2.38668 41.6382

2.33174 2-27935

2.22930 2.18145

2-I357I

41.4778 41.3120 41.1432 40.9708 40.7947 40.6150

2.09190 2.04995 40.4317 2.00969 40.2448

S i n h f l / 4 ¿ / 0 / 4 ¿ tanh θ / 4 5° / θ /45° Size Size Slope Slope numeric degrees numeric degrees 1.00000 0.0000 ι.00000 0.0000 1.00000 0.0009 ι.00000 0.0019 ι. 00000 0.0037 1.00000 0.0077 ι.00000 0.0084 1.00000 0 . 0 1 7 3 ι.00000 0.0151 1.00000 0.0307 ι.00000 0.0236 I.00000 0.0480 ι.00000 0.0342 1.00000 0.0690 ι.00000 0.0465 0.99999 0.0938 ι.00000 0.0608 0.99999 0 . 1 2 2 5 ι . 00000 0.0770 0.99999 0 . 1 5 5 0 ι. 00000 0.0952 0.99999 O . 1 9 1 3 ι.00000 0.1152 0.99998 0 . 2 3 1 4 ι.00000 0.1372 0.99998 0 . 2 7 5 3 1.00000 0.1611 0.99997 0.3230 1.00000 o.i86g 0.99996 03745 1.00000 0.2146 0.99995 0 4 2 9 9 1.00000 0.2442 0.99994 0.4891 0.2757 1.00001 0.99992 0.5521 I.OOOOI I.OOOOI

0.3092

0.3445

1.00001 0.3817 1.00002 1.00002 1.00002 1.00002 1.00003 1.00003 1.00003 1.00004 1.00004 1.00005 1.00005 1.00006 1.00006 1.00007 1.00008 1.00009

0.4208 0.4619 0.5048

0.5497 05965

0.6453 0.6959

07485

0.8029 0.8592 0.9174

0-9775 1.0396 1.1036 1.1694 1-2372 1.3069

1.00010 1.00011 1.3786

ι.00013

I.OOOI 4

1.00016

1.00017

1.00019 1.00021 1.00023 1.00025 1.00027 1.00030 1.00032 1.00035

1.4522 1.5278 1.6053 1.6846 1.7658 1.848g

1-9338

2.0206 2.1093 2.2000 2.2926 2.3872

Negative quantities are in heavy type. tanh 0.39 ¿4¿ ^ Examples. = = = 0.99819 \2 .9006. Q-39/45 coth 0.50 /45° = 2.00969 \4o".2448. Note.

[225]

0.99991 0.99989 0.99987 0.99984 0.99981 0.99978

0.6189 0.6896 0.7641 0.8424

09245 1.0104

0.9997s I.IOOI 0.99971 0.99966 0.99960 0.99953 0.99945 0.99937 0.99928 0.99919 0.99908 0.99896 0.99883 0.99869 0.99854 0.99837 0.99819 0.99800 0.99780

0-99759

0.99736 0.99711 0.99684 0.99655 0.99624 0.99591

0-99555 0.99516

1.1936 1.2908 1.3919 1.4968 1.6055 X.7I8I 1.8346 1.9548 2.0787 3.2063

2-3377 2.4728 2.6116

2.754a

3.9006 3.0506 3.2043 3.3618 3.5230 3.6880

345568

4.0292 4 2053 4-3850 4.5683

4-7552

TABLE XXIII.

HYPERBOLIC FUNCTION

FORMULAS

(from Smithsonian Mathematical Tables No. 1871.of 1909, Becker and van Orstrand's "Hyperbolic Functions," by permission.) A.

RELATIONS BETWEEN HYPERBOLIC AND CIRCULAR

1.

sinh u — — i sin iu = tan gd u.

2.

cosh u = cos iu = sec gd u.

3.

tanh u = — i tan iu = sin gd u.

4.

tanh \ u=

5.

e" = (1 + sin gd u) H- cos gd u,

tan

FUNCTIONS

\gdu.

= [ 1 — cos ( I π + gd = tan ( j x + 6.

sinh iu = i sin u.

7.

cosh iu = cos u.

8.

tanh iu = i tan u.

-r- sin Q τ + gd u),

ìgdu).

8a. sin u = — i sinh iu = tanh (gdr1 u). 8b. cos u = cosh iu = sech ( g d - 1 «). 8c. tan u = — i tanh iu = sinh (gd~l u). 9.

sinh (u ± iv) = ± i sin (ν

iu),

= sinh u cos ν ± i cosh u sin v. 10.

cosh (u ± iv) = cos (v =F iu), = cosh u cos ν ± i sinh u sin v.

10a. sin (M ± iv) = ± i sinh (ν ± iu) = sin u cosh ν ± i cos u sinh v. ι ob. cos (m ± iv) = cosh (v

iu) = cos m cosh ν =F i sin m sinh ι».

11.

cosh ( m π) = cos w x .

12.

sinh (2 w + 1) J î 7Γ = i sin (2 m + 1) J 71-.

B.

(m is an integer.) (m is an integer.)

RELATIONS AMONG THE HYPERBOLIC

FUNCTIONS

13.

sinh u = i (eu — e~u) = — sinh ( — «) = (csch u) = 2 tanh | ® T (1 — tanh 2 £ «) = tanh M -4- (1 — t a n h 2 «)*·

14.

cosh M = i (eu + e~") = cosh ( — u) = (sech M) -1 , = (1 + tanh 2 5 « ) t ( i - tanh 2 5«) = 1 ν ( 1 - tanh 2 w)*.

15.

tanh u = (eu - e - " ) -s- (eu + r"") = - tanh ( - u), = (coth M)-1 = sinh M -f- cosh M = (1 — sech 2 «)*.

16.

sech m = sech ( — « ) = ( 1 — tanh 2 «)*. [226]

FORMULAS 17.

csch m = — csch (— m) = (coth2 u — ι)*.

18.

coth m = — coth (— u) = (csch2 u + i) } .

19.

cosh2 u — sinh2 u = 1.

20.

sinh \ u = V 3 (cosh m — 1).

21.

cosh è « = V J (cosh m + 1).

22.

tanh 5 « = (cosh « — 1) -f- sinh m,

_

= sinh u τ (ι + cosh «) = V (cosh m — 1) -r- (cosh m + 1) 23.

sinh 2U = 2 sinh m cosh m = 2 tanh w -s- (1 — tanh2 u).

24.

cosh 2W = cosh2 m + sinh2 « = 2 cosh2 u — i, = 1 + 2 sinh2 m = (1 + tanh2 m) -f- (1 — tanh2 u).

25.

tanh 2M = 2 tanh« -5- (1 + tanh 2 «).

26.

sinh 3m = 3 sinh « + 4 sinh3 m.

27.

cosh 3« = 4 cosh3 u — 3 cosh u.

28.

tanh 3« = (3 tanh m + tanh3 u) -s- (1 + 3 tanh2 u).

28a* w cosh u + tt sinh u — 5 {m + ri) eu + 3 (m, — n) e - ". 28b. me" ± n>e~" = (m ±n) cosh u + (m ψ n) sinh u. 29.

30.

sinh nu = Í^J jj η cosh" -1 u sinh u + —-———y6 cosh nu = cosh" u H

η (M — j j

3.1. sinh « + sinh » = 2 sinh

2)

cosh"-3 u sinh3 u + . . .

cosh "~2 m sinh2 m +

+ v) cosh \{u— v).

32.

sinh u — sinh κ = 2 cosh i (u + v) sinh 3 (u — s).

33.

cosh « + cosh 0 = 2 cosh £ (w + ») cosh 3 (u — v).

34.

cosh « — cosh u = 2 sinh 3 (m + s i n h 3 (m — !>).

35.

sinh u + cosh u = (1 + tanh | m) -r (1 - tanh J m).

36.

(sinh u + cosh m)" = cosh nu + sinh nu.

36a. a sinh u + b cosh u = Λ]a2 — b2 sinh (u + tanh -1 f). = Vè2 - a2 cosh (u + coth"1 £). 36b. a cosh u±b

sinh « = Va2 — ô2 cosh (w db tanh -1 f).

37.

tanh « + tanh v = sinh (m + ») -f- cosh u cosh v.

38.

tanh « — tanh v — sinh (w — v) -5- cosh u cosh v.

39.

coth m + coth t> = sinh (u + v) -s- sinh u sinh v. [227]

FORMULAS 40. coth u — coth ν = — sinh (u — v) 4- sinh u sinh v. 41.

sinh (« ± ») = sinh u cosh ν ± cosh u sinh v.

42.

cosh (u ± v) = cosh u cosh υ ± sinh u sinh v.

43.

tanh ( u ± v ) = (tanh u ± tanh s) H- (1 ± tanh u tanh s).

44.

coth (u ± v) = (coth M coth ν ± 1) -f- (coth

45.

sinh (u + v) + sinh (u — v) = 2 sinh « cosh 11.

46.

sinh (u + v) — sinh (u — v) = 2 cosh M sinh v.

47.

cosh (« + !>)+ cosh (u — v) = 2 cosh M cosh u.

48.

cosh (M + v) — cosh (u — v) = 2 sinh M sinh A.

49.

tanh 5 (u +

50.

tanh \ (u — v) = (sinh M — sinh v) -F- (cosh M + cosh v).

51.

coth i (u + v) = (sinh M — sinh n)

52.

coth I (M — v) = (sinh M + sinh v) H- (cosh u — cosh v).

ς-2

tanh u + tanh v = sinh (M + v) —. tanh u — tanh v sinh (u — v)

± coth M).

= (sinh u + sinh v) -H (cosh M + cosh I>).

coth u + coth v _ coth u — coth v

(cosh u — cosh v).

sinh (u + v) sinh (u — v)

55.

sinh (u + v) + cosh (u + v) = (cosh u + sinh u) (cpsh v + sinh v).

56.

sinh (M + v) sinh (u — v) = sinh 2 u — sinh 2 v, = cosh2 u — cosh2 v.

57.

cosh (M + v) cosh (u — v) = cosh2 u + sinh2 v, = sinh 2 u + cosh2 v.

58.

sinh (mi τ ) = o.

59.

cosh (mi τ) = ( — i) m .

60.

tanh (mi x) = o.

61.

sinh (u + mi π) = (— i) m sinhw.

62. 63.

cosh (u + mini) — (— i ) m cosh u. sinh (2 m + 1) £ i τ = ± i.

64.

cosh (2 m + 1) 5 i ir = o.

(m is an integer.)

Π 228 j

FORMULAS 66a. sinh { (u + iv) + i ? } = sinh \ (u + iv) + i ¿ ]

= i cosh (u +

iv).

66b. cosh j (M + iv) + i f £ = cosh { ( « + iv) + i ¿ ^ = i sinh (u +

iv).

66c. tanh { ( « + iv) + i § j = tanh \ {u + iv) + i i_ \ = coth (u + 66d. sinh j ( « + iv) + iv\

= sinh { (u + iv) + i2

iv).

—sinh (u +

} =

66e. cosh ^ (« + iv) + i ir ^ = cosh { (u + iv) + i 2 } =

iv).

— c o s h (u +

66f.

tanh \ (u + iv) + ιπ \ = tanh \ (u + iv) + Í2_ \ = tanh (« +

67.

tanh (Μ + ΜΓ) = t a n h u.

iv). iv).

67a. If sinh { ( « + i (1 — q) | = χ + iy;

then sinh $ u + i (1 + q) j = — χ +

67b. If cosh \ (u + i (1 — q) È — χ + iy:

then cosh ^ u + i (1 + q) } = — X +

67c. If tanh { u + i (1 — q) ^ = χ + iy:

C.

then tanh J u + i (1 + q) } = x —

INVERSE H Y P E R B O L I C FUNCTIONS

68.

s i n h - 1 « = l o g (u +

V« 2 + ι ) = cosh" 1 V M 2 + 1

69.

c o s h - 1 u = log (m +

V« 2 -

70.

t a n h - 1 u = i log (ι + u) — è log (ι — u)

=

71.

c o t h - 1 u = J log (1 + u) — è log (m — ι )

=

72.

s e c h - 1 « = log ( ï +

73.

CSCh- M = log ( ì +

74.

sin - 1 u = — i s i n h - 1 iu = — i log (iu +

75.

c o s - 1 u = — i c o s h - 1 u=

76.

t a n - 1 u = — i t a n h - 1 iu =

77.

COt - 1 M = i coth

78.

s i n - 1 IM = i s i n h - 1 u = i log (u +

79.

c o s - 1 iu=

1

1) = sinh -

=

+ ι) =

V« 2 -

1

- f

=

u ' d

u ( l

1

=

«(I -

" u i ) i -= c o s h --1 - · M5)» u

¿M - / —2 — ¿ ϊ u(u + l)i V

1 —

M2)·

— i log (u + i y/τ — M 2 ).

2 í

log (1 + iu)

— — . log (1 — 21

iu = — . log (íM — i ) 2 l V1 +

Î-; log (lU + i ) . u2).

— i c o s h - 1 iu = - — i log (M + C 2293

V i + M2)·

iu).

iy.

iy. iy.

FORMULAS So.

Τ % tan -1 tu = i tanh -1 u = - log (i + «) — - log (i — u).

•81.

cot - 1 iu=—

%

i coth -1 m =

2

log (μ + 1) H

%

2

log (« — 1),

•82. cosh -1 5 i « + - ) = sinh -1 ì fu — -) = tanh~lW„ . I , V «/ \ «/ w2 + 1 = 2 tanh -1 83.

tanh -1 tan u = \ gd 2 u.

84.

tan - 1 tanh u = §

•85.

cosh -1

W , 1 = log u. u + ι

2 «.

csc 2 « = — sinh -1 cot 2 m = — tanh - 1 cos 2 » = log tan w.

•86. tanh -1 tan2 tan2

ir + J u) = J log csc w.

87.

tanh -1

\ u = % log sec u.

88.

cosh -1 « ± cosh -1 ν = cosh -1 [uv ± V (M2 - 1) (v2 - 1)].

89.

sinh -1 « ± sinh -1 D = sinh -1

V 1 + ν2 =fc ν V ι + w2].

D.

SERIES

90.

e« = ι + 11 + -. + —. + -.+ 2! 3! 4!

91.

log u = (w — 1) — - (w — i) 2 + — (« — i) 3 — . . . 2 3

94.

log (1 + w) = u — — W2 + — M3 — — « 4 + . . . 2 3 4

95· log

97.

sinhu = u-\

=

2

[M + y

u%

. . .

+ y

(u2< cc)

+ J

¡- + —J + —¡- + . . . [230]

+ · · · ]

(2> u> o)

(w2 < 1)

(«2 < i )

(M2

ι

« V«2 + ι

¿gd« , — s — = sech «.

du d gd-1 u — 5 du

=

sec«. [232]

FORMULAS

F.

INTEGRALS.

(INTEGRATION CONSTANTS ARE OMITTED.)

129.

J*SINHudu

= COSH«.

130.

J*COSH udu

= SINH u.

131.

J * TANH udu

= LOG COSH u.

132.

J*COTH udu

= LOG SINHu.

133.

J*SECH u du = 2 TAN -1 EU = GD M.

134.

J*CSCH udu

135.

J * SINH" udu

— LOG TANH"·

= ^ SINH

W. COSH « — *

I = — ; — SINH N + L u COSH u

J"COSH"udu

= ^ SINHW. COSH N_1 I

η

+

* J * SINH N - 2 « ¿Κ,

M+ 2 C ; — I SINH N + 2 u du.

η + ι

136.

W

η + i J

cos M

u+

^

M+ 2 • SINH u COSHN+1 « + — ; — I COSH"+2 u du. » + i J Ι

137.

J " u SINH udu

138.

J*u

139.

J%u2SINH

udu

140.

^un

u du — un COSH u — nun~l SINH u + η (n —

141.

J * SINH2 udu

142.

143.

= uCOSHu

— SINHu.

COSH u çlu = u SINH u — COSH u.

SINH

= (M2 + 2) COSH u — 2

= I (SINH « COSH « — « ) .

SINH « . COSH

J*COSH2udu

uSINHu.

udu=

I

COSH (2

u).

= i (SINHuCOSHu +

«).

imi

1) J'M""J

SINH

udu.

FORMULAS 144. J*tanh2 udu = « — tanh u. 145. J*coth2 udu = u — coth u. 146. J*sech2udu

= tanhu.

147. J"sech3 udu = § sech u tanhu + } gd «. 148. J*csch2udu 149. J'sinh-1 150.

= — cothu.

udu = u sinh -1 m — (1 + m2)*.

cosh -1 udu = ucosh-1 m — (m2 — ι)*.

151. J*tanh - 1 udu = m tanh -1 u + i log (1 — w2). 152. J*u sinh -1 u du = £

(2 m2 + i) sinh -1 « — « (1 + «2)J J .

153. J ' u cosh -1 u du = i

(2 m2 — 1) cosh -1 u — u(u2 — i)i J .

154. J*(cosh

a + cosh u)~l du = 2 csch a. tanh - 1 (tanh £ m. tanh J a), = csch a [ log cosh J (u + a) — log cosh f (m — a) J.

155.

(cos a + cosh w) -1 du = 2 csc a. tan -1 ( tanh J «. tan J ¿; ^ ,, β positive, ac < b2; a negative.

F O R M U L A S

du

χ

au

,

+

δ

/ { a u *

+

2

b

+

u

c)

= (δ

2

(δ

2

=

¿ I

7

6

Γ

·

( a c -

=

~ -

1

-

1

t a n h

ac)i

-

( β - « )

c o t h - i

^

_

2

^

£

ΐ

Λ

- 1

( μ

1 7 9 .

J * ( a

1 8 0 .

J "

— —

a

2

- 1

t a n

2

) »

d u =

$

u ( u

-

w

2

)*

d u

J

m ( a

+

α

2

)*

á « = i « ( «

d «

=

2

δ

I tí β

_

'

Γ

1 8 3 .

/

« e °

u

«

e°

m

¿ M

u

r