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HYPERBOLIC FUNCTIONS.



MATHEMATICAL MONOGRAPHS

J.DITKD BY

The Late Mansfield Merriman and Robert S. Woodwerd

Octavo, Cloth

No. 1. History of Modern Mathematics.

Liy David Kli.;ene S.mith. $1.:.'o itet.
N . 2. Synthetic Projective Geometry.

l!y tlie Late CiEOiiGE Bruce Iialsted. SI, 25 net.
No. 3. Determinants.

By tlie Late Laenas Gifford Weld. S1.2,j net.
ITo. 4. Hyperbolic Functions.

By the Late James ^Ic^L^IIO^■. $L2o net.
ITo. 5. Harmonic Functions.

By William L. Byerlv. S1.2o net.
No. 6. Grassmann's Space Analysis.

15y Edward W. Hyde. SI. 25 7iet.
No. 7. Probability and Theory of Errors.

By UoBEKT .-. Woodward. $L2o net.
No. 8. Vector Analysis and Quaternions.

By tlie Late Alex.\.ndeu A1.\cf.\rlane. SL25 net.
No. 9. Differential Equations.

By \\ illiam Woolsey Johnson. §1.25 iu\'.
No. 10. The Solution of Equations.

By the Late Mansfield Merri.man. SI.l'6 net.
No. 11. Functions of a Complex Variable.

By Tho.mas S. Fiske. $1.25 net.
No. 12. The Theory of Relativity.

By Robert D. Carmicu.^el. $1.50 net.
ITo. r3. The Theory of Numbers.

By Robert L). Car.michael. $1.25 net.
ITo. 14. Algebraic Invariant>.

By Leonard ];. Lick-SON. $1.50 ?iet.
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By Robert D. CAR^:I^HAEL. $1.50 net.
No. 17. Ten British Mathematicians.

By tlie Late Alexander Mackarlane. $1.50 nci.
ITo. 13. Elliptic Integrals.

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ITo. 10. Empirical Formulas.

By Theodore R. Running. $2.00 net.
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PUBLISHED BY
JOHN WILEY & SONS, Inc., NEW YORK

CHAP.MAN & HALL. Limited. LONDON



MATHEMATICAL MONOGRAPHS.



EDITED BY



MANSFIELD MERRIMAN and ROBERT S. WOODWARD.



No. 4.



HYPERBOLIC FUNCTIONS.



JAMES McMAHON,

LaIE I'ROFEsb^K OF iU A 1 H EM Al ICb IN CoKNELL UNIVERSITY.



FOURTH EDITION. ENLARGED.



NEW YORK:

JOHX WILEY & SONS.

London: CHAPMAN e^ HALL, Limited.






(X-cUl-



Copyright, 1S96,

BY

MANSFIELD MF.RRIMAN and ROBERT S. WOODWARD

UNDF.K THE T ] LE

HIGHER MATHEMATICS.

First Edition, September, 1896.
Sjecond Edition, January, 1898.
Third Edition, August, 1900.
Fourth Edition, January, 1906.



9/^7



Printed in U. S. A.



PRESS OF

BRAUNWOHTH & CO . INC

eOOK MANUFACTURERS

BROOKLVN. NEW VCrtK



EDITORS' PREFACE.



The volume called Higher Mathematics, the first edition
of which was published in 1896, contained eleven chapters by
eleven authors, each chapter being independent of the others,
but all supposing the reader to have at least a mathematical
training ecjuivalent to that given in classical and engineering
colleges. The pubHcation of that volume is now^ discontinued
and the chapters are issued in separate form. In these reissues
it will generally be found that the monographs are enlarged
by additional articles or appendices which either amplify the
former presentation or record recent advances. This plan of
publication has been arranged in order to meet the demand of
teachers and the convenience of classes, but it is also thought
that it may prove advantageous to readers in special lines of
mathematical literature.

It is the intention of the publishers and editors to add other
monographs to the series from time to time, if the call for the
same seems to warrant it. Among the topics which are under
consideration are those of eUiptic functions, the theory of num-
bers, the group theory, the calculus of variations, and non-
Euchdean geometry; possibly also monographs on branches of
astronomy, mechanics, and mathematical physics may be included.
It is the hope of the editors that this form of pubhcation may
tend to promote mathematical study and research over a wider
field than that which the former volume has occupied.

December, 1905.



742995



AUTHOR'S PREFACE.



This compendium of hyperbolic trigonometry was first published
as a chapter in Merriman and Woodward's Higher Mathematics.
There is reason to believe that it supplies a need, being adaj)ted to
two or three ditTerent types of readers. College students who have
had elementary courses in trigonometry, analytic geometry, and differ-
ential and integral calculus, and who wish to know .something of the
hyperbolic trigonometry on account of its important and historic rela-
tions to each of those branches, will, it is hoped, find these relations
presented in a simple and comprehensive way in the first half of the
work. Readers who have some interest in imaginaries are then intro-
duced to the more general trigonometry of the complex plane, where
the circular and hyperbolic functions merge into one class of transcend-
ents, the singly periodic functions, having either a real or a pure imag-
inary period. For those who abso wish to view the subject in some of
its practical relations, numerous applications have been selected so as
to illustrate the various parts of the theory, and to show its use to the
physicist and engineer, appropriate numerical tables being supplied for
these purposes.

With all these things in mind, much thought has been given to the
mode ot approaching the subject, and to the presentation of funda-
mental notions, and it is hoped that some improvements are discerni-
ble. For instance, it has been customary to define the hyperbolic
functions in relation to a sector of the rectangular hyperbola, and to
take the initial radius of the sector coincident with the principal radius
of the curve, in the present work, these and similar restrictions are
discarded in the interest of analogy and generality, with a gain in sym-
metry and simplicity, and the functions are defined as certain charac-
teristic ratios belonging to any sector of any hyperbola. Such defini-
tions, in connection with the fruitful notion of correspondence of points
on comes, lead to simple and general proofs of the addition-theorems,
from which easily follow the conversion-formulas, the derivatives, the
Maclaurin expansions, and the ex{)onential expressions. The proofs
are .so arranged as to apj)ly equally to the circular functions, regarded
as the characteristic ratios belonging to any elliptic sector. For th(j.se,
however, who mav wish to start with the exponential expressions as
the definitions of the hyperl)olic functions, the appropriate order of
procedure is indicated on page 25. and a direct mode of l)ringing such
exponential definitions into geometrical relation with the hvperbolic
sector is shown in the Appendix.

December. n)Oz,.



CONTENTS„



Art. I. Correspondence of Points on Conics Page ?

2. Areas of Corresponding Triangles g

3. Areas of Corresponding Sectors 9

4. Characteristic Ratios of Sectorial Measures . . . . 10

5. Ratios Expressed as Triangle-measures 10

6. Functional Relations for Ellipse 11

7. Functional Relations for Hyperbola 11

8. Relations between Hyperbolic Functions 12

9. Variations of the Hyperbolic P'unctions .,..,., 14

10. Anti hyperbolic Functions . . .16

11. Functions of Sums and Differences 16

12. Conversion Formulas ,18

13. Limiting Ratios . . 19

14. Derivatives of Hyperbolic Functions 20

15. Derivatives of Anti-hyperbolic Functions 22

16. Expansion of Hyperbolic Functions 23

17. Exponential Expressions 24

18. Expansion of Anti-functions 25

19. Logarithmic Expression of Anti-functions 27

20. The Gudermanian Function 28

21. Circular Functions of Gudermanian 28

22. Gudermanian Angle 29

23. Derivatives of Gudermanian and Inverse .... -30

24. Series for Gudermanian and its Lnverse 31

25. Graphs of Hyperbolic Functions ^2

26. Elementary Integrals ... ... 3^

27. Functions of Complex Numbers . . 38

28. Addition Theorems for Complexes .,..,.. 40

29. Functions of Pure Imaginaries . .41

30. Functions of x+ty in the Form X ^iV 43

31. The Catenary' .... . . 47

32 The Catenary of Uniform Strength . . 49

33. The Elastic Catenary 50

34. The Tr.actory . . 51

35. The Loxodrome .....,..,,,, . 52



6 CONTENTS.

Art. 36 Combined Flexure and Tension 53

37. Alternating Currents 55

38. Miscellaneous Applications 60

39. Explanation of Tables 62

Table I. Hyperbolic Functions 64

II. Values of cosh {x^iy) and sinh (x+iy) 06

III. Values of gdu and 0'^ 70

IV. \'ALUES of gdw, LOG SINH U, LOG COSH U 70

Appendix. Historical and Bibliographical 71

Exponential Expressions as Definitions .... 72



Index



73



HYPERBOLIC FUNCTIONS.



Art. 1. CORRESPONDENXE OF POINTS ON CONICS.

To prepare the way for a general treatment of the hyper-
bolic functions a preliminary discussion is given on the relations
between hyperbolic sectors. The method adopted is such as
to apply at the same time to sectors of the ellipse, including
the circle; and the analogy of the hyperbolic and circular
functions will be obvious at every step, since the same set of
equations can be read in connection with either the h}'perbola
or the ellipse.* It is convenient to begin with the theory of
correspondence of points on two central conies of like species,
i.e. either both ellipses or both hyperbolas.

To obtain a definition of corresponding points, let (9,/4,,
0J\ be conjugate radii of a central conic, and O^A^, O^B^
conjugate radii of any other central conic of the same species;
let /'j , /*, be two points on the curves; and let their coordi-
nates referred to the respective pairs of conjugate directions
be (^, , J',), (.1', , J',); tlien, by analytic geometry,

*The hyperbolic functions are not so named on account of any analogy
with what are termed Elliptic Functions. " The elliptic integrals, and thence
the elliptic functions, derive their name from the early attempts of mathemati-
cians at the rectification of the ellipse. ... To a certain extent this is a
disadvantage; . . . because we employ the name hyperbolic function to de-
note cosh M sinh «, etc., by analogy with which the elliptic functions would be
merely the circular functions cos


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Online LibraryJames McMahonHyperbolic functions → online text (page 1 of 6)