9Ji'!;i:'il|'!iU:|!iiii il!!:i;:ii!;!^l?iii.!:i c 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. 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(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 nding areas it is convenient to adopt the following use of the word " measure": The measure of an\' area connected with a given central conic is the ratio which it bears to the constant area of the triangle formed by two conjngite diameters of the same conic. i'^or example, the measure of the sector yl^OJ\ is the ratio sector A,0,P^ triangle .r/?^, AREAS OK CORRESPONDING SECTORS. 9 and is to be regarded as positive or negative according as A^OJ^^ and A^O^B^ are at the same or opposite sides of their common initial Hne. Art. 2. Areas of Cokresponding Triangles. The areas of corresponding triangles have equal measures. For, let the coordinates of P^, Q^ be (-f , , Ji), (-t'/, j/)' ^^^^ ^^t those of their correspondents/',, Q^ be (;f,, jj, (-^V- j/); let the tY\a.ng\cs P^O.Q,, I\O^Q^hc Z, , T^, and let the measuring tri- angles A^0^6\, A^O^Bj be A',, A',, and their angles &?, , a\] then, by analytic geometr}% taking account of both magnitude and direction of angles, areas, and lines, T, ^ i{A\v!- x/y,) sin a?, ^ .r, j/ _ .r/ y^^ 7^ ^ i.rj'/-,r/j,) sin (k?^ _x^ j/ _ £^'_^^ A', i^?/, sin &7, rt, <^, tf, b^ T T Therefore, by (2), -f = — \ (3) Art. 3. Areas of Corresponding Sectors. The areas of corresponding sectors have equal measures. For conceive the sectors 5,, 5, divided up into infinitesimal corresponding sectors : then the respective infinitesimal corre- sponding triangles have equal measures (Art. 2); but the given sectors are the limits of the sums of these infinitesimal triangles, hence 5, 5, In particular, the sectors A^O.P^, A^O^P^ have equal m.eas- ures ; for the initial points A^, A^ are corresponding points. It may be proved conversely by an obvious reductio ad absurdum that if the initial points of two equal-measured sectors correspond, then their terminal points correspond. Thus if any radii 0,A^, O^A^ be the initial lines of two equal-measured sectors whose tei-minal radii are O^P^, O^P^^ 10 HYPERBOLIC FUNCTIONS. then /",, P^ are corresponding points referred respectively to the pairs of conjugate directions 6^,^,, (9,Z),, and O.^A^, 0^B^\ tliat is, •^ _ ^2 y^ ^yj_ a^ ~ a,' d, ~ b: Prob. I. Prove that the sector P^O^Q^ is bisected by the Hne joining O^ to tlie mid-point of P ^Q^^. (Refer the points P ^, Q^, re- spectively, to the median as common axis of .v, and to the two opposite conjugate directions as axis of y, and show that P^, Q^ are then corresjjonding points.) Prob. 2. Prove that the measure of a circuhTr sector is equal to the radian measure of its angle. Prob. 3. Find the measure of an elliptic quadrant, and of the sector included by conjugate radii. Art. 4, Char.acteristic Ratios of Sectorial Measures. Let A^O^P^ = 5, be any sector of a central conic; draw P^M, ordinate to 0,A^, i.e. parallel to the tangent at A^; let 0,AI, = -r,, J/,/^, =J\, O^A^ = «> tanh//. (14) 14 HYPERBOLIC FUNCTIONS. Prol). 7. Express all the hyperbolic functions in terms ot sinh u. Given cosh u = 2, find the values of the other functions. Prob. 8. Prove from ecjs. 10, 11, that coslw/> sinh//, cosh//'>i, tanh // < I, sech // < i. Prob. 9. In the figure of Art. i, let OA — 2, OB=i, AOB — 60", and area of sector AOP = 3; find the sectorial measure, and the two characteristic ratios, in the elliptic sector, and also in the hyper- bolic sector; and find the area of the triangle AOP. (Use tables of cos, sin, cosh, sinh.) Prob. 10. Show that coth ii, sech le, csch u may each be ex- pressed as the ratio of two lines, as follows: Let the tangent at P make on the conjugate axes OA, OB, intercepts OS = w, OT — n\ let the tangent at B, to the conjugate hyperbola, meet OP in R^ making BR = /; then coth // = //- (18) tanh (— ii) = — tanh u, coth (— //) = — coth //. j Prob. 12. Trace the changes in sech //, coth 7/, csch //, as 1/ passes from — CO to + 00 . Show that sinh u, cosh // are infinites of the same order when u is infinite. (It will appear in Art. 17 that sinh u, cosh t/ are infinites of an order infinitely higher than the order ofu.) Prob. 13. Applying eq. (12) to figure, page 14, prove tanh i/, = tan A OF. 16 Hyperbolic functions. Art. 10. Anti-hyperbolic Functions. X y ■ ^ Tlie equations - = cosh u, -j = sinh //, 7 = tanli u, etc., * a a b may also be expressed by the inverse notation ?^ = cosh"^ — , _ y t u = sinh ^-7, u ^= tanh '— , etc., which may be read: " ;^ is the sectorial measure whose hyperbolic cosine is the ratio x to «," etc. ; or " u is the anti-h-cosine of x/a'' etc. Since there are two values of 7i, with opposite signs, that correspond to a given value of cosh u, it follows that if u be determined from the equation cosh ti = m, where m is a given number greater than unity, u is a two-valued function of w. The symbol cosh ' m will be used to denote the positive value of // that satisfies the equation cosh u — vi. Similarly the symbol sech"* vi will stand for the positive value of 11 that satisfies the equation sech 21 = ;//. The signs of the other functions sinii"'w, tanh"';;/, coth~' ;;/, csch"' ;;/, are the same as the sign of ;;/. Hence all of the anti-hyperbolic functions of real numbers are one-valued. Prob. 14. Prove the following relations: cosh"';// = sinh"' V m^ — i, sinh"';;/ = ± cosh'' V;;/" -j- i, 'he. upper or lower sign being used according as ;;/ is positive or negative. Modify these relations for sin "' , cos"' . Prob. 15. In figure, Art. i,let OA = 2,0B = i,AOB = 60°; find the area of the hyperbolic sector A OP, and of the segment AMP, if the abscissa of P is 3. (Find cosh"' from the tables for cosh.) Art. 11. Functions of Sums and Differences. (a) To prove the difference-formulas sinh (// — 7') = sinh // cosh t> — cosh // sinh 7>, ) ( (19) cosh (7/ — 7') = cosh ;/ cosh 7' — sinh // sinh 1'. ) Let OA be any radius of a hyperbola, and let the sectors AOP, AOQ have the measures //, v\ then // — v is the measure of the sector QOP. Let OB, OQ' be the radii conjugate to OA, OQ; and let the coordinates of P, Q, Q' be (^, ,J,), i-^, y), (^'> j') with reference to the axes OA, OB; then FUNCTIONS OF SUMS AND DIFFERENCES. 17 . , , , . , sector (9(9/^ trianHe (9(9/' .. ^ Sinn {ti — V) = siiih -^ — ■ = ^ — [Art. 5. ii^Ji— -i'jO sin 00 j\ X y x^ ^aj)^ sin 00 b^ a^ b^ a^ = sinh u cosh v — cosh ii sinh v\ cosh (?^ — ■z') = cosh sector OOP trianorle POO' K K K't'y-^.'O sin 03 _ y' ,r, y^x' [Art. 5. but 2,aJ^^ sin gj? y' X — > X' _y a. b! b. a. b. a, ' (20) since Q, Q' are extremities of conjugate radii ; hence cosli {11 — 7') = cosh n cosh v — sinh u sinh v. In the figures 11 is positive and v is positive or negative. Other figures may be drawn with u negative, and the language in the text will apply to all. In the case of elliptic sectors, similar figures may be drawn, and the same language will apply, except that the second equation of (20) will be x' /a^ = — //^,; therefore sin (;/ — v) = sin ?/ cos 2' — cos ?/ sin t>, cos {?( — I') = cos 7/ cos V -\- sin u sin v. (b) To prove the sum-formulas sinh (7/ --\- v) =: sinh u cosh v -{- cosh 7/ sinh ?', cosh (// -j- v) = cosh u cosh t -{- sinh ?/ sinh 2'. These equations follow from (19) by changing v into — v, (21) 18 HYPERBOLIC FUNCTIONS. and then for sinh (— f), cosh (— z'), writing — sinh t^, cosh z> (Art. 9, eqs. (i8)j. (c) To prove that tanh {u ±_ v) = tanh 21 ± tanh v I ±tanh 71 tanh v (22) Writing tanh (u ± v) = ■ -— ^, expanding and dividing cosh {n ± T'j ^ ^ ^ numerator and denominator by cosh ?/ cosh v, eq. (22) is ob- tained. ^ Prob. 16. Given cosh ?/ — 2, cosh v = 3, find cosh (// + v). Prob. 17. Prove the following identities: ^ I. sinh 2// = 2 sinh // cosli ?/. 1^ 2. cosh 2u = cosh'/c + sinh'/^ = i -|- 2 sinh'' // = 2 cosh^ u — i. '■^ 3. I + cosh // = 2 cosh' 4«, cosh /^ — i = 2 sinh^ -^u. sinh // _ cosh ?' — i _ /cosh ?/ — i\* I + cosh // sinh u \cosh // -\- 1/ . , 2 tanh ?/ , T -*- tanh^ « 5. sinh 2U = 4. tanh ^u = cosh 2tt I — tanh'' «' '" 1 — tunh' u' 6. sinh 3/(' = 3 sinh /^ + 4 sinh" u, cosh 3// = 4 cosh'« —3 cosh a. , . , I + tanh ^u 7. cosh u 4- sinh ?/ = , : -. ' I — tanh ^u 8. (cosh u -f- sinh //)(cosh Z' + sinh Z') = cosh {u -\- ?') -f- sinh {u -f 7')- 9. Generalize (8); and show also vvhatrit becomes when u = v^ . . , 10. sinh^v cosj' + cosh^v sin^ = sinhV -\- sin^'j'. 11. cosh"'w ± cosli"';/ = cosh~'Lw« ± V (w' — i)(«'— i)j. 12. sinh"' w ± sinh"'// = sinh''| w y i -f- «' ± ;/ y i + m'j. Prob. 18. What modifications of signs are required in (21), (22), in order to pass to circular functions ? Prob. 19. Modify the identities of Prob. 17 for the same purpose. Art. 12. Conversion Formulas. To prove that cosh 7/,-|- cosh ?{, — 2 cosh ^(//.-f" ''j) cosh K//,— ?/,)» cosh 71,— cosh 7/ J = 2 sinh f(//, -j-^/Jsinh i{7/,— ?/,), sinh 71, -\- sinh //, = 2 sinh ^(//, -f '0 cosh ^u,— ?/,), j shih «. — sinh //, = 2 cosh ^(//, -[" '^) ■'^'"'1 aC'^ — ''''o)- J (23) LIMITING RATIOS. 19 From the addition formulas it follows that cosh {u -\- v) -j- cosh (// — v) = 2 cosh ji cosh v, cosh [h -\- v) — cosh {u — f) = 2 sinh u sinh v, siiih [h -{- v) ~\- sinh {u — 7') = 2 sinh u cosh v^ ■ sinh (// -{-v) — sinh {u — z/) = 2 cosh Ji sinh ?', and then by writing u -\- v = //, , u — v zz^ n^ , u = ^(;/, -f~ ^^)> ^1 = ^(;/, — z/^), these equations take the form required. Prob. 20. In passing to circular functions, show that the only modification to be made in the conversion formulas is in tlie alge- braic sign of the right-hand member of the second formula. _. , ^. ... cosh 2U + cosh AV cosh 2U -\- cosh 4?^ Prob. 21. Simplify -r— ; ; r-; , , : ■• sinh 2U -\- smh 47; cosh 2« — cosh 4^ Prob. 22. Prove sinh^x — sinh^'j^ = sinh (.v -\-y) sinh {x — y). Prob. 23. Simplify cosh^v cosh^j' ± sinhlv sinh'j'. Prob. 24. Simplify cosh^a* cos^>' -f- sinh^x sin'_y. Art. 13. Limiting Ratios. To find the limit, as u approaches zero, of sinh u tanh ii U II which are then indeterminate in form. By eq. (14), sinh ii^ u~> tanh ti ; and if sinh ;/ and tanh Ji be successively divided by each term of these inequalities, it follows that sinh II . I < < cosh u, u . tanh u sech II < < L* ti but when u-^O, cosh u ^ i, sech ti ^ i, hence lim. sinlw/^^^ ][,Ti. tanh u ^ ^ .. u = o u ' u ^o II 20 HYPERBOLIC FUNCTIONS. Art. 14. Derivatives of Hyperbolic Functions. To prove that ^(sinh u) {d) du ^^(cosh //) du \ 3 4! hence cosh u = ^i^" -\- e "), sinh u = ^(r" — e'"), e" — €" . 2 r (30) tanh u = sech 2i = -, etc. The analogous exponential expressions for sin ?/, cos u are cos u = \e"' -i-e^"'), sin u = —{e"' — ^-'"■), {i = V — i) where the symbol r"' stands for the result of substituting 7^z for X in the exponential development This will be more fully explained in treating of complex numbers, Arts. 28, 2p. EXPANSION OF ANTI-FUNCTIONS. tb Prob. 37. Show that the properties of the liyperbolic functions could be placed on a purely algebraic basis by starting with equa- tions (30) as their definitions ; for example, verify the identities : sinh (—//) = — sinh n, cosh (—//) = cosh //, cosh^ u — sinh^ //= 1, sinh {u -(-<') = sinh // cosh v -f cosh u sinh z', ^/^(cosh mil) ^/'"(sinh mu) , . , r-j = m cosh mil, -— = m^ sinh mu. du dll Prob. 38. Prove (cosh 11 -\- sinh 11)" = cosh nu -)- sinh 7iu. Prob. 39. Assuming from Art. 14 that cosh «, sinh u satisfy the differential equation //V/^/«'^ =i', whose general solution may be written y — ^e" + Be'", where yl, B are arbitrary constants ; show how to determine A, B in order to derive the expressions for cosh //, sinh //, respectively. [Use eq. (15).] Prob. 40. Show how to construct a table of exponential func- tions from a table of hyperbolic sines and cosines, and vice versa. Prob. 41. Prove u = log^ (cosh u -\- sinh //). Prob. 42. Sliow that the area of any hyperbolic sector is infinite when its terminal line is one of the asymptotes. Prob. 43. From the relation 2 cosh u — e" -f- e'" prove 2''~'(cosh //)" = cosh //// + ;/ cosh {11 — 2)11 + hi{/i—i) cosh (//— 4)« + ..., and examine the last term when « is odd or even. Find also the corresponding expression for 2""' (sinh 11)". Art. 18. Expansion of Anfi-Functions. <'/(sinh ' -t') _ I Since -'^ ~, — - — - = — = (i 4- x)-^ ax ./- . .2 / • 12,134 1356- = I x^ A-- ~ x' -i I x' -f- 2 24 246 hence, by integration, 23^245 2467^ ^^ ^ the integration-constant being zero, since sinh ' x vanishes with X. This series is convergent, and can be used in compu- 26 HYPERBOLIC FUNCTIONS. tation, only when x < i. Another series, convergent when X > I, is obtained by writing the above derivative in the form 4sinh-' x) , 5 , , . i( , i\'"* 1 1+-^ 1 _L3 5 2 x" 24 x' 246a- '•+•■•]• .,1 ^11 |II I3I1I35I /x .-. sinh-' ^= C+log^H ^ — ^_-^^_-2.1 . (32) ' 2 2.r' 2 4 4,v* 2 4 6 6x' ' where C is the integration-constant, which will be shown in Art. 19 to be equal to log, 2. A development of similar form is obtained for cosh~'.r; for ^(cosh-' x) , , . , I / I \-* dx X\. ^ 2 X'^ 2 AX'^ 2 Afi x'^' S hence I I 4 I 3 I 4 T 3 5 I cosh-;.'=r+log^'--^,--^-,--^^-.-..., (33) in which C is again equal to log, 2 [Art. 19, Prob. 46]. In order that the function cosh"'.i' maybe real, ;ir must not be less than unity; but when x exceeds unity, this series is con- vergent, hence it is always available for computation. Again, '!^':'Jl = _i_, = ,+.' + .-H ^-^ + ... , '^ dx I — X and hence tanh"' x = x -\- - x' -\- -x' -\- ~x' -{-... , (34) 3 5 7 From (32), (33), (34) are derived : .-1 I :c\\~' X = coslr 2,2 2.4.4 2 .4.6.6 (35) LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS. 27 csch-'^ = sinh-i = l-i-i-3 + i.l-L-i3i_L + ..., X X 2 TyX 2 \ ^X" 246 7^' ^ ^2.2 2. 4.4^2.4.6.6 • ' ^-5 ^ coth-' .r = tanh-' l = i4--L-fJ_4. _L + (77) X X ^ ix' ^ t^x- ^ 7x' ^ ^^^^ Prob. 44. Show that the series for tanh" Vx", coth~* .r, sech~* jr, are always available for computation. Prob. 45. Show that one or other of the two developments of the inverse hyperbolic cosecant is available. Art. 19. Logarithmic Expression of Anti Fun'ctions. Let X = cosh //, then Vx'' — i = siiih u\ therefore x -\- Vx^ — i = cosh u -\- sinh // = e", and u, = cosh"*.r, = log (x -|- \^x'' — i). (38) Similarly, sinli"*^ = log (^x -\- Vx'' -\- i). (39) Also sech"'.r = cosh"*- = log — — -, (40) 1-1 • t -ii I I -{- V\ A- x^ / , csch ^x = sinh - = log — ■ •. (41) X X Again, let x = tanh u = ^" + ^-"' therefore — — = -^ = e , I — X e' 2u =.\o%-^^ — , tanh ^x^^\ log "*" ; (42) I X -A- I and coth~';f = tanh"'- = \ log — ! — . (43) X X — I Prob. 46. Show from (38), (39), that, when .r^ 00, sinh~'jc — log :V:^ log 2, cosh"'jc - log x -i log 2, and hence show that the integration-constants in (32), {^^) are each equal to log 2. 28 HYPERBOLIC FUNCTIONS. Prob. 47. Derive from (42) the series for tanh '.v given in (34). Prob. 48. Prove the identities: logA- = 2 tanh"'" =:tanh'* — — =sinh'M(j:— jc"')=cosh"H(A- + A-"'): a- + 1 x' + i " " ^ " log sec .r = 2 tanh"' tan" ^x; log esc x = 2 tanh' ' tan'(j7r -|- -kx); log tan .r = — tanh'' cos 2X = — sinh"' cot 2x = cosh"' esc 2X. Art. 20. The Gudermanian Function. The coirespondence of sectors of the same species was dis- cussed in Arts. 1-4. It is now convenient to treat of the correspondence that may exist between sectors of different species. Two points P^,P^, on any h)-perbola and enipse,are said to correspond with reference to two pairs of conjugates O^A^, O^B^ , and O^A^, O^B^, respectively, when -t'i/^> = ^^/'^'.> (44) and when J, jjj/j have the same sign. The sectors A^O,P., A^O^P^ are then also said to correspond. Thus corresponding sectors of central conies of different species are of the same sign and have their primary characteristic ratios reciprocal. Hence there is a fixed functional relation between their re- spective measures. The elliptic sectorial measure is called the gudermanian of the corresponding hyperbolic sectorial measure, and the latter the anti-gudermanian of the former. This relation is expressed by SJK, = gd SJK, or z> =■ gd //, and 11 = gd"'t'. (45) Art. 21. Circular Functions of Gudermanian. The six hyperbolic functions of 11 are expressible in terms of the six circular functions of its gudermanian ; for since — = cosh u, — = cos 7', (see Arts. 6, 7) in which //, t- are the measures of corresponding h)-perbolic and elliptic sectors, hence GUDERMANtAN ANGL£. cosh u = sec z>, [eq. (44)] sinh H = v'secV — i = tan 7', 29 (46) / tanh u = tan t'/sec v = sin v, COth H = CSC v, sech // = COS7', csch u = cot ^'. The gudei-maiiian is sometimes useful in computation ; for instance, if sinh u be given, i' can be found from a table of natural tarigents, and the other circular functions of z' will give the remaining hxperbolic functions of //. Other uses of this function are given in Arts. 22-26, 32-36. Prob. 49. Prove that gd u — sec~'(cos]i ti) = tan" '(sinh u) ^ = COS"" '(sech u) = sin" '(tanh u), Prob. 50. Prove gd "' Z' = cosh" '(sec z') = sinh"' (tan z') = sech"'(cos z') = tanh" '(sin ?'). Prob. 51. Prove gd o = o, gd 00 = ^-;r, gd(— 00 ) = — ^^r. gd"' 0=0, gd~'(4;r) =00, gd '(— ^/7)= — 00. Prob 52. Show that gd // and gd" ' z' are odd functions of //, z'. Prob. 53. From the first identity in 4, Prob. 17, derive the rela- tion tanh iu — tan ^z'. J Prob. 54. Prove tanh" '(tan //) = 4 gd 21/, and tan" '(tanh x) = 4 gd"'2A-. Art. 22. Gudermanian Angle If a circle be used instead of the ellipse of Art. 20, the gudermanian of the hyperbolic sectorial measure will be equal to the radian measure of the angle of the corresponding circular sector (see eq. (6), and Art. 3, Prob. 2). This angle will be called the gudermanian angle ; but the gudermanian function z', as above defined, is merely a number, or ratio ; and this number is equal to the radian measure of the gudermanian angle 6, which is itself usually tabulated in degree measure ; thus 6 = i8o°t'/n- (47) :^o HYPERBOLIC FONCTlONS. Prob. 55. Show that the gudermanian angle of// may be construct- ed as follows: Take the principal radius OA of an equilateral hyperbola, as the initial line, and OP as the terminal line, of the sector whose measure is u\ from M, the foot of the ordinate of P, draw MT tangent to the circle wliose diameter is the transverse axis; then A07'\?> the angle required.* Prob. 56. Show that the angle B never exceeds 90°. Prob. 57. The bisector of angle AOT bisects the sector AOP (see Prob. 13, Art. 9, and Prob. 53, Art. 21), and the line AP. (See Prob. i, Art. 3.) Prob. 58. This bisector is parallel to TP, and the points 7', /* are in line with the point diametrically opposite to A. Prob. 59. The tangent at '' passes through the foot of the oidinate of T, and intersects TM ow the tangent at A. Prob. 60. The angle AP M is half the gudermanian angle. Art. 23. Derivatives of Gudermanian and Inverse. Let V = gd u, 71 ■= gd~' z/, then sec v = cosh u, sec V tan vdv = sinh n du, sec 7'di> = ////, therefore ^(gd"' 7-) = sec 7> dv. (48) Again, //t' = cos ■:> di/ =: sech 71 dn, therefore //(gd 7/) — sech // du. (49) Prob. 61. Differentiate: y = sinh // — gd //, y = sin ?• + gd~' 7>, y = tanh // sech // + gd //, y = tan 7' sec v -\- gd~' v. *This angle was called by Gudermann the longitude of u. and denoted by lu. His inverse symbol was li, ; thus « = ILU")- (Crelle's Journal, vol. 6, 1S30.) Lambert, who introduced the angle 5, named it the transcendent angle. (Hist, de I'acad roy de Kerlin, 1761). Hoiiel (Nouvelles Annales, vol. 3, 1864) called it the hvperbolic amplitude of //, and wrote it amh n, in analogy with the amplitude of an elliptic function, as shown in Prob. 62. Cayley (Elliptic Functions. 1876) made the usage uniform by attaching to the angle the name of the mathemaiician who had used it extensively in tabulation and in the theory of elliptic functions of modulus unity., SERIES FOR GUDERMANIAN AND ITS INVERSE. 31 Prob. 62. Writing the "elliptic integral of the first kind" in the form J Vi — K^ sin''' 0' X" being called the modulus, and (p the amplitude; that is, = am //, (mod. k), show that, in the special case when k = i, u = gd~^ 0, am It = gd u, sin am u = tanh «, cos am // = sech ?/, tan am // = sinh //; and that thus the elliptic functions sin am //, etc., degenerate into the hyperbolic functions, when the modulus is unity.* Art. 24. Series for Guderm.\nian and its Inverse. Substitute for sech //, sec t' in (49), (48) their expansions, Art. 16, and integrate, then gd ;/ = ;/- iu' + ^\u' - ^^^^//' + . . . (50) gd-'z' = v + Iz-' + ^V^.^ +^tio^'' + • . . (51) No constants of integration appear, since gd u vanishes with u, and gd'^z> with 7>. These series are seldom used in compu- tation, as gd u is best found and tabulated by means of tables of natural tangents and hyperbolic sines, from the equation gd !( = tan~'(sinh n), and a table of the direct function can be used to furnLsh the numerical values of the inverse function ; or the latter can be obtained from the equation, gd"'z^ = sinh "'(tan Z') = cosh~'(sec z'). To obtain a logarithmic expression for gd"':', let gd""'t^ = u, z' = gd i(, * The relation gd u — am u, (mod. i), led Hoiiel to name the function gd u, the hyperbolic amplitude of m, and to write it amh // (see note, Art. 22). In this connection Cayley expressed the functions tanh «, sech u. sinli u in the form sin gd u, cos gd u. tan gd u, and wrote them sg «, eg u, tg tt, to correspond with the abbreviations sn u, en u, dn u for sin am it, cos am «. tan am u. Thus tanh « = sg « = sn u, (mod. i); etc. It is well to note that neither the elliptic nor the hyperbole functions received their names on account of the relation existing between them in a special case. (See foot-note, p. 7 ) 32 HYPERBOLIC FUNCTIONS. therefore sec v = cosli ?/, tan v = sinh u, sec V -f- tan v = cosh u -\- sinh u =■ e", I -\- sin V _i — cos (^;r -|- z/) e" = cos t^ sin {^rr -\- v) 21, = gd 'v, = log, tan (i^ + |t'). tan (iTT + ^j;), (52) Prob. 6^. Evaluate gd u — u gd 'z' — v' Prob. 64. Prove that gd u — sin u is an infinitesimal of the fifth order, when // = o. Prob. 65. Prove the relations Itt + h'= tan"V', i^r — h; — tan"V~". Art. 25. Graphs of Hyperbolic Functions. Drawing two rectangular axes, and laying down a series of points whose abscissas represent, on any convenient scale, suc- cessive values of the sectorial measure, and whose ordinates represent, preferably on the same scale, the corre- sponding values of the function to be plotted, the locus traced out by this series of points will be a graphical representation of the variation of the func- tion as the sectorial meas- GRAPHS OF THE HYPERBOLIC FUNCTIONS. 33 ure varies. The equations of the curves in the ordinary carte- sian notation are : Fig. Full Lines. Dotted Lines. A y =^ cosh X, y = sech x ; B y = sinh x, y = csch x ; C y ^ tanh x, y = coth x ; D ^ = gd X. Here x is written for the sectorial measure //, and j for the numerical value of cosh li, etc. It is thus to be noted that the variables x, y are numbers, or ratios, and that the equation y = cosh X merely expresses that the relation between the numbers x and j is taken to be the same as the relation be- tween a sectorial measure and its characteristic ratio. The numerical values of cosh u, sinh u, tanh u are given in the tables at the end of this chapter for values of u between o and 4. For greater values they may be computed from the devel- opments of Art. 16. The curves exhibit graphically the relations : sech u = — : — , csch 7f = -— - — , coth u cosh !(' sinh u tanh //' cosh u < I, sech u > i, tanh // > i, gd ;/ <^;r, etc. ; sinh (— !/) = — sinh u, cosh (— //) = cosh //, tanh (— «) = — tanh ?/, gd {— ?/) = — gd //, etc.; cosh 0=1, sinh = 0, tanh = 0, csch (o) =00 , etc.; cosh (i 00 ) = CO , sinh (it 00 ) = ^oo , tanh (± 00 ) = ± i, etc. The slope of the curve j' = sinh x is given by the equation dy/dx = cosh x, showing that it is always positive, and that the curve becomes more nearly vertical as x becomes infinite. Its direction of curvature is obtained from d'^y/dx'^ — sinh x, proving that the curve is concave downward when x is nega- tive, and upward when x is positive. The point of inflexion is at the origin, and the inflexional tangent bisects the angle between the axes. 34 HYPERBOLIC FUNCTIONS. The direction of curvature of the locus j = sech x is given by dy/dx' — sech x{2 tanh'jr — i), and thus the curve is con- cave downwards or upwards according as 2 tanh' ^ — i is negative or positive. The in- ""'~ flexions occur at the points X = ± tanh-'.707, = ± .881, y = .707 ; and the slopes of the inflexional tangents are =Fi/2. The curve y = csch x is asymptotic to both axes, but approaches the axis of x more rapidly than it approaches the axis of )', for when x := 3, j is onh' .1, but it is not till _j' = 10 -I - that X is so small as .T, The curves j' cross at the points ^ = ± .881, j = ± i. csch X, y = sinh x Prob. 66. Find the direction of curvature, the inflexional tan- gent, and tlie asymptotes of the curves jr = gd .v, v — tanh .v. Prob. 67. Show that there is no inflexion-point on the curves y z= cosh X, y = coth x. Prob. 68. Show that any line _v = mx + // meets the curve y = tanh x in either three real points or one. Hence prove that the equation tanh x = f/ix -(- n has either three real roots or one. From the figure give an approximate solution of the equation tanh .V = .V — i. ELEMENTARY INTEGRALS. 35 Prob. 69. Solve the equations: cosh ;v — ■ .v -(- 2; sinh x = ^x; gd X = X — ^TT. Prob. 70. Show which of the graphs represent even functions, and which of them represent odd ones. Art. 26. Elementary Integrals. The following useful indefinite integrals follow from Arts. 14. 15. 23: Hyperbolic. Circular. 1. / sinh It du = cosh //, / sin 11 dii = — cos u, 2. I cosh 11 dti = sinh u, j cos ?/ du = sin k, 3. / tanh u du = log cosh u, I tan u du = — log cos //, 4. / coth ;/ du = log sinh u, j cot // du = log sin u, 5. y^csch udu = log tanh - , /esc u du = log tan -, = — sinh-'(csch //), = — cosh-'(csc u), 6. Aech u du = gd u, I sec u du = gd-' u, 7. / = sinh- -/ ./ , , = sin-' -, r dx , X- r —dx ,x 8. / , = cosh-' - , / COS" Q. / -^ i =-tanh-'-, / -,— — I =-tan- — , * Forms 7-12 are preferable to the respective logarithmic expressions (Art. 19), on account of the close analogy with the circular forms, and also because they involve functions that are directly tabulated. This advantage appears more clearly in 13-20. 36 HYPERBOLIC FUNCTIONS. lO. I I. /—dx ~| I ^ X r — dx I .X ~ i =-coth-'-, / = — cot"' — a ^ a -\- X a a 12 r — dx I , , t' /* dx I / — — — sccn~ — / — ^i: — «/ ;r 4/^^ _ x' a a' ^ X s/x" -a' a X P — dx I see" ./ = - csch-' X Vd' -\- x' a a = — csc~ — , a' ^ X \ ' x^ - d^ a a From these fundamental integrals the following may be derived : .3./ dx I ax -\- b = — ^ sinh~ —- , tf positive, tfr> 3"; Vax' -j-2dx-\-c Va Vac- d' I , , ax 4-/7 = — =cosh ^ — , ^? positive, ^f b'; Vb — ac Vb —ac ' — I , ax 4- b , coth-' ,^--^ , ac < /;^ .?a- + ^^ > Vb' - ac ; = — coth--'(^-— 2) Vb'-ac Inus, / — 1 j— ',./ ^'-4^-4-3 = tanh-'(.5)-tanh-'(.3333) = . 5494-. 3466=. 2028,* :coth-'2 — coth-'3 / - , ^'^ , =-tanh-'(x-2) =tanh-'o-tanh-'(.5) t/g ;ir —/\x-\-l A-i = - •5494- (By interpreting these two integrals as areas, show graph- ically that the first is positive, and the second negative.) 5. C ' "^ , = —^^^^ tanh-' K /- J {a-x\Vx-b Va-b V' a-F *For tanh-' (.5) interpolate between tanh (.54) = .4930, tanh (.56) = .5080 (see tables, pp. 6^, 65); and similarly for tanh-' (.3333). ELEMENTARY INTEGRALS. 37 / '^-^ 2 Ix — b tan ~ \ —, , or — , cotn~ \'b-a V ^'-(^ Va-b V '^-f^ the real form to be taken. (Put x — b = s", and apply 9, 10.) ^ r dx 2 b—x 16. /; --^==:— ==tanh-'A /t t/ {a—x\ Vb—x {a—x)\U-)—x \'b — a \l b-a' 2 j b — x —2 Ib—x or ,- coth" A / -, , or — — - tan "' a / ; ; \/b—a V '^'-^ \^a-b V ^~^ the real form to be taken. (.1-' — rt-')-^/,;- = --t'(^r — <'?')^ ^/' cosh-'-. By means of a reduction-formula this integral is easily made to depend on 8. It may also be obtained by transforming the expression into hyperbolic functions by the assumption X = a cosh u, when the integral takes the form rt^ / sinh' udu=z — / (cosh 2u — \)du = -f^^(sinh 2u — 211) = |rt'(sinh u cosh u — ii), which gives 17 on replacing a cosh 71 by .r, and a sinh u by (,t"' — rt^)i. The geometrical interpretation of the result is evident, as it expresses that the area of a rectangular-hyper- bolic segment AMP is the difference between a triangle OMP and a sector OAP. 18. J^{a' - .x'fdx = ~x{a' - x'f + ~a' sin"' -. 19. fix' + a'fdx = ^.r(.r' + a^f -f -^a^ sinh"' ^. 20. / sec' 0^/0 = / (I -[- tan- 0)v/ tan = ^ tan 0(1 4- tan' 0)^ -\- ^ sinh"' (tan 0) = ^ sec tan 0+2 gd"' 0. 21. / sech'?/ b\ Prob. 73. Prove / sinh u tanh // du — sinh u — gd //, J t 2/ cosh u coth // (/u = cosh u -\- log tanh — . 2 Prob. 74. Integrate (a-' + 2-v + 5)-W, (a-'^ + 2.v + 5)-VA-, Cv^ + 2x + s)V>. Prob. 75. In the parabola ^ = 4px, if j- be the length of arc measured from the vertex, and (p the angle which the tangent line makes with the vertical tangent, prove that the intrinsic equation of the curve is ds/d

/ = O. The meanings assigned should also, if possible, be such as to permit the addition-formulas of Art. 1 1 to be made general, with all the consequences that flow from them. Such definitions are furnished by the algebraic develop- ments in Art. i6, which are convergent for all values of u, real or complex. Thus the definitions of cosh {x -\- ij), sinh [x -f- iy) are to be cosh {x + iy) = I + ^{x + tyy + l-(x + iyy + 2 ! 4 • sinh {x + /» = {x + iy) -f -(.r + /jf + (52) From these series the numerical values of cosh {x -{- iy), sinh {x-\-iy) could be computed to any degree of approxima- tion, when X and 7 are given. In general the results will come out in the complex form* cosh {x -f- iy) = a-\- ib, sinh (.V -|- iy) = c -\- id. The other functions are defined as in Art. 7, eq. (9). Prob. 79. Prove from these definitions that, whatever u may be, cosh (—//) = cosh u, ■ sinh (—//)=— sinh «, , lilt cosh // = sinh //, du sinh u = cosh u, 7 2 72 ^^cosh mil = ;;/' cosh w//, j-^ sinh w// = w' sinli /////.f du du' *It is to be borne in mind that the symbols cosh, sinh, here stand for alge- braic operators which convert one number into another; or which, in the lan- guage of vector-analysis, change one vector into another, by stretching and turning. f The generalized hyperbolic functions usually present themselves in Mathe- matical Physics as the solution of the differential equation d''(p/dn'^ = fi^, w, u are complex numbers, the measures of vector quantities. (See Art. 37.) 40 HYPERBOLIC FUNCTIONS. Art. 28. Addition-Theorems eor Complexes. The addition-theorems for cosh {/i -\- 7'), etc., where 7i, v are complex numbers, may be derived as follows. First take u,v as real numbers, then, by Art. Ii, cosh {h -\- v) — cosh 7c cosh v -\- sinh u sinh v, hence I + ^',(» + r)' +, ..=(.+ ^W + ...)(. + ^^'+. . .) + („ + _L^„. + ...)(„+±y+...) This equation is true when n, v are any real numbers. It must, then, be an algebraic identity. For, compare the terms of the rt\\ degree in the letters //, z' on each side. Those on the left are — (/^-|- t')'; and those on the right, when collected, form an rth-degree function which is numerically equal to the former for more than r values of // when v is constant, and for more than r values of v when u is constant. Hence the terms of the rth degree on each side are algebraically identical func- tions of // and z'.* Similarly for the terms of any other degree. Thus the equation above written is an algebraic identity, and is true for all values of u, v, whether real or complex. Then writing for each side its symbol, it follows that cosh {u -\- 7') = cosh ;/ cosh 7' -|- sinh ii sinh v\ (53) and by changing 7' into — 7', cosh {h — 7') — cosh // cosh v — sinh // sinh 7'. (54) In a similar manner is found sinh {u ± 1') = sinh u cosh v ± cosh 71 sinh v. (55) In particular, for a complex argument, cosh (x ± ij) = cosh x cosh ij' ± sinh x sinh /)', ) [ (56) sinh {x ± /r) = sinh x cosh ly ± cosh x sinh ?j'. ) * " If two ;'lh-degree functions of a single variable be equal for more than r values of the variable, then they are equal for all values of the variable, and are algebraically identical." fUNCTlONS OF PURE IMAGINARIES. 41 Prob. 79. Show, by a similar process of generalization,* that if sin //■, cos //, exp ti \ be defined by their developments in powers of ti, then, whatever u may be, sin (// -\- v) ^^ sin u cos v + cos u sin z', cos (// -|- ^') = cos // cos V — sin // sin v, ^-"^ exp (/^ -\- 7') = exp // exp ?'. Prob. 80. Prove that the following are identities: cosh'' // — sinh* /^ = i, cosh // -f- sinh // = exp //, cosh // — sinh u = exp ( — //), cosh // = o[exp // 4" tx]:) ( — //)], sinh // = i[exp // — ex])(— //)]. Art. 29. Functions of Pure Imaginaries. In the defining identities cosh ?( = !-[- ~ii^ A -//* -I- . . ., 2! 4! ' ' sinh 11 ■= 21 -\ — -^11' -J — /'^ -f- . . ., 3- 5- put for // the pure iniaginary //, then cosh iy ^ \ — --/ -I- - / - . . . = COS7, (57) z. 4 sinh iy = iy ^ -,(/»' + -,(?»' + . . . {/-^y+^/ = /sin/, (58) and, by division, tanh iy = / tan y. (59) * This method of generalization is sometimes called the principle of the " permanence of equivalence of forms." It is not, however, strictly speaiving, a " priiiciple," but a method; for, the validity of the generalization has to be demonstrated, for any particular form, by means of the principle of the alge- braic identity of polynomials enunciated in the preceding foot-note. (See Annals of Mathematics, Vol. 6, p. 81.) f The symbol exp u stands for "exponential function of u," which is identi- cal with e'< when it is real. 4^ HYPERBOLIC FUNCTIONS. These formulas serve to interchange hyperbohc and circular functions. The hyperbolic cosine of a pure imaginary is real, and the ii\-perbolic sine and tangent are pure imaginaries. The following table exhibits the variation of sinh u, cosh u, tanh II, exp u, as u takes a succession of pure imaginary values. tt sinh u cosh u tanh u exp u o O I I \^^ .yi .7* i •7(1+0 ^171 i CO / i lirr .7/ -•7 — i •7(1 - i) — I — I \iTt -.ji -.7 i -.7(1+0 — i pTT — i 00 /' liTt -.7i •7 — i -.7(1-0 2iit 1 I * In this table .7 is written for \ \/i, = .707 . . Prob. 81. Prove the following identities : cos J = cosh /)■ = i>[exp /)■ + exp (— /v)], sin J' = - sinh /)' = -[exp iy — exf) (— /v)], cos }' -\- i sin y = cosh iy + sinh /)' = exp iy, cos^y — / sin_)' = cosh iy — sinh iy = exp (— ^V), cos iy = cosh y, sin iy = i sinh v. Prob. 82 Equating the respective real and imaginary parts on each side of the equation cos ny -f i sin f/y = (cos y + i sin _>')", express cos //y in powers of cos_v, sin v ; and lience derive the cor- responding expression for cosh ny. Prol). 83. SIiow that, in the identities (57) and (58), y may be replaced by a general complex, and hence that sinh (x ± iy) = ± i sin {y T /v), FUNCTIONS OF .V -f /)' IN THE FORM A' -(- I V. 43 cosh (.v ± iv) = COS ( v ^ is), sin {x ± /)■) = ± /sinh (r ^ ix), cos (.r ± /V) = cosh ( )' =F ix). Prob. 84. From the product-series for sin .v derive that for sinh X : ( ^t^v ^'-^ V ■^■ sin vT = a- I r, I — — , II — 7f/\ 2'n'')\ T^'n Art. 30. Functions of x ^ iy in the Form X -[- iY. By the addition-formulas, cosh (.r -(- iy) = cosh x cosh iy -\- sinh x sinh iy, sinh (,t' -j- /j) = sinh x cosh z/ -f- cosh ,r sinh z/, but cosh iy = cos y, sinh iy = / sin y, hence cosh {x -\- iy) = cosh x cos y -\- i sinh x sin y, ^n ... . . (60) sinh (x -\- iy) = sinh x cos y -|- 1 cosh ,t' sin j. Thus if cosh (x -\- iy) = a-}- id, sinh {x -\- iy) = c -\- it/, then a = cosh X cos ;/, /? = sinh ,r sin y, (61) ^ = sinh X cos jj/, ^/ = cosh x sin j' From these expressions the complex tables at the end of this chapter have been computed. Writing cosh s =Z, where :: = x -^ iy, Z = XA^ iV; let the complex numbers s, Z he represented on Argand diagrams, in the usual way, by the points whose coordinates are (x, y), {X, F); and let the point z move parallel to the j-axis, on a given line x = ;//, then the point Z will describe an ellipGe whose equation, obtained by eliminating y between the equa- tions X =^ cosh ;// cos y, Y= sinh vi sin y, is X' V (cosh my (sinh mf and which, as the parameter m varies, represents a series of confocal ellipses, the distance between whose foci is unity. 44' HYPERBOLIC FUNCTIONS. Similarly, if the point z move parallel to the ;tr-axis, on a given line J = «, the point Z will describe an liyperbola whose equa- tion, obtained by eliminating the variable x from the equations A'= cosh X cos ;/, Y = sinh x sin n, is _JC^ F^ _ (cos //)' (sin ny and which, as the parameter n varies, represents a series uf hyperbolas con focal with the former series of ellipses. These two systems of curves, when accurately drawn at close intervals on the Z plane, constitute a chart of the hyper- bolic cosine; and the numerical value of cosh (;// -j- /;/) can be- read off at the intersection of the ellipse whose parameter is vi with the hyperbola whose parameter is «.* A similar chart can be draw^n for sinh {x-\-iy), as indicated in Prob. 85. Periodicity of Hyperbolic Functions. — The functions sinh m and cosh u have the pure imaginary period 2/-. For sinh (M + 2/;r) =sinh u cos 27r + ? cosh u sin 27: = sinh w, cosh {u\2iTi) =cosh u cos 2ti-\-i sinh u sin 2;: = cosh w. The functions sinh u and cosh u each change sign when the argument u is increased by the half period irr. For sinh (w + /r:) =sinli u cos ;: + i cosh w sin ;:= —sinh w, hd tt ' cosh (« + /;:)= cosh u cos 7r + i sinh w sin ;r= —cosh u. The function tanh u has the period iit. For, it follows from the last two identities, by dividing member by member, that tanh {u-^iTz) =tanh u. By a similar use of the addition formulas it is shown that sinh {u\\iiz) =i cosh u, cosh {u + ^ir:) =i sinh u. By means of these periodic, half-periodic, and quarter-periodic relations, the hyperbolic functions of x-\-iy are easily expressible in terms of functions of x -f iy', in which y' is less than ^iz. * Such a chart is given by Kennelly, Proc. A. I. E. E., April 1895, and is used by him to obtain the numerical values of cosh {x -\- iy) sinh (.r-|- (r), which present themselves as the measures of certain vector quantities in the theory of alternating currents. (See Art. 37.) The chart is constructed for values of x and of 1' between o and 1.2; but it is available for all values of r, on account of the periodicity of the functions. FUNCTIONS OF xi-iy IN THE FORM X+IY. 45 The hyperbolic functions are classed in the modern function- theory of a complex variable as functions that are singly periodic with a pure imaginary period, just as the circular functions are singly periodic with a real period, and the elliptic functions are doubly periodic with both a real and a pure imaginary period. Multiple Values of Inverse HyperboHc Functions. — It fol- lows from the periodicity of the direct functions that the inverse functions sinh~^ m and cosh~i m have each an indefinite number of values arranged in a series at intervals of 2/;r. That partic- ular value of sinh~^w which has the coefficient of i not greater than |7r nor less than —^n is called the principal value of sinh~i w; and that particular value of cosh"^ m which has the coefficient of i not greater than n nor less than zero is called the principal value of cosh~^w. When it is necessary to distinguish between the general value and the principal value the symbol of the former will be capitalized ; thus Sinh~i m = sinh~^ m + 2ir7i, Cosh~^ m = cosh~^ m + 2/>7r, Tanh~^ m = tanh~i m + irrc, in which r is any integer, positive or negative. Complex Roots of Cubic Equations. — It is well known that when the roots of a cubic equation are all real they are expressible in terms of circular functions. Analogous hyperbolic expressions are easily found when two of the roots are complex. Let the cubic, with second term removed, be written X^±7,bx=2C. Consider first the case in which b has the positive sign. Let x = r sinh u, substitute, and divide by r^, then . , , 3^ . , 2C smh u + ~ smh u-^-t,. r^ r gives Comparison with the formula s!nh^ 7/+f sinh u = \ sinh 3W 3^ 3 2C sinh 2>u 11-1 ^ I ^ whence r=20*, smh3w = -T^, w = -smh~^Tg; / I . c therefore x=2h^ sinh - sinh~^T5 \3 b' 46 HYPERBOLIC FUNCTIONS. in which the sign of b^ is to be taken the same as the sign of c. Now let the principal vakic of sinh^^Ty, found from the tables, be n; then two of the imaginary values are n±2i~, hence the three values of x are 20- smh - and 20- sm..(-± — ). The 3 \3 3 / last two reduce to — /;Msinh — ±/\'^3 cosh -j. Next, let the coeflicient of .v be negative and equal to —T,b. It miy then be shown similarly that the substitution x = r sin d leads to the three solutions ,, . « ,, / . n /— w\ , c — 20* sin-, Ml sm -±v 3 cos-J, where » = sm~^ rr. These roots are all real when f"%&-. If c>b^, the substitution x = rcosh7< leads to the solution :v = 2&*cosh (-cosh~iTyj, which gives the three roots ft / It f7\ ^ 2^ cosh -, — /'■ ( cosh - ± / V 3^ sinh - I , '.vherein n = cosh"'* tj , 3 \ 3 ' 3/ b' in which the sign of b^ is to be taken the same as the sign of c. Prob. 85. Show that the chart of cosh (.r + ;)) can he adapted to sinh {x -f- /v), by turning through a right angle; also to sin (.v +/V)- , „, ^ , ., • , / , -s sinli 2 ^ "+ ' sin 2r Prob. 80. Prove the identity tanli (.v -f- t\) = ; '- . cosli 2.V -j- cos 2J' Prob. 87. If cosh (x -\- iv), = a -\- ib, be written ii'i the " modulus and amplitude" form as r(cos B -\- /sin (^), ~ r exp W, then /-' = a"^ -\- b'^ =1 cosh^ .V — sin^j' = cos'^'j' — sinh^ .r, tan 6 = b/a = tanh x tan 7. Prob. 88. Find the modulus and amplitude of sinh {x -\- ty). Prob. 89. Show that the periotl of exp is id. a Prol). 90. When ;// is real and > i, cos~' ffi = / cosh~* m, sin~' ;// = — — /cosh ' m. 2 When m is real and < i, cosh"' ;// = / cos~' m. THE CATENARY. 4'^ Art. 31. The Catenary. A flexible inextensible string is suspended from two fixed points, and takes up a position of equilibrium under the action of gravity. It is required to find the equation of the curve in which it hangs. Let w be the weight of unit length, and s the length of arc ^/'measured from the lowest point A ; then zus is the weight of the portion AP. This is balanced by the terminal tensions, T acting in the tangent line at P, and H in the horizontal tangent. Resolving horizontally and vertically gives T cos (p = //, T s\n (p = ws, in which is the inclination of the tangent at P\ hence U'S s tan0 = ^=:-, wheie c is written for ///7i>, the length whose weight is the constant horizontal tension ; therefore dy s lis / s" dx ds dx c' dx Y ' c" c \^s'' -f- c""' X . , , -f • , '^ s dy y x — = smh~ — , smh — = — = 3—, — = cosh -, c c c c dx V c which is the required equation of the catenary, referred to an axis of x drawn at a distance c below A. The following trigonometric method illustrates the use of the gudermanian : The " intrinsic equation," s ^^ c tan 0, gives ds = c sec'' <^/0; hence dx, = ds cos cp, = c sec (pd(p; dy,^=ds sin 0, = r sec tan d(p ; thus x=c gd"' 0, y = c sec 0; whence y/c = sec = sec gd x/c = cosh x/c ; and s/c = tan gd x/c = sinh x/c. Numerical Exercise. — A chain whose length is 30 feet is suspended from two points 20 feet apart in the same hori- zontal ; find the parameter c, and the depth of the lowest point. 48 HYPERBOLIC FUNCTIONS. The equation s/c ■=■ sinh x/c gives I'^/c = siiih lo/c, which, by putting lo/c = c, may be written i.5,c = sinh ^. By exam- ining the intersection of the graphs of;- = sinli;;, y = 1.5^, it appears that the root of this equation is ;? = 1.6, nearly. To find a closer approximation to the root, write the equation in the iovm /[,a) = sinh 2 — i.^^ = o, then, by tiie tables, /(1.60) = 2.3756 — 2.4000 = — .0244, /(1.62) = 2.4276 — 2.4300 =: — .0024, /(1.64) = 2.4806 — 2.4600 = -f -0206; whence, by interpolation, it is found that y(i.622i) = o, and z = 1.622 1, c = lo/s = 6.1649. The ordinate of either of the fixed points is given by the equation j'/c = cosh x/c = cosh 10/^ = cosh 1.6221 = 2.6306, from tables; hence j' = 16.2174, and required depth of the vertex = j — r = 10.0525 feet.* Prob. 91. In tlie above numerical problem, find the inclination of the terminal tangent to the horizon. Prob. 92. If a perpendicular AfJV he drawn from the foot of the ordinate to the tangent at P, prove that A/iV is equal to the con- stant r, and that JVP is equal to the arc A P. Hence show that the locus of JV is the involute of the catenary, and has the prop- erty that the length of the tangent, from the point of contact to the axis of .V, is constant. (This is the characteristic property of the tractory). Prob. 93. The tension Tat any point is ecjual to the weight of a portion of the string whose length is equal to the ordinate j' of that point. Prob. 94 An nrch in the form of an inverted catenary f is 30 feet wide and 10 feet higli; show that the length of the arch can be obtained from the ecp'.ations cosh 5 — — s =1, 2S ^= "^ sinh z. 3 2 * See a similar problem in Cha[). I, Art. 7. f For the theory of this form of arch, sec "Arch" in the Encyclopaedia Britannica. CATENARY OF UNIFORM STRENGTH. 49 Art. 32. Catenary of Uniform Strength. If the area of tlie normal section at any point be made proportional to the tension at that point, there will then be a constant tension per unit of area, and the tendency to break will be the same at all points. To find the equation of the curve of equilibrium under gravity, consider the equilibrium of an element PP' whose length is c/5, and whose weight \% g poods, where 00 is the section at P, and p the uniform density. This weight is balanced by the difference of the vertical components of the tensions at /'and P\ hence ^(/sin (p) = gpojds, d{ T cos 0) = o ; therefore T cos (p =z H, the tension at the lowest point, and T = H sec 0. Again, if oo^ be the section at the lowest point, then by hypothesis 00/ co^ = T/ H = sec cf), and the first equation becomes Hd(sec (p sin (p) = gpco^ sec ((yds, or c d id^n = sec c/)ds, where c stands for the constant H/gpoj^, the length of string (of section co^) whose weight is equal to the tension at the lowest point ; hence, ds = c sec 0^/0, s/c = gd~'0, the intrinsic equation of the catenary of uniform strength. Also dx = ds cos = c(^(p, dy = ds sin ^ = c tan d(p ; hence .r = C(p, y = c log sec 0, and thus the Cartesian equation is ■y/c = log sec x/c, in which the axis of x is the tangent at the lowest point. Prob. 95. Using the same data as in Art. 3i» find the parameter ^ and the depth of the lowest point. (The equation x/c = gd s/c gives lo/c = gd i^/c, which, by putting i^/i' = z, becomes 50 HYPERBOLIC FUNCTIONS. gd s = fz. From the grapli it is seen that z is nearly 1.8. If f(z) =: gd 2 — §2, then, from the tables of the gudermanian at the end of this chapter, /(1.80) = 1.2432 — 1.2000 = + -0432, /(1.90) — 1.2739 — 1.^667 = + .0072, /(i'95) — i-288i — 1.3000 = — .0119, whence, by interpolation, 2 = 1.91S9 and c— 78170. Again, yjc = logc sec x/c ; but xjc = 10/^ = 1.2793; ^"d 1-2793 radians = 73° 17' 55"; hence^ = 7.8170 X .54153X2.3026 = 9.7472, the required depth.) Prob. 96. Find the inclination of the terminal tangent. Prob. 97. Show that the curve has two vertical asymptotes. Prob. 98. Prove that the law of the tension T, and of the section a?, at a distance 5, measured from the lowest point along the curve, is T 00 , J — = — = cosh -; H G), c and show that in the above numerical example the terminal section is 3.48 times the minimum section. Prob. 99. Prove that the radius of curvature is given by o z= c cosh s/c. Also that the weight of the arc s is given by /F" = H smh. s/c, in which s is measured from the vertex.. Art. 33, The Elastic Catenary. An elastic string of uniform section and densitj- in its natu- ral state is suspended from two points. Find its equation of equilibrium. Let the element da stretch into ds ; then, by Hooke's law, ds = d(T{\ -\- XT), where X is the elastic constant of tlie string; hence the weight of the stretched element ds, = jpoodcr, = goa)ds/{i -{-XT}. Accordingly, as before, ^r sin 0) = gpoods/{\ -\- XT), and T cos (p =z H = gpcoc, hence )d(p, s/c = tan -(- ^/<(sec cp tan + gd~^ 0), [prob. 20, p. 37 which is the intrinsic eqnation of the curve, and reduces to that of the common catenary when /,i — o. The coordinates x, y may be expressed in terms of the single parameter by put- ting dx = ds cos = ^(sec 4~ /< sec^ (p)d(p, dy = ds sin = r(sec'' + /< sec' 0) sin dcp. Whence x/c = gd"' (p -\- ju tan 0, j/f = sec + 2/' tan' 0. These equations are more convenient than the result of eliminating 0, which is somewhat complicated. Art. 34. The Tractory.* To find the equation of the curve which possesses the property that the length of the tangent from the point of con- tact to the axis of x is con- stant. Let FT, P'T' be two con- secutive tangents such that PT= P'T' = c, and let OT = /; draw TS perpendicular to P'T'; then U PP' = ds, it is evident that ST' differs ' ''^ ^ ^' from ds by an infinitesimal of a higher order. Let PT make an angle with OA, the axis of y ; then (to the first order of infinitesimals) PTdcp = TS = TT' cos 0; that is, Cif(f) = cos (pdf, / = r gd~'0, X = ^ — c sin 0, = r(gd~' — sin 0), y = c cos 0. This is a convenient single-parameter form, which gives all * This curve is used in Schieie's anti-friction pivot (Minchin's Statics, Vol. i, p. 242) ; and in the theory of the skew circular arch, the horizontal projection of the joints being a tractory. (See "Arch," Encyclopedia Britannica.) The equation = gd tjc furnishes a convenient method of plotting the curve. 6^ HYPERBOLIC FUNCTIONS. values of x, y 3.s (p increases from o to ^,t. The value of s, ex- pressed in the same form, is found from the relation ds = ST' = dt sin = ^ tan , - gd " >, ; hence gd"'/^ — gd~'j', -{- {.\\ — x^) tan 01, from which the final latitude can be found when the initial latitude and the differ- ence of longitude are given. The distance sailed is equal to {y^ — y,) CSC a radii, a radius being 60 X i8o/;r nautical miles. Mercator's Chart. — In this projection the meridians are parallel straight lines, and the loxodrome becomes the straight line y' = x tan a, hence the relations between the coordinates of corresponding points on the plane and sphere are x' = x, y' = gd~ y. Thus the latitude y is magnified into gd ~ 'y, which is tabulated under the name of " meridional part for latitude j" ; the values of j/ and of 7' being given in minutes. A chart constructed accurately from the tables can be used to furnish graphical solutions of problems like the one proposed above. Prob. 103. Find the distance on a rhumb line between the points (30° N, 20° E) and (30° S, 40" E). Art. 36. Combined Flexure and Tension. A beam that is built-in at one end carries a load P at the other, and is also subjected to a horizontal tensile force Q ap- plied at the same point; to find the equation of the curve assumed by its neutral surface: Let x, y he any point of the 64 HYPERBOLIC FUNCTIONS. elastic curve, referred to the free end as origin, then the bend- ing moment for this point is Qy — Px. Hence, with the usual notation of the theory of flexure,* ax ax P Q Ef 'vhich, on putting/ — vix = ;/, audcPj/dx'^ = (Pu/c/x'', becomes d^u dx , = n'u, whence that is, u =- A cosh nx -\- B sinh nx, [probs. 28, 30 y =. Dix -)- A cosh nx -\- B sinh nx. The arbitrary constants A, B are to be determined by the terminal conditions. At the free end a' = o, j = O ; hence /i must be zero, and y = inx -|- B sinh nx, — =. ni -\- hB cosh nx ; dx but at the fixed end, x = /, and dy/dx = o, hence i> = — jji/n cosh «/, and accordingly y = mx in sinh nx n cosh ;// To obtain the deflection of the loaded end, find the ordinate of the fixed end by putting x = I, giving deflection = mil— -tanh;//). n ' Prob. 104. Compute the deflection of a cast-iron beam, 2X2 inches section, and 6 feet span, buik-in at one end and carrying a load of 100 pounds at the other end, the beam being subjected to a horizontal tension of 8000 pounds. [In this case / = 4/3, ^=15X10', Q = 8000, /* = 100 ; hence n = 1/50, w = 1/80, deflection = ^17(72 — 50 tanh 1.44) — ^^(72 — 4469) = -341 inches.] ^Ier^iman, Mechanics of Materials ^New York, 1895), pp. 70-77, 267-269 ALTERNATING CURRENTS. 55 Prob. 105, If the load be uniformly distributed over the beam, say 7U per linear unit, prove that the differential equation is EI^, = Qv - hux\ or '-A = ,i\v - nix'), 2 VI and that the solution is_)' = ^ cosh nx -\- B sinh ux \- tiix' ^ ^. n Show also how to determine the arbitrary constants. Art. 37. Altp:rnating Currents.* In the general problem treated the cable or wire is regarded as having resistance, distributed capacity, self-induction, and leakage ; although some of these may be zero in special cases. Tile line will also be considered to feed into a receiver circuit of an}' description ; and the general solution will in- clude the particular cases in which the receiving end is either grounded or insulated. The electromotive force may, without loss of generality, be taken as a simple harmonic function of the time, because any periodic function can be expressed in a Fourier series of simple harmonics. f The E.M.F. and the current, which may differ in phase by any angle, will be supposed to have given values at the terminals of the receiver circuit ; and the problem then is to determine the E.M.F. and current that must be kept up at the generator terminals ; and also to express the values of these quantities at any inter- mediate point, distant x from the receiving end ; the four line-constants being supposed known, viz.: r = resistance, in ohms per mile, / = coefificient of self-induction, in henrys per mile, c = capacity, in farads per mile, g = coefificient of leakage, in mhos per mile. J It is shown in standard works§ that if any simple harmonic * See references in footnote, Art. 27. f Byerly, Harmonic Functions. t This article follows the notation of Kennelly's Treatise on the Application of Hyperbolic Functions to Electrical Engineering Problems, p. 70. § Thomson and Tait, Natural Philosophy, Vol. I. p. 40; Raleigh, Theo'y of Sound, Vol. I. p. 20; Bedell and Crehore, Alternating Currents, p. 214. 56 HYPERBOLIC FUNCTIONS. function a sin (&»/ -(- S) be represented by a vector of length a and angle d, then two simple harmonics of the same period 2n/cj, but having different values of the phase-angle 0, can be combined by adding their representative vectors. Now the E.M.F. and the current at any point of the circuit, distant x from the receiving end, are of the form e = e^ sin {cot -{- H), i = /, sin {oot -)- B'), (64) in which the maximum values <',, /,, and the phase-angles B, B', are all functions of x. These simple harmonics will be repre- sented by the vectors eJB, ijd' ; whose numerical measures are the complexes r, (cos B -f-y' sin ^)*, /, (cos B' -\- j sin B'), which will be denoted hye,i. The relations between /and i may be obtained from the ordinary equations f di de de di for, since de/dt = ooe^ cos (w/ -\- 6) = wCi sin (co/ + ^ + §7r), then de/dt will be represented by the vector oiei/d-\- ^ir] and di/'dx by the sum of the two vectors gex/d, Cijie^/d -\-\ir\ whose numerical measures are the complexes ge, juce; and similarly for de/dx in the second equation ; thus the relations between the complexes e, i are ^ = (^ + icoOe, ;£. = ('' + i"0*'- (66)t * In electrical theory the symbol j is used, instead of /, for '♦^ — i. t Bedell and Crehore, Alternating Currents, p. 181. The sign of dx is changed, because .v is measured from the reccivmg end. The coefficient of leakage, g, is usually taken zero, but is here retained for generality and sym- metry. I These relations have the advantage of not involving the time. Steinmetz derives them from first principles without using the variable /. For instance, he regards r -\- joil as a generalized resistance-coeflicicnt, which, when applied to i, gives an E.M.F., part of which is in phase with /, and part in quadrature with /. Kennelly calls r + j^^l the conductor impedance; and g -\- juc the dielectric admittance; the reciprocal of which is the dielectric impedance. ALTERNATING CURRENTS 57 Differentiating and substituting give ^2= (.'' + i^Oig -^jc^c)e, dH dx :. = (r + J^Ois + j^<^)i' (67) and thus e, I are similar functions of x, to be distinguished only by their terminal values. It is now convenient to define two constants a, So by the equations * «2 ^(^r + ju^l) (g + jo^c) , z, = a/{g + ji^c) ; (68) and the differential equations may then be written C?2g dH -1—., — a-e, -r-, = a-t, dx- dx- (69) the solutions of which are f e =^ A cosh ax + ^ sinh ax, i = A' cosh ax -\- B' sinh ax, wherein only two of the four constants are arbitrary; for substituting in either of the equations (66), and equating coefficients, give {g-^io:c)A=aB', {g-^ jooc)B = aA', whence B' = A/zo, A' = B/z^. Next let the assigned terminal values of e, t, at the re- ceiver be denoted hy E, /; then putting x = O gives E= A, I = A', whence B = zj, B' = E/zq] and thus the general so- lution is e = E cosh ax + ZqI sinh ax, i = I cosh ax H E sinh ax, 2o (70) * Professor Kennelly calls a the attenuation-constant, and So the surge- impedance of the line. t See Art. 14, Probs. 28-30; and Art. 27, foot-note. 58 Hyperbolic functions. If desired, these expressions could be thrown into the ordi- nary com})Iex form X -\- jY, X' -\-jV', by putting for the let- ters their complex values, and applying the addition-theorems for the hyperbolic sine and cosine. The quantities X, Y, X', Y' would then be expressed as functions of x ; and the repre sentative vectors of e, i, would be i\/0, z, /8\ where ^/ = A'^-[~ ^S /; = X" + Y'\ tan = Y/X, tan"^ =~Y'/X. For purposes of numerical computation, however, the for- mulas (70) are the most convenient, when either a chart,* or a table, f of cosh //, sinh u, is available, for complex values of ?/. Prob. 106. J Given the four line-constants: r = 2 ohms per mile, f = 20 millihenrys per mile, c = 1/2 microfarad per mile, g = o; and given co, the angular velocity of E.M.F. to be 2000 radians per second; then 0)1 = 40 ohms, conductor reactance per mile; r + /co/ = 2 + 40/ ohms, conductor impedance per mile; uc = .GDI mho, dielectric susceptance per mile; g + juc = .001; mho, dielectric admittance per mile; (g -|- /aj'~)~* = — 1000/ ohms, dielectric impedance per mile; a- = (r+ j'cjoI) (g + /wc) =.04 +.002/, which is the measure of .04005 177° 8'; therefore a = measure of .2001 88° 34' = .0050 + .2000;, an ab- stract coeflficient per mile, of dimensions [length]" S z^ = a/{g + /coc) = 200 — 5/ ohms. Next let the assigned tenninal conditions at the receiver be ; 7 = (line insulated); and E = 1000 volts, whose phase may be taken as the standard (or zero) phase ; then at any distance x, by (70), E e = E cosh ax, ^ = ~ sinh ax, in which ax is an abstract complex. Suppose it is required to find the E.M.F. and current that must be kept up at a generator 100 miles away; then * Art. 30, foot-note. t See Table II. X The data for this example are taken from Kennelly's articif' (1. c. p. 38). ALTERNATING CURRENTS. 69 e — looo cosh (.5 -f- 20/), I = 200(40 — jY^ sinh (.5 -\- 20J), but, by page 44, cosh (.5 + 2oj) = cosh (.5 + 2oy — GttJ) = cosh (.5 + 1. 15/) = .4600 + .4750/' obtained from Table II, by interpolation between cosh (.5 + i.y) and cosh (.5 -(- 1.27); hence e — 460 + 475/'= -^.(cos 6^4- /sin /^), where log tan ^ = log 475 - log 460 = .0139, (^ = 45° 55', and e^ — 460 sec = 661.2 volts, the required E.M.F. Similarly sinh (.5 + 207) = sinh (.5 + i-iSy) = .2126+ 1.0280/, and hence ^'"^ 7a°£(4o + /)(.2i26 + 1.028/) = -7— (1495 + 8266/) lOOI lOOI = /,(cos 0' -{- J sin 6'), where log tan 6' = 10.7427, 6' = 79° 45', /, = 1495 sec ^'/i6oi — 5.25 amperes, the phase and magnitude of required current. Next let it be required to find ^ at .v = 8; then ^= 1000 cosh (.04 -(- i.6y) = 1000/ sinh (.04+ -os/)* by subtracting ^tt/, and applying page 44. Interpolation be- tween sinh (0 + 0;) and sinh (o -f- .1/) gives sinh (o -f- -03/) = 00000 + .02995/. Similarly sinh (.1 -f -03/) = .10004-)- ■03oo47' Interpolation between the last two gives sinh (.04 -[- .03/) = .04002 -\- .02999/ Hence r =y(^o. 02 +29.99/)= — 29.994-40.02/ =^, (cos B-{-j?>\n H), where log tan 6 = .12530, ^ = 126° 51',^, = — 29.99 s^c 126° 51' = 50.01 volts. Again, let it be retpiired to find e Vit x = 16; here e — lOoo cosh (.08 + 3.2/) = — 1000 cosh (.08 -\- .o6j), but cosh (o -|- .o6y) = .997° + o/, cosh (. i -j- .06/) = 1.0020 -|- .006/; hence cosh (.08 -)- .06/) = 1.0010 -{-.0048/, and ^= — iooi-|-4.8/= c'll^cos ^-f-ysin ^), where ^ — 180° 17', e^ = looi volts. Thus at a distance of about 16 miles the E.M.P'. is the same as at the receiver, but in opposite 60 HYPERBOLIC FUNCTIONS. phase. Since c is proporlional to cosh (.005 -|- •2j)x, the value of X for which the phase is exactly 180° is tt/.z — 15.7. Similarly the phase of the E.AI.F. at x = 7.85 is 90°. There is agreement in phase at any two points whose distance apart is 31.4 miles. In conclusion take the more general terminal conditions in which the line feeds into a receiver circuit, and suppose the current is to be kept at 50 amperes, in a phase 40° in advance of the elec- tromotive force; tiien / — 5o(cos 40° +/ sin 40°) = 38.30 + 32-i4/> and substituting the constants in (70) gives e =z icoo cosh (.005 -\- .2j)x -\- (7821 -j- 6216J) sinh (.005 -f -V)-^ — 4604 4757 -4748+93667= -4288+984iy=r,(cos ^-fy sin ^), where ^= 113° ZZ'y^x — '°73° volts, the E.M.F. at sending end, This is 17 times what was required when the other end was insulated. Prob. 107. If / = o, g = o, 7=0; then a={i-{-j)n, Zo = (i +y)Mi, where n^ = core '2, w,- = r/2c«jc; and the solution is ^1 — T7=£ t^cosh 2nx + cos 2nx, tan 6 — tan nx tanh nx, ^ 2 ii = — E I cosh 2nx — cos 2nx, tan 6' = tan nx coth nx. Prob. 108. If self-induction and capacity be zero, and the re- ceiving end be insulated, show that the graph of the electromotive force is a catenar}^ if g ?^ o, a line if g = o. Prob. log. Neglecting leakage and capacity, prove that the solution of equations (66) isi — I,e — E-\-(r-\- juljix. Prob. no. If a; be measured from the sending end, show how equations (65), (66) are to be modified; and prove that e = Eo cosh ax — zJo sinh ax, I — h cosh ax En sinh ax, _ _ •^0 where £c lo refer to the sending end. Art. 38. Miscellaneous Applications. 1. The length of the arc of the logaritlimic curve y = <7* is S = M{cosh //-(-logtanli |?/), in which Al= i/log a, sinh u — y/M. 2. The length of arc of the spiral of Archimedes r =^ a^ xs, s = i(7(sinh 2ti -j- 2//), where sinh 21 = 6>. 3. In the hyperbola x^ /a" — y' /b' = i the radius of curva- ture is p = {a' sinh' u -\- h' cosh' iif/ab; in which u is the measure of the sector AOP, i.e. cosh u = x/a, sinh // z=y/b. 4. In an oblate spheroid, the superficial area of the zone MISCELLANEOUS APPLICATIONS. 61 between the equator and a parallel plane at a distance j is 5 = iTT/iXsinh 2u -|- 2u)/2e, wherein b is the axial radius, e eccen- tricity, sinh u = ey/p, and / parameter of generating ellipse. 5. The length of the arc of the parabola jj/' = 2px, measured from the vertex of the curve, is / = 5/'(sinh 2u-\-2u\ in which sinh u ^=y/p = tan 0, where is the inclination of the termi- nal tangent to the initial one. 6. The centre of gravity of this arc is given b^' 2,lx =r/'(^cosh' u — i), 64/]' =: p" (sinh 411 — 4//) ; and the surface of a paraboloid of revolution is 5 = 2ti yl. 7. The moment of inertia of the same arc about its ter. minal ordinate is /= ;/[,r/(^ — 2^) -f ^'^/^W], where /< is the mass of unit length, and JSl =i II — '^ sinh 2n — ^ sinh 4?{-\~ y^^ sinh 6//. 8. The centre of gravit)' of the arc of a catenary measured from the lowest point is given by 4/y= ^'(sinli 2?/ -\- 211), !x = c^{ii sinh ii — cosh ?/ -f~ i)j • in which ?( =x/c; and the moment of inertia of this arc about its terminal abscissa is / = J^c'Xj^iy sinh 3// -\- f sinh ?/ — 7/ cosh ?i). 9. Applications to the vibrations of bars are given in Ray- leigh, Theory of Sound, Vol. I, art, 170: to the torsion of prisms in Love, Elasticity, pp. 166-74; to the flow of heat and electricity in Byerly, Fourier Series, pp. 75-81 ; to wave motion in fluids in Rayleigh, Vol. I, Appendix, p. 477, and in Bassett, Hydrodynamics, arts. 120, 384; to the theory of potential in Byerly p. 135, and in Maxwell, Electricity, arts. 172-4; to Non-Euclidian geometry and many other subjects in Giinther, Hyperbelfunktionen, Chaps. V and VI. Several numerical examples are worked out in Laisant, Essai sur les fonctions hyperboliques. b'-J HYPKRHOLIC FUNCTKjNS. Art. 39. Explanation of Tables. In Table I the numerical values of the hyperbolic functions sinh II, cosh n, tanh u are tabulated for values of u increasing from o to 4 at intervals of .02. When ti exceeds 4, Table IV may be used. Table II gives hyperbolic functions of complex arguments, in which cosh {x ± iy) = « ± ib, sinh [x ± iy) = c ±_ id, and the values of a, b, c, d are tabulated for values of x and o[ y ranging separately from o to 1.5 at intervals of .1. When interpolation is necessary it may be performed in three stages. For example, to find cosh (.82 -[- 1-340 • Fh'st find cosh (.82 -j- 1.3/), by kecpingj'at 1.3 and interpolating between the entries under a" = .8 and.r = .9 ; next find cosh (.82 -f 1. 4/), by keeping ^^^ at 1.4 and interpolating between the entries under ;ir = .8 and x ^ .9, as before; then by interpolation between cosh (.82 -(- 1.3/) and cosh (.82 -|- i-40 ^""^^ cosh( .82 -f 1-340' in which x is kept at .82. The table is available for all values of _^, however great, b\- means of the formulas on page 44: sinh (.r -]- 2/'T ) = sinh a', cosh {x A^ 2/t) = cosh x, etc. It does not apply when x is greater than 1.5, but this case sel- dom occurs in practice. This tabic can also be used as a com plex table of circular functions, for cos {y ■±_ ix) = cr =p //;, sin {y ± ix) ^ d ±_ic ', and, moreover, the cxponenticd function is given by exp {±x ± ty) z=a±c ± ;(/; ± d), in which the signs of c ami ^/are to be taken the same as the sign of X, and the sign of i on the right i^; to be the product of the signs of x and of i on the left. (See A[)pendix, C.) Table III gives the values of v— gd n, and of the guder- manian angle 6= 180° I'/n, as ti changes from o to I at inter- EXPLANATION OF TABLES. ti3 vals of .02, from i to 2 at intervals of .05, and froiri 2 to 4 at intervals of .1. In Table IV are given the values of gd u, log sinh ?/, log cosh u, as u increases from 4 to 6 at intervals of .1, from 6 to 7 at intervals of .2, and from 7 to 9 at intervals of .5. In the rare cases in which more extensive tables are neces- sary, reference may be made to the tables* of Gudermann, Glaisher, and Geipel and Kilgour. In the first the Guderman- ian angle (written k) is taken as the independent variable, and increases from o to 100 grades at itUervals of .01, the corre- sponding value of u (written Lk) being tabulated. In the usual case, in which the table is entered with the value of //, it gives by interpolation the value of the gudermanian arigle, whose circular functions would then give the hyperbolic functions of u. When it is large, this angle is so nearl\' right that inter- polation is not reliable. To remedy this inconvenience Gu- dermann's second table gives directly log sinli //, log cosh;/, log tanh //, to nine figures, for values of?/ var}'ing by .OOI from 2 to 5, and by .01 from 5 to 12. Glaisher has tabulated the values of r* and c'", to nine sig- nificant figures, as x varies by .001 from o to .[, by .01 from O to 2, by .1 from o to 10, and by i from o to 500. From these the values of cosh x, sinh x are easily obtained. Geipel and Kilgour's handbook gives the values of coshjt, sinh X, to seven figures, as x varies by .01 from o to 4. There are also extensive tables by Forti, Gronau, Vassal, Callet, and Hoiiel ; and there are four-place tables in Byerly's Fourier Series, and in Wheeler's Trigonometry, (See Ap- pendix, C.) In the following tables a dash over a final digit indicates that the number has been increased, *Gudermann in Crelle's Journal, vols. 6-9, 1831-2 (published separately under ihe title Theorie der hyperbolischen Functionen, Berlin, 1S33). Glaisher in Cambridge Phil. Trans., vol. 13, 1881. Geipel and Kilgour's Electrical Hand- book. 64 HYPERBOLIC FUNCTIONS. Table I. — Hyperbolic Functions. u. sinh u. cosh u. tanh ». u. sinh «. cosh u. tanh u. .00 .0000 1.0000 .0000 1.00 1.1752 1.5431 .7616 02 0200 1.0002 0200 1.02 1.20 3 1 5669 7699 04 0400 1.00C8 0400 1.04 1 2379 1.5913 7779 06 0600 1.0018 0599 1.06 1 2700 1 6164 7857 08 0801 1.0032 0798 1.08 1.302) 1.6421 7932 .10 .1002 1.0050 .0997 1.10 1 3356 1 6685 .8005 12 1203 1 0072 1194 1.12 1 3693 1.6956 8076 14 1405 1 .0098 1391 1.14 1 40:!5 1.7233 8144 16 1607 1.01 ••8 1586 1.16 1.4382 1.7517 8210 18 1810 1.0102 1781 1.18 1.4735 1.78U8 8275 .20 .2013 1.0201 .1974 120 1 5095 1 8107 .8337 22 2218 1.0243 2165 1.22 1.54.0 1.8412 8397 24 2423 1.0289 2355 1.24 1.5831 1.8720 8455 26 2029 1.0340 2543 1.26 1.6209 1.9045 8511 28 2837 1.0395 2729 1.28 1.6593 1.9373 856.5 .30 .3045 1 0453 .2913 1.30 1.6984 1.9709 .8617 32 3255 1 0516 3095 1.32 1.7381 2.0053 8668 34 3466 1.0584 3275 1 34 1.77^6 2 0404 8717 m 3618 1.0655 3452 1.36 1.8198 2.0764 8764 38 3892 1.0731 3627 1.38 1.8617 2.1132 8810 .40 .4108 1.0811 .3799 1.40 1.9043 2 1509 .8854 42 43-.J5 1.0895 3969 1.42 1.9477 2 1894 8896 44 4543 1.0984 4136 1.44 19919 2.228S 8937 46- 47(54 1.1077 4301 1.46 2.0369 2.2691 8977 48 '4986 1.1174 4462 1.48 2.0827 2.3103 9015 .50 .5211 1.1276 .4621 1.50 2.1293 2.3524 .9051 52 5)38 1.1383 4777 1 52 2 1768 2.3955 9087 54 5666 1 1494 4930 1.54 2.2251 2.4395 9121 56 5897 1.1609 5080 1.56 2.2743 2.4845 9154 58 6131 1.1730 5227 1.58 2.3245 2.5305 9186 .60 .6367 1 1855 .5370 1.60 2.3756 2.5775 9217 62 6605 1.1984 5511 1.62 2.4276 2 6255 9246 64 6846 1 2119 5649 1.64 2.4806 2.6746 9275 66 7090 1.2258 5784 1.66 2 5346 2.7247 9302 68 7336 1.2402 5915 1.68 2.5896 2.7760 9329 .70 .7586 1.2552 .6044 1.70 2.6456 2.8283 .9354 72 7888 1.2706 6169 1.72 2.7027 2.8818 9379 74 8094 1 2S65 6-.' 9 1 1 74 2.7609 2.9364 9402 76 8353 1.3030 6411 1.76 2.8202 2 9922 9425 78 8615 1.3199 6527 1.78 2.8806 3.0492 9447 .80 .8881 1.3374 .6040 1.80 2.9422 3 1075 .9468 82 9150 1.3555 6751 1.82 3.0049 3.1669 9488 84 9423 1 3740 685S 1.84 3.0689 3 2277 9508 86 9700 1 3932 6963 1.86 3.1340 3 2897 95':7 88 9981 1.4128 7004 1.88 3.2005 3 3530 9545 .90 1.0265 1.4331 .716:1 1.90 3.2682 3.4177 .9562 92 1 . 0554 1.4539 7259 1.92 3.3372 3 4S38 9579 94 1 0S47 1 4753 7352 1 94 3.4075 3.5512 9595 96 1.1144 1.4973 7443 1.96 3 4792 3.6201 961 i 98 1 1446 1.5199 7531 1.98 3.5523 3 6904 9626 TABLES, 65 Table I. Hyperbolic Functions. ». sinh u. cosh «. tanh u. u. sinl) u. cosh u. tanh u. 2.00 3.6269 3.7622 .9640 300 10.0179 100677 .99505 2.02 8.7028 3.88.-)5 9654 3.02 10.2212 10.2700 99524 2.04 3.78U8 89108 9667 8.04 10.4287 10.4765 99.543 2.06 3.8598 8.9867 9680 3 06 10.6408 10 6872 99561 2.08 8 9898 4.0647 9698 3.08 10.8562 10.9022 99578 2.10 4.0219 4 1443 .9705 3.10 11.0765 11.1215 .99.594 2.12 4.1056 4.2256 97 It; 3 12 11.8011 11.3453 99610 2 14 4 1909 4.3085 9727 3.14 11 5803 11 57 £,6 99626 2.16 4 2779 4.3982 9787 3.16 11 7641 11.8065 99640 2.18 4.3666 4.4797 9748 3.18 12.0026 12.0442 99654 2.20 4 4571 4.5679 . 9757 8.20 12.2459 12 2866 .99668 2.22 4.5494 4.6510 4.7499 9767 8.22 12.4941 12.5340 99681 2.24 4.6484 9776 3.24 12.7473 : 2 7^ 64 99693 2.26 4 7894 4.8487 978.5 3 26 18 0056 18.0440 99705 2 28 4.8872 4 9895 9793 3.28 13.2691 13.3067 99717 2.80 4.9370 5.0.^.72 .9801 3.80 13.5379 13 5748 .99728 2 82 5 0887 5.1370 9809 3 82 18.8121 13 H483 99788 2.84 5.1425 5 2888 9816 3.84 14.0918 14.1278 99749 2.86 5 2488 5.3427 9828 3.86 14.8772 i4.4i::o 99758 2.38 5.3562 5 4487 9880 3.88 14.6684 14.7024 99768 2.40 5.4662 5.5569 .9887 3.40 14 96.54 14.9987 .99777 2.42 5.5785 5.6674 9843 3.42 15.2684 15 80U 99786 2.44 5.6929 5.7801 9849 3.44 15. .5774 15 6095 99794 2.46 5 8097 5 8951 9855 3 46 15.8928 15.9242 99802 2.48 5.9288 6.0125 9861 3.48 16.2144 16.2453 99810 2.50 6 0502 6 1323 .9866 3 50 16.5426 16. .5728 .99817 2 52 6.1741 6.2545 9871 3 52 16.8774 1 6 9070 99824 2.54 6.8004 6.8793 9876 3.54 17.2190 17,2480 99831 2.56 6.4293 6.5066 9881 3.56 17.5674 17.59,58 99838 2.58 6.5607 6.6364 9886 3.58 17.9228 17.9507 99844 2.60 6 6947 6.7690 .9890 3.60 18.2854 18.8128 .99850 2.62 6.8815 6.9043 989.5 3.62 18.6554 18.6822 99856 2.64 6 9709 7.04 3 9899. 3.64 19.0328 19.0590 99862 2.66 7.1132 7 1882 9903 3.66 19.4178 19.4435 99867 2.68 7.2583 7.3268 9906 3.68 19.8106 19.83-,8 99872 2.70 7.4068 7.4735 .9910 3.70 20.2113 20.2360 .99877 2.72 7.5572 7.6281 9914 3.72 20 6201 20.6443 99882 2.74 7.7112 7.7758 9917 3.74 21.0871 21.0609 99887 2.76 7.8683 7.9816 9920 3.76 21.4626 21.4859 99891 2 78 8 0285 8.0905 9923 3.78 21.8966 21.9194 99896 2.80 8.1919 8.2527 .9926 3.80 22.3394 22 8618 .99900 2 82 8 3.">86 8.4182 9929 3.82 23.7911 22 8181 99904 2.84 8.5287 8.5871 9932 3 84 23.2.520 28.2735 99907 2.86 8 7021 8.7.594 9985 3.86 28.7221 28.7432 99911 2 88 8 8791 8.9352 99:!7 3 88 24.2018 24.2224 99915 2 90 9 0596 9.1146 .9940 3 90 24 6911 24.7113 .99918 2.92 9.2487 9.2976 9942 8.92 25 1908 25.2101 99921 2.94 9.481.5 9 4844 9944 3.94 25.6996 25.7190 99924 2 96 9 6281 9 6749 9947 3.96 26 2191 26.2382 99927 2.98 9 8185 9.8693 9949 3.98 26.7492 26.7679 99930 66 HYPERBOLIC FUNCTIONS. Table II. V^alues of cosh (x + ij') and sinh (x -f I'v). X = o X = .\ y a b c d a b c d 1.0000 0000 0000 .0000 1.0050 .00000 .10017 .0000 .1 ().'J9.-.0 " 0998 1 . 0000 01000 09967 1003 .3 0.9801 " 19S7 0.9850 0199(1 0;)817 1997 .3 0.9.j5:j " 2955 0.9G01 02960 09570 2970 .4 .9211 .' .3 m .9257 .03901 .09226 .39)4 .5 8776 " 4? 94 8820 04802 08791 4S1H .6 82.-)3 " 5046 82J5 05(;50 08-J67 5675 .7 7648 " 6442 7687 06453 07661 U74 .8 .6967 .< .7174 .7002 .07186 .06979 .7200 .9 6216 " 7S33 0247 07847 06227 7872 1.0 5403 " 8415 5430 08429 05412 8457 1.1 4536 " 8912 4559 08927 04544 8957 1.2 .3624 " .9330 .3642 .09336 .03630 9367 1.3 2675 " 9()36 2688 09652 02().so 0.9684 1.4 1700 " 9851 1708 09871 01703 . 9904 1.5 0707 ' ' 99 7o 0711 09992 00709 1.0025 \^ 0000 " 1.0000 0000 10017 00000 1.0050 y X = • 4 X = •5 a b c d a b c ,/ 1 0811 .0000 .4108 .0000 1.1276 .0000 .5211 .0000 .1 1.0 :5() 0410 4087 1079 1.1220 0520 518.5 1126 .2 1.0595 0816 4026 2148 1.1051 1025 5107 2240 .3 1.0328 1214 3924 3195 1.0773 1540 4978 3332 .4 .9957 .1600 .3783 .4210 1.0386 .2029 .4800 .4391 .5 9487 1969 361)5 5183 0.9896 2498 4573 5406 .6 8922 2319 3390 6104 0.9306 2942 4301 6367 .7 8268 2646 3142 6964 0.8624 3357 3986 7264 .8 .7532 .2947 .2862 .7755 .7856 .3738 .3631" 0.8089 .9 6720 3218 2553 8468 7009 4082 3239 0.S833 1.0 5841 3456 2219 9097 6093 4385 2815 0.9489 1.1 4904 3661 1863 9(i3.5 5115 4644 23(i4 1.0050 1.2 .3917 .3829 .1488 1.0076 .4086 .4857 .18S8 1.0510 1.3 2892 3958 1099 1.0417 3016 5021 1394 1.0865 1.4 1838 4048 0698 1.0653 1917 5135 0886 1.1163 1.5 0765 4097 0291 1.0784 0798 5198 0369 1.1248 \Tl 0000 4108 0000 l.OSli 0000 5211 0000 1.1276 TABLES. Table II. Values of cosh {x -f ijy) and sinh (x -|- jy). 67 X = .2. X = •3 a ^ C d a b c d y 1.0201 .0000 .2013 .0000 1.0453 .0000 . 3045 .0000 1.0150 0201 2003 1.018 1.0401 0304 3o;,o 1044 .1 0.9997 0100 1973 2027 1.0245 0605 298.5 2077 o 0.974.5 0595 1923 3014 9987 0900 2909 3089 !3 .9:J9.-) .0784 . 1854 .3972 .9628 .1186 .280.5 .4071 .4 8953 0965 1767 4890 9174 1460 2672 5012 .5 8-119 1137 1662 5760 8627 1719 2513 5903 .6 7802 1297 1540 6571 7995 1962 2329 6734 .7 .7107 .1444 .1403 .7318 .7283 .2184 2122 .7498 .8 6341 1577 1252 7990 6498 2385 189§ 8188 .9 5511 1694 1088 8.584 5648 2.562 1645 8796 1.0 4627 1795 0913 9091 4742 2714 1381 9316 1.1 .3696 .1877 .0730 0.9.507 .3788 .2838 .1103 0.9743 1.2 2729 1940 0539 0.9^29 2796 2934 081.5 1.0072 1.3 1734 1984 0342 1.0052 1777 3001 0518 1.0301 1.4 0722 2008 0142 1 0175 0739 3038 0215 1.0427 1.5 0000 2013 0000 1.0201 0000 3045 0000 1.04.53 '.n X = .6 X = • 7 a i c d a b c d y 1.185f) .0000 . 6367 .0000 1.2552 .0000 .75S6 .0000 1.1795 0636 6335 1183 1.2489 0757 7518 12.53 .1 1.1618 1265 6240 2355 1.2301 1507 743.5 2494 .2 1 . 1325 1881 6082 3503 1.1991 2242 7247 3709 .3 1.0918 .2479 .5864 .4617 1.1561 .29.54 .6987 .4888 .4 1.0403 3052 5587 5684 1.1015 3637 6657 6018 .5 0.9784 35! 5 52.15 6694 1.0:'..59 4283 6261 7087 .6 9067 4101 4869 7637 0.9600 4887 5802 8086 .7 .8259 .4567 .4436 0.8.504 .874.5 .5442 .5285 0.9004 .8 7369 4987 3957 0.9286 7802 5942 4715 0.9832 .9 6405 5357 344(1 0.9975 6782 6383 4099 1.0.562 1.0 5377 5674 2888 1.0.56.5 5693 6760 3441 1.1186 1.1 .4296 5934 .2307 1.1049 .4.548 .7070 .2749 1.1699 1.2 3171 613.5 1703 1.1422 33.58 7309 3029 1 2094 1.3 2015 6274 10S2 1 1682 2133 7475 1289 1 . 2369 14 0839 6:;5i 04.50 1.1825 0888 7567 0537 1.2.520 1.5 0000 6367 0000 1.1855 0000 7586 0000 1.2552 \n G8 HYPERBOLIC FUNCTIONS. Table II. Values of cosh (x + iy) and sinh (jt + iy). X = .8 X = •9 y a b c d a b c d 1.3374 0000 .8881 .0000 1.4:131 .0000 1.0265 .0000 .1 1 . 3:508 0887 8837 1335 1.4259 1 025 1.0214 1431 .2 1.3108 1764 8:u4 26:) 7 1.4045 2031» 1.0061 2847 .3 l.'iTTG 2625 8481 3952 1.3691 3034 0.9807 4235 .4 l.?319 .3458 .8180 .5208 1.3200 .3997 .9455 .5581 .5 1.1737 4258 7794 6412 1 2577 4921 9008 6b7i .6 1 . 1038 5015 73o0 7552 1 18-28 5796 8472 8092 .7 1.0229 5721 6793 8616 1.0961 6613 7851 9232 .8 .9:118 .6371 .6188 0.9595 .9984 .7364 .7152 1.0280 .9 8314 6957 5521 1.0476 8908 8041 6381 1.1226 1.0 7226 7472 4798 1.1254 7743 8638 5546 1.2059 1.1 6067 7915 40-28 1.1919 6:)00 9148 4656 1.2772 1.2 .4S46 .8-278 .3218 1.2465 .5193 0.9568 .3720 1.3357 1.3 3r)78 8557 2:176 1.2887 3834 0.9891 2746 1 3809 1.4 2273 8752 1510 1 3180 2436 1.0124 1745 1.4122 1.5 0946 8859 0628 1.3341 1014 1.0239 0726 1.4295 in 0000 .8881 0000 1.3374 0000 1.0265 0000 1.4331 X = 1.2 X = I 3 y a b C (/ a b c d 1.8107 .0000 1.5095 .0000 1.9709 .0000 1.6984 .0000 .1 1.8(tl6 1507 1.5019 18(iS 1.9611 1696 1 6899 1968 .2 1.7746 29i)9 1.4794 35! IS 1.9:116 3374 1.6645 3916 .3 1.7298 4461 1.4420 5351 1.88-29 5019 1.6225 5824 .4 1.6677 .587S 1.3903 0.70.-,l 1.8153 .6614 1.5643 0.7675 .5 1.5890 7237 1.3-247 8681 1.7296 8142 1 4905 9449 .6 1.4944 8523 1.2458 1.0-224 1 6267 9590 1 4017 1.1131 .7 1.3849 9724 1.1 54f) 1.1 66f) 1 5074 1.0941 1.2990 1.2697 .8 1.2615 1.08-28 1.0.-)17 1.2989 1.3731 1.2183 1.1833 1.4139 .9 1.12")-, 1.18-24 9:58:^ 1.4IS:{ 1 22.-.1 1.3304 1.0557 1.5439 1.0 0.978:? 1.2702 0.8156 1.5-236 i.oc.iij 1 4291 0.9176 1.6585 1.1 0.8213 1.3452 0.6847 1.6137 0.8940 1.5i:J6 0.7704 1.7565 1.2 .6-)61 1.4069 .547(1 1 6S76 .7142 1.58:!0 .6154 1 8370 1.3 4844 1.454 i 4088 1.7447 5272 1 6365 4.-)43 1 8991 1.4 3078 1.4S75 2566 1.7S43 33.10 1 6737 28S7 1.9422 1.5 1281 1.5057 1068 1.80(il 13!»4 1.6941 1201 1.9660 \Tt 0000 1.5095 0000 1.8107 0000 1.6984 0000 1.9709 TABLES. Table II. Values of cosh (x -|- iv) and sink (jt -)- iy.) 69 X = I.O X = I.I a i> c J a d c d y 1.5431 .0000 1.1752 .0000 1.6685 .0000 1.3356 .0000 1.5354 1173 1.1693 1541 1.6602 1333 1 3--'90 1666 .1 1.5128 2335 1.1518 3066 1.6353 2654 1.3090 3315 .2 1.4742 3473 1.1227 4560 1.5940 3946 1.2760 4931 .3 1.4213 .4576 1.0824 .6009 1.5368 .5201 1 2302 6498 .4 1.3542 5634 1.0314 7398 1.4643 6403 1.1721 0.7099 .5 1.2736 6636 0.9699 8718 1.3771 7542 1 1024 0.9421 .6 1 . 1803 7571 0.8988 9941 1.2762 8604 1.0216 1.0749 .7 1.0751 0.8430 .8188 1 1069 1.1625 9581 .9306 1 1969 .8 9592 9206 7305 1.2087 1.0372 1.0462 8302 1.3(»70 .9 0.8337 0.9,s89 6350 1.298.5 0.9015 1.1239 7217 1.4040 1.0 0.6999 1.0473 5331 1.3752 7568 1.1903 6058 1.4870 1.1 .5592 1.0953 .4258 1.4382 .6046 1 2449 .4840 1.5551 1.2 4128 1.1324 3144 1.4869 4463 1.2870 3)73 1.6077 1.3 2623 1.1581 1998 1.5213 2836 1.3162 2270 1.6442 1.4 1092 1.1723 0831 1 5392 1180 1.3323 0945 1.6643 1.5 0000 1.1752 0000 1 5431 0000 1.3356 0000 1.6685 \n X = 1-4 X = 1.5- a d 53, 63, 70. Gunther's Die Lehre, etc., 61, 71. Haskell on fundamental notions, 72. Houel's notation, etc., 30, 31, 71. Hyperbola, 7 et seq., 30, 37, 44, 60. Hyperbolic functions, defined, 11. addition-theorems for, 16. applications of, 46 et seq. derivatives of, 20. expansions of, 23. exponential expressions for, 24. graphs of, 32. integrals involving, 35. Hyperbolic functions of complex num- bers, 38 el seq. relations among, 12. relations to gudermanian, 29. relations to circular functions, 29, 42. tables of, 64 et seq. variation of, 20. Imaginary, see complex. Impedance, 34. Integrals, 35. Interchange of hyperbolic and circular functions, 42. Interpolation, 30, 48, 50, 59, 62. Intrinsic equation, 38, 47, 49, 51. Involute of catenary, 48. of tractory, 50. Jones' Trigonomet.y, 52. Kennclly on alternating currents, 38, 58. Kcnnelly's chart, 46, 58; treatise, 73. Liisant's Essai, etc., 61, 71. Lambert's notation, 30. place in the history, 70. Leakage of conductor, 55. Limiting ratios, 19, 23, 32. Logarithmic curve, 60. expressions, 27, 32. Love's elasticity, 61. Loxodrome, 52. Macfarlane on definitions, 72. Maxwell's Electricity, 61. Measure, defined, 8; of sector, 9 et seq. IMercator's chart, 53. Miller, W. E., Tables, etc., 73 Modulus, 31, 46. Moment of inertia, 61. Multiple values, 13, 16, 45. Newton, reference to, 71. Numbers, complex, 38 et seq. Ohm, reference to, 71. Operators, generalized, 39, 56. Parabola, 38, 61. Periodicity, 44, 62. INDEX. Ti Permanence of equivalence, 41. Phase angle, 56, 59. Physical problems, 21, 38, 47 et seq. Potential theory, 61. Product -series, 43. Pure imaginary, 41. Ratios, characteristic, 10. limiting, 19. Rayleigh's Theory of Sound, 61. Reactance of conductor, 58. Reduction formula, 37, 38. Relations among functions, 12, 29, 42. Resistance of conductor, 56. Rhumb line, 53. Riccati's place in the history, 71. Schellbach, reference to, 71. Sectors of conies, 9, 28. Self-induction of conductor, 55. Series, 23, 31. Spheroid, area of oblate, 58 Spiral of Archimedes, 60. Steinmetz on alternating currents, 38. Susceptancc of dielectric, 58. Tables, 62, 73. Terminal conditions, 54, 58, 60. Tractory, 48, 51. Van Orstrand, C. E., Tables, 73. Variation of hyperbolic functions, lA. Vassal's Tables, 63. Vectors, 38, 56. Vibrations of bars, 61, Wheeler's Trigonometry, 65. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. MQ ;aO 1842 DEC 14 A9^^ OCT 4 ^m i:s ^^ -prr^ FEB 8 19^"^ f\U G 91985 R - ^Ci^1VEP £ ^ JUL i '^ 13H ^UWAtiON PP g^ -i*r LD 21-100m-7, '40 (6936s) GENERAL UBBARY- U.C. BERKELEY BDomopaoD ^42995 ; ■ o<. 1 A - UNIVERSITY OF CALIFORNIA LIBRARY ,— .^*i;* s.-i- ,^j.^ . . mm iiii mm iiiii