m v','!. «;>;■ li .'' M^3 i^iU CORNELL UNIVERSITY LIBRARY 3 1924 064 123 478 DATE : DUE >«^^ FfB 0^ ^002 hi A n -1 IIAR ) APR 2^ UOQ? GAYLOflO l**INTeOIN U.t.A. Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924064123478 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1992. Olorn^U llntu^rfittu Ktbrarg BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF 1891 ■ATHEaiATIOi i V.^3<^ ; ';' :„. „,. 9306 MATHEMATICAL MONOGRAPHS EDITED BY 1 Mansfield Merriman and Robert S. Woodward. | Octavo, Cloth. No. '• History of Modern Mathematics. By Davu) Eugene Smith. Ji.oo net. No. 2. Synthetic Projective Geometry. By George Bruce Halsted. Ji.oo net No. 3. Determinants. By Laenas Gifford Weid. Ji.oo net. No. 4. Hyperbolic Functions. By James Mc- Mahon. Ji.oond. No. 5. Harmonic Functions. By William E. Byekly. Ji.oo net. No. 6. Qrassmann's Space Analysis. By Edward W. Hyde. Ji.oo net. No. 7. Probability and Theory of Errors. By Robert S. Woodward. Ji.oo net. No. 8. Vector Analysis and Quaternions. By Alexander Macfarlane. Ji.oo net. No. 9. Differential Equations. Bv William Woolsey Johnson. Ji.oo net. No. 10. The Solution of Equations. By Mansfield Merriman. Ji.oo net. No. U. Functions of a Complex Variable. By Thomas S. Fiske. $i.oo net. No. 12. The Theory of Relativity. By Robert D. Carmicuael. Jj.oo net. No. 13. The Theory of Numbers. By Robert D. Carmichael. Ii.oo net. No. 14. Algebraic Invariants. By Leonard E. Dickson. $1.25 net. No. IS. Mortality Laws and Statistics. By Robert Henderson, #1.25 net. No. 16. Diophantine Analysis. By Robert D. Carmichael. *i.2S net. No. 17. Ten British Mathematicians. By Alsz- ANDER Macfarlane. Jt.2s net. No. 18. Elliptic Integrals. By Harris Hancock. J1.25 net. PUBLISHED BY JOHN WILEY & SONS, Inc., NEW YORK. | CHAPMAN & HALL, Limited, LONDON. 1 MATHEMATICAL MONOGRAPHS EDITED BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD No. 18 ELLIPTIC INTEGRALS HARRIS HANCOCK Professor of Mathematics in the University of Cincinnati FIRST EDITION FIRST THOUSAND NEW YORK JOHN Wn.EY & SONS, Inc. London: CHAPMAN & HALL, Limited 1917 Copyright, 1917 BY HARRIS HANCOCK PRESB OF BRAUNWORTH & CO. BOOK MANUFACTURERS BROOKLYN, N. V. CONTENTS PAGE Introduction S CH.VPTER I Elliptic Integrals of the First, Second, and Third Kinds. The Legendre Transformations g CHAPTER II The Elliptic Functions 24 CHAPTER III Elliptic Integrals of the First Kind Reduced to Legendre's Normal FORil 41 CHAPTER IV Numerical Computation of the Elliptic Integrals of the First and Second Kinds. Landen's Transformations 65 CHAPTER V Miscellaneous Examples and Problems 88 CHAPTER VI Five-place Tables 92 Index 103 INTRODUCTION The editors of the present series of mathematical mono- graphs have requested me to write a work on elliptic integrals which " shall relate almost entirely to the three well-known elliptic integrals, with tables and examples showing practical appHcations, and which shall fill about one hundred octavo pages." In complying with their request, I shall limit the monograph to what is known as the Legendre-Jacobi theory; and to keep the work within the desired number of pages I must confine the discussion almost entirely to what is known as the elUptic integrals of the first and second kinds. In the elementary calculus are found methods of integrating any rational expression involving under a square root sign a- quadratic in one variable; in the present work, which may be regarded as a somewhat more advanced calculus, we have to integrate similar expressions where cubics and quartics in one variable occur under the root sign. Whatever be the nature of these cubics and quartics, it will be seen that the integrals may be transformed into standard normal forms. Tables are given of these normal forms, so that the integral in question may be calculated to any degree of exactness re- quired. With the trigonometric sine function is associated its inverse function, an integral; and similarly with the normal forms of elliptic integrals there are associated elliptic functions. A short account is given of these functions which emphasizes their doubly periodic properties. By making suitable trans- formations and using the inverse of these functions, it is found that the integrals in question may be expressed more con- cisely through the normal forms and in a manner that indi- cates the transformation employed. 5 6 ELLIPTIC INTEGRALS The underlying theory, the philosophy of the subject, I have attempted to give in my larger work on elliptic functions. Vol. I. In the preparation of the present monograph much use has been made of Greenhill's Application of Elliptic Func- tions, a work which cannot be commended too highly; one may also read with great advantage Cayley's Elliptic Functions. The standard works of Legendre, Abel and Jacobi are briefly considered ih the text. It may also be of interest to note briefly the earlier mathematicians who made possible the writings just mentioned. The difference of two arcs of an ellipse that do not over- lap may be expressed through the difference of two lengths on a straight line; in other words, this difference may be expressed in an algebraic manner. This is the geometrical signification of a theorem due to an Italian mathematician, Fagnano, which theorem is published in the twenty-sixth vol- ume of the Giornale de' letterari d'ltalia, 1716, and later with numerous other mathematical papers in two volumes under the title Produzioni mathematiche del Marchese Giulio Carlo de' Toschi di Fagnano, 1750. The great French mathematician Hermite {Cours, redige par Andoyer, Paris, 1882) writes " Ce resultat doit etre cite avec admiration comme ayant ouvert le premier la voie a la theorie des fonctions elliptiques." Maclaurin in his celebrated work A Treatise on Fluxions, Edinburgh, 1742, Vol. II, p. 745, shows " how the elastic curce may be constructed in all cases by the rectification of the conic sections." On p. 744 he gives Jacob Bernoulli " as the cele- brated author who first resolved this as well as several other curious problems " (see Acta Eruditorium, 1694, p. 274). It is thus seen that the elUptic integrals made their appearance in the formative period of the integral calculus. The results that are given in Maclaurin's work were sim- plified and extended by d'Alembert in his treatise Recherches sur le calcul integral. Histoire de I'Ac. de Berlin, Annee 1746, pp. 182-224. The second part of this work, Des diferentielles qui se rapportent a la rectification de Vellipse ou de I'hyperbole, INTRODUCTION 7 treats of a number of differentials whose integrals through simple substitutions reduce to the integrals through which the arc of an ellipse or hyperbola may be expressed. It was also known through the works of Fagnano, Jacob BernouUi and others that the expressions for sin (a+^S), sin (a — /3) etc., gave a means of adding or subtracting the arcs of circles, and that between the limits of two integrals that express lengths of arc of a lemniscate an algebraic relation exists, such that the arc of a lemniscate, although a transcendent of higher order, may be doubled or halved just as the arc of a circle by means of geometric construction. It was natural to inquire if the ellipse, hyperbola, etc., did not have similar properties. Investigating such properties, Euler made the remarkable discovery of the addition-theorem of elliptic integrals (see Nov. Comm. Petrop., VI, pp. 58-84, 1761; and VII, p. 3; VIII, p. 83). A direct proof of this theorem was later given by Lagrange and in a manner which elicited the great admiration of Euler (see Serret's CEuvres de Lagrange, T. II, p. 533). The addition-theorem for elliptic integrals gave to the elliptic functions a meaning in higher analysis similar to that which the cyclometric and logarithmic functions had enjoyed for a long time. I regret that space does not permit the derivation of these addition-theorems and that the reader must be referred to a larger work. The above mathematicians are the ones to whom Legendre refers in the introduction of his Traiie des fondions elliptiques, published in three quarto volumes, Paris, 1825. This work must always be regarded as the foundation of the theory of elliptic integrals and their associated fimctions; and Legendre must be regarded as the founder of this theory, for upon his investigations were established the doubly periodic properties of these functions by Abel and Jacobi and indeed in the very form given by Legendre. Short accounts of these theories are found in the sequel. For more extended works the reader is referred to Appell a ELLIPTIC INTEGRALS et Lacour, Fonctions elliptiques, and to Enneper, Elliptische Funktionen, where in particular the historical notes and list of authors cited on pp. 500-598 are valuable. Fricke in the article " Elliptische Funktionen," Encylcopddie der mathematischen Wissensckafien, Vol. II, gives a fairly complete list of books and monographs that have been written on this subject. To Dr. Mansfield Merriman I am indebted for suggesting many of the problems of Chapter V and also for valuable assistance in editing this work. I have pleasure also in thanking my colleague, Dr. Edward S. Smith, for drawing the figures carefully to scale. Harris Hancock. 2365 Auburn Ave., Cincinnati, Ohio, October 3, 1916. ELLIPTIC INTEGRALS CHAPTER I ELLIPTIC INTEGRALS OF THE FIRST, SECOND AND THIRD KINDS. THE LEGENDRE TRANSFORMATION Art. I. In the elementary calculus are studied such integrals as I — , I -. 7--, etc., where s^ = ax^ + 2bx+c. In general J s J {ax+b)s the integral of any rational function of x and 5 can be trans- formed into other typical integrals, which are readily integrable. Such types of integrals are n dx n dx p dx ^^^ J Vl-x2' Jo Vl-X"' X Vx^ + 1 In the present theory instead of, as above, writing s^ equal to a quadratic in x, we shall put s^ equal to a cubic or quartic in X. Suppose further that F{x, s) is any rational function of X and 5 and consider the integral | F{x, s)dx. Such an integral may be made to depend upon three types of integral of the form /dx Cx^dx J r dx These three types of integral, in somewhat different notation, were designated by Legendre, the founder of this theory, as elliptic integrals of the first, second, and third kinds respect- ively, while the general term " elliptic integral " was given by him to any integral of the form | F{x, s)dx The method of expressing the general integral through the three types of inte- 9 10 ELLIPTIC INTEGRALS gral as first indicated by Legendre, may be found in my Elliptic Functions, Vol. I, p. i8o. Art. 2. First consider integrals of the form ex f- VRix)' ^'^ which, as will be shown, reduce to a definite typical normal form,* when R(x) is either of the third or fourth degree in x. Suppose that R(x) is of the fourth degree, and write R{x) =aoX*+aiX^+a2X^+a3X+ai, where ao, ai, . . ., are real constants. It is seen that (i) may be written _i_ C^ (2) where X, when decomposed into its factors, is X=±{x-a){x-^){x--y)(x- 6), and Vao is a real quantity. If the roots are all real, suppose that q:>/3>7>5; if two are complex, take a and /3 real and write 7 = p+zV, 5 = p — i(r, where i = V—i; and if all four of the roots are complex, denote them by a = ij.+ii', fi-ii — iv, 'Y = p-\-ia, b = p — i(T. In the present work the variable is taken real unless it is stated to the contrary or is otherwise evident. We shall first so transform the expression X that only even powers of the variable appear. With Legendre (loc. cit., p. 7), write x = P±^ (3) 1+3' ^^' It follows at once that dx _ {q—p)dy (4) * See Legendre, Traite des fonciions elUpliques, T. I., p. ii, et seq.; Richelot, Crelle, Bd. 34, p. i; Enneper, EllipHsche Functionen, p. 14. INTEGRALS OP FIRST, SECOND AND THIRD KINDS 11 where Y=\p-a+{q-a)y\[p-^+{q-^)y\[p-y + {q-'r)y\[p-b+{q-5)y\. (5) As all the results must be real, it will be seen that real values may be given to p and q in such a way that only even powers of y appear on the right-hand side of (s). If in this expres- sion we multiply the first and second factors together, we have {p~a){p-^) + {q-a){q-fi)y^ provided {p-a){q-ei) + {p-0){q-a)=o; .... (6) and similarly if {p-y){q-b) + {p-5){q-y)=o, .... (7) the product of the third and fourth factors of (5) is (/'-7)(/'-5) + (9-7)(?-5)/. From (6) and (7) it follows that pq+aP=t±l{a+fi), 2 and 2 From the last two equations, it also follows that p+q _ aP-y8 al3{y+d) -ydja+H) ,^. 2 a+/3-7-5' ^^ a+p-y-S From (8) it is seen that the sum and quotient of p and q are real quantities whatever the nature of the four roots a, 0, y, and S may be; and further from (8) it is seen that g-pY_ (a-y){a- 8)i0-y)(0- 8) _ _ ^^^ 2 / (a-fjS — 7-5)2 which is always a positive quantity. It follows that q—p is a real quantity, and that p and q are real. The equations (8) and (9) cannot be used if a+0 = y+S. 12 ELLIPTIC INTEGRALS In this case, as is readily shown, instead of the substitution (3), we may write .a+0 .y+S x = y-\ = y+- . 2 2 It follows that (5) takes the form F = (±w2±M2j,2)(±r2±Z2y2), where m, n, r, and / are real quantities. The expression (4) then becomes dx ^{q — p)dy _ dy VX~ V^' ~ fV±ii±g^y^){idzh^fy (10) where/, g, and h are essentially real quantities. In the expression on the right-hand side, suppose that h>g and put hy = t, and 7 = c, where c < i. h It follows that dx dt VX /^V±(l±/2)(i±c2/2) (11) It is seen that under the radical there are eight combi- nations of sign. With Legendre, loc. cit.. Chap. II, and Enneper, p. 17, a table will be given below from which it is seen that the corresponding functions may be expressed by means of trigonometric substitutions in the one normal form dx \ d I dv / X VX JW"Vi-yfe2sin2 MV(i-z)2)(i-Pz,2) where M is a real quantity and v = sin . The quantity k, called the modulus, is also real, and sit- uated within the interval o^k^i. Of the expressions under the root sign ^ —{i-\-f){i+c^t^) may be neglected, since R{x), assumed to be positive for at least some real value of the original x, cannot be transformed into a function that is always negative by a real substitution. Art. 3. Writing A<> = Vi — ^2 gjjj2 ^ and defining the com- INTEGRALS OF FIRST, SECOND AND THIRD KINDS 13 plementary modulus k' by the relation k'^-\-k'^ = i, the fol- lowing table results: dt d(t> II. III. IV. V. VI. via. V(l+/2)(l+c2/2) A.^' t = tan 0, F = I-C2 dt -k'd4> t = cos (^, I+C2 V(l-/2)(l+c2i2) A, ^^- I 2 I+c2 V(f2_i)(i+c2/2) A./, ' dt -kd

sec C2 ^ ~I+C2 V(l+/2)(c2;2_j) A./.' dl d(i> V(l-/2)(i_c2/2) A0' t = sin 0, F = C2 I t = ^, . F = c2 V(/2_i)(c2/2_i) A0' csin0 VII. -=4_=-^, t- = sin^^+'^, k^ = .-c^ V{t2-i)(i-cH^) A' ^ c2 The formulas VI and Via have the same form; in VI it is necessary that /£i, while in Via it is required that / = -. Art. 4. It is seen that the eight transformations in the table are all of the form A +B sin2 4) ^ ~C+D sm^ MV(l-I)2)(i_;fe2,,2)' 14 ELLIPTIC INTEGRALS where z; = sin <^. These substitutions and reductions are given in full in Chap. III. dv The radical in — is real for real values of V that are i° less than unity and 2° greater than t- In the latter case, write J' = l~' a-^d then dv ds V(l-Z)^)(l-yfeV) V(l-52)(l-/fe252) In this substitution as v passes from - to 00 , the variable 5 k passes from i to o. It is therefore concluded that by making the real sub- stitution (i), the differential expression * dt may be reduced to the form I dv MV{i-v^){i-kWj' where the variable v lies within the interval o . . . i. Such transformations fail if the expression under the root contains only even powers of t, the two roots in i^ being imaginary, i.e., if R{x)=Ax^+2Bx^+C, where B^—AC = A{k, 4>), there results the Legendre notation as normal integrals of the first kind of the second kind, Eik, )=] \{k,,f>)d, and of the third kind, "^"'^''^^=Jo (i+«sin2 0)A(^,,»- The modulus k is omitted from the notation when no particular emphasis is put upon it. The evaluation of these integrals is reserved for Chap. rV. However, the nature of the first two integrals may be studied by observing the graphs in the next article. 16 ELLIPTIC INTEGRALS Art. 6. Graphs of the integrals F(k, (t>) and E{k, 4>). In Fig. I there are traced the curves y=- Let values of be laid off upon the A''-axis. It is seen that the areas of these curves included between the x-axis and the ordinates corresponding' to the abscissa <^ will represent the integrals F{k, 4>) and E{k, 4>). See Cayley, Elliptic Functions, p. 41. If ^ = 0, then A0 = i, and the curves y = A become the straight line y = i; while the corresponding integrals Fig. I. F{(t>), E{4>) are both equal to and are represented by rectangles upon the sides and i. When o<^ lies below it. As (j) increases from zero, the integrals F{(t>) and £(<^) increase from zero in a continuous manner, the integral F{^) being always the larger. Further, for a given value of 4>, as k in- creases the integral F{4>) increases and E(4>) diminishes; and conversely as k decreases, F{) increases. li Flk,-\ be denoted by Fi{k), or Fi, and if we put INTEGRALS OF FIRST, SECOND AND THIRD KINDS 17 Ei=E(k,-], it is seen that when k = o, f(o,-]=Fi{o)=- =Ei{o). When k has a fixed value, it is often omitted in the notation. Fi and £i are called complete integrals. It is evident that both curves are symmetric about the line y=^Tr and that for a fixed value of k, it is sufficient to 0.0 1 ■ / 1 / 1 / p/ ^ ^ / ^ // •§/ / / / / %/ / /J y / '/ / / A Y / '/ A / /. // // / i i v/ V/ / A V/ A Y \^^ t r l/ / r / r— 10 20 30 40 50 60 70 Fig. 2. The Elliptic Integral F{e, from o to |x. For F{ir) = 2Fi, and for any value =a, F{a)=F{ir)—F{T-a), or F{ir—a) = 2Fi-F{a). In the latter formula, as a diminishes from - to o, F{tt>) increases 2 from - to TT. 2 18 ELLIPTIC INTEGRALS Further noting that F{—a) = —F{a), the formula F{a)=F(T)+F{a-n), = 2Fi+F{a-T), a.u / / y y / 5^ '■i^ / / . ^ ^ y ^ ^ 1.5 — ■ >^ ^ ^ __^ ^ -!« ^ ^ I — -^ 1 - ^ 1 • 1 1.0 ___ 'Qi ^ ^ ' ! 1 1 j 1 1 4" 1 1 i i 1 ' i ; ' i 1 I ?(, 0..5 ■" -"; 1 1 j 1 20- ! \y\ nn 0' 10 20 30 40 50 CO TO Fig. 3. The Elliptic Integral F(9, 0). k =sin e. 80 90 gives the values of F( = 27r, etc. In general, F{mvd=a) = 2mFidzF{a), E{nnr±a) = 2mEi±E{a). Art. 7. When ^ = i, the graphs of the two curves in Fig. i are entirely changed, the curve >' = A0 becoming 3) = cos = \i^, and between the values \-k and \-k there is a branch lying wholly below the line y= — i, the ordinates for the values (^ = §7r and <^ = fir being = — oo . For the values fr and fir there is a branch lying wholly 2.0 r 10 20 30 40 50 60 TO Fig. 4. The Elliptic Integral £(e, >= +i, the ordinates for fir and fir being + oo and so on. Corresponding to the first curve, £(<^) = I cos <^ and consequently £i = i . This, taken in connection with what was given above, shows that as k increases from o to i, £i decreases from ^tt to i . 20 ELLIPTIC INTEGRALS For the second curve, ^(<^)=r sec (t>d(t> = log tan ( — | — 4 2 so that Fi is logarithmically infinite when k =i; and this taken in connection with what was given above, shows that 1.0 ao / / / < w /: / ^> / «^^ ^ /. '/ ^ L^ Q^90 /, V y ■" — /> % y^ ^ ^ i> 4 ^ . J* r > r / r / 10 20 30 40 50 CO VO Fig. 5. The Elliptic Integral £(9, 1^). Iz =sin B. 80 90 -^0 as ^ increases from o to i, Fi increases from \-k to logarithmic infinity. Art. 8. In Figs. 2-5 are added other graphs of the integrals F{k, 4>) and E{k, ) which require no further explanation. At the end of the book are found tables which give the values of these integrals for fixed values of k and 0. INTEGRALS OF FIRST, SECOND AND THIRD KINDS 21 EXAMPLES I. A quartic function with real coefficients is always equal to the product of two factors M = l-\-2mx+nx'', N=\+2iJix-\-i'x^, where all the coefficients are real. Remove the coefficient of * in Af and N in the integral dx /; Vmn' and thereby reduce this integral to {q-p)dy J V{ay'+b){a'y^+b'y p-\-qy by a substitution a;= — ; — . and show that p and q are real. Legendre, i+y' Vol. I., Chap. II. ^f(x)dx 2. Show that fi Vs{x) may be reduced to the integral g{z)dz h V4z' — g^z — g3 where / and g are rational functions of their arguments and S{x) = ax'-\-sbx^+iCX+d. b The substitution required is x—mz+n, where n= — - , am' =4. Appell et Lacour, p. 247. 7,. Knowing a real root a 6f R(x) . find the form of , , when x=a-\ — . •^ * Vr{x) y Write R(x) = {x—a)(cx'+Cix'+CiX+C3). Leuy, p. 77. 4. Show that the substitution /- (i+sin 0) + Vc(i — sin *) y/ cx= 7= (i— sin 0)+ V c(i+sin ■/>) transforms dx . (i+VJyd^- mto V(x^-i){i-cV) 2Vi-A2sin2.fr' 22 ELLIPTIC INTEGRALS where i+Vc 5. Show that by the substitution x= — ; — a/-, the integral in which R{x) has the form \'+2\n cose x^+m'^'', is transformed into one which has under the radical an expression of the form w^(i+gy)(i+^'y^)- Legendre, Vol. I, Chap. XI. 6. If the four roots of X are all real, such that a > /3 > 7 > 5, show that the substitution T(5-5)-s(/3-7)sin2)sin2 dx . 2 d(j> mto Vx V(a-7)(^-5) Vi-zfe^sin^^' 3 — y ct — 5 *2 = and y^>7, show that the substitution y=y+{0—y) sin^ transforms dy , 2 d mto Vf Va-yVi-kHm''' where /3— 7 *' = and 7<>' x—li N i+cos transforms dx . I d(t> mto V{x-a)ix-i3)[{x-py+ where I {M+Ny-(,a-0)^ 2 2MN and 00 >a:>a: or |3>x> — 00. INTEGRALS OF FIRST, SECOND AND THIRD KINDS 23 9. Show that the substitution s—ei transforms the integral r dt_ J V(/-e0(/-6 -e2){t-ez) into itself. 10. Show that the substitutions c c ^ z—ai 02 — at Ui — at at—ai t = , k^ = ; 2 — 02 ai—di di — di 02— ai 1 I o transform =. mto dz ±V(a n dt (ai-a2){ai-a,) j Jai v(z-ai)(z-C2)(z-a3)(2-ai) II. Prove that the substitution z — ai fls — fli t — a2 at — flj 2—02 O3 — <32 ' — fll 04- fli transforms r ), n(w, k, 4>) were called by Legendre elliptic functions; these quantities are, however, elliptic integrals. It was Abel * who, about 1823, pointed out that if one studied the integral m as a function of X in C' dx r* d ■ ^ r \ u= I — = I =, a; = sm = amplitude of u, and written cj> = amu. Considered as a function of u, we have a; = sin <^ = sinawM, and associated with this function are the two other elliptic functions cos <^ = cos am u and A4> = Aamu = Vi^ k^ sin^ = sn u. Vi —x^ = cos = cnu. Vi — k^x^ =A4> = dnu, ' Abel (CEuvres, Sylow and Lie edition, T. I., p. 263 and p. 518, 1827-30). 24 ELLIPTIC FUNCTIONS 25 It follows at once that dn^u-\-k'^sn^u = i. From (i) results -3- = — or -^ = A<^, so that -— amu=Aam,u = dnu. a4> A^- = cnudnu, du du du -^cn ti= —snu dnu, du -r-dn u= — k^sn u en u. du Further, if m = o, then the upper limit <^ = o, so that am 0=0, and consequently, sn = 0, en = 1, dn = 1. If (j) be changed into — ^, it is seen that 11 changes its sign, so that am{ — u) = —am u, and sn{ — u) = ~snu, en{ — u)=enu, dn{ — u)=dnu. Art. 10. In the theory of circular functions there is found the numerical transcendent -k, a quantity such that sin - = i , 2 cos - = o. Writing C' dx . _, Jo Vi—x^ we have a; = sinM. Thus - may be defined as the complete 2 integral T_ n dx 2 Jo vr^ 26 ELLIPTIC INTEGRALS Similarly a real positive quantity K (Jacobi) may be defined through Jo V(i-a;2)(i-yfeV) Jo ^'^ r = F(k " (Art. 6). Associated with K is the transcendental quantity K', which is the same function of the complementary modulus k' as K is of k. The transcendental nature of these two functions of k and k' may be observed by considering the following infinite series through which they are expressed. If {i—k^ sin^ (t>)~' be expanded in a series, then Jn Vi- k^ sin^ (j) 2.4. . . . 271 where V2n = I sin'''^ d4>. In particular, if =mr-e, ELLIPTIC FUNCTIONS 27 then it becomes 2 n + l 2 'd4> I — = K: and if in the integral i we put = mr+e, then this integral is de Jo Ad It follows that £ nd^^ nd^^ pd0^ ^ r' ^ Jo A<^ jo A0 j^ A0 ■ ■ " j A r''^d , r''-+» d ^ , Jo A0 jo A-/* j.r A0 j„ A,^ Jo Ae' further, since any arc a may be put =Mir±/3, where /3 is an arc between o and -, we may always write J 2 a = mr±0 = am{2nK±u), or 2n flw K±am u = am{2nK±.u). Art. 12. Making use of the formula just written, it is seen that am K=-, 2 snK = i, cnK = o, dnK = k'. sn{u±2K) = -mu, cn{ii±2K) = -cnu, dn{u±2K)=dnu; sn{u±4K)=snu, cn{u:h4K)=cnu, dn{u±4K)=dnu. Note that 4K is a period of the three elliptic transcendents snu, cnu and (iraw; in fact, it is seen that 2ii: is a period of dn u and of ^— = tn u. Also note that cnu 28 ELLIPTIC INTEGRALS sn2K = o, cn2K= — I, dn2K = i, sn4K = o, cn4K = i, dn^K^i. Of course, the modulus of the above functions is k; and, since K' is the same function of k' as K is of k, we also have sn{u±2K' , k') = —sn{u, k'), sn(u±4K', k')=sn{u, k'), etc. Art. 13. The Gudermannian. As introductory to the Jacobi imaginary transformation of the following article, there is a particular case * where ^ = i . Then M = F(i, \4 2/ Here 0, considered as a function of u, may be called the Guder- mannian and written =gd u, the functions corresponding to sn u and en u being sg u and eg u. Then "" = tan /-+-^ = ^+^^'^ '^/^ _ i+sin < /> _ cos (^ V4 2/ I — tan<^/2 cos i — sin^' or, CgM ' 1+SgU CgU It follows that <^g"^ « I -u == ^= — r— = sechM, e +e cos ZM cosh m and €" — € " .sin iu sinh M sgu= — -—zz = - i = — T— = tanh u. e +e costu coshw These formulas may be written sgu= —i tan iu, cgu = i/cos iu, tgu= —i sin iu ; sin iu = itgu, cos iu = i/cgu, ta.n iu = isg u. *See Gudermann, Crelle, Bd. i8, pp. i, et seq.; see also Cayley, loc. cit. p. 56; Weierstrass, Math. Werke I, pp. 1-49 and the remark p. 50. ELLIPTIC FUNCTIONS 29 The above relations may also be derived by considering two angles 6 and connected by the equation cos Scos , cos d = i/cos 4>, tan d = ism (j>, Further, there results, cosedd = isec^(i>d(f), or de=i — ~. cos ct> It follows that e = nogtan('^+^y Then, by assuming that cj)=gdu,vfe have e = iu, and conse- quently the foregoing relations. Art. 14. Jacobfs Imaginary Transformations* Writing sine=ztan<^, cos 6 = -^—, sin0=-ztane, A(e, k)= ^^'^' \ cos cos (/) , ,„ . del) J, r^ de . r* dd> we have dd = z and . 1 v- Jo A(e, ^) Jo cos Jo A(e, ^) Jo A(0, ^')' * d,p T^=u, SO that <^ = aw(M, ^'). there results I I —, — TT=iu, and 6 = am iu. '0 A(», ^) These expressions, substituted in the above relations, give sn{iu, k) =i tn{u, k'), cn{iu, k) = A' cn{u, k') dniiu,k)J-^^. cn(u, k ) From this it is evident that the two functions en and dn have real values for imaginary values of the argument, while sn{iu) is an imaginary quantity. * Jacobi, Fundamcnta Nova, § ig. See also Abel, CEuvres, T. I., p. 272. 30 ELLIPTIC INTEGRALS Among the trigonometric and exponential functions, we have, for example, the relation cos m — , where the argument of the trigonometric function is real while that of the exponential function is real. We note that an elliptic function with imaginary argument may be expressed through an elliptic function with real argument, whose modulus is the complement of the original modulus. Art. 15. From the formulas of the preceding article it follows at once sw[i(M+4iir'), k]=itn{u+4K', k')=sn{iu, k), and also cn{iu+4iK', k) =cn{iu, k), dniiu+^iK' , k) =dn{iu, k). If in these formulas iu be changed into u, we have sniu±4iK', k)=sn{u, k), cn{u±4iK', k)=cn{u, k), dn{u±4iK' , k)=dn{u, k). It also follows that sn{u±4iK, k')=sn(u, k'), etc. If in the formula sn{iu) =i tn{u, k'), we put U + 2K' in the place of u, then sn{iu+2iK', k) =i tn{u+2K\ k')=itn{u, k') =sn iu. Changing iu to u, we have sn{u±2iK')=snu, cn{u±2iK') = —cnu, dn{u±2iK') =^ —dnu, and sn(2iK')=o, cn{2iK') = — i, dii{2iK') = — i. The modulus k is always understood, imless another modulus is indicated. ELLIPTIC FUNCTIONS 31 It follows at once that sn{ud=4iK')=sn u, cfi{udz4iK')=cn u, dn{u±4iK')=dn ii. and sti UiK') = o, en {4iK') = i . dn{4iK') = i . It is also seen that sn{u±2K±2iK') = —sn u, S}i{uzi=4K±4iK')=s>i u, etc. In particular, notice that the periods of smi are 4K and 21 K', the periods of cnu are 4A" and 2K+2iK', the periods of dmi are 2K and 4/A". Art. 16. Periodic Functions. Consider the simple case of the exponential function c" and suppose that u = x + i\. It may be shown that f""^-'"=f" for all values of u; for it is seen that e'' = c''"'"''' = e''(cos v+/ sin v),. If we increase u by 27r/. then y is increased by 2ir and consequently e"+2'< = e^[cos {y+2T)+is\n (>'+27r)] =e^(cos y+i sin >')=£''. It follows that if it is desired to examine the function e", then clearly this function need not be studied in the whole w-plane, but only within a strip which lies above the A'-axis and has the breadth 2t\ for we see at once that to every point mq vvhich lies without this period-strip there corresponds a point wi within the strip and in such a way that the function has the same value and the same properties at mo and xii. Similarly it is seen that the two functions sin u and cos 11 have the real period air, and consequently it is necessary to study these functions only within a period-strip which lies adjacent to the I'-axis with a breadth 2ir. As already noted, Abel and Jacobi found that the elliptic functions had two periods. In the preceding article it was seen that snu had the real period 4K and the imaginary period 2iK'. 32 ELLIPTIC INTEGRALS On the Z-axis lay off a distance 4K and on the F-axis a distance 2K' and construct the rectangle on these two sides. Further suppose that the whole plane is filled out with such rectangles. 4K 2iK' 2iK 4K |8K Fig. 6. Then it will be seen that the function sn u behaves in every rectangle precisely as it does in the initial rectangle. Similar parallelograms may be constructed for the functions en u and dnu. See Art. 21. A-x XT ^ •.. • cose ^, ^ ^' sin e Art. 17. Next write sm.d> = , so that cos = , AS ' Ai9 ' and A0 = — . It follows that — = — — and consequently AS A4, Ad M J- r^= p^= p^_ r^=K-u, Jo A0 Je Ad Jo Ae Jo A0 if we put u -f: de Ad' or 6 = am u. It follows that (t> = am(K—u), and from the above formulas / „ s cnu ,,, , k' snu sn[K—u)=- — , cn{K—u)^ . an u dnu ' dn{K — u) = dnu In these formulas change - m to m and note that sn{~u) = —snu, etc. ELLIPTIC FUNCTIONS 33 It is seen that , ^ ^. cnu ^ sn{u±K) = ±-j — snK = i, dnu cn{u±K) = ^-- , cnK = o, dnu dn{u±K) = +-^, dnK^k'. dnu For the calculation of the elliptic functions, the above relations permit the reduction of the argument so that it is always comprised between o and ^K, just as in trigonometry the angle may be reduced so as to lie between o and 45° for the calculation of the circular functions. Art. 18. In the above formulas put iu in the place of u, and it is seen that en iu I sn{iu±K) = ± cn{iu±K) = =F dn{iu±K) dniu dn{u, k')' ik'sn(u, k') dn{u, k') k'cn{u, k') dn{u, k') Further, in the formulas sn iu = i tn{u, k'), etc., write u±.iK for u and it is seen that i cniu, k') sn{iuztiK', k) =i tg am{u±K' , k') cn{iu±iK' , k) = dn{iu±iK', k) = =F k sn{u, k')' dn(u, k') sn{u, k'Y I sn{u, k') In the above formulas change iu to u. We then have sniu±:iK')=-r , k snu cn{u±tK') = ^- , k snu dn{udtziK') = ^i cot am u. 34 ELLIPTIC INTEGRALS If in these formulas u = o, then sn(±iK') = 00 , cn{±iK') = oo , dn{±.iK') = oo . Further, if in the preceding formulas u-\-K be put in the place of u, then II I dnii sn{u+K±iK') k sn{u+K) k Lti ^<, cn{u+K±iK') = ^-^, kcnu dn{u+K±iK') = ±ik'tgamu; and from these formulas, writing, m = o, there results T ih' sn{K±iK')=\, cniK±iK') = =F^, dn{K±iK')=o. k R Art. 19. Note the analogy of the transcendent K of the elliptic functions to - of the circular functions. Due to the ■K 2 TT relation am{K~u)= — amu (Art. 11) Jacobi called the ampli- 2 tude olK — u the co-amplitude of u and wrote am{K — u)— coam u. It follows at once from the above formulas that cnu sin coam u = - — , dtiu k'snu cos coam u = — , dnu A coam u = - — . dnu sin coam(zM, k)=-—. — --, etc. an{u, k ) Art. 20. Remark. The results obtained for the imaginary argument have been derived by making use of Jacobi's imaginary transformation; and by changing iu into u we have implicitly made the assumption (proved in my Elliptic Functions, Vol. I, ELLIPTIC FUNCTIONS 35 Chaps X and XI) that the elliptic functions have the same properties for real and imaginary arguments. Art. 21. By a zero of a function, snu for example, we mean that value of u which, when substituted for u in sn ii, causes this function to be zero, while an infinity of a function is a value of u which causes the function to become infinite. In studying the following graphs note that on the bound- aries of the period parallelogram of sn u, there are six points at which this function becomes zero; but if the adjacent period parallelograms be constructed, it will be seen that only two zeros belong to each parallelogram. In fact, in each period-parallelogram there are two values of u which cause the function to take any fixed value; that is, any value being fixed, there are always two values of u which cause the function to take this value. From the following graphs it is seen that any real value situated within the interval — oo to -|- 00 is taken twice by each of the three functions sn u, en u, dn u. u = 2iK' u-iK ^ = V^l y-0 !/=-! y=o J/ = a3 '-i y--o ^=-^ S/=«> V-0 u = K 11=1 u=2K J-3K Fig. 7. y =sn(H, k). ZEROS 2mK+2niK' where m and n are any integers. INFINITIES 2mK+{2n + i)iK' PERIODS 4K, 2iK' 36 ELLIPTIC INTEGRALS u+iK' Fig. 9. y = sn{u-{-iK ). In Fig. 9, the value iK' coincides with the origin. K K + iK' K+2iK' Fig. loa. y = sn{iu+K). 3K 3K -1 +iK' 3K + 2iK' -1 Fig. 106. y = sa.{iu+2iK). u-,2iK' ELLIPTIC FUNCTIONS 2 K + 2 1 K' 37 V-+1 I I I ,l/=iA' V=+l u=K U-2K u=3K U = 4K I/=0 I/--1 V=0 f=l Fig. II. y=cn{u). ZEROS INFINITIES {2m+i)K+2niK' 2}nK+(2n + i)iK' where m and n are any integers. PERIODS 4A', 2K + 2iK' iK' 2tK' Fig. 12. y = cn{iu); y = dn(iu). 38 ELLIPTIC INTEGRALS SiK 4,-K' 2K + 4iK' y=i l/=fc' 3/=! y^-a V-0 y=-o: V = -l 1/— f !/=-! !/=0 !/_oo u=K 2K Fig. 13. y = dn(M). ZEROS {2m+i)K+{2n + i)iK' where m and n are integers. INFINITIES 2mK+{2n+i)iK PERIODS 2K, 4iK' Fig. 14. y=dn(K+iu). ELLIPTIC FUNCTIONS 39 EXAMPLES I. In the formulas of Art. 17 put u= ~, and show first that dn—=^vk' , , „K i-k' 1 K k' K _li and then iH — = — ■rr- = — —-„ en — = — — -, aw— = tan \ 77. 2 k^ i+k 2 i+k 2 \k 2. Prove that X i+k' -' \ i+k' 3. Prove that iK' 7 /A" V7+k , iK'_ /—— sn — = — ^, en — = -— , an — — Vi + ^. 2 \ k 2 \'k 2 4. Show that sn{K+liK') = -^, cn{K+\iK') = -i-—~, dn{K+liK')==Vj-k. Vk v'/t Show that \ 2k cni^K+UK') = - '"""'^ ^ X 2k dn{iK+?jiK') = --{Vi+k'+iVi-k'). 6. Show that du u i«(«+A + 3JA )=- , k en II ■ cniu+iK+iK') = -^, ken u )=F{k,sm-^x) (i) In particular, it is seen from this formula that the substitu- rdx V{i-x'^){i-k--x^) * Clifford, Mathematical Papers, p. 207. 41 42 ELLIPTIC INTEGRALS into the normal form I =.P(^, )■ Further, from the tables given at the end of the book, which we shall learn later to construct and use, the integral is known as soon as X is fixed. Similarly, if there be put x = cnu, Vi—x-=snu, V^'2+^2^2 = dnu, — -= — ; — = —snudnu= —"^{i—x^){k'^+k^x^),itiol\oyvs du du that n dx X V(7 z = u =cn~^x=sn~^V 1 ~ x~ = dn~^V k'^-\-k^x^ x^){k''^+k^x^) ==F(k, )=F{k, cos-i;x:) = Fik,sm-Wi-x^) (2) It is seen also that the substitution x = cos transforms the integral on the right-hand side into the normal form. Tf , Vi-x^ Vx^-k"^ dx ii x = dnu, = sn u, ; =cnu,---—— k'^sn ucnu k k du = - y/{i-x^){x:'-k'^), we have X dx J _. ^jVi—x^ = u = dn ^x = sn ' X V{l-X^){x^-k'^) =cn-n^^^-^)=F{k,4.) =F\k, si sin ' (3) Further, writing x = tan amu, it follows that snu= — ^ Vi+x^' Vi+a;2 \/i+a;2 du cn^u " and Jo V(i+;c2)(i+/fe'2a;2) Wi+W = F{k, tan-1 x). (4) FIRST KIND REDUCED TO LEGENDRE'S FORM 43 Art. 23. I. If a>b>x>o, write a: = 6 sin in the integral dx V(a^-x^){b-^-x^y and we have, ii k^ = ~, = - I -7 — = -sn X b .&' a isa) 2. If ca >x>a, write x = - , and it is seen that sin I" dx V(x--a-0(x2-62) a sn a b Lx' a If a>b>x>o. dx =cn' (56) (6a) {6b) V(a^+x^)(6--.v-0 Vd^+b^ [b' Vd~ + b^. (see IV, in Art. 3), and also r^" dx I ,[b a ' Jc V{d^+x^){x^-b^) Va^ + b^ lx Vd^ + b-i (see V in Art. 3). It is almost superfluous to add that for example in (6a) X the substitution t = cos transforms the integral dx ^+x^){b^-x^) into /a^+bVo d(t> V^ b- sin-^ a^+b^ It is also seen that if a>x>b>o, dx Jx V(a2-a;2)(:c2-62) a La a J 44 ELLIPTIC INTEGRALS that is, the integral on the left-hand side becomes ajo •^i ^— sin-^c^ ^2 for the substitution - = \ I 5— sm2(^. Further if a > 6 Jo \/(x2+a2)(x2+62) a lb M a^ \ (See I in Art. 3.) Art. 24. In the formulas (i), (2), (3) and (4) above, sub- stitute X for x^, and it is seen that- Jo Vx{i—x){i—k-'x) = 2dn~^iVi-k^x,k), .... (9) C '^'^ ^orn-Uy'r. k) , (10) Jx Vx{l-x){k'^ + k^x) C ^"^ = W».-l(Vr h\, (11) Jx Vx{i-x){x-k'2) C ^"^ = ^in-Uyx,h^ (12) Jo \/x{i+x){i+k"^x) Art. 25. Suppose that a, /3, and 7 are real quantities such that a>^>7; further write M = ^^^^, k,^ = ^^^ and k2^ = ^^, 2 a — 7 a — y where ^1^4-^2^ = i, so that the one is the complementary modulus of the other. Put A' = {x—a){x— C) (x — 7) . If CO >x>a> p>y, write a; — 7 = (a — 7) cosec2<^ and we have FIRST KIND REDUCED TO LEGENDRE S FORM When 00 >x>a> p>y, it is seen that and when p>x>y, we have X 45 (14) " ; (a-7)(/3-4 A/(3-7)(a-x)' ' = cn' J ia-l3){x-y) {fi-y){a-xy ki (15) Further if P>x>y, then 4'^^'"-WS''-)="-W"-^y'' =(/«-mJ^-^", /fei). (16) a — y Art. 26. As above write a-l3 M = ^^-\ k,^ = . X = {x-a){x-fi,{x-y). 2 a. — y For the interval a> x> fi> y, it is seen that "X°^='«-'WS'^="-[n/:-^'^' (•" and for the same interval JffV-X l^ia-ff){x-y) -cn' Further, if 7>:c> — 00 , then Jx V-X l^^-x J L\i3-a; , (19) and for the same interval <7lv=»-te 'h-'i^^, 4 (») 46 ELLIPTIC INTEGRALS Art. 27. From formula (14) itis seen that, if 00 > a; > --, k- J^^ dx —^f^ ' ^ 1 Vx{x-i){k^x-i) ^ji_Vx{x-i){x-i/k'^ ■■ 2sn' .Lkzil^ .=,..-!. Uz^ k]=2cn'A.l , „ k , (21) x — i ' I \yk'{x—iy and from formula (13) for the same interval, J. Vx{i-x){i-k-^x) \\k^x / \\ k^x (22). Using formula (17), it follows that, if —> x> 1, f"' dx ■ I i — k'^x ,, 7x Vx(i-x)(i-k^x) \^ i-k- i;(i —x){i—k^x) and for the same interval (see formula (18)) = 2icn-H^j^-^^.k'}. (23) Ji Va;(i-x)(i-/fe2^) V\x(i-F) ^ If o> a;> — CO , the formula (19) offers f ^^ -2isn-^[ in k'\ Jx Va;(i-a:)(i-yfe%) \Vi-x' / = 21 CM" while for the same interval it follows from formula (20) that r -=J^= = 2isn-^(JIH, k'\ J-Wx{i-x){i-kH) V\i-Fx / = 2icn-'(J^l^,k'). . (26) \ ^i I — Px FIRST KIND REDUCED TO LEGENDRE's FORM 47 Art. 28. Next let X={x-a){x-p){x-y)(x- d) and further put V(a-y){P~8) (p-y){a-8) (a-^)(y-S) '' ia-yXP-Sy^'-ia-yXp-Sy and note that k3^+ki'^ = i. If then CO >x>a, there results, supposing always that a> |8> T>5, -en' I (a-/3)(a:-5) L\(a-6)(x-/J)' and if ci>x> ^ ()3-5)(a-x) (a-0)(x-5)' {a-b){x-fi) \(a-ei){x-b) ,ki lia>X>P, J0 V-X L^Ca- ■y)ix-i3) ■p){x-yy = cn' l (p-y)(a-x) ^ia-ayx-yy"- while if fi>x>y, i VA'"'" [^{p-y){a-xy' = cn~^ I (a-l3)ix-y) ^(B-yVa-xy 'iP-y){a-xy When x lies within the interval ^> x> y, kz €^.-i4 (0-8)(x--y) {p-y){x-sy -cn' ■(V {y-b){&-x) {p-y){x-by (27) {21 (29) (30) (31) 48 ELLIPTIC INTEGRALS and when 7>a;> 5, it is seen that -»-'^/f^|^'*<)- ■ <3'' If 7>ic>5, — (^/J7Z^.M, . (33) and if 5>x> — t» „ C^ dx _,/ (a-7)(5-x) , Jo VX \^{a-&){y-x) -■cn' \\((x-6){y-x) I Art. 29. By means of the above formulas it is possible to integrate the reciprocal of the square root of any cubic or biquadratic which has real roots; for example (see Byerly, Integral Calculus, 1902, p. 276), dx n dx ^ r" Jo y/{2ax — x^){d^—x-) Jx V{2a—x}{a — x)x{a+x} Ja V{2a-x){a-x)x{a+x) ^L \ 2 / 2 -sn-Y^,^^)] [cf.(3o)] \ 3 2 FIRST KIND REDUCED TO LEGENDRE's FORM 49 Remark. — In the above integrals it is well to note that (34), for example, may be written showing that each factor under the root sign is positive for the interval in question. Art. 30. It is seen that the substitution a — Y y~y or x—a_fi—y or x—P a—y x-y 13-y' x-y (3-7 x—y a—y changes r dx .^^^ p dy J. ^{x-a){x-^){x-y) Jy V(3,-a)(j.-/3)(^,_^)' or (13) into (16). For example, r dj^ ^ n dy Ja y/{x-a){x-fi){x-y) Jy V(^'-a)(y-ii)(J^ [y-a){y-i5){y-y) 2K Va- (35) R y where k^= , see (16). a — Y By the same substitution (14) becomes (15). Similarly the substitution a — x a — y X-~I3 7 — 7 X — y B — y = ', or =- — -, or = - — - a — j3 a — y a — B a — y a — y 01 — y changes (17) into (20) and shows that ra dx r^ dv X V{a-x){x-ii){x-y) J— V{a-y){p-y)iy-y) 2K' V a — 7 where = k^. a — Y (36) 50 ELLIPTIC INTEGRALS By the same substitution (i8) becomes (19). Art. 31. Let the roots of the cubic be one real and two imaginary, so that X has the form {x — a)[{x — pY + c^]. Make the substitution ,2 X (x-p)2 + '2 + 4(p-q:)j-4(72 = 0. The roots of this equation are, say, (yi, y2) = -2(p-a)±2V(p — a)2 + o-2. It is evident that ji is positive and yo is negative. If we eliminate y from (i) and (2), we have the biquadratic [(x-p)2 + oand }'2, so that d amu = dnudu, dsnu = A(i> cnudnudu, dcnu= —snudnudu, ddnu= —Pcnusnudu. We further note that sn^u+cn^u = i, dn^u — k'- = k^cn^n, dn^u + k^sn^u = i. We have without difficulty - k'^sn u en udu x f dv /, I r — k^sniicnudu i f snudu^--\ -^^ =-^ (if v = dn u). The last integral is I dnu - cosh~i -7= — - cosh~i . k Further since dnK = k', Art. 17, we have , /"'^ , , ,/dnu\ . , ,/,cmi\ . dnu+kcnu k I snudti = CQsn~^[ -— - I =sinh~M k-— I =log —, . Similarly it may be proved that k\ cnu du — cos~^{d7tu) =sm~^ {ksnu), and r dnudu = 4> = am u = sin ^snu = cos ^ en u Art. 37. The following integrals should be noted: ^sn ucnu dn it du \ f dv /dii _ Cs sn u J -f- , (if v = sn^u). snhicnudnu 2 J z'v(i — d)(i— yfe-^uj Further writing V ( I — z))(i — li^v)=(\ — v)z, the last integral becomes — log V(i-d)(i-Fd) + i I+F" '0 2 cnudnu-\-\ i+^^1 - log — 2 2 W = _i log p— "•;"- -^^ I +c 2 L iW^M 2 J 1 , [ 2cn u dn u+cn^u+dn^u\ , ^ = — log 5 +C, 2 L 25?rM J so that, omitting C, r du _. r snu 1 J 5«M \_cnii-\-dnu\' FIRST KIND REDUCED TO LEGENDRE's FORM 59 where the arbitrary constant is omitted. Similarly it may be shown that / du _ I , ^ r^'5WM+) = I A4> d4> (cf . Art. 5), or since 4) = am u and d am u = dnudii, E{am m) = J dn^u du. It follows that and X sv?udu=y^[u—E{amu, k)], cn^udu=-r-[E{amu, k)—k'hi\. Art. 38. Reduction formulas. The following is a very useful and a very general reduction formula.* Consider the identity T" d (w+sin2 ^y sin (^ cos <^A(/)= I — {(wi+sin^ A(f>\dcl> Jo d4> = j *i2M(w+sin2 c^)"-' sin2 cos^A^4> + (w+sin2,^)''[cos2 = v, so that sin^ =v—m, cos^ Jo '*(w+sin2 A4> * See, for example, Durege, Elliptiscke Funktionen, § 4, Second edition. 60 ELLIPTIC INTEGRALS then there is found (m+sin^ <^)''sin cos 4>A), F (tan^cjb), where i^ is a rational function of its argument, may be expressed through an aggregate of terms of the form M{m+sm^ 4>y, where yu is a positive or nega- tive integer or zero and where M and m are real or imaginary constants. Further by writing x= , where 2 = sin , . , namely, I , may be put in the form I — -, which in turn may be expressed through integrals that correspond to the integrals I'o, Vi and V-i of the preceding article. Art. 40. Returning to formula {i) above, make m= — i, and note that if 771 =0, we have A=o, B = i; the formula becomes /■ \ i-jA^ r ^ d4>] ,^ Csm- 4> d(t> Next let 7)1 = —I, so that ^ =0, B= —k'^, and we have (b) -tanM^=-/fe'2 f-^^-k^^ rcol^4> ^ ^ J cos2 A4> J A4> finally let wj=— — , so that A=o, B=—, and the reduction -, -, _ ^ formula is _Fsin), where A(l> = Vi^^Wsin^ = A, '0 A(j) X j Ad4>=E{k, ), or j dnhidu=E{u), since d cos /o J^" du _E{ii) k^snucnu )o dn^u k'~ k'H7iu i ''^-^^ = ^^F{k,)-E{k,4>)lox 62 ELLIPTIC INTEGRALS J I srfiudu = r-^^ , C" 2 J -k'^u+E{u) I cn^udu= — — —^. Jo k^ rX^r^=Pi[^ t^^ 0+^'^i^(^, 4>)-E{k, 4>)l or Ja A COS'' 4> R-' C" du _tnudnu + k'^u — E{u) Jo ^^l P2 ' J'~'^ d _ A tan <)>-E{k, ) . A P '°' , , , dnutnu — Eiu) tn^u du -J- — . r J'~'''d4> cos^ 4> Ite'/i. ,\ ir/T. ,ST , sin (/> COS (^ J^*(f sin^ (/> I ,. , . 7/21?/!, _,M sin I A tan20)-2£(/fe, 0), A3£i<^ = -A sin cos » + ^ £(^, )-— F{k, 4,), Jo 3 3 3 1 A sm2 0(i0= A sin cos 0+_-i£(^, ^)+:l^^F(yfe, ,^), 3 3^ 3^ 1 A cos2 4>d4, = ^A sin <^ cos 0+^4!^£(A!, 0) -~F{k, 0). 3 3^ 3^ To these may be added J^'-^jJ^= cot +K-Ei-F{k, ^)+E{k, 0), or FIRST KIND REDUCED TO LEGENDRE'S FORM 63 I — — = cotamudnu+K—E\—u+E{u), Ju sn^u JrK g^ I — — = col amndnu—Ei+E(u), or r IT J'''^ d4> _ r - i-sin^0 d^ ^ tan^ A J« sin- A^' EXAMPLES 1. Show that f— £^ = iVIcn-'(:c,iV;), 2. Show that I y/\—x*dx-2^ 2 \ {dnH—dn*x)dx-—^K[ mod )=o.874oi... 3. Show that J^ ^^ dx= 2aJ i«'^x-4^ and G0B = +4', p = R cos {+4^)+lcos{(l>-i), or r=iR+l) cos <^cos rp-(R-l) sin sin he n, so that r = {R+l) cos M, or cos At = —^, sinM= — ^ ' • Denote the ratio yr— by Am, so that Am=^ — ,; then since Afi^ = i — k^ sin^ /u, it is seen that k^ = (R+iy-r^' Returning to the figure, it is seen that AM^=AQ^-MQ^ = R^+P+2Rl cos 2ct>-r^ = (i2+/)2-/-2_4;i?sin2 0; or r2 AM^ = {(R+iy-r'^\A^; and similarly BM^ = \{R+iy-r^W^I^. If the tangent is varied, its new position becoming A'B\ consecutive to the initial position, then clearly we have AA' :BB'=AM : BM; or AM BM ' and if for AM and BM their values be substituted, it follows that dd) , dxlf A0 AiA Suppose that the smaller circle is varied, the centre moving along the X-axis while r and I are subjected to the condition ^^ = (ry_^i\2_ 2 ' ^ ^^^^ constant. NUMERICAL COMPUTATION 67 In particular when the smaller circle reduces to the point circle at L, as in Fig. i6, then f = o, 0L=/and/fe2=- 4^^ {R+lf Fig. i6. Let 6 represent the angle XLA . It is seen that 2 and consequently dd = d-\-d4'- It is also seen that the angle LAO = 6 — 2(i) and GOZ = 0+i/'. From the triangle ALO it follows at once that lsme=Rsm{2^-d) (i) The relation —-^+—^=o, becomes here AM BM d4> _ d4/ _ de ^ 'AM~BM~2AG' or, since it follows that d de (R+l) A 2VR^-Pam^9 (2) 68 ELLIPTIC INTEGRALS Formula (i) may be regarded as the algebraic integral* of (2), or (2) may be considered as being produced by the trans- formation (i). Write ^1 =^ and put <(>i in the place of 6. It is seen that '^~R+ri+kr i+/fei' ' j+k" ■ • ^^' and — rr— -r=-(l+Ai) -rr j^, (2) ^1 sin <^i=sin (2i — 4> + (j>)=sm {4>-i + 4>), from which we have at once tan (, ... (3) i+«i or (i+^')tan0 . , (1+^') sin A{k, yfei,it r i+yfei follows that ^>^i. From (i') it is seen that o<<^<0i, if 4>^- 2 From (2') it is seen that 2 = (i+^i)(i+fe) . . . (i+^„)&i^, . {A) 2 * John Landen, An investigation of a general theorem for finding the length of an arc of any conic, etc., Phil. Trans. 65 (1775), pp. 283, et. seq.; or Mathe- matical Memoirs I, p. 32 of John Landen (London, 1780). An article by Cayley on John Landen is given in the Encyc. Brit., Eleventh Edition, Vol. XVI, p. 153. See also Lagrange, (Euvres, II, p. 253; Legendre, Traite, etc., I, p. 89. NUMERICAL COMPUTATION 69 where the moduli are decreasing and the amplitudes are increasing. It is also seen that fif) ~— _ I — Vl— Fr-l IV=1,2, . I + Vl— yfe2^_i \ ^0 = ^ tan (<^„-0c_i) = V'i-^2___j l-a^n <^„_i. . (i) It is further evident that F{kn, n) approaches the limit I dtj)^^, where $ is the limiting value of <^ as w increases. If <>=-, it follows at once from (i), see also Art. 49, that 2 01= IT, 02 = 27r, . . ., ) =9.8613464 i =0.6034084 log tan (<^2 — )=^Mki,i), (i) I+k i+k sin {2(t>i — 4>)=k sin , where ki>k and (t>i<. Applying the formula (i) n times, there results 2 2 I+* 1+^1 I+^n-l or, since 2 ^1 2 k2 . =—-^=, etc.. 1+^ V/fe' i+^i V^i it is seen that F{k ,) = k„^ ^'^' ■-■^'^-' F{K^), where 2V^,-l h= r^, sin (2 <^,- (/>,-! ) I+Kc-l = ^„_i sin(/)r-i(j; = i,2, . . . ; ^0 = ^, <^o = <^)- It follows also that = r ^'^ rsec0(i.i. = logetan(-+ > V'i-sin2 ^ Jj \4 72 ELLIPTIC INTEGRALS and Art. 46. The method of the preceding articles may also be used to evaluate ^^(30°, 40°), thus ^= .5 log ^ = 9.6989700 1+^ = 1.5 log (1+^) =0.1760913 log V^ = 9.8494850 log 2=0.3010300 colog (1+^) =9.8239087 log ^1 = 9.9744237 ^1= .942809 log All = 9.9744237 i+^i = 1.942809 log (i+^i) =0.2884301 log V^ = 9.9872ii8 log 2 =0.3010300 colog (i+*i) =9.7115699 log/b2 = 9-9998ii7 k2= .999567 log ^2 = 9-9998117 1+^2 = 1.999567 log (1+^2) =0.3009359 log V/fez = 9.9999059 log 2 =0.3010300 colog (1+^2) =9.6990641 ki = : log ki = 0.0000000 log ^ = 9.6989700 log sin <^ = 9.8080675 log sin (2i) = 9.6650640 202-01 = 27° 32' 43. "08 202 = 56° 54' 68."i3 02 = 28° 27' 34."o6 log ^2 = 9.9998117 log sin 02 = 9.6780866 log sin (2 0.j- 02) =9.6778983 203-02 = 28° 26' 45. "53 203 = 56° 54' i9."59 03 = 28° 27' 9."78 When ^3 = 1, then sin (2 04-0,) = sin 03, or 04 = 03. ••• 04 = 28° 27' 9."78 T = ^4° 13' 34."89 ^ = 7+^=59° 13' 34."89 * = S9° 13' 34."89 logiotan$= .225120S log log tan* = 9.3524156 cologM = o.3622i57 (*see below) log V^i =9.9872118 log Vife2 = 9.9999059 colog ^^ = 0.1505150 log F(3o°, 40°) =9.8522640 F(30°,4o°)= .711647 Art. 47. Cayley, Elliptic Functions, p. 324, introduced instead of the standard form of the radical, a new form Va^cos- (p+b'^ sin- (a>b); * Division is made by the modulus M to change from the natural to the com- mon logarithm, where J/=. 43429448. 74 ELLIPTIC INTEGRALS and he further wrote F{a, b, ^ cos^ (p+b^ sin^ 4> E{a, b, 4>) = rVa2 cos^ ,p+b^ sin^ <^. (i) (2) It is clear that Va^ cos^ +b^ sin^ (j) = aVi — P sin^ (^, where *-.-»^,*'=^. The functions (i) and (2) are consequently -F{k, 0) and aE{k, ). Fig. 17. In the figure let P be a point on the circle, whose centre is and let Q be any point on the diameter AB. Further let QA=a,QB = b, ZAQP = i, ZA0P = 2cj,, £ABP = . Write ai=\{a+b), bi = Vab, ci=^{a — b). It follows at once that OA=OB=OP = ai,OQ = ai-b=^{a-b)=Ci, QP sin i=ai sin 20, QP cos 0i=ci+ai cos 20. NUMERICAL COMPUTATION 75 On the other hand Qp = ci2+2Ciai cos 20+ai2=^(a2+62)+|(a2-62) cos 2 =|(a2+&2)(cos2 0+sin2 <^) +§ (a2 - ^2) (cos2 0-sin2 ) = a- cos2 (t>+b~ sm2 0. Therefore it follows that ai sin 24> , ci+fli cos 2+b~ sin2 cos (20 — 0i)=— Vai2 cos2 0i+6i2 sin2 0i . ai If in the figure we consider the point P' consecutive to P, then, PQd(l>i=PP' sin PP'Q = 2ai cos (20-0i)(i0; or, writing for PQ its value from above, there results 2d4> i Va^ cos2 0+62 sin2 V'ai2 cos2 0i+6i2 sin2 0i Integrating, this expression becomes F{a, b, 0)=§F(ai, 61, 0i), or F{k, 0)=i -F{k', 0') = -^ ^(^1. <^i). 2 fli i+« 76 ELLIPTIC INTEGRALS where . ^ ^{i+k') sin 2 sin i = , . - • V I — ^^ sin^ (/) a? a a^ \a-\-hJ \i+k / or, yfei = i^^, and ^' = ^ — r^, as given at the end of Art. 42. i+k i+«i Art. 48. Cayley derives a similar formula for the integrals of the second kind as follows, his work being here in places con- siderably simplified. From the relation of Art. 42, we have sin {2 4> — 4>i) =ki sin !, or sin 24, cos i =ki sin i; it follows that cos 20= —^1 sin^ 01-1-cos 0i A0i, and consequently 2 cos^ 0= I —^1 sin- 01 + cos 0iA0i, 2 sin^ = 1+^1 sin^ 0i — cos 0iA0i. From these two relations it is seen at once that 2 (a2 cos2 + ^2 sin2 0) =a^+b^-{a^-b^)ki sin^ 0i + {a^-b^)cos 0iA0i = (a2 + 62)(cos2 0i+sin2 0i) -ia^-b^)ki sin2 0, + (a2-^)2)cos 0iA0i = 4(01^ cos2 0i+&r sin2 0i) — 2bi-+4Ci cos 0iV ai^ cos2 0i+&i2sin2 0i. Multiply this expression by the differential relation given above, viz., 2J0 _ dtj)i Va^ cos2 0+62 sin2 Var cos^ 0i+6i2 sin^ 0i' NUMERICAL COMPUTATION 77 and integrating, there results E{a, b, )='^Eiku 0i) -- —F{ku 00+^ sin 0i, " 2 flai a or £(^, 0)=^(i+y)£(^^, 0i)__A_/7(^j_ 4>i)+ki-k')sm and '_. ,,^ i+^'tan-0 cos^ 0i d^~^^'^ \i-/fe'tan2 0)2 cos2 ' a positive quantity, it appears that i increases with . It is further evident that tan 0i =0 when tan = 00 . It is clear from (i) that when = o, )= — and £(a„, &„, ^) =a„(/>; an * Gauss, Werke, III, pp. 361-404. NUMERICAL COMPUTATION 79 further if <^=|ir, it is seen that F{an, b„) = — and £(a„, b„) =- o„, where a„=n. 2a„ 2 The equation F{a, b, <^) =| F{ai, bi, 0i) gives F{a, b, 4>)=iF{ai, bi, <^i)=^ F{a2, 62, 2) 2*' 2" 2"a„ where the (f>'s are to be calculated from the formula ai sin 2 sin i = V a^ cos^ 0+6^ sin^ (Z2 sin 2<^i sin (j)2=- \ ai^ cos^ 01 +61^ sin^ (pi Art. 52. r/je integrals of the second kind. Note that, since Fia,b, )=E{ai, bi, i)-ai^F{ai, bi, 4>i) +F{ai, bi, 0i)(ai2-|a2_ijj2)_|_cjsin0i; or, since ar-^a^-^bi^= -l{a^-b^) = -aici, the above equation is E{a, b, 4>)-a^F{a, b, )=E{ai, bi, i)-ai'^F{ai, bi, dn) — a\CiF{ai, bi, 4>i)+C\ sin (pi. Observing that, as n increases, hm [E{an, b„, )-a^F{a, b, 0) = -[2aiCi+4a2C2+SazC3+ ■ ■ ■ ]F{a, b, ) +ci sin 01+C2 sin 02+C3 sin (^3+ • • • ; or finally E{a, b, 0) = [a2-2CiCi-4a2C2-8a3C3 - . . . ]F{a, b, ) +ci sin 4>i+C2 sin 02+C3 sin 3+ . . . 80 ELLIPTIC INTEGRALS In particular, if <#> =|ir, we have Art. 49, <^i =ir, 02 = 2%, and then £(a, 6) = [(2^ — 2aiCi — 432^2— . . . ] — . 2 an It also follows immediately that F{k, 0) H — ^sin 01 H — ? sin 02+ a a or, noting that aiCi I ,, a2C2 I , (X3C3 I r = ^^, = fti, = k2, . 02 4 aiCi 4 a2C2 4 Cl ki c% k2 a\ I fli i+fe' a i+^i' C3 ^3 02 I fli 1 02 1+^3' di 1+^2' a i+^i' ■ ■ ■ ' the equation becomes, E{k, ) = [i-\Hi+\ki+\kik2-¥\kik2h+ . . . )]F{k, 0) , ^1 . , , k2 ■ , ■ Sin 01 +;^ -— r Sm 02 I+^l (l+^l)(l+>fe2)' + (l+/fel)(lA)(l+/fe3) '^ *' + Further since I fe I ^ =, or I+^i 2Vyfei 1+^1 zV/fei I ^1 I ^V^i ■= — ^, or - 1+^2 2V/b2' (l+^l)(l+^2) 4Vyfe2' I _ ^2 I _^v'^i^2 T+kz~2s/Yz °^ (l+*l)(l+^2)(l+^3)~ 8V^ ' NUMERICAL COMPUTATI0\ 81 the last line of the above expression may be written k[lV^ sin 4>i+l\^kik2 sin <^2-Fiv^^i^2*j sin s+ . . . ]. In particular if <^ = §ir, we have Art. 53. As a numerical example (see Legendre, Traite etc., T. I, p. 91) , let a = I, 6 = I "^2 — V3 = cos 75°, and let tan = -J -7=. \ V3 TO It follows that k^ = i ^ = sin 75°. a~ The following table may be at once constructed. Index (o) (i) (2) (3) (4) I .0000000: 0.6204095 0.5690761 0.56747^4 0.5674713' 0.2588190 0.5087426 0.565S68S 0.5674701 0.5674713 0.3705905 0.0603334 0.0016037 O.OOOOOII 0.965925S, 0.258S190 47 3 31 0.5SS790S 0.1060200 0.002S260 0.0000020 0.S082856 62 36 3 0.9943636 119 55 48 0.9999959240 o o o. 9990900480 o o (See Cayley, loc. cit., p. 335.) The complete integral fi = = 2.768063 . . . and 2 Ui i^(75°, 47°3'3i")=t'v = o.9226877 • • • Note that the first integral is three times the second. It is also seen that Ui-j-]= aici = .2332532 + 2a2C2 = .0686686 +4a3C3 = .0036402 +8a4C4 = .0000051 = •3055671 and £1 = 1.0764051 82 ELLIPTIC INTEGRALS The computation of E{k, ) is found in the next article. Art. 54. To establish in a somewhat different manner the results that were given in the preceding article, consider * a function G{k, (j)) composed of an integral of the first and of an integral of the second kind, such that Ju Vl— ^2sin2 (f, where a and are constants. Making in this integral the substitutions of Arts. 42 and 48, namely d^^i+h d^^ gj^2 ^ = i(i+;fei sin2 4>i-A4>i cos <^i), it is seen that G(^, 0) = l±^-i[G(^,, <^i)-i^sin0i], . . . (i) 2 where .*0=f'^ G{K4>.)=1 ^ii±^ii>Ili^d<^ A4)i the constants ai and /3i being defined by the relations We saw in Art. 48 that , i-Vi-F ^ ^ , — - Ki = . , tan {i — , I + V I — ^2 where ^1 <^ and 4>i> 4>. It follows directly from (i) that 22 2 iTi+^i . i+^i 1+^2, . _, , — /3 sin <)!)iH jSi sm 4)2+ 2L 2 22 i+^i 1+^2 i+^"a • . 1 22 2 J * See also Legendre, Traile, etc., I, p. 108. NUMERICAL COMPUTATION 83 where k\k2 and /3p = /3 a„=a+|/3(l ^1 k\k2 k\k2 2 2^ 2"- Since /3„ becomes o with kn, it is seen that lim G{h X* a„ d(i)=a„4)n. From Art. 43 we had i+^i 1+^2 I+^n n=F{k, ), and, see Art. 42, 2 k ' 2 k\ ' It follows that the above formula becomes G(^,^)=f(^,<^)[.+^i+^+^^+'-^^+ . . .)_ — T sin <^iH 5— sin <^2H n — sm z+ . k\ 2 2^ 2^^ If in this formula we put a = i, /3= — A^, it becomes 1^2 , k-[k2kz £(^,.^) = F(^,<^)[i-f(i+^^+^^ "^'^i_:_ ^ I ^kik2, 2 +jfep-^sin «/>i+ ".;"" sin <)2- 5 — sm 03+ . where /Cp "^ 1 — Vi— A;Vi i + Vi-)feVi and tan (0p— <^p-i) = v^i-'feVi tan <^p-i. 84 ELLIPTIC INTEGRALS These results verify those of Art. 52. With Legendre, Fonct. Ellip., T. I., p. 114, we may find E{k, 4>) where ^ = sin 75° and tan 2=— .C013888 kv kik2k.:ki . Sin (j>i = .0000010 16 Writing sum = .3799180 24 8 16 ' it is found that Z, = . 3888658 . . . In Art. 53 it was seen that F{k, 4>) = .9226877 . . . It follows that E{k,(j))=F{k,<^))L+.s^ggI8o 0.7387196 . . . Further since E there follows '■t)-i'-t)'- £1=1.0764049 . . . Art. 55. Inverse order of transformation. If the modulus k is nearer unity than zero, the following method is preferable. The equation (i) of the preceding article may be written G{ki, 4>i)=^G{k, 0)+|i sin <^i, since fi = ^. NUMERICAL COMPUTATION 85 If in this formula the suffixes be interchanged, then where now 2 B G{k, (i>)=——-G{ki, 0i)+-sin0, i+k k .2^ ^ ki= r, sin (201 — 0) =^ sin (^, i+« ki>k, <^i<)=^ sin 4>+-^ sin 0iH — -=^^2 sin 4,2 k Vk Vk -f — ;=— |33Sin<^jH 7= /3„-i sin 0„-i Vk where kki . . . kp-i and ^\ ki k\k2 ' ' ' k\k2 ■ ■ ■ kp-i Since K approaches unity (rapidly) as n increases, ^ r*f.a„+/3„sin2 , limG(*„, ^^ = (a„ + ^n) loge tan(- + Y) -^n S^ <#>»• In Art. 45 it was shown that ^ k^^bEI^lo, tan(^+^) =F{k, 0) 86 ELLIPTIC INTEGRALS We may consequently write the above formula G{k, )-F{k, )[a-^-(^z+^+-^+ . . ^1^2 • • ■ ^n-1 ,2 2^ Vk Vkki V/feyfeiyfe2 +- sin^H — ^sin<^iH — ^==sm4>2-i — , ^_ , sin 3+ ■ ■ . ^sin0„-i — Vkkl . . . k„-2 Vkki . . . k Writing a = i , /3 = — yfe^ in this formula, it becomes E{k, 4,)=F{k, 4>)^i+k(i+^+~+ . . . 2"-^ 2" y klk2 . . ■ kn-1 ^1^2 ■ • • kn-\l . / 2 2^ — ^|sm(/)H — -=sin<>iH — p=sm<^2+ ■ • • \ Vk Vkkl = sin„-!- = sin<^„), V ^^1^2 . • • ^n-2 vkkl ■ where kp = — — ^ — and sin {2p — p~i)=kp-i sin p-i. i+kp-i Taking the example of the preceding article, and using the values given in Art. 53, it is seen that — k sin = —0.7071070 — 2 Vk sin 01 = — 1 .4 146540 Vk +4—= sin 4)2 = 2.8293085 Vki F{k, 0) = .9226877 and F(^, (^) 1+^-— = 0.0311720 E{k, )= 0.7387195 . . . NUMERICAL COMPUTATION 87 Art. 56. Two of the principal problems that appear in practice will now be given. Problem i. When u and k are given, calculate the values oj snu, cnu, dn u. 1. Computation of snu. In the Table II, p. 96, is found an imme- diate answer to the problem. For when u and ^ = sin e are known, the value 4> may be found in the table and then sn u from the formula sn !< = sin (i>. If, for example, ^ = 5 = sin d, and m = .47S5i, it is seen that for 9=30°, M = .47SSi, we have =2-]°, and sin 0=. 45399 = 5;! u. 2. The computation of en u and dn u are had from the formulas cnu= ±V (i— in u){i-\-snu), dnu= :ii^{i—ksnu){i+ksnu). Problem 2. Having give^i the elliptic Junction, calculate the argument. 1. If snu is known, find u. Table II furnishes the solution. Sup- pose that a is the given value of snu, and suppose that k = s\n B is also known. Hence, since snu = sva.^=a, we may determine <(>. With e and known, we find the value of u from the table. Denote this value by Mo- From the relation snu = sn Uo, we have (Art. 21), M = Mo +4tnK ■\-2m'iK'. Further in the formula (Art. 12). sn u= —sn{u-]-2K), substitute u=—uo, and then we have —snuo=—sn{2K—Uo), so that u may also have the form « = 2K—Uo+4mK+ 2m'iK'. 2. li en u and dn u are given, snu and then u may be found as above. CHAPTER V MISCELLANEOUS EXAMPLES AND PROBLEMS I. The rectification of the lemniscate. The equation of the curve is (y2+i-2)2+a2(/-a:2) = o; or, writing x=r cos 6, y = r sin 9, the equation becomes )-2 = (j2 cos 2S. From the expression ds''- = dr'^-\-r-d9-, the differential of arc is ^ a p( T- ^ Jo Vi-jsin^e V2J0 Vi-isin^,^ V2 VVI' j which may be calculated at once from the tables when a and 9 (or 0) are given. A quadrant of the lemniscate is Jo Vi-2sin2e "^zjo Vi-isin2 V^ ^^2 2. r/ze rectification of the ellipse. Let the equation b From the integral X^ -y2 Let the equation be — \-— = i,a>b a^ b^ "IM. dx, a'^ — f we have, by writing k^= — - — , x=at, a' -kV)dt ^ P (i-kV)d Jo V(l-/2)(i_ kV) Fig. 18. Finally writing / = sin (see Art. 3) and that is x=a sin 0, we have j= I Ad(t>=aE{). Jo SS MISCELLANEOUS EXAMPLES AND PROBLEMS 89 Here k is the numerical eccentricity of the ellipse. The angle 4, = C0Y = 90— CO^, where in astronomy the angle COA is known as the eccentric anomaly of the point P Writing = Tr/2, it is seen that the quadrant of the ellipse is aE, where E is the complete integral of the second kind. If the equation of the ellipse is taken in the form *=osin0, y = 6coS(^, it follows at once that , ds'^ = a-{i — k'^sm'^ (t>)d'^, or s=aE{). 3. The major and minor axes of an ellipse are 100 and 50 centimeters respectively. Find the length of the arc between the points (o, 25) and (48, 7). Find also the length of the arc between the points (48, 7) and (50, o). Determine the length of its quadrant. 4. If X denotes the latitude of a point P on the earth's surface, the equation of the ellipse through this point as indicated in the figure, may be written in the form a cos X x = - It follows at once that tf2(l-t.2 y= a(i—e^) sin X Vi — e'^sia^ X (i- ds''=dx^+dy-' = so that s=aii-e') I Jo (i- e^ sin- X)'' d\ Fig. 19. -e- sin^ X) This integral may be at once evaluated by the third formula in Art. 41. Compute the lengths of arc of the ellipse between 10° and 11° and between 79° and 80° where a =6378278 meters and €- = 0.0067686. Compare these distances with the length of an arc that subtends i" upon a circle with radius=6378278 meters. 5. Plot the curves, the elastic curves, which are defined through the differential equation , y-dv d^=± ; , Va"— y* for the values a= i, 2, 4, 9. 6. The axes of two right cylinders of radii a and b respectively (a > b) intersect at right angles. Find the volume common to both. Let the z-axis be that of the larger cylinder and the y-axis that of the smaller, so that the equations of the cyhnders are x'^+y^=a- and x^+z^=b^ respectively. The volume in question is V -j: Va^-x^ Vb'^-x'^dx. 90 ELLIPTIC INTEGRALS Writing t = sn~H j-,-)> (see formula 5a, Art. 23), then x=bsnt, b^—x^= hhnH, a.'^—x^=aHnH, d = b cnidnldt. It follows that F=8a62 I i- ° „ snH-\-—snH\di. (SeeByerly,/«^ Cai., i902,p. 276.) Joting (see sixth formula of Art. 41, and (ii) of Art. 48) that Jr-K I (-K I k" Jo ' Noting (see sixth formula of Art. 41, and (ii) of Art. 48) that it follows at once that Compute V when a = 60 and 6=12 centimeters respectively; also find the volume common to both when the shortest distance between the axes is 8 centimeters. 7. The differential equation of motion of the simple pendulum is dh dy dfi^~^Js' or multipljnng by -3- and mtegratmg, dt |y=-2gy+C. If the pendulum bob starts from the lowest point of its circular path with the initial velocity that would be acquired by a particle falling freely in a vacuum through the distance yo, so that iio^=2gyo (Byerly, loc. cit., p. 215), it is seen that this is the value of C, and consequently Further taking the starting-point as the origin (see figure) the equation of the circular path is x^+y'^—2ay=o, so that fds\ ' a' l^y\ ^ \dt/ 2ay—y'^\dt and consequently dy t = '^2gJo ^/(yo-y)(2ay-/)' which is the time required to reach that point of the path whose ordinate is y. ting k^= 2a ya yo y Writing fe^= — and sin^ =—, this integral becomes at once / ^. = \-Fik,,p). Vi-Ai^sin^* S MISCELLANEOUS EXAMPLES AND PROBLEMS 91 Let OC=CA = ahe the length of the pendulum. Let A be the highest point reached by it in the oscillation so that the ordinate of A is y^. Let the angle ACO be a, and let e be the angle PCO, where P is the point reached at the expiration of the time t. It is seen that yo — =1 — COSc, a so that iV V2=x/K:-c< and similarly, V — =sm-. It follows also that V-^ sin - yo . a sin - 2 When e=a, sin 0=1, or = -, and consequently, the time of a half-oscilla- , ■ ■ .P lation IS \- i F(sin-,^ 2 2 Show by Table I that when = 36°, the time of oscillation is 1.0253 • • ■ la times greater than that given by the approximate formula / = "V - •"-. g The following problems taken from Byerly's Calculus are instructive: 8. A pendulum swings through an angle of 180°; required, the time of oscillation. Ans. 3.708V-. 9. The time of vibration of a pendulum swinging in an arc of 72° is observed to be 2 seconds; how long does it take it to fall through an arc of s", beginning at a point 20° from the highest point of the arc of swing? Ans. 0.095 • ■ second. 10. A pendulum for which [a. . vibrates through an arc of 180°; through what arc does it rise in the first half second after it has passed its lowest point? In the first | of a second? A7ts. 69"; 20° 6'. 1 1 . Show that a pendulum which beats seconds when swinging through an angle of 6°, will lose 11 to 12 seconds a day if made to swing through 8° and 26 seconds a day if made to swing through 10°. (Simpson's Fluxions, § 464.) CHAPTER VI FIVE-PLACE TABLES The following tables of integrals are given in Levy's Theorie des fonctions elliptiques. As stated by Professor Levy, he was assisted by Professor G. Humbert in compiling these tables from the ten-place tables that are found in the second volume of Legendre's Treatise. Table I gives values of the integrals K= P'— =ii=== and E= C'd4>Vi-sm^ 6 sin^ . Jo Vi — sin^ d sin^ Ja For example, if 6 = 78° 30', then iT = 3.01918 and £ = 1.05024. Table II gives values of the integral d F{k,4>) = i v^ - sin^ d sin^ For example, if ^ = 65° and <^ = 8i°, then F(k,) = 1.94377. Table III gives values of the integral E{k, V I -siii^ 9sin2 0. For example, if 0=40° and <^ = 34°, then E{k, <>) =0.57972. 92 FIVE-PLACE TABLES 93 I. -THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS 8 K E 6 K E 6 K E o° 1.57080 1 . 5 7080 50° 1-93558 1-30554 82° 0' 3-36987 I .02784 I 092 068 51 5386 29628 12 9457 670 2 127 032 52 7288 28695 24 3-41994 SS8 3 187 1.56972 53 9267 27757 36 4601 447 4 271 888 54 2.01327 2681S 48 7282 338 5 379 781 55 3472 25868 83 3 . 50042 231 6 511 650 S6 5706 24918 12 2884 126 7 668 495 57 8036 23966 24 5814 023 8 849 296 58 2 . 10466 23013 36 8837 1921 9 I ■ 58034 114 59 3002 22059 48 3-61959 821 10 284 1.55880 60 2.15652 2 1 106 84 3-65186 I. 01 724 II 539 640 61 8421 20154 12 8525 628 12 820 368 62 2.21319 19205 24 3 71984 534 13 I 59125 073 63 4355 18259 36 5572 443 M 457 I -54755 64 7538 17318 48 9298 354 IS 814 415 65 2.30879 16383 85 3-83174 266 i6 I 60IQ8 052 66 4390 15455 12 7211 181 17 608 1-53667 67 8087 14535 ^■i 3-91423 099 i8 1.6104s 260 68 2.41984 13624 36 5827 018 19 510 t -52831 69 6100 12725 48 4.00437 0940 20 1.62003 380 70° 0' 2 ■ 50455 11838 86 5276 86s 21 523 I. 5 1908 30 2729 "399 12 4.10366 792 22 I 63073 415 71 5073 10964 24 5736 721 23 6.S2 I . 50901 30 749° 10533 36 4.21416 653 24 1.64260 366 72 9982 106 48 7444 588 25 900 1.49811 30 2-62555 09683 87 4-33865 526 26 1-65570 237 73 5214 265 12 40733 466 27 6272 1.48643 30 7962 8851 24 8115 410 2S 7006 029 74 2 . 70807 443 36 56190 356 29 7773 1-47397 30 3752 039 48 64765 306 3° I -68375 I .46746 75 6806 7641 88 74272 258 31 041 1 077 30 9975 248 12 84785 215 32 I 70284 I 45391 76 2.83267 6861 24 96542 174 33 1192 44687 30 6691 480 36 5-09876 137 34 2139 43966 77 2.90256 106 48 25274 J04 3S 3125 229 30 3974 5738 89 43491 07s 36 4150 42476 78 7857 378 6 54020 062 37 5217 41707 30 3,01918 024 12 65792 050 38 6326 40924 79 6173 4679 18 79140 049 39 7479 126 30 3 . 10640 4341 24 94550 030 40 1.78677 I -39314 80 5339 oil 30 6.12778 021 41 9922 38489 12 7288 3882 36 35038 014 42 1.81216 37650 24 9280 754 42 63854 008 43 2560 36800 36 3-21317 628 48 7-04398 004 44 3957 35938 48 3400 503 54 737" on 45 5407 35064 81 5530 379 90 00 000 46 6915 34x81 12 77" 257 47 8481 33287 24 9945 126 48 I. 90108 32384 36 332234 017 49 1800 31473 48 4580 2900 94 ELLIPTIC INTEGRALS II.— ELLIPTIC INTEGRALS OF THE FIRST KIND e 0° 5° 10° 15° 20° 25° 30° 3S" 40° 45° ~I~ 0.0174s 0.01745 0.0174s 0.0174s 0.01745 0.01745 0.01745 0.01745 0.0174s 0.01745 2 03491 03491 03491 03491 03491 03491 03491 03491 03491 03491 3 05236 05236 05236 05236 05236 05236 05237 05237 05237 05237 4 06981 06981 06981 06982 06982 06982 06983 06983 06984 06984 5 08727 08727 08727 08727 08728 08729 08729 0S730 08731 08732 6 10472 10472 10473 10473 10474 10475 10477 10478 10480 10482 7 12217 12218 12218 12219 12221 12223 12225 12227 12230 12233 8 13963 13963 13964 13966 13968 13971 13974 13978 13981 13985 9 15708 15708 15710 15712 15715 15719 15724 15729 IS735 15740 lO 17453 17454 17456 17459 17464 17469 17475 17482 17490 17498 II 19199 19200 19202 19206 19212 19220 1922S 19237 19247 19258 12 20944 20945 20949 20954 20962 20971 20982 20994 21007 21021 13 22689 22691 22695 22702 22712 22724 22738 22753 22770 22787 14 24435 24436 24442 24451 24463 24478 24495 24514 24535 24556 15 26180 26182 26189 26200 2621S 26233 26254 26278 26303 26330 i6 27925 27928 27936 27949 27967 27989 28015 28044 28075 28107 17 29671 29674 29684 29699 29721 29748 29779 29813 29850 29889 i8 31416 31420 31431 31450 31475 31507 31544 31585 31629 31675 19 33161 33166 33179 33201 33231 33268 33312 33360 33412 33466 20 34907 34912 34927 34953 34988 35031 35082 35138 35199 35262 21 36652 36658 36676 36706 36746 36796 36855 36920 36990 37063 22 38397 3S404 38425 38459 38505 38563 38630 38705 387S6 38871 =3 40143 40151 40174 40213 40266 40331 40408 40494 40587 40683 24 41888 41897 41924 41968 42027 42102 42189 42287 42392 42503 25 43633 43643 43674 43723 43791 43875 43973 44084 44203 44328 26 45379 45390 45424 45479 45555 45650 45761 45885 46020 46161 27 47124 47137 47174 47236 47321 47427 47551 47690 47841 48000 28 48869 48883 48925 48994 49089 49207 49345 49500 49669 49846 29 5061 s 50630 50677 50753 50858 50988 51142 51315 51503 S1700 30 52360 52377 52428 52513 52628 52773 52943 53134 53343 53562 31 54105 54124 54181 54273 54401 54560 54747 S4959 55189 55432 32 55851 55871 55933 5603 s 56175 56349 56555 56788 57042 57310 33 57596 57619 57686 57797 579SO 58141 58367 58623 58902 59197 34 S934I 59366 59439 59561 59727 59936 60183 60463 60769 61093 35 61087 61113 61193 61325 61506 61734 62003 62308 62643 62998 36 62832 62861 62948 63090 63287 63534 63827 64159 64524 64912 37 64577 64609 64702 64857 65070 65337 65655 66016 66413 66836 38 66323 66356 66457 66624 66854 67144 67487 67879 68309 68769 39 68068 68104 68213 68393 68641 68953 69324 69747 70214 70713 40 69813 69852 69969 70162 70429 70765 71165 71622 72126 72667 41 71558 71600 71726 71933 72219 72580 73010 73502 74047 74632 42 73304 73349 73483 73704 74011 74398 74860 75389 7,5976 76608 43 75049 75097 75240 75477 75805 76219 76714 77282 77914 78594 44 76794 76846 76998 77251 77600 78043 78573 79182 79860 80592 45 0.78540 0.78594 0.78756 0.79025 0.79398 0.79871 0.80437 0.81088 0.81815 0.82602 FIVE-PLACE TABLES 95 n.— ELLIPTIC INTEGRALS OF THE FIRST KIND * e 50° 55° 60° 65° 70° 75° 80° 85° 90° T° 0. 01 745 0.0174s 0.01745 0.01745 0.01745 3.01745 0.01745 0.01745 0.01745 2 03491 03491 03491 03491 03491 03491 03491 03491 03491 3 05237 05232 05238 05238 05238 05238 05238 05238 05238 4 0698s 06985 06986 06986 06986 06987 06987 06987 06987 S 08733 08734 0873s 08736 08736 08737 08737 08738 08738 6 10483 10485 10486 104S8 10489 10490 1 049 1 10491 10491 7 1^235 12238 12240 12242 12244 12246 12247 12248 12248 8 139S9 13993 13997 14000 14003 14005 14007 14008 14008 9 15746 15751 15757 15761 15765 15769 15771 15772 15773 lO 17505 17513 17520 17526 17532 17536 17540 17542 17543 II 19268 19278 19288 19296 19304 19310 19314 19317 19318 12 21034 21047 21059 21071 21080 21088 21094 21098 21099 13 22804 22821 22836 22851 22863 22873 22880 22885 22886 14 24578 24599 24618 24636 24652 24664 24674 24680 24681 IS 26356 26382 26406 26428 26448 26463 26475 26482 26484 i6 28139 28171 28200 28227 282S1 28270 28284 28293 2829s 17 29927 20965 30001 30034 30062 30085 30102 30112 30116 i8 31721 31766 31809 31848 31881 31909 31929 31942 31946 19 33520 33574 33624 33670 33710 33742 33766 33781 33786 20 35326 35388 35447 3S50I 35548 35586 35615 35632 35638 21 37137 37210 37279 37342 37396 37441 37474 37494 37501 22 38956 39040 39119 39192 39255 39307 39346 39369 39377 23 40782 40878 40969 41053 41126 41186 41230 41257 41266 24 42614 42724 42829 42925 43008 43077 43128 43159 43169 25 44455 44580 44699 44808 44904 44982 45040 4507s 45088 26 46304 46445 46580 46704 46812 46901 46967 47008 47021 27 48161 48320 48472 48612 48735 4883 s 48910 48956 48972 28 50027 50206 50377 50534 50672 50785 50870 50922 50939 29 51902 52102 52293 5247c 52624 52752 52847 52905 52925 30 53787 54009 54223 54420 54593 54736 54843 54908 S493I 31 55681 55928 56166 S6386 56579 56739 56858 56931 55956 32 33 34 3S 57586 57860 58123 58367 58582 58760 58893 5897s 59003 59501 59803 60095 60365 60604 60802 60950 61042 61073 61427 61760 62082 62381 62646 62865 63029 63131 63166 63364 63730 64085 64415 64707 64950 65132 6524s 65284 36 37 65313 65715 66104 66468 66790 67058 67260 67385 67428 67273 67713 68141 68540 68895 69131 69414 69552 69599 ?8 69246 69727 70195 70633 71023 71340 7,1594 71747 71799 39 40 71232 73231 71756 73801 72267 74358 72746 74882 73175 75352 73533 75745 73804 76043 73972 76228 74029 76291 41 42 43 44 4S 75243 77269 79308 81362 0.83431 75862 77940 8003 s 82149 0.84281 76469 78600 80752 82926 0.85122 77041 79224 81432 83665 0.85925 77555 79786 82045 84333 0.86653 77987 80258 82562 84898 0.87270 78313 80617 82954 85329 0.87741 78517 80841 83200 85598 0.88037 78586 80917 83284 85690 0.88137 96 ELLIPTIC INTEGRALS II.— ELLIPTIC INTEGRALS OF THE FIRST KIND e 0° 5" 10° 15" 20° 25° 30° 3S° 40° 45° 46° 0.8028s 0.80343 0.80515 0.80801 0.81198 0.81701 0.82305 0.83001 0.83779 0.84623 47 82030 82092 82275 82578 82999 83535 84178 84920 85752 86656 48 83776 83841 84035 84356 84803 85371 86055 86846 87734 88701 49 85521 85590 85795 8613s 86609 87211 87937 88779 89725 90759 50 87266 87339 87556 87915 88416 89054 89825 90719 91725 92829 SI 89012 89088 89317 89697 90226 90901 91716 92665 93735 94912 52 90757 90838 91078 91479 92037 92750 93613 946 1 8 9S7SS 97007 S3 92502 92587 92841 93262 93850 94603 95514 96578 97784 0.991 1 5 54 94248 94337 94603 95047 95666 96458 97420 0-98545 0.99822 101237 SS 95993 96086 96366 96832 97483 0.98317 0.99331 1.00519 1.01871 03371 S6 97738 97836 98130 0.98618 0.99302 1. 001 79 1. 01 24 7 02499 03928 05519 57 0.99484 0.99586 0.99894 1 .00406 1. 01 1 23 0204^ 03167 04487 05996 07680 58 1.01229 1.01336 1.0165? 02194 02946 03912 05092 06481 08073 09854 59 02974 03086 03423 03984 04770 05783 07021 08482 10159 12042 60 04720 04837 05188 05774 06597 07657 08955 10490 12256 14243 61 06465 06587 06954 07566 08425 09534 10894 12504 14361 16457 62 08210 08338 08720 09358 10255 11414 12837 14525 16476 18685 63 09956 10088 10486 11151 12087 13296 14784 16552 18601 20926 64 1 1 701 1 1 839 12253 12945 13920 15182 16735 18586 2073s 23180 6S 13446 13590 14020 14740 I57S5 17070 18691 20626 22877 25447 66 15192 15340 15787 16536 17592 1 896 1 20651 22672 25029 27727 67 16937 17091 17555 18333 19430 20854 2261S 24724 27190 30020 68 18682 18842 19324 20130 21269 22750 24583 26782 29359 32325 69 20428 20593 21092 21928 23110 24648 26555 28846 31537 34642 70 22173 22345 22861 23727 24953 26548 28530 30915 33723 36972 71 23918 24096 24630 25527 26796 28451 30509 32990 35917 393^3 72 25664 25847 26400 27328 28641 30356 32491 35070 38118 41666 73 27409 27599 28169 29129 30488 32263 34477 37155 40328 44030 74 29154 29350 29939 30930 32335 34172 36466 39244 42544 46404 75 30900 31102 31710 32733 34184 36083 38457 41339 44767 48788 76 32645 32853 33480 34535 36034 37996 40452 43437 46997 51183 77 34390 34605 35251 36339 37884 39911 42449 4S540 49232 53586 78 36136 36356 37022 38143 39736 41827 44449 47647 51474 55999 79 37881 38108 38793 39947 41588 43744 46451 49757 53721 58419 80 39626 39860 40565 41752 43442 45663 48455 51870 55973 60848 81 41372 41612 42336 43S57 45296 47583 50462 53987 58230 63283 82 43117 43364 44108 45362 47150 49504 52470 56106 60491 65725 83 44862 45115 45879 47168 49005 51426 54479 58228 62756 68172 84 46608 46867 47651 48974 50861 53350 56490 60352 65024 7062^ 8S 48353 48619 49423 50781 52717 55273 58503 62478 67295 73082 86 50098 50371 51195 52587 54574 57198 60516 64605 69569 75542 87 51844 52123 52968 54394 56431 59123 62530 66734 71844 78006 88 53589 53875 54740 56200 58288 61048 64545 68864 74121 80472 89 55334 55627 56512 58007 60145 62974 66560 70994 76399 82939 90 1.57080 1-57379 1.58284 1.59814 1.62003 1 .64900 1,68575 1-73125 1.78677 1.85407 FIVE-PLACE TABLES 97 U .—ELLIPTIC INTEGRALS OF THE FIRST KIND * 50° 55° 60° 65° a 70° 75° 80° 85° 90° 46° 0.85515 0.86431 0.87342 0.88213 . 89005 0.89678 0.90193 0.90517 0.90628 47 87614 88601 89585 90529 91390 92224 92687 93042 93163 48 89729 90791 91853 92875 93811 94610 95226 95614 95747 49 91860 93001 94146 95252 96267 97139 0.97810 0.98235 0.98381 SO 94008 95232 96465 0.97660 0.98762 0.99711 I . 00444 1 . 00909 I .01068 SI 96171 97484 0.98811 I .00102 1.01297 I .02329 03129 03638 03812 S2 0.98352 0.99759 1.01185 02578 03872 0499s 05868 06425 06616 S3 1.00550 1.02055 03587 05089 06491 07711 08665 09274 09483 54 02765 04374 06018 07637 09155 10481 11521 12188 12418 SS 04998 06716 08479 10223 11865 13307 14442 15171 15423 S6 07248 09082 10971 12848 14624 16190 17430 18229 18505 57 09517 11472 13494 15513 17433 19136 20488 21364 21667 S8 1 1803 13886 16050 18220 20295 22145 23623 24582 24916 59 14108 16325 18638 20970 23212 25223 26837 27890 28257 60 16432 18788 21254 23764 26186 28371 30135 31292 31696 61 18773 21277 23916 26604 29219 31594 33524 34795 35240 62 21134 23792 26606 29490 32314 34897 37008 38407 38899 63 23513 26332 29332 32425 35473 38281 40594 42135 42679 64 25910 28898 32094 3S409 38699 41753 44288 45989 46591 6S 28326 31491 34893 38443 41994 45316 48098 49977 50645 66 30760 34109 37728 41529 45360 48976 52031 541 1 2 54855 67 33212 36753 40600 44668 48800 52738 56096 58404 59232 68 35683 39423 43510 47860 52317 56606 60303 62868 63794 69 38171 42119 46457 51107 55913 60586 64661 67518 68557 70 40677 44840 49441 S44IO S9S9I 64684 69181 72372 73542 71 43200 47587 52463 57768 63352 68905 73877 77450 78771 72 45739 50359 55522 61182 67198 73256 78759 82774 84273 73 48296 53155 58618 64653 71132 77743 83844 88370 90079 74 50867 55974 61750 68180 75155 82371 89146 1.94267 1.96226 7S 53455 58817 64918 71763 79269 87145 1.94682 2.00499 2.02759 76 56056 61682 68120 75401 83473 92073 2 . 00470 07106 09732 77 58672 64569 71356 79094 87768 1.97157 06529 14136 17212 78 61302 67476 74625 82840 92154 2.02403 12878 21644 25280 79 63943 70403 77924 86637 1.96630 07813 19538 29694 34040 80 66597 73347 81253 90484 2.01193 13390 26527 38365 43625 81 69261 76309 84609 94377 05840 19131 33866 47748 54209 82 7193s 79286 87991 1.98313 10568 25035 41569 57954 66031 83 74618 82278 91395 2.02290 15371 31097 49648 69109 79422 84 8S 77309 85281 94821 06303 20244 37309 58105 81362 2 .94870 80006 88296 1.98264 10348 25178 43658 66935 2.94869 3 13130 86 82710 91320 2.01723 14421 30166 50129 76116 3.09782 35467 87 88 85418 94351 05194 18515 35198 56703 85612 26198 3 64253 88129 1.97388 08674 22627 40265 63357 2.95366 44116 4.04813 89 90843 2.00429 12161 26750 45354 70068 3 05304 63279 4.7413s 90 1.93558 2.03472 2.15652 2.30879 2.50455 2 . 76806 3-15339 3.83174 00 98 ELLIPTIC INTEGRALS III— ELLIPTIC INTEGRALS OF THE SECOND KIND * e 0° 5° 10° 15" 20° 25° 30° 35° 40° 45" I^ 0.01745 0.01745 0.01745 0.01745 0.01745 0.01745 0.01745 0.01745 0.01745 0.0174s 2 03491 03491 03491 03491 03491 03491 03490 03490 03490 03490 3 05236 05236 05236 05236 05236 05236 0523s 05235 05235 05235 4 06981 06981 06981 06981 06981 06980 06980 06979 06979 06978 5 08727 08727 08726 08726 08725 08725 08744 08723 08722 08721 6 10472 10472 1047 1 10471 10470 10469 10467 10466 10464 10462 7 12217 12217 12216 12215 12214 12212 12210 12207 12205 12202 8 13963 13962 13961 13960 13957 13955 13951 13948 13944 13940 9 15708 15707 15706 15704 15700 15696 15692 15687 15681 15676 10 17453 17453 17451 17447 17443 17438 1 743 1 17427 17417 17409 II 19199 19198 19195 19191 19185 19178 19169 19160 19150 19140 12 20944 20943 20939 20934 20926 20917 20906 20894 20881 20868 13 22689 22688 22683 22676 22667 22655 22641 22626 22609 22593 14 24435 24433 24427 24419 24406 24392 24374 24355 24335 24314 15 26180 26178 26171 26160 26145 26127 26106 26083 26058 26032 i6 27925 27923 27914 27901 27883 27S61 27836 27807 27777 27746 17 29671 29667 29658 29642 29620 29594 29563 29529 29493 294SS i8 31416 31412 31401 31382 31357 31325 31289 31248 31205 31161 19 33161 33157 33143 33121 33092 33055 33012 32965 32914 32862 20 34907 34901 34886 34860 34825 34783 34733 34678 34619 34558 21 36652 36646 36628 36598 36558 36509 36451 36387 36319 36249 22 38397 38390 38370 38336 38290 38233 38167 38094 38015 37934 23 40143 40135 40111 40073 40020 39955 39880 39796 39707 39614 24 41888 41879 41852 41809 41749 41676 41590 41496 41394 41289 25 43633 43623 43593 43544 43477 43394 43298 43191 43076 42958 26 45379 45367 45333 45278 45203 45110 45002 44882 447S3 44620 27 47124 47111 47074 47012 46928 46824 46703 46569 46425 46276 28 48869 48855 48813 48745 48651 48536 48402 48252 48092 47926 29 50615 50599 50553 50477 50373 50245 50097 49931 49753 49569 30 52360 52343 52292 52208 52094 51953 51788 SI 605 51409 5 1 205 31 54105 54086 54030 53938 53813 53657 53476 S3275 S3059 52834 32 55851 55830 55768 55667 55530 55360 55161 54940 54703 54456 33 57596 57573 57506 57396 57245 57059 56842 56600 56341 56070 34 59341 59317 59243 59123 58959 58756 58520 58256 57972 57677 35 61087 61060 60980 60850 60672 60451 60194 59907 59598 59276 36 62832 62803 62716 62575 62382 62143 61864 61552 61217 60868 37 64577 64546 64452 64300 64091 63832 63530 63193 62830 62451 38 66323 66289 66188 66023 65798 65519 65193 64828 64436 64027 39 68068 68031 67923 67746 67503 67203 66851 66459 66035 65594 40 69813 69774 69658 69467 69207 68884 68506 68084 67628 67153 41 71558 71517 71392 71188 70909 70562 70157 69703 69214 68703 42 73304 73259 73126 72907 72609 72238 71804 71318 70793 70245 43 75049 75001 74859 74626 74307 73910 73446 72927 72365 71778 44 76794 76744 76592 76343 76003 75580 75085 74530 73931 73303 45 0.78540 0.78486 0.78324 0.78059 0.77697 0.77247 0.76720 0.76128 0.75489 0.74819 FIVE-PLACE TABLES gg III.-ELLIPTIC INTEGRALS OF THE SECOND KIND e So° 55° 60° 65° 70° 75° 80° 85° 90° X °o.oi74 50,0174 50.0174 ; 0.01745 0.0174s 0.01745 0.0174^ 0.0174'; o.oi7^c 2 0349< 3 0349( 3 0349( 3 034g( 3 03490 03490 03490 03490 03490 3 4 S 0523. 0697^ 0872c ) 0523. i o697f ) 0871C ^ 0523. i 0697 ) 0871! i 0523^ 7 0697 i 0871S t 05234 05234 05234 05234 05234 ! 06976 06976 06976 06976 06976 ! 08717 08716 08716 08716 08716 6 10461 1045? ) 1045S i I045f I045i 1045, i 104S3 1045; 10453 7 1219s 1219- 1219; 12192 I2igc 1218c ) 12188 1218; 12187 13917 15643 17365 8 I393t 13932 13925 13925 13923 1392c > 13919 1391J 9 IS67C 1566 = 15 66c 15655 15651 1564? ' 15645 15644 lo 17401 17394 17387 17381 17375 17371 17367 17365 II 1913c 1912c 19110 19102 1909s 19085 19084 19082 19081 20791 12 208SS 20842 20830 20819 20809 20801 20796 20792 13 22576 22SS9 22544 22530 225x8 22508 22501 22497 22495 14 24293 24272 24253 24236 24221 24209 24200 24194 24192 IS 26006 25981 25957 25936 25917 25902 25891 25884 25882 i6 27714 27684 2765s 27629 27606 27588 27575 27567 27564 17 29418 29381 29347 293 1 5 29288 29267 29250 29241 29237 i8 31116 31073 31032 3099s 30963 30937 30917 30906 30902 IQ 32809 32758 32710 32666 32629 32598 32575 32561 32557 20 34496 34437 34381 34330 34286 34250 34224 34207 34202 21 3617S 36109 36044 3S98s 3 5934 35892 35862 35843 3S837 22 37853 37773 37699 37631 37572 37525 37490 37468 37461 23 39521 39431 39345 39268 39201 39146 39106 39081 39073 24 41183 41080 40983 40895 40819 40757 40711 40683 40674 25 42838 42722 42612 42513 42426 42356 42304 42273 42262 26 44486 44355 44232 44120 44023 43944 4388s 43849 43837 27 46126 45980 45842 45716 45607 45518 45453 45413 45399 28 47759 47595 47441 47301 47180 47081 47007 46962 46947 29 49383 49202 49031 4887s 48740 48629 48548 48498 48481 3° Siooo 50799 50609 50437 50287 50165 S0074 S0019 50000 31 52608 52386 52177 51986 51821 51686 51586 51525 SI 504 32 54207 53964 53733 53524 53341 S3 193 53082 53015 52992 33 55798 5S53I SS278 55048 54848 54684 54563 54489 54464 34 57379 57087 56811 56559 56340 56161 56028 55947 55919 35 58952 58634 58332 58057 57818 57622 57477 57388 57358 36 60515 60169 59841 59541 59280 59067 58909 S8811 58779 37 62068 61693 61337 61011 60727 60495 60323 60217 60182 38 63612 63206 62820 62467 62159 61907 61720 61605 61566 39 65146 64707 64290 63908 63574 63302 63099 62974 62932 4° 66671 66197 65746 65334 64974 64679 64459 64324 64279 41 68185 . 67675 67189 66745 66356 66038 65801 65655 65606 42 69688 69140 68619 68140 67722 67379 67124 66966 66913 43 71182 70594 70034 69520 69070 68701 68426 68257 68200 44 72665 72036 7143s 70884 70401 70005 69710 69527 69466 45 c •74137 c ■ 73465 c (.72822 c .72232 c ■7171S c .71289 c ). 70972 c ■70777 c 3.70711 100 ELLIPTIC INTEGRALS III.— ELLIPTIC INTEGRALS OF THE SECOND KIND 0" 5° 10° ■ 15° 20° 7 25° 30° 35° 40° 45° 46° 0.80285 0.80228 0.80056 0.79775 0.79390 0.78911 0.78350 0.77721 0.77040 0.76326 47 82030 81969 81787 81489 81081 80573 79977 79308 78584 77824 48 83776 837" 83518 83202 82770 .82231 81599 80890 80121 79313 49 85521 85453 85249 84914 84457 83887 83217 82466 81651 80794 50 87266 87194 86979 86626 86142 85539 84832 84036 83173 82265 SI 89012 88936 88709 88336 87826 87189 86442 85601 84689 83728 S2 9°757 90677 90438 90045 89507 88836 88048 87161 86197 85182 53 92502 92418 92166 91753 91187 90481 89650 88715 87698 86627 54 94248 94159 93895 93450 92865 92122 91248 90264 89193 88063 55 95993 95900 95622 95166 94S4I 93761 92843 91807 90680 89490 S6 97738 97641 97350 96872 96216 95397 94433 93345 92160 90908 57 0.99484 0.99381 0.99077 0.98576 97889 97030 96019 94878 93634 92318 S8 1. 01 229 1.01122 1.00803 1.00279 0.99560 0.98661 97602 96405 95100 93719 59 02974 02863 02529 01981 1.01229 1.00289 0.99180 97928 96560 95111 60 04720 04603 04255 03683 02897 01915 1.00756 0.99445 98013 9649s 61 06465 06343 05980 05383 04563 03538 02327 1.00957 0.99460 97871 62 08210 08084 07705 07083 06228 05158 03895 02465 1.00900 0.99238 63 09956 09824 09430 08781 07891 06776 054S9 03967 02334 1.00598 64 1 1 701 1 1 564 11154 10479 09553 08392 07020 05465 03762 01949 65 13446 13304 12878 12176 11213 10005 08577 06958 05183 03293 66 15192 15043 14601 13873 12871 11616 10132 08447 06599 04629 67 16937 16783 16324 15568 14529 13225 1 1 683 09932 08009 05957 68 18682 18523 18047 17263 16185 14832 13231 11412 09413 07279 69 20428 20262 19769 18957 17839 16437 14776 12888 10812 08593 70 22173 22002 21491 20650 19493 18040 16318 14360 12205 09901 71 23918 23741 23213 22343 21145 19640 17857 15828 I3S94 11202 72 25664 25481 2493s 24034 22796 21239 19394 17293 14977 12497 73 27409 27220 26656 25726 24446 22837 20928 18754 16356 13786 74 29154 28959 28377 27417 26094 24432 22459 20211 17731 15068 75 30900 30698 30097 29107 27742 26026 23989 21666 19101 16346 76 32645 32437 31818 30796 29389 27619 25516 23117 20467 17618 77 3439° 34176 33538 32486 31035 29210 27041 24566 21830 1888s 78 36136 35915 35258 34174 32680 30800 28565 26012 23189 20148 79 37881 37654 36978 35862 34325 32389 30086 27456 24544 21407 80 39626 39393 38698 37S50 35968 33976 31606 28897 25897 22661 81 41372 41132 40417 39238 37611 35563 33124 30336 27246 23912 82 431 1 7 42871 42137 40925 ,39254 37148 34641 31773 28594 25159 83 44862 44610 43856 42612 40896 38733 36157 33209 29939 26404 84 46608 46349 45575 44299 42537 40317 37672 34643 31282 27646 8S 48353 48087 47294 45985 44178 41900 39186 36076 32623 28886 86 50098 49826 49013 47671 45819 43483 40699 37508 33963 30124 87 S1844 51565 50732 49357 47459 45066 42211 38939 35302 31360 88 53589 53304 52451 5 1043 49100 46648 43723 40369 36640 32596 89 55334 55042 54170 52729 50740 48230 45235 41799 37977 33830 90 1.57080 1.56781 1.55889 1-54415 1.52380 1. 498 1 1 1.46746 1.43229 1-39314 1.35064 FIVE-PLACE TABLES 101 III.— ELLIPTIC INTEGRALS OF THE SECOND KIND ♦ so" 55° 60° 65° 70° 75° 80° 85° 90° 46° -75599 0.74881 0-74195 0.73564 0.73010 0.72554 0.72215 0.72005 0.71934 47 77050 76285 75553 74879 74287 73800 73436 73211 73135 48 78490 77676 76896 76177 75546 75025 74636 74396 74314 49 79920 79054 78225 77459 76786 76230 75815 75558 75471 SO 81338 80419 79538 78724 78007 77414 76971 76697 76604 51 82746 81772 80836 79971 79208 78578 78106 77814 77715 52 84143 83111 8212c 81202 80391 79720 79218 78907 78801 S3 85529 84438 8338S 82415 81554 80842 80307 79976 79864 S4 86904 85752 84641 83610 82698 81941 81374 81021 80902 SS 88269 87052 85879 8478S 83822 8302c 82417 82042 8191S S6 89622 88340 87101 85949 84926 84076 83436 83039 82904 S7 90965 89614 88308 87092 8601 1 85110 84432 84010 83867 58 92297 90876 89500 88217 87075 86122 85404 84957 84805 S9 93619 92125 90677 89325 88119 87112 86352 85878 85717 60 94930 93362 91839 904 1 5 89144 88080 87276 86773 86603 61 96231 94586 92986 91488 90148 89025 88175 87643 87462 62 97521 95797 94118 92543 91132 89948 89049 88486 88295 63 0.98802 96996 95236 93581 92096 90848 89898 89303 89101 64 1.00072 98183 96339 94602 93041 91725 90273 90094 89879 6S 01333 0.99358 97427 95606 93965 92580 91523 90858 90631 66 02585 1.00522 98502 96593 94870 93412 92297 91595 9135s 67 03827 01674 0.99562 97564 95756 94222 93047 92305 92050 68 05060 02815 1.00609 98518 96622 95010 93771 92987 92718 69 06284 03945 01643 0.99456 97469 95775 94470 93642 93358 70 07500 05064 02664 1.00379 98298 96519 95144 94270 93969 71 08707 06173 03672 01286 99108 97240 95793 94870 94552 72 09907 07272 04668 02178 0.99900 97940 96417 95442 95106 73 1 1098 08362 05651 03056 1 .00674 98619 97016 95987 95630 74 12283 09442 06624 03919 01431 99278 97590 96503 96126 75 13460 10513 07586 04769 02172 0.99916 98141 96992 96593 76 14631 11577 08537 05607 02896 I 00534 98667 97453 97030 77 15795 12632 09478 06432 03605 01133 99170 97887 97437 78 16954 13680 10410 07245 04300 01714 0.99650 98293 9781S 79 18107 14721 11333 08047 04981 02277 1.00107 98671 98163 80 1925s 15755 12249 08839 05648 02823 00543 99023 98481 81 20399 16784 13156 09621 06304 03354 00958 99348 98769 82 21538 17807 14057 10395 06948 03870 01 3 54 99646 99027 83 22673 18825 14952 11161 07582 04372 01731 0.99920 99255 84 23805 19839 15841 11920 08207 04863 02091 I. 00168 99452 85 24934 20850 16726 12673 08825 05343 02436 00394 99619 86 26061 21857 17606 13421 09435 05813 02768 00598 99256 87 27186 22862 18484 1416s 10041 06277 03089 00784 99863 88 28310 23865 19359 14906 10642 0673s 03401 00954 99939 89 29432 24867 20233 15645 11241 07188 03708 01113 0.99985 90 I 30554 1.25868 I .21106 1.16383 1. 11838 I .07641 1.04011 1,01266 I. 00000 INDEX Abel, Niels, 6, 7, 24, 29, 31 Addition-Theorem, 7 d'Alembert, Jean, 6 Amplitude, 24 Andoyer, H., 6 Appell and Lacour, Fonctions Ellip- tiques, 7, 14, 21 Arithmetico-Geometrical Mean, 78 Bernoulli, Jacob, 6, 7 Burnside, W. S., 54 Byerly, Integral Calculus, 48, 63, 90, 91 Calculation of elliptic functions, 33 Calculation of elliptic function, when modulus and argument are given, 87 Calculation of argument, when k and the function are given, 87 Cayley, Arthur, 6, 14, 16, 28, 65, 68, 73, 76 Clifford, W. K., 41 Co-amplitude, 35 Computation, see Numerical computa- tion Cosine-amplitude = CH, 24 Delta-amplitude = dH, 24 Durege, Elliplische Funklionen, 59 E{k, ), see Integral E{k, 4,) El, the complete integral, 16, 17, 81 Tables for £1, 93 Elastic curves, 6, 89 Eimepei, EUiptische Funklionen, 8, 10, 12 Euler, Leonhard, 7 Fik, h), see Integral F{k, h) F,, the complete integral, 16, 17, 81 Tables for Fi, 93 Fagnano, G. C, 6, 7 Fricke, Elliplische Funktionen, 8 Functions, elliptic, 22-39 Gauss, C. F., 78 Graphs of the integrals of the first and second kinds, 16, 17, 18, 19, 20 Graphs of the functions sn, en and dn, 36,37,38 Greenhill, A. G., 6, 52 Gudermann, M., 24, 28 Gudermannian, The, 28, 29 Hancock, Harris, 6, lo, 15 Hermite, Chas., 6 Humbert, G., 92 Hyperbola, rectification of, 6 Infinities, 35, 36, 37, 38 Integrals, the F{k, ) and E{k, 0), their graphs, 15-20 Integrals, tables for F{k, ) and £(*, ) , 94-101 Integral, the n (h, k, ip), 15 Integrals, the general elliptic, 9, 14 Integrals of certain elliptic functions, 58-63 Integrals of first kind reduced to normal form, 41-63 Integrals, normal, 10, 15 Integrals, of the first, second and third kinds, 9, 60 103 104 INDEX Jacobi, C. G. J., s, 6, 7, 24, 25, 26, 29. 31, 34, 6s Jacobi's imaginary transformation, 25, 34 K, K', tables for, 93 K, K', the transcendents, 26 Lacour, E., see Paul Appell Lagrange, Joseph L., 7, 68 Landen, John, 68 Landen's transformations, 65—73 Latitude, 89 Legendre, Adrien Marie, 5, 6, 7, g, 10, II, 12, 15, 21, 22, 23, 24, 61, 68, 71, 81, 82, 84,92 Legendre's transformation, 10-13 Lemniscate, 7, 88 Levy, Fonclions ellipiiques, 21, 92 Maclaurin, Colin, 6 Modulus, 12 Molk, Jules, see Jules Tannery Numerical computation of integrals of ist and 2d kinds, 69-87 Pendulum, simple, 90, 91 Periods, 27, 31, 35-38 Periodic functions, property of, 7, 31 Period-parallelogram, 32 Period-parallelogram of sn u, p.35, of en u, p. 37, of in u, p. 38. Period-strip, 31 Quadrant of lemniscate, 88; of ellipse, 89 Rectification of ellipse, 88; of lemnis- cate, 88 Reduction formulas, 59-61 Richelot, F. S., 10 Serret, J. A., 7 Simpson's Fluxions, 91 Sine-amplitude = iK, 24 Tables of complete integrals of ist and 2d kinds, 93 Tables of elliptic integrals of ist kind, 94-97 Tables of elliptic integrals of 2d kind, 98-101 Tables of useful integrals, 61-63 Tannery and Molk, Fonctions Eilip- tiques, 14 Volume common to two cylinders, 89 Wallis's formula, 26, 60 Weierstrass, Karl, 24, 28 Zeroes, 35-38 ComBitHVERsnYUBfwnr 4UL 1 r, 1992 wsmtfmmfRt