ELLIPTIC FUNCTIONS. N and P being rational integral functions of x. Whence Eq. (3) becomes (4) V= I * / = Ndx- i/ *J Eq. (4) shows that the most general form of Fcan be made to depend upon the expressions (5) V'= iy and fNdx. This last form is rational, and needs no discussion here. We can write * + .. )* Multiplying both numerator and denominator by we have a new mnaerator which contains only powers of x* The result takes the following form : p=^ x) . x. Equation (5) thus becomes ELLIPTIC INTEGRALS. ^ We shall see presently that R can always be assumed to be of the form Therefore, putting x* = 2, the second integral in Eq. (6) takes the form I r Y(z\ . dz which can be integrated by the well-known methods of Integral Calculus, resulting in logarithmic and circular transcendentals. There remains, therefore, only the form J R to be determined. We will now show that R can always be assumed to be in the form We have R V Ax* + Bx* + Cx* + Dx -f E = VG(x - d)(x - b](x - c](x - d\ a, b, c, and d being the roots of the polynomial of the fourth degree, and G any number, real or imaginary, depending upon the coefficients in the given polynomial. Substituting in equation (i) _ X i we have (7) V THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES ELLIPTIC FUNCTIONS. AN ELEMENTARY TEXT-BOOK FOR STUDENTS OF MATHEMATICS. BY ARTHUR L. BAKER, C.E., PH.D., PROFESSOR OF MATHEMATICS IN THE STEVENS SCHOOL OF THE STEVENS INSTITUTE OF TECHNOLOGY, HOBOKEN. N. J. ; FORMERLY PROFESSOR IN THE PARDEE SCIENTIFIC DEPARTMENT, LAFAYETTE COLLEGE, EASTON, PA. r H(u) sin am = -?-. r. 4/A &(U) NEW YORK: JOHN WILEY & SONS, 53 EAST TENTH STREET. 1890. Copyright, 1890, BY ARTHUR L. BAKER. ROBERT DRTTMMOND, FERRIS BROS., Electrotyper, Printers, 444 & 446 Pearl Street, 326 Pearl Street, New York. Hew York. Library 34-3 PlU PREFACE. IN the works of Abel, Euler, Jacobi, Legendre, and others, the student of Mathematics has a most abundant supply of material for the study of the subject of Elliptic Functions. These works, however, are not accessible to the general student, and, in addition to being very technical in their treat- ment of the subject, are moreover in a foreign language. It is in the hope of smoothing the road to this interesting and increasingly important branch of Mathematics, and of putting within reach of the English student a tolerably com- plete outline of the subject, clothed in simple mathematical language and methods, that the present work has been com- piled. New or original methods of treatment are not to be looked for. The most that can be expected will be the simplifying of methods and the reduction of them to such as will be intelligi- ble to the average student of, Higher Mathematics. I have endeavored throughout to use only such methods as are familiar to the ordinary student of Calculus, avoiding those methods of discussion dependent upon the properties of double periodicity, and also those depending upon Functions of Com- plex Variables. For the same reason I have not carried the discussion of the @ and H functions further. IV PREFACE. Among the minor helps to simplicity is the use of zero subscripts to indicate decreasing series in the Landen Trans- formation, and of numerical subscripts to indicate increasing series. I have adopted the notation of Gudermann, as being more simple than that of Jacobi. I have made free use of the following works : jACOBl's Fundamenta Nova Theoriae Func. Ellip.; HOUEL'S Calcul Infinitesimal ; LEGENDRE's Trait des Fonctions Elliptiques ; DUREGE'S Theorie der Elliptischen Functionen ; HERMITE'S Theorie des Fonctions Elliptiques ; VERHULST'S Theorie des Functions Elliptiques ; BERTRAND'S Calcul Integral ; LAU- RENT'S Th6orie des Fonctions Elliptiques ; CAYLEY'S Elliptic Functions ; BYERLY'S Integral Calculus ; SCHLOMlLCH's Die Hoheren Analysis; BRIOT ET BOUQUET'S Fonctions Ellip. tiques. I have refrained from any reference to the Gudermann or Weierstrass functions as not within the scope of this work, though the Gudermannians might have been interesting examples of verification formulae. The arithmetico-geometrical mean, the march of the functions, and other interesting investi- gations have been left out for want of room. CONTENTS. PAGE INTRODUCTORY CHAPTER, ....... i CHAP. I. ELLIPTIC INTEGRALS, ...... 4 II. ELLIPTIC FUNCTIONS, . . . . . .16 III. PERIODICITY OF THE FUNCTIONS, ... .22 IV. LANDEN'S TRANSFORMATION, ..... 30 V. COMPLETE FUNCTIONS, ...... 45 VI. EVALUATION FOR (f>, . . . . .48 VII. FACTORIZATION OF ELLIPTIC FUNCTIONS, . . .51 VIII. THE FUNCTION, ...... 66 IX. THE & AND H FUNCTIONS, ..... 6g X. ELLIPTIC INTEGRALS OF THE SECOND ORDER, . . 81 XI. ELLIPTIC INTEGRALS OF THE THIRD ORDER, . . .90 XII. NUMERICAL CALCULATIONS, q, . .94 XIII. NUMERICAL CALCULATIONS, K, . . . 98 XIV. NUMERICAL CALCULATIONS, , . . . . . 102 XV. NUMERICAL CALCULATIONS, 0, . . . . . 108 XVI. NUMERICAL CALCULATIONS, E(k, 0), . . . in XVII. APPLICATIONS, . . . . , . .115 ELLIPTIC FUNCTIONS. INTRODUCTORY CHAPTER* THE first step taken in the theory of Elliptic Functions was the determination of a relation between the amplitudes of three functions of either order, such that there should exist an algebraic relation between the three functions themselves of which these were the amplitudes. It is one of the most re- markable discoveries which science owes to Euler. In 1761 he gave to the world the complete integration of an equation of two terms, each an elliptic function of the first or second order, not separately integrable. This integration introduced an arbitrary constant in the form of a third function, related to the first two by a given equation between the amplitudes of the three. In 1775 Landen, an English mathematician, published his celebrated theorem showing that any arc of a hyperbola may be measured by two arcs of an ellipse, an important element of the theory of Elliptic Functions, but then an isolated result. The great problem of comparison of Elliptic Functions of dif- ferent moduli remained unsolved, though Euler, in a measure, exhausted the comparison of functions of the same modulus. It was completed in 1784 by Lagrange, and for the computation * Condensed from an article by Rev. Henry Moseley, M.A., F. R.S. , Prof, of Nat. Phil, and Ast., King's College, London. 2 ELLIPTIC FUNCTIONS. of numerical results leaves little to be desired. The value of a function may be determined by it, in terms of increasing or diminishing moduli, until at length it depends upon a function having a modulus of zero, or unity. For all practical purposes this was sufficient. The enor- mous task of calculating tables was undertaken by Legendre. His labors did not end here, however. There is none of the discoveries of his predecessors which has not received some perfection at his hands ; and it was he who first supplied to the whole that connection and arrangement which have made it an independent science. The theory of Elliptic Integrals remained at a standstill from 1786, the year when Legendre took it up, until the year 1827, when the second volume of his Traite" des Fonctions Elliptiques appeared. Scarcely so, however, when there ap- peared the researches of Jacobi, a Professor of Mathematics in Konigsberg, in the I2^d number of the Journal of Schumacher, and those of Abel, Professor of Mathematics at Christiania, in the 3d number of Crelle's Journal for 1827. These publications put the theory of Elliptic Functions upon an entirely new basis. The researches of Jacobi have for their principal object the development of that general relation of functions of the first order having different moduli, of which the scales of Legrange and Legendre are particular cases. It was to Abel that the idea first occurred of treating the Elliptic Integral as a function of its amplitude. Proceeding from this new point of view, he embraced in his speculations all the principal results of Jacobi. Having undertaken to de- velop the principle upon which rests the fundamental proposi- tion of Euler establishing an algebraic relation between three functions which have the same moduli, dependent upon a cer- tain relation of their amplitudes, he has extended it from three to an indefinite number of functions; and from Elliptic Func- tions to an infinite number of other functions embraced under an indefinite number of classes, of which that of Elliptic Func- INTRODUCTORY CHAPTER. 3 tions is but one ; and each class having a division analogous to that of Elliptic Functions into three orders having common properties. The discovery of Abel is of infinite moment as presenting the first step of approach towards a more complete theory of the infinite class of ultra elliptic functions, destined probably ere long to constitute one of the most important of the branches of transcendental analysis, and to include among the integrals of which it effects the solution some of those which at present arrest the researches of the philosopher in the very elements of physics. CHAPTER I. ELLIPTIC INTEGRALS. THE integration of irrational expressions of the form Xdx VA + Bx + Cx\ or Xdx VA 4- BX + v ' ^f being a rational function of x, is fully illustrated in most ele- mentary works on Integral Calculus, and shown to depend upon the transcendentals known as logarithms and circular functions, which can be calculated by the proper logarithmic and trigono- metric tables. When, however, we undertake to integrate irrational expres- sions containing higher powers of x than the square, we meet with insurmountable difficulties. This arises from the fact that the integral sought depends upon a new set of transcendentals, to which has been given the name of elliptic functions, and whose characteristics we will learn hereafter. The name of Elliptic Integrals has been given to the simple integral forms to which can be reduced all integrals of the form (i) V=fF(X,R)dx, where F(X, R) designates a rational function of x and R, and R represents a radical of the form R = VAx* 4- Bx 3 -f- Cx* -\- Dx -f- E, ELLIPTIC INTEGRALS. 5 where A, B, C, D, E indicate constant coefficients. We will show presently that all cases of Eq. (i) can be reduced to the three typical forms r*_ dx / i/O dx which are called elliptic integrals of the first, second, and third order. Why they are called Elliptic Integrals we will learn further on. The transcendental functions which depend upon these integrals, and which will be discussed in Chapter IV, are called Elliptic Functions. The most general form of Eq. (i) is (3) r= where A, B, C, and D stand for rational integral functions of x. A + BR -^. ~r- can be written L -f- UK. A + BR _ AC- BDR* (AD - CB}R* \_ C'-ITJ? ' C*-D'l? ' R ELLIPTIC FUNCTIONS. N and P being rational integral functions of x. Whence Eq. (3) becomes (4) 17 Cw C Pd * V= Ndx-- - jr . IX I/ Eq. (4) shows that the most general form of Fcan be made to depend upon the expressions (5) V '= IX and fNdx. This last form is rational, and needs no discussion here. We can write Multiplying both numerator and denominator by we have a new mHaerator which contains only powers of The result takes the following form : . . = ^) _|_ W(x*} . x. Equation (5) thus becomes ELLIPTIC INTEGRALS. ^ We shall see presently that R can always be assumed to be of the form Therefore, putting x* = z, the second integral in Eq. (6) takes the form I C W & dz V V(i-z}(i- which can be integrated by the well-known methods of Integral Calculus, resulting in logarithmic and circular transcendentals. There remains, therefore, only the form J R to be determined. We will now show that R can always be assumed to be in the form We have R= V Ax* + Bx* + Cx* -f Dx + E = VG(x - a\x - b}(x - c}(x - d\ a, b, c, and d being the roots of the polynomial of the fourth degree, and G any number, real or imaginary, depending upon the coefficients in the given polynomial. Substituting in equation (i) x i we have (7) v 8 ELLIPTIC FUNCTIONS. p designating the radical P= tfG\p-a + (q-a}y\ [p-b+(q-b)y\ \j>- c +(g-c)y\. . . . In order that the odd powers of y under the radical may disappear we must have their coefficients equal to zero; i.e., -a = O, (P -c}(q-d} + (p- d) (q - .c] = O ; whence 2 pq (p + q)(a -f b) + lab = O, zpq (p + q\ c + d) + 2cd = O, and _ ab(c + d)- cd(a + b) q ~~ 2(lb - 2cd Equation (8) shows that p and q are real quantities, whether the roots a, b, c, and d are real or imaginary ; a, b, and c, d being the conjugate pairs. Hence equation (i) can always be reduced to the form of equation (7), which contains only the second and fourth powers of the variable. This transformation seems to fail when a-\-b (c-\-d) = o; but in that case we have R= VG\x* (a+ b)x + ab\ [x* and substituting a + b x = y -- ^ 2 will cause the odd powers of y to disappear as before. If the radical should have the form - d](x - V)(x - r), ELLIPTIC INTEGRALS. \ placing x = y* -f- a, we get designating a rational function of y and p. Thus all integrals of the form contained in equation (i), in which R stands for a quadratic surd of the third or fourth degree, can be reduced to the form (9) r J? designating a radical of the form VG(i + and n designating constants. It is evident that if we put n m we can reduce the radical to the form We shall see later on that the quantity & 1 , to which has been given the name modulus, can always be considered real and less than unity. Combining these results with equation (6), we see that the integration of equation (i) depends finally upon the integration of the expression do) R IO ELLIPTIC FUNCTIONS. The most general form of 0(^ 2 ) is L Hence /, / , /i^2W .V-j^ / CLX -jT + ztj (?+iifR / / , / , depends upon / -^- and / 5-, which can ^ J K J K be shown as follows : Differentiating Rx 2m ~ 3 , we have a + fix* (2m $)x-*dx V a-\- fix* -f- yx* x* m Integrating and collecting, we get Cx-*dx ; . _ Cx-*d -* (2m 3) / - ^ -- h (2* - 2 )^ / ^ . Cxdx -\-(2m- \}y I ^~ " 4 fx~*dx Cxdx_ R YY R ELLIPTIC INTEGRALS. II Whence we get, by taking m 2, Cdx , Cx*dx , Cx*dx (13) & fi= J 1[ + ft J + r j< irt /x im dx R can be found by successive calculations, when we are able to integrate the expressions /dx Cx^dx -R and J -IT' the first and second of equation (2). We will now consider the second class of terms in eq. (i i), Ldx V17 M (** + a} n K This second term is as follows : i \ ^ C L f Adx C Bdx J (x* + a}R-J (x*+aYR^J (** + <#-*& /?+ / V"* I c * ) ^~ Each of these terms can be shown to depend ultimately upon terms of the form x*dx dx dx -, -5-, and , , , , p . R ' R The two former will be recognized as the two ultimate forms already discussed, the first and second of equation (2). The third is the third one of equation (2). This dependence of equation (14) can be shown as follows: 12 ELLIPTIC FUNCTIONS. We have _ '(*'+*)* - \xdR+Rdx) - 2x>R(n+ 1 )(*'+)* - *dx (*+*) x) 2x*R(n -\}dx Substituting the value of * and ^ = (fix we get (*+> X 37 x 6 + 2? x* + 2aft x* +a 2(n i)y h 3^7 Hh 2(n - i)P - 2(n - \]a dx ~R (zn 3)or dx or, by substituting in the numerator^ 3 = z a, (2n - (2n 3)0- -\-(2n dx ELLIPTIC INTEGRALS. 13 or, after resubstituting z x* -}- ft, and integrating, xR C dx C dx /dx C dx C dx (x* + ay-*R^ P *J (S + ar^ Y J &+* r ^ (^+) Making n = 2, we have / ^ ^ (^ + tf ) Equation (16) shows that r ^r J (S+afR depends upon the three forms /x*dx Cdx C dx - r , ^ ^-, and J p^^, 14 ELLIPTIC FUNCTIONS. the three types of equation (2), and equation (15) shows that the general form depends ultimately upon the same three types. We have now discussed every form which the general equa- tion (i) can assume, and shown that they all depend ultimately upon one or more of the three types contained in equation (2). These three types are called the three Elliptic Integrals of the first, second, and third kind, respectively. Legendre puts x = sin 0, and reduces the three integrals to the following forms : (, 7 ) F(k, = 75 ; cos' , E (iP)F - sn / / / o /^sec 8 A tan + (i - k^F - E */0 / l J.L l ( r ^ sin cos 0\ -d = - i -\E ^ )', /^sin 11 i IE (i k*}F sin cos / cos a F E sin cos ^ - C. A \ -&- & .- -o CHAPTER II. ELLIPTIC FUNCTIONS. LET u = tf sin 2 0* is called the amplitude corresponding to faz. argument u, and is written = am (u, k] = am u. The quantity k is called the modulus, and the expression I/I k* sin 2 is written * l/i /P sin 2 = A am u and is called the delta function of the amplitude o* u, or afc/te 0/~ 0, or simply delta 0. can be written u = F(k, 0). The following abbreviations are used : , sin = sin am u = sn f ; cos = cos am u = en f ; /^0 = /4 am u dn f //; = du ; tan = tan am u = tn u. Let and ^ be any two arbitrary angles, and put = am u ; ^> = am v. * Legendre. \ Gudermann, in his " Theorie der Modularfunctionen " : Crelle's Journal, Bd. 18. 16 ELLIPTIC FUNCTIONS. I'J In the spherical triangle ABC we have from ^ B Trigonometry, c and C being constant, d(f> dtp cos B ' cos A Since C and c are constant, denoting by k an arbitrary con- stant, we have sin C __ , sin // But sin B sin C sin -<4 = sin ib . -7 = sin t^ = k sin t&. r sin sin yw Whence cos A = Vi sin 2 A = Vi & sin a In the same manner cos 5 = Vi sin" .5 = Vi J? sin 2 0. Substituting these values, we get (2) + = o. Vi k 1 sin 2 Vi k* sin" ^> Integrating this, there results /** C, which requires that one of the angles of the triangle shall be obtuse and the other two acute. In the figure, let C be an acute angle of the triangle ABC, and PQ the equatorial great circle of which C is the pole. The arc PQ will be the measure of the an- gleC Let AG and AH be the arcs of two great circles perpendicular respectively to CQ and CP. They will of course be shorter than PQ. Hence AB = j* must intersect CQ in points between CG and HQ, since j^>(C = PQ}. In any case either A or B will be obtuse according as B falls between QH or CG respectively ; and the other angle will be acute. In the case where C is an obtuse angle, it will be easily seen that the angle at A must be acute, since the great circle AD^ perpendicular to CP, intersects PQ in D, PD being a quadrant. The same remarks apply to the angle B. Hence, in eithei ELLIPTIC FUNCTIONS. 1 9 case, one of the angles of the triangle is obtuse and the other two are acute, as a result of the condition sin C - =k< . i. sin j.i From Trigonometry we have ^ ^~~p cos /u = cos cos ^ -|- sin sin ^> cos C\ and since the angle C is obtuse, cos C = Vi sin 2 C = 1/1 Xr sin and (5) cos /* = cos cos ^ sin sin ^ rl k 1 sin 2 //, the relation sought. The spherical triangle likewise gives the following relations between the sides: cos = cos j.i cos ip -f- sin /* sin ip Vi k* sin cos >p = cos // cos -|- sin fj. sin V i / 2 sin 2 These give, by eliminating cos /*, cos 2 if? cos 2 sin w sin cos ip A y sin ty cos /j ' Avhich, after multiplying by the sum of the terms in the de- nominator and substituting cos 2 = i sin", can be written (sin 2 sin 2 /?)(sin cos rfi A ty -f- sin ^ cos 0/40 Sill A^ - - o 7 5 I yf 2 i 2 i o "7 Tb & sin sin ?/> cos cos ?/? J/< = ;~s . o i i o i k sin sin 2O ELLIPTIC FUNCTIONS. These equations can also be written as follows : sin am (u v) = cos am ( v) = A am (u v) sin am u cos am r /7 am v sin am v cos am ^/ am u i * sin 4 am u sin 2 am v cos am cos am v T sin am z< sin am * ^ am u A am v I k 1 sin 1 ' am u sin* am j' // am u A am f T /' 2 sin am w sin am v cos am u cos am y I k' 2 sin- am sin 4 am v or (8) sn (u v) = cn (u T') = dn ( v) = sn u cn K dn v sn r cn u dn i k* sn 2 sn'" v cn u cn v ^F sn u sn y dn u dn y i k* sn 2 sn 2 v dn u dn K =p < 2 sn z^ sn r cn u cn i k sn sn" y Making u = v, we get from the upper sign 2 sn u cn & dn u (9) sn 2u = cn 2u = dn 2u = cn 2 w sn 2 u dn a 7* I 2 sn a sn u i _ ' sn 1 u i - ^ sn 4 u ' dn 2 u t? sn 2 u cn 2 u i 2^ 2 sn 2 u -f- >^ 2 sn 4 From these (10) i k sn n I cn 211 = I -|- cn 2u = i dn u = i + dn u = i sn 2 cn 2 w dn a u I ^ sn" u ' 2 cn 2 u I fc* sn 4 w ' 2& 1 sn 2 ?/ cn 2 i k 1 sn 4 2 dn s TS i i k sn u ELLIPTIC FUNCTIONS. 21 and therefore sn u = en u = dn 3 u = I en 2u I -f- dn 2u ' dn 2ft -f- en 2 I -)- dn 2 i k* -f- dn -f- a en i -(- dn 2 and by analogy (12) sn /I 2=V T en -j- dn ' u /en u -f- dn I -{- dn dn = > 3 -J- dn w -}- ^ a en u I -f- dn u In equations (7) making u v, and taking the lower sign, \ve have f" sn = 0; (13) j cno= i; [ dno = I. Likewise, we get by making u o, f sn ( w) = sn u ; (14) \ cn( 7/) = + cn ; |_ dn ( ) = dn u. CHAPTER III. PERIODICITY OF THE FUNCTIONS. WHEN the elliptic integral d Vi k* sin 2 has for its amplitude -, it is called, following the notation of Legendre, the complete function, and is indicated by K, thus : K=. . vi k sin When k becomes the complementary modulus, k', (see eq. 4, Chap. IV,) the corresponding complete function is indicated by AT', thus: |/i -k'* sin 3 From these, evidently, am (K, k} = -, am (A", k'} = -. 2 2 'sn (K, k) i ; (i) -j en (AT, ) = o ; dn (AT, ^) = k'. PERIODICITY OF THE FUNCTIONS. 23 From eqs. (7), (8), and (9), Chap. II, we have, by the sub- stitution of the values of sn (K) = i, en (K} = o, dn (K} = k', (2) sn 2K = o ; en 2K= i ; These equations, by means of (i), (2), and (3) of Chap. II, give f sn (ft -j- 2A~) = sn u ; (3) -j en (ft + 2 AT) = en u ; [ dn (u -\- 2K) = dn u ; and these, by changing u into u -f- 2K, give f sn (ft -f- 4^f ) = sn u ; (4) \ en ( + 4AT) = en ; (__ dn (ft -f- 4AT) = dn ft. From these equations it is seen that the elliptic functions sn, en, dn, are periodic functions having for their period ^K. Unlike the period of trigonometric functions, this period is not a fixed one, but depends upon the value of k, the modulus. From the Integral Calculus we have /* xw tf . / 2 #0 , / ' r~ / r~. . . -f- / A / - = 2AT / Thus we see that the Integral with the general amplitude a can be made to depend upon the complete integral K and an Integral whose amplitude lies between o and -. Put now This gives / or am (2nK u) = mt ft (5) = mt am u (6) = 2n . am /T am u ; or, since am ( #) = am z, am (# 2wA") = am u mr = am u 2# . am A". Taking the sine and cosine of both sides, we have sn (it + 2nK) sn ?/ ; en (# -f~ 2^T) = en 11 ; the upper or the lower sign being taken according as n is even or odd. By giving the proper values to n we can get the same results as in equations (3) and (4). Putting n = i in eq. (5), we have sn (2.K u} = sin n en u cos n sn u (7) = sn u. 26 ELLIPTIC FUNCTIONS, Elliptic functions also have an imaginary period. In order to show this we will, in the integral assume the amplitude as imaginary. Put sin = i tan $. From this we get i (8) cos = cos COS ' COS d Ad>~ ~ ' * sn ( u) = i tn (iu, k'\ i en ( u) = dn = en (iu, k') ' en (/, k) ' or, from eq. (14), Chap. II, sn = i tn (/', >^') : (10) en 11 = dn z^ = en (iu, k'} ' dn (i, Q -- p ~ . en (iu, k ) From eqs. (7), Chap. II, making v = K, we get, since sn K = i, en ^T = o, dn K ', en (u -4- K~\ -1- en u di 1 U en u Sn \li 1\ ) It rn d/ -4- K"\ i k* sn a u ~ sn u dn uk' L dn u ' _&'snu dn (u K}= + dn 2 u k' dn u ' In these equations, changing u into iu, we get, by means of eqs. (9), sn (iu K) = - -. TTT ; dn (u. k) (12) en sn .. dn (, Q 28 ELLIPTIC FUNCTIONS. Putting now in eqs. (9) u K' instead of u, and making use of eqs. (10), and interchanging k and k', we have (13) sn ( tK) = - en (in iK>) = k sn (21, k') ' i dn (iu iK') = T - TTT . sn (w, /r) Substituting in these iu in place of u, we get, by means of eqs. (9) and eqs. (14) of Chap. II, (14) sn (# iK') = k sn ' en (u iK') =F T dn u sn u dn (u iK') = q= i cot am u. In these equations, putting u -\- K in place of , we get dn u 05) sn ( -f- K iK'} = en ft? 1 k en ' dn (u-\- K iK') = ^' tn u. Whence for = o we get sn (K iK) = i; (16) PERIODICITY OF THE FUNCTIONS. 2$ If in eqs. (14) we put u = o, we see that as u approaches zero, the expressions sn ( iK'\ en ( iK'\ dn ( iK'} approach infinity. We see from what has preceded that Elliptic Functions have two periods, one a real period, and one an imaginary period. In the former characteristic they resemble Trigonometric Functions, and in the latter Logarithmic Functions. On account of these two periods they are often called Doubly Periodic Functions. Some authors make this double periodicity the starting-point of their investigations. This method of investigation gives some very beautiful results and processes, but not of a kind adapted for an elementary work. It will be noticed that the Elliptic Functions sn #, en u, and dn u have a very close analogy to trigonometric functions, in which, however, the independent variable u is not an angle, as it is in the case of trigonometric functions. Like Trigonometric Functions, these Elliptic Functions can be arranged in tables. These tables, however, require a double argument, viz., u and k. In Chap. IX these functions are de- veloped into series, from which their values may be computed and tables formed. No complete tables have yet been published, though they are in process of computation. Then CHAPTER IV. LANDEN'S TRANSFORMATION. LET AB be the diameter of a circle, with the centre at O, the radius A O = r, and C a fixed point situated upon OB, and OC = k^r. Denote the angle PBC by 0, and the angle PCO by t . Let P' be a point indefinitely near to P. sin PCP' sin PCP But P. therefore PC sin PP'C P = 2?Y/0, and sin / 277/0 <3 cos OPC 1 t CP f = PCP - 1 ' 1 But also r* cos 2 Therefore 2 -f 21* k t cos 20 cos a + (r - rktf sin 8 0; C = r 2 - r 3 sin 2 OPT = r 2 r 2 /^ 2 sin 2 0,. 4/(r -I- r/fe.) 9 cos which can be written 2 r>& ) sin 2 r/ sn 30 LAN DEN'S TRANSFORMATION. 3 1 Putting 4*.*-' 4*. _ , (r+njr--(i-f jjr we have r d = i + k r 4 " *fo . (2) J^ \/T^~tf sin* 2 ^ i/i _ o ' sin 1 0, ' no constant being added because and 0, vanish simulta- neously ; and 0, being connected by the equation sin OPC _ sin (20 0,) _ r _ ^ sin <9CY> = sin 0, : T := "' From the value of J? we have k' is called the complementary modulus, and is evidently the minimum value of J0, the value of A

> , for, putting eq. (i) in the form we see that if = i, then k = , but as ^ < i, always, as is evident from the figure, k must be greater than . It is also evident, from the figure, that t > 0. Or it may be deduced directly from eq. (3). Since k < i, we can write k = sin 0, k' = Vi & = cos 0. 32 ELLIPTIC FUNCTIONS. Substituting in eq. (5), we have and we can write = sin 6 , k, f = Vi-k: = cos 0, . From eq. (5) we have Substituting the value of in that for /, we get We also have 20 0, = (0, 0) and, eq. (3), sn (20 0,) = k. sin 0, , becomes sin cos (0 : 0) cos sin (0, 0) = sin cos (0, 0) + k cos sin (0, = sin ^ = tan' *0 ; LANDEN'S TRANSFORMATION. 33 (8) *,' = (9) * = , x 7 2 __ __ ' * ~ "i + k' ~~ k '- ^/p ~ cos 2 (i i) sin (20 0,) = sin 0, ; (12) tan (0 t 0) = k' tan ; (13) (14) * = tT- ^' 2 , Upon examination it will easily appear that k and , and and , are the first two terms of a decreasing series of moduli and angles; k' and kf, and and 0, , of an increasing series; the law connecting the different terms of the series being de- duced from eqs. (6) to (12). By repeated applications of these equations we would get the following series of moduli and amplitudes : ^00 V 0, *. K 0, k k' The upper limit of the one series of moduli is i, and the lower limit of the other series is o, as is indicated, k and k', 34 ELLIPTIC FUNCTIONS. which are bound by the relation 1? -\- k'* = I, are called the primitives of the series. NOTE. It will be noticed that the successive terms of a decreasing series are indicated by the sub-accents o, oo, 03, 04, ... on; and the successive terms of an increasing series by the sub-accents i, 2, 3, . . . n. Again, by application of these equations, we can form a new series running up from k, viz., ,,,,&,,.../&= i (M=00) ; and also a new series running down from k', viz., k a ', k' ' 00 , . . . k an = O (M=00 ). So also with 0. Collecting these series, we have k w = k n ' =\ n V 0, *,' 0i # . . . ... *; 00 &<* 00, NOTE. In practice it will be found that generally n will not need to be very large in order to reach the limiting values of the terms, often only two or three terms being needed. Applying eqs. (7), (12), (13), and (14) repeatedly, we get (H.) = sin i-k' , = tan 2 $0 = sin , = tan 2 tan 4 = sn _ 1} = sn OB = cos = COS = COS = COS LAN DEN'S TRANSFORMATION. tan (#, 0) = ' tan ; tan (0, 0,) = // tan 0, ; tan (0 3 a ) = / tan 0, ; ^ tan (0 M 0._ I ) = '(_!) tan W _, 35 (14.) Multiplying these latter equations together, member by member, we have ?, = - ; ; Or Sin 9. = -- ; - : - 3. 1 i -j- k i + sin LA N DEN'S TRA NSFORMA TION, Solving this equation for sin 8, we get sin 8 = tan 2 f0, . Hence we can write k = sin = tan 3 J0, ; k = sin = tan" ; 37 k n = sin M . From equation (12) we get (18.) sin (20 ) =. k sin ; * sin (20 00 ) = k l sin ; sin (20 OM 0( _ I) ) = k n _, sin 0( _ I} . * When sin = i nearly, is best determined as follows: From eq. (12) we have tan (

' tan = ' tan nearly; whence = i?^ ' tan nearly, ^? being the radian in seconds, viz. 206264". 806, ad log R = 5.3144251. Substituting the approximate value of 0o , we can get a new approximation. Example. 0o = 82 30' 'oo = log" 1 5.8757219 tan 82 30' k 00 R 10.8805709 5-8757219 5-314-1251 2.0707179 117". 684 = i'.g6i4 0oo = i'.g6i4 0oo = 82 28'.0386 ist approximation. ELLIPTIC FUNCTIONS. To determine /' , /' , etc., we have k' = sin 77, > = cos 77 ; k' = - 7 = tan 2 iw = sin 77,,, k. cos w n ; (i8 3 ) -j I-M j k n tan 2 -|-77 = sin 77 oe , k^ = cos 77 00 ; etc. etc. etc. Or, since i + ' = - , i 4- ^' = etc., cos f cos 2 417.' we can put eq. (18) in the following form : ( ,9) F(k, 0) = . cos l/ cos . . , . cos . From equation (13) we have (19)* whence By repeated applications this gives, after combining, This value gives 0o 0oo = 117". 1675 = i'. 95279 . ' . 0oo = 82 28'. 04721 2d approximation. This value gives 0o 0oo = 117". 1698 = i'.gs283 0oo = 82 28'. 04717 3d approximation. LANDEN'S TRANSFORMATION. 39 k k k 1 * " (20) F(k, 0) = .y' > ' k " . F(k, , 4,,,) k l , , , etc., being determined by repeated applications of . , , r I -\-k or by equations (i8j). In equation (19)* let us change k l and into k' and re- spectively, so that the first member may have for its complete function K' = F(k', 0). Upon examination of eq. (19)* we see that the modulus in the second member must be the one next less than the one in the first member, that is, /; and likewise that the amplitude must be the one next greater than the amplitude in the first member, viz., 0, ; hence we get F(k>, 0) = / 2 Indicating the complete functions by K' and K^ we have, 7t since = when 0, = ?r (see Chap. V), and in the same manner, 4 ELLIPTIC FUNCTIONS. whence K> = (i + *.0(i + k'^ . . . (i + Since rr d(j> = - , (n = limit,) 2 lave (20)* we have ~ #v * (I + ^I + ^'oo) .-*(!.+ *'o) - ~ From eq. (19)* we have, since [eq. (10), Chap IV] *>/ */ o whence also, since for = -, = TT, and (I + ^.0( or Kn K_ , 2-(l+ ')(l+' 00 )... (21) =-K. LAN DEN'S TRANSFORMATION. 41 Let us find the limiting value of F(k on , n ) in eq. (15). In the equation tan (0 M M _,) = >_, tan M _, , we see that when >_! reaches the limit i, then

on 0,, Y " cos e ' 2*' n i being the number which makes k' n _^ = i. In these last three equation , k on are determined by eqs. (14.) ; 0, , 0, , etc., by eqs. (14,) * ; 0, , etc., by eqs. (14,) ; and k', k /, /& 2 , etc., for use in eq. (14^) by eqs. (14,). * Taking for 0i $>, etc., not always the least angle given by the tables, but that which is nearest to . 42 ELLIPTIC FUNCTIONS. BISECTED AMPLITUDES. We have identically u dam - 2 U U / f U\ - = 2 . - = 2F\k, am ] ; 2 \ ' ' u = F(k, am u} = 2 n F\k, am ) \ fifl u = 2" . am- (n limit.) n u am being determined by repeated applications of eq. (12) of Chap. II, as follows : , u i en u 2 sin" \ am u sn = : = : j - ; 2 i + dn u i -f- dn am z* sin u sin \ am u 2 (24) sn - - - 2 being an angle determined by the equation (25) cos ft = dn u = Vi ft? sn a u, and n being the number which makes u 2" am = constant. n u am - is found by repeated applications of eq. (24). LAN DEN'S TRANSFORMATION. 4$ Indicating the amplitudes as follows: am u = (26) F(k, 0) = 20 oa ; w being the limiting value. In eq. (18), when k n reaches its limit I, we have F(k n , 0..) = -- = log, tan (45 + i0 ) , j/, cos V'ow and eqs. (18) and (19) become (27) F(k, 0) = (i + ')(i + ^ 00 ) . . . (i + ' OH )log e tan(45+i0 ) (28) cos" f 77 cos f// . . . cos f rj on I J cos 8 \rj cos* ^v cos2 2"^o -^ being the number which renders k H = I. Eq. (20) becomes 44 ELLIPTIC FUNCTIONS. (29) F(k, 0) - - lo ^ tan (45 tan _ /cos ,,. cos ,?..... cos' _ I j tan (4 " y cos rj M In these equations ^/, y^',,,,, etc., are determined by eqs. (i8 3 ); 77, 77,,, etc., by eqs. (:8 3 ); , 00 , etc., by eqs. (18,) ; k^ k t , etc., by eqs. (18,). Substituting in eq. (27) from eq. (20)*, we have 2.K F(k, 0) = log e tan (45 + ^0 OB ) 2.K' (30) = =M lo S tan (45 + i0J- CHAPTER V. COMPLETE FUNCTIONS. INDICATE by K the complete integral d i-tfsin'0 and by K the complete integral and in a similar manner K 00 , K 03 , etc. From eq. (12), Chap. IV, we have tan (0, 0) = k' tan tan 0, tan I -j- tan 0, tan ' whence (i + k'} tan tan 0, = ^ tan> k' tan tan0 45 46 ELLIPTIC FUNCTIONS. From this equation we see that when

n can be calculated, ^ , k 00 , etc., being found by means of equations (14,), Chap. IV. Then, having

) & on sin

(-i) sm 0-i> sin (20 0,) = k, sin 0, ; whence we can get the angle 0. When /> V^ the following formulas will generally be found to work more rapidly : From eq. (29), Chap. IV, we have 7/ 1W (2) log tan (45 + 40..) = EVALUATION FOR (p. 49 from which we can get , ; ,,,,, etc., being calculated from eqs. (18,), Chap. IV, and being calculated from the follow- ing equations : tan (0, ( _ x > ) = k n tan OM , tan (0 00 ) = a tan , , tan (0 0,) = tan ; whence we get 0. This gives a method of solving the equation where n and and the moduli are known, and if; is the required quantity, n and give F$, and then ^ can be deter- mined by the foregoing methods. When k = i nearly, equation (2) takes a special form, i. When tan is very much less than -77. In this case d r _ ^0 _ _ r "J Vcos' + ^ /a sin 3 ~J V(i "tan0)cos'0 COS whence we can find 0. 2. When tan and -p approach somewhat the same value, and k' tan cannot be neglected, F(k, 0) must be transposed into another where K shall be much smaller, so that k' tan can be neglected. 5O ELLIPTIC FUNCTIONS. 7t These methods for finding apply only when 0< -, that 12 is, u < K. In the opposite case (u > K) put u = 2nK v, the upper or the lower sign being taken according as K is con- tinued in u an even or an odd number of times. In either case v < K, and we can find v by the preceding methods. Having found v, we have from eq. (5), Chap. Ill, am u = am (2nK r) = mt am v. CHAPTER VII. DEVELOPMENT OF ELLIPTIC FUNCTIONS INTO FACTORS. FROM eq. (12), Chap. IV, we readily get sin (20 0) k sin ; ^ sin 2< A> sin = ----- ~" _ VI -\-j? -\- 2k COS 20 __ sin 20 _ " 4/(i+) 4/sin a _ i -f/fe/ sin 20 2 Vi k? sin" (since -r k^ and \^-k , eqs. (6) and (10), Chap. *~r* I ~r^o and thence (i ^ sin cos {,) s,n = From eq. (13), Chap. IV, we have and from eq. (4), Chap. V, passing up the scale of moduli one step, 51 52 ELLIPTIC FUNCTIONS. whence Put ^"(0o> ^i) = u i an d -^(05 ^) ^> whence u =^u. Furthermore, = am (, ^) ; 0, = am (a;, , /,) = am ( -^>?^, j. Substituting these values in eq. (i), we have sn (, = dn IrT' k But from eq. (11), Chap. Ill, we have dn (K, ^ or + K DEVELOPMENT INTO FACTORS. 53 whence (2) sn (, k] = (i + .') sn ^ sn [^ ( + 2 (Mod. = k, .) From this equation, evidently, we have generally (2)* sn (r, k n } = (i + *.', + ) sn ^ v sn [^ (v + 2 AT,)]. (Mod. =*,+..) Applying this general formula to the two factors of eq. (2), we have K, Kji sn = J * sn sn (Mod. >&,) = (I + ' 00 ) sn sn ( + 4^) ; (Mod. ^ ;) < 3 ) snl^-^+2^ K. . sn The last argument in this equation is equal to and since, eq. (7), Chap. Ill, sn (, 4) = sn f The analogous formula in Trigonometry is sin = ^ sin ^0 sin ^(0 -}- it}. 54 ELLIPTIC FUNCTIONS. we can put in place of this, whence eq. (3) becomes sn X ^ + 2K ^ * = + ^..) sn E" (2K-u\ Substituting these values in eq. (2), we have (4) sn (*,,) = (i + / I ' s _ . sn -^ (2K ) sn ^ (4^+ ), (Mod. k,,) in which the double sign indicates two separate factors which are to be multiplied together. By the application of the general equation (2)* we find that the arguments in the second member of eq. (4) will each give rise to two new arguments, as follows : and K, K K" u) gives - DEVELOPMENT INTO FACTORS. 55 and (2K ) + 2 TV" gves and Subtracting (a) and (b) from 2A" 3 , by which the sine of the amplitudes will not be changed [eq. (7), Chap. Ill], and since our new modulus is &, , we have for the expressions (a) and (b}> Substituting these values in eq. (4), and remembering the factor (i -{- '03) introduced by each application of eq. (2)*, we have sn 2f IT JL\. n -**< . sn -^ (2.K 11) sn -^ (&K u) ; . sn -y^ (6K ) sn --^ (SK -(- w). (Mod. ^ s . $6 ELLIPTIC FUNCTIONS. From this the law governing the arguments is clear, and we can write for the general equation (5) sn (, k] = (i + 0(i + k' M }\i + k\$ . . . (i + k' m j n ~ l K n u K n . . sn 2", Indicate the continued product of the binomial factors by A', and we have A' = (i + e O(i + k'J(i + k'J(i + ' 04 ) 8 Since the limit of //, k' w , etc., is zero, it is evident that these factors converge toward the value unity. It can be shown that the functional factors also converge toward the value unity. Thus the argument of the last factor can be written From eq. (n), Chap. Ill, we get then , K n u\ sn dn , -^ 2 n K But since k n at its limit is equal to unity, en = dn ; whence the last factor of eq. (5) is unity. DEVELOPMENT INTO FACTORS. 57 From eq. (21), Chap. IV, we have K n 27t limit 2 n K ~ 2K'' Therefore for n = oo , eq. (5) becomes sn (u, k) = A' sn sn -, (zK u) 7T 7T . sn v?7 (4A zi) sn ^7 (6K ),.... (Mod. i,) or r o i (6) sn (, /) = ^' sn ~^- f \II*\sn -, (zhK ), (Mod. i,) where the sign [IT] indicates the continued product in the same manner as 2 indicates the continued sum. r When k = i, I / r (0, /) becomes %y o / i 4- sin ci - = -A- log : '- ; cos ' i sin whence i -j- sin ~ I sin 0' and 58 ELLIPTIC FUNCTIONS. Hence in equation (6) iru iru nu _ e* K> e * K> sm ;rrF7 = 2K' ' ** _ L ' f K '+e 2/r/ hirK iru hnK iru sn grf (zhK M) = -^r^ ^r^ Put (6)* q = e~ K , q' = e~ K> , and the last expression becomes ir r sn -^ ( 2 ^ J ST ) = q - -^- - ; /-* 2^' , , k q e +q e sn 7 (2^^+ ) sn 7 From plane trigonometry we have the equations g* e~* -- : - = z tan ix, e x -4- e ~ x = 2 cos zx e x -\-e~* DEVELOPMENT INTO FACTORS. 59 where i = V i : which gives nu niu sn T^7 = - * tan -=-, ; (Mod. i;) sn 7 (2/;A~+ w) sn 7 (2MT- ~- cos From eq. (10), Chap. Ill, we have sn (u, k] = i tn (m, k'}. Substituting these values in eq. (6), we have tn (iu, k')=A' tan ^ Now in place of the series of moduli k', k^ and the cor- responding complete integral K 1 ', we are at liberty to substitute the parallel series of moduli k, k Q and the corresponding com- plete integral K; calling the new integral n, we have nu nu l ~ 2 ^ C S ~K + ^ (8) tn (u, fy^Atan^n- - I -f- 2^* COS -^ -\- {f* ELLIPTIC FUNCTIONS. 1 , 4 I 20 cos -jy -4- a . . nu A ' * A tan r 2K nu I nu 2,7 4 COS -^ 6 , 12 I - 2? 6 COS -^ -f ?' 2 nu 2q cos -77 + q where (9) Now in equation (6) put u -(- AT for &, and we have, since en u [eq. (I i), Chap. Ill] sn (u + K) = j, '111 ft- ] sn ^K 2 ^ + i)AT+] dn 2A . sn 777 K 2 ^ O-^" u ~\' (Mod. i.) Now from 2h i and 2/2 -{- i we have the following series of numbers respectively : 2h i : i, 3, 5, 7, 9, etc. 2/2 + i : 3, 5> 7> 9> etc - It will be observed that the factor outside of the sign [77], viz., sin am ^7 ^,would, if placed under the sign [77], supply 2A DEVELOPMENT INTO FACTORS. 6 1 the missing first term of the second series. Hence, placing this factor within the sign, we have en u n (10) -: =A\n\ sn ^ [(2/1 i}K-4-u~] ami L J 2A LV . sn j^r, [(2/1 i)K u]. (Mod. i.) Comparing this with equation (7), we see that the factors herein differ from those in equation (7) only in having 2/1 i in place of 2k ; hence we have sn --, [(2A - i)AT+ ] sn -^ [(2/1 - i)K - u] I 2q' 2fl ~ l cos -v?7 + ^ /4A ~ 2 . (Mod. i.) I ~\~ 2(j' 2 ^ I COS \ (J 1 4^~ 2 From eqs. (10), Chap. Ill, we have en (u, k] _ i dn (u, k) dn (tu, k'} ' whence eq. (10) becomes 1-7^ 7^ = ^' \n \ Ji ' cos ~ and when in place of tu, k', K', g', A', we substitute //, k, K, q and A, and invert the equation, we have (12) dn (, ^) = -1 [77] TtU I -f- 2? 2 *- 1 cos -^ -4- q&- J\. L 7tU . L 2g 2A ~ I cos -=>.- -f- ^~ : 62 ELLIPTIC FUNCTIONS. Bearing in mind the remarkable property (Chap. Ill, p. 29) that the functions sn u and dn u approach infinity for the same value of , we see that both these functions, except as to the factor independent of u, must have the same denominator. Furthermore, since sn u and tn u disappear for the same value of M, they must, except for the independent factor, have the same numerator. Hence, indicating by B a new quantity, dependent upon k but independent of u, we have - -f- q^ (15) sn (, k) = B sin i 2q* k ~ l cos pr -j- cos-^ (.^. / J\ -' I I 1*- 16) en (, ^) = -4 cos -= J COS vjr -j- DEVELOPMENT INTO FACTORS. 63 . , nil I-f 2? 2 *-'COS-^- + TtU i 20 2 *- 1 cos -r^r -f- a**-* K. To ascertain the values of A and B, we proceed as follows : In eq. (17) we make u = o, whence, by eq. (13), Chap. II, we have ~A whence In equation (17), making u = /T, we get, by equation (i), Chap. Ill, ,y- ~ (19) .-. We have identically whence To calculate B, put e* K = v; if we change ^ into 2J\. 64 ELLIPTIC FUNCTIONS. ^ -4- ~- , v will change into v \fg, and sn u will become, by 2K. 2K. eq. (14), Chap. Ill, sn (u -4- iK'} = -, . R sn u rtU TtU Now, replacing sin ^ and cos ~ by their exponential val- ues, and observing that I 2q" cos -jf we have B v Sn U = - ' -=: 2 i/_ Changing into -j- z^T', and consequently v into v V^, we have I B vVv sn u ~~ 2 Multiplying these equations together, member by member, and observing that v Vq we get l_ _ i -qv* _ k ~ 4 ' v Vq DEVELOPMENT INTO FACTORS. 65 4V? (i - cos ' + etc. Since cos = this becomes (5) 0() = ^ + i^^ 2 '* + \C + \Be~** + i^-^ + i-^- 6 - + . . . ; whence (6) - e-** (u} = -- e-** --- --- e 21 * --- e#* q q 2q 2q 2q _ _ P~V X _ _ f~^ x _ _ e~^ ix _ 2q 2q 2q 68 ELLIPTIC FUNCTIONS. Now in equation (5) put u -f 2iK' in place of 11, remember- ing that e* lx and e~ 2ix are thereby changed respectively into ^V'* and q~ 2 e~ 2ix , and we have (7) 2iK'} = C Since equations (6) and (7) are equal, we have ~^ = ^> B=-2gA; _C _B 2q ~ 2 ' ' ' 2 ^ ' _ D __ Cq" 2q ~ 2 ' " whence (8) 7TU = A(l 20 cos 27TU COS ^r 2^ COS ::L -^r A A^ The series in the second member has been designated by Jacobi and subsequent writers by (?/), thus: t\ (9) = I 2q cos nu cos - ... CHAPTER IX. THE AND H FUNCTIONS. IN equation (20), Chap. VII, viz., UTt sn , = _ sm iJi, 2K\ , TtU L J I 2? 2 *-' COS -pr + ^ 4 *- 2 the numerator and the denominator have been considered sep- arately by Jacobi, who gave them a special notation and de- veloped from them a theory second only in importance to the elliptic functions themselves. Put [see equation (8), Chap. VIII] - 2 -' cos 7TW , (2) H(u) =2- VfSMjx _ l ~ 2q C S K ^ being a constant whose value is to be determined later. From these we have (T\ sn C M _ _ ( ' } " Vk 69 7O ELLIPTIC FUNCTIONS. The functions sn u and en u can also be expressed in terms of the new functions ; thus we have TtU I -f- 2q 2h cos -7? -f- q* /, AN A/ K 4/~ I TT I A (4) en (, k) = \ - . 2 Vq cos -= or, since sin x = cos f x -f- J and cos ;r = cos (x -f- - and putting ?< = /2^ir \ //&' L * V ' 2 en , *i = \ 7T Replacing - by its value, zt, we have (5) en ( Furthermore, TtU - x COS -r (6) dn (, ^) = , , _ , ^ - 1 cos-^ + ^*- THE & AND H FUNCTIONS. gives in the same manner 7T 7T or If we put (8) fl (9) e the three elliptic functions can be expressed by the following formulas : (10) k' H(u) These functions and -H" can be expressed in terms of each other. By definition, 72 ELLIPTIC FUNCTIONS. but 2 cos - I \/ . vr-!x \\i q h e K \ TTlll 7TIU 7TU sin T> = and consequently (13) H(u)=Cy'ge '' V Now, changing u into u -\- iK', and remembering that -TtK' e K = q, we have (14) H(u + iK') and reuniting the factors two by two, this becomes (15) H(u + iIT) iriu -2 cos and finally, according to equation (i), (16) H(u -j- /A"') = -/ i^ * THE Q AND H FUNCTIONS. 73 In the same manner, we can get (17) e(-H'AT') = V-12 *e 2 H(u). Substituting u -(- 2K for u in equations (i) and (2), we get (18) @(u + 2 AT) = (), (19) ff( + 2 AO = -#(), 7T 7T# 7T 7TW since cos -^.(u -\- 2K) = cos -^ and sin ^(u-\-2K)=sm j,. The comparison of these four equations with equations (10), (u), and (12) shows the periodicity of the elliptic func- tions. For example, comparing eqs. (10) and (16) and (17), we see that changing u into u -}- iK' simply multiplies the nu- merator and denominator of the second member of eq. (10) by the same number, and does not change their ratio. The addition of 2K changes the sign of the function, but not its value. We will define @, and H^ as follows : (20) ,(*)= (* + tf); (21) H,(x) = H(x Hence we get, from equation (17), since e 2 = cos -- V^~i sin - = V i ; 2 2 74 ELLIPTIC FUNCTIONS. whence - -( + ,*") (22) e^-H^o^fiifcy In a similar manner we get - (2-r+iAT') (22)* H l (x+iK') = ,(*)' 2JK* In eq. (9), Chap. VIII, put u = - , and we get /2A^\ (23) I - - j = I 2^ COS 2^ -j- 2^ 4 COS 42 ... Now, in this equation, changing z into z -| , and observing 2 eq. (20), we get (24) i(-^r) = I + 2? cos 2^ + 2q" cos 4? + . . . Applying eq. (22) to this, we have [ .. THE & AND H FUNCTIONS. 75 + <-* 4- /<-* + = ? J O 1 '' + 0'* 31 '* + ?"' 5 " + + ^-' + ^V-3' z 4- f = 2^ [cos z + g* cos 3^ + ^ 6 cos 5-sr + ...]; whence ( 2 5) HI ( j = 2 Vq cos z -f- 2 y/ cos 3^ -|- 2 r cos In this equation, changing 2- into z - , and applying eq. (21), we get (26) flff J = 2 V^sin z 2 V0* sin 3^ + 2 Vq sin 5^ . . . , since We will now determine the constant A of eq. (8), Chap. VIII, and eqs. (i) and (2) of this chapter. Denote A by f(q\ and we have Tttl K Substituting herein u o and u , we have 76 ELLIPTIC FUNCTIONS. From eq. (9), Chap. VIII, we get (27) 0(0) = 1-2^ + 2^- 2f + 2? 16 - . . . ; (28) (~) - I - 2,?< + 2?" - 2 20 > 28 > 36; 8>fc, 8, 1 6, 24, 32. Hence, the three expressions taken together contain all the even numbers, and Therefore, multiplying eq. (29) by we have THE AND H FUNCTIONS. 77 Now in this equation, by successive substitutions of q* for g t we get /(?") " I - Now <7 being less than i, ^" tends towards the limit o as n increases, and consequently i q n tends towards the limit I. Also, from eq. (8), Chap. VIII, we see that f(o) = i. Hence, multiplying the above equations together member by member, we have (30) or (3i) A = Substituting this value in equation (8), Chap. VIII, we have, after making u = o, 6(0) (See equation (9), Chap. VIII.) Transposing one of the series of products from the left- hand member, we get 78 ELLIPTIC FUNCTIONS. Introducing on both sides of the equation the factors I 4% I q\ i g K , etc., we get (I -,7)(I -/)(!- ?')(!- ?<) whence (32) @( o) = (i^j (i + )( Resuming equation (20), Chap. VII, and dividing both members of the equation by u, we have 7TU TCU 4,- sin r > I 2q 2 cos -j= - l - "** sn u 2 \ a 2 A r^n & r U 4/A Z^ . 7T?/ I 2^ 2A "' COS-^ A sn u This, for ?* = o, since the limiting value of for u = o is i, TtU sin v> and of for x = o is -^ becomes w 2A or ' \ :7 / \ :/ / Further, from equation (21), Chap. VII, for u O, we have \,/ b f 1//7 L V* tf )\ 9 )\ 2 / * " * ' r A- f ( / ' THE AND H FUNCTIONS. 79 The quotient of these two equations gives or, substituting the value of i 7 ^'' from eqs. (18) and (19), Chap. VII, _ * ii Comparing this with equation (32), we easily get (37) From equation (9), Chap. VIII, making u = K, we get Making z = o in equation (24), Chap. IX, we have (39) ,(0) = i 4 This might also have been derived from eq. (38) by observ in that Knowing (o), it is easy to deduce &(K) an From equation (7) we have (u + K) dn ti = K ' . . Making u = o, we have, since dn (o) = I, (40) 80 ELLIPTIC FUNCTIONS. From equation (5) we get, in the same manner, (41) H(K] = From eq. (12), Chap. IX, we have (41)* dn u = V'\ -P sin" = Vk' ; (y(u) Till and putting x = f>, we have dn u \-\-2q cos 2x -\- 2q* cos ^ -(- 2q 9 cos 6x + . . . ^"' 1/^7 i 2q cos 2^r -]- 2^ 4 cos ^x 2q" cos 6x -j- . . .' Putting dn & ( 4 2)* -^=r == COt y, we have cot y I cos 2.r+/(4 cos 3 2x 3 cos 2^)4- - whence tan (45 - r)[l + y 4 (4 cos 2 2^r - 2)] (43) cos 2x = ^~ g\4 cos 3 2x 3 cos 2x) and approximately, tan (45 - r) (44) cos 2X ~- ^ q -- From equations (37) and (40), Chap. IX, we have n (45) * - p~j ' whence (46) * = x&(K\ CHAPTER X. ELLIPTIC INTEGRALS OF THE SECOND ORDER. FROM Chap. I, equation (19), we have E(k, 0) = r 1/i - tf sin 2 . dtf> = From this we have (0) -|- E(ifi = FAQ . d + t/o t/o Put (i) Differentiating, we get (2) A< But we have, Chap. II, equation (2), or (3) 4$ . d$ + J0 . ^ = o. Adding equations (2) and (3), we get (4) (J0 + ^)(rf0 + ^) = dS. 81 82 ELLIPTIC FUNCTIONS. Substituting cos ^ from eq. (5), in eq. (5)*, Chap. II, we get sin cos tyA}* -|- cos sin sin u (5) sin tl> cos 0z7/^ -f- cos ^ sin a 3= : : sm jj. whence A).i i i (6) ^J0 dip = sin (0 ^). Substituting in equation (4), we have ds = sn (7) 7T^ Integrating equation (7), we have The constant of integration, C, is determined by making = o; in this case ^ = /<, -"0 = o, ^ = "yu, and 5 = EH ; whence *-75rjr &-*>* and by subtraction, E p = - (cos ^ cos cos ^ + sin sin ^>). But, Chap. II, eq. (5), cos i*. cos cos ip = sin sin ELLIPTIC INTEGRALS OF THE SECOND ORDER. 83 whence

-j- Etp = E}JL -}- k? sin sin ip sin >w. When = ^, we have (9) ->u == 2^0 ^ sin 2 sin //. But in that case (10) cos p = cos 2 sin 2 whence i cos Let 0, 0i, 0j, etc., be such values as will satisfy the equa- tions (12) E

k 1 sin 2 0j sin etc. etc. Assume an auxiliary angle y, such that {13) sin Y k sin 0; whence J0 = cos ^, and Chap. IV, eq. (24), sin 4-0 (14) sm0^=- p.. cos 84 ELLIPTIC FUNCTIONS. Applying eqs. (13) and (14) successively, we get (15) sin sin 0i = -- r-. sin y\ = k sin 0i , cos sin sin 0i = - r- , sin y = & sin 0i ; cos sn sin - cos whence (16) E(f> = 2 n E4>- L (sin sin* ^ + 2 s i n 0* si 11 " Xi 2* + 2 s sin 0j sin 2 y + . . 2"' 1 sin A sin 2 >/ i \ 2 2"-' / To find the limiting value, E<>\ , we have, by the Binomial s *0 5 Theorem, since sin = i 4^- - -- etc., \~~ A= (i ~ t? sin 5 0)^ Whence (17) 2 W i/o a" k* , I 2O ELLIPTIC INTEGRALS OF THE SECOND ORDER. 85 Substituting in eq. (16) the numerical values derived from equations (15) and (17), we are enabled to determine the value of 0. Landen's Transformation can also be applied to Elliptic Integrals of this class. From eq. (11), Chap. IV, we get, by easy transformation, (18) sin' 20 = sin 2 4 (i -f- k a -\- 2k t cos 20). From this we easily get 2k t cos 20 sin 2 0, == sin 2 20 sin* 0, k* sin 2 0, = i cos* 20 sin* 0, k* sin 2 t = ^f* 0, sin" 0j cos 2 20 ; whence cos 2 20 -}- 2k sin* 0, cos 20 = ^ 2 ^ 0, sin* 0, ; and from this, cos 20 = k sin 2 0, 1rfkjp t sin* t + k* sin 4 0, (19) = cos 0, J t k sin* 0, ; whence, also, i cos* 20 = i cos* 1 /P0 1 -[-2/ sin* 0, cos

__ k a cos 0, -f- J/ 86 ELLIPTIC FUNCTIONS. But from (19), and eq. (6), Chap. IV, t?(\ cos 20) sin" = & ?k i + k sin z 0, cos 0, whence and /f AM d $- - & COS 0. + W'dfr* (1+ and ,,,, . ^ (^. cos 0. a n , where k on approxi- mates to zero and Ek w (f> n to H . Or, by reversing, it may be made to depend upon Ek n (f> on> where k n approximates to unity and Ek n (f) on to cos,0 OM . ELLIPTIC INTEGRALS OF THE SECOND ORDER. 8/ To facilitate this, assume Gk = Ek$ Fk = -Fkrf l (see eq. (13), Chap. IV), m we have sn Repeated applications of this give Gk & ~ T I A 1 ~r K n 2Fk 88 ELLIPTIC FUNCTIONS. and since, also, (compare eq. (6), Chap. IV,) we have " r h \ "t*-') r 77 * ~TT ^ L_ -J "^on _ ^o ^ ^o ^o(-i) r I TT | r "5?- "' J" " J" ' ^ K \ 11 ro(-i ^o K oo K vn I I Substituting these values in equation (22), and neglect- ing the term containing Gk w

n F& on

n 4> = o, (n = limiting value,) we have sn (25) 2 n , . . sm "~~ sm sm ...] 2 whence (26) Ek = Ft* i - - sm -I . . .J. ELLIPTIC INTEGRALS OF THE SECOND ORDER. 89 From eq. (3), Chap. V, we see that when (/> = -, Substituting these values in equation (26), we have for a complete Elliptic Integral of the second class, In a similar manner we could have found the formula for E(k, 0) in terms of an increasing modulus, viz., (28) r -j -- .- sin

Her = n sin cr / ^ / >4 l/fSBO 1 ** whence A 6" = IZ"cJ- -|- sin cr / Jo A ~ (7) J70 + 77^ = Tier + sin cr dq 2Bq -\- Cq* or /*? j -4 - 2Bq + But we have AC B* + (Cq B}* CM dq AC-B* , Cq-B Cdq M , + (-^=f-,)' VAC -B* where M = n s'm cr. The integral of the second member is _ M Cg-B _ _ " ~ _ -- j ^^ "-" * ELLIPTIC INTEGRALS OF THE THIRD ORDER. 93 whence A M f CqB B -i / ttS=S.= , .= tan- 1 Z =r-f-tan-' = ; / V^cZ^L V^c-^^ t<4CWZ?J ty o or, since x -\- y ~ I y = tan" 1 *-, i xy Substituting the values of y2, -5, 6" and M, we have ^C ^ 3 = (i + ? //V)(i + w sin a o-) ' cos a o- = w(i -f- w JV -j- w(i -f- w) sin 3 cr #JV = w(i -j- w)(i JV + n si n * ") = w(i + )(^ 2 + w ) si n * " ; and putting d + X^ + ) n we have VAC B* = n Vn sin cr. Substituting these values in eq. (7), we have II(n, k, 0) -{- -?!(, ^, ^) H(n, k, i *JL q~~ ? = ' ' whence log - log - = M*T? = 1.8615228, (6) log log - + log log -, = 0.2698684, by which we can deduce q from q'. EXAMPLE. Let = 79 36' 14". To find q. 90 -0= 10 23' 46". By eq. (5) we get log q' = 7.3 1 563 1 6, log -> = 2.6843684, and log log = .4288421 ; NUMERICAL CALCULATIONS, f. 97 and by eq. (6), log log - = 9-8410263 ; whence logy= 1.3065321. When f = = cos 45 = i V^, eq. (6) becomes (7) 10^=^=1.3643763; (* = ^0 whence log q 2.6356237, ? = 0.0432138. (*=*) EXAMPLE. Given ^ = 10 23' 46". Find q. Ams. log q = 7.3156316. EXAMPLE. Given = 82 45'. Find q. Ans. logy = 9-379l9- CHAPTER XIII. NUMERICAL CALCULATIONS. K. CALCULATION OF THE VALUE OF K. WE have already found from eq. (37), Chap. IX, 6(0)=., , and from eq. (40), same chapter, But, eqs. (38) and (27), Chap. IX, (K) = i+ 2 q+2? + 2q + 24" + . . . , 0(0) = I - 2q + 2q" - 2q* + 2q l - . . . ; whence, eq. (2), <3) K j ( l + 2 ? + 2 ^ 4 + 2 ^ 9 + )' By adding eqs. (i) and (2) we get whence K= 2\ 2 L i -\- Vk r -i NUMERICAL CALCULATIONS. K. QQ EXAMPLE. Let k = sin B sin 19 30'. Required K. First Method. By eq. (3). By eq. (5), Chap. XII, we find log q 8.6356236. Apply- ing eq. (3), using only two terms of the series, we have I -f- 2q 1.0147662 log (i -\- 2q) = 0.0063660 2 log (i -(- 2q] = 0.0127320 log - = 0.1961 199 log K 0.2088519 K 1.615101 Second Method. By eq. (4). Equation (4) may be written, neglecting q*, _ 7T/I -f- VcOS # K= 2\ T~ whence log cos e = 9.9743466, log^cos # = 9-9871733, Vcos V = 1.9708973, t/COS d = 0.98 544865; and log K =. 0.2088519, K= 1.615101, the same result as above. TOO ELLIPTIC FUNCTIONS. Third Method. By eq. (7), Chap. V. = 19 30' = 9 45' log tan = 9.235103 log cos |0 = 9.993681 log tan 2 log sm = 8.470206 0. = i" 41' 31".! = o 50' 45".5 log cos |0 = 9.999953 log cos 2 = 9.987362 log cos 2 1= 9.999906 9.987268 log - = 0.196120 \ogK= 0.208852 00 is not calculated, as it is evident that its cosine will be i. EXAMPLE. Given k = sin 75. Find K. By eq. (7), Chap. V. From eqs. (14,), Chap. IV, we find / = sin = sin 75 tan 2 = tan 2 37 30' sin sin 36 4' i6".47 I ( tan 2 (sin - tan 2 18 2 = sin 6 5' 9^38 ' 8".2 35 ) ' 9^38 f (tan 2 i0 00 =tan 2 3 2' 34". 69 ) 03 ~ ( sin 03 = sin 9' 42 /x .9O ) log = 9.9849438 9.7699610 9.0253880 7.4511672 NUMERICAL CALCULATIONS. K. IOI log 2 log a. c. 2 log cos \B cos 37 30' 9.8994667 9.7989334 0.2010666 cos %V Q = cos 1 8 2'.i3725 9.9781184 9.9562368 0.0437632 cos^ oa = cos 3 2'.578i7 9-9993873 9-9987746 0.0012254 cos M = cos 4'-8575 9.9999995 9.9999990 o.oooooio 0.2460562 n * - - .1961199 log K = 0.4421761 K = 2.768064 Ans. EXAMPLE. Given k = sin 45. Find K. Method of eq. (7), Chap. V. From eqs. (14,), Chap. IV, we have log ( tan 2 6 tan 7 22 3 * == |sin. =sin 9 5 ( tan' i0. = tan 1 4 56'.3/84l *- = sin i = sin 2' ^8733009 tan 2 i# 00 = tan 3 I2'. 3395 sin(? M =sin o'.o 5 5-1445523 a. c. log cos 8 0.0687694 a. c. log cos" # 0.0032320 a. c. log cos 2 # 00 0.0000060 log- 0.1961199 log AT = 0.2681273 K = 1.8540747 Ans. EXAMPLE. Given 6 = 63 30'. Find K. Ans. log K= 0.3539686. EXAMPLE. Given = 34 30'. Find K. Ans. K= 1.72627. CHAPTER XIV. NUMERICAL CALCULATIONS, u. CALCULATION OF THE VALUE OF U. WHEN 6 = sin-' < 45. EXAMPLE. Let = 30, k = sin 45. Find u. First Method. Eq. (23), Chap. IV, and eqs. (14,), (i4 s ), (I 43 ), Chap. IV. By equations (14,), 2 = 22" 30' ; log tan - =9.6172243 ; a log tan 2 - = 9.2344486 log k, = log sin ; = 9 5 2 ' 45 ".4i; log tan - = 8.9366506 ; n log tan 2 - = 7.8733012 = log 00 = log sin 00 ; log tan 2 - = 5.144552 = log os . 102 NUMERICAL CALCULATIONS. U. IO3 By equations (14.,), log tan = 9.761439 log cos = 9.849485 log tan (0j 0) = 9.610924 s _ 0= 22 12' 2f'.$6 0, = 52 12' 2f. 56 log tan 0, = o.i 10438 log cos O g = 9.993512 log tan (0, - 0.) = 0.103949 0,-0, = 5i 47' 32"-59 2 = I04o'o".i5 log tan a = 0.603228 log cos 00 = 9 999988 log tan (0 3 2 ) = 0.603216 s -0 2 = 104 o' i ".5 3 =2o8o' i ".65 Since ^ = 26 o r o".O4 and = 26 o' o".2l, 4 o we need not calculate 4 . ~ = 936oo 7/ .2i. Reducing this to radians, we have log -~ = 9.656852. -' IO4 ELLIPTIC FUNCTIONS. Substituting in eq. (23), Chap. IV, we have, since cos 03 = I, a. c. log cos = o. 1 505 1 5 log cos # = 9.993512 log cos # 9.999988 0.144014 cos cos 6 = log | 3 = 9.656852 log u = 9 728859 u = 0.535623, Ans. When 6= sin '>45. EXAMPLE. Given k = sin 75, tan = \ / - . To find 4/3 First Method. Bisected Amplitudes. By equations (24) and (25), Chap. IV, we get =47 3'3o".9i, 0i =25 36' 5 ".6 4 , /3 ^45; 0j =13 6' 3 o // . 9 8, /? = 24 40' 10". gt; 4 = 6 35' 4 o".74, Ac = 12 39' I 5 // -83 ; 0A= 3 18' 8".75, As- 6 22' 8". 4 o; ^= i& 7 /7 .43, A,- Substituting in equation (26), Chap. IV, we have F(k t 0) = 32X i 39' 7" A3 = 525i / 58 // .o 3 = 0.9226878. NUMERICAL CALCULATIONS. U. 105 Second Method. Equation (29), Chap. IV. From equations (i8 3 ), Chap. IV, we have log k = cos r) = cos 15 o' C/'.OO 9.9849438 k' = sin TI = sin 15 o' o r/ .oo 9.4129962 k , = \ tan 3 i ff = tan' 7 30' o".oo ) g 88 82 1 sin % = sin o 59' 35"-25 ) k, = cos / - cos o 59' 35".25 9.9999348 , _ ( tan 2 i ; /0 - tan 5 o 29' 4 7 // -6 2 ) ^- |sin, /00 =sin o o'i 5 ".49i 5 757 ^ a = cos /; = cos o o' is" -49 o.ooooooo )' 1.1493838 From equations (i8 a ), Chap. IV, we get 20 = 45; = 4 6 i'45"475; Substituting these values in eq. (29), Chap. IV, we get fk i F(k, 0) = . log tan 68 o' 4 4 // -7O5 = 0.9226877. T/>^ Method. Equation (23)*, Chap. IV. IO6 ELLIPTIC FUNCTIONS. From equations (14,), Chap. IV, we have k = sin 6 = sin 75 o' o" log = 9.9849438 k' = cos = cos 75 9.4129962 ( tan 2 i = tan 2 37 30' ) k = \ a io I *// 9.7699610 ( sin = sin 36 4 16 .47 ) / = cos 9.9075648 n'i0 = tan'i8 2'8".2 35 ) oo ,-, 88o n^ = sin 6 S ' 9 ".38 f /&/ = cos e w 9.9975452 , ( tan 2 ^^ 00 = tan 8 3 2' 34". 6g ^= a > n ( sin 03 = sin 9 42 .90 k\ = cos <9 OS 9.9999982 ^4 = (i ^ 3 ) 2 4-3002761 k{ = o.ooooooo From equations (i4 s ), Chap. IV, we have = 47 3' 3o".94; I= 62 36' 3 ".io; 0, = 119 55'47"-67; 3 = 240 o' o".i9; 4 = 480 o' o". Therefore the limit of 0, -- , , or ^ is 30 = J ^. ^ O Substituting these values in eq. (23)*, Chap. IV, we have Tf . ? = 0.9226874. EXAMPLE. Given = 30, /^ sin 89. Find u. ^Method of eq. (28), Chap. IV. NUMERICAL CALCULATIONS. U, IO? From eqs. (18,) we find k^ = sin 0j and tan 3 ^ 0, = k = sin 0, from which we find that /, = I as far as seven decimal places. From eqs. (i8 8 ) we have sin = 9.698970x3 k = 9-9999338 sin (20 0) = 9.6989038 20 - = 29 59'.6 9 733 20o = 59 59'-69733 ' 45 + i0.* = 59 59'-92433 log (45 + i0.) = 0.2385385 From eqs. (i8 3 ), Chap. IV, we have k = cos // = cos i, ^rf = 30' Substituting in eq. (28), Chap. IV, we have a. c. log cos ?; 0.0000330 log log (45+i0,) 9-3775585 a. c. log M 0.3622157 log F(k, 0) = 9.7398072 F(k, 0) 0.549297. EXAMPLE. Given = 79, k = 0.25882. Find u. Ans. u = 0.39947. EXAMPLE. Given = 37, k'= 0.86603. Find u. Ans. u = 0.68141. * Since ki = i, o = 4>o, and we need not carry the calculation further. CHAPTER XV. NUMERICAL CALCULATIONS. 3 24 ; 2 ! 4 1 7.5488952 ( sin 00 = 12 .16659 i cos e oo 9-9999974 ( tan 2 # 00 = tan 1 6'.o8329 / *- = | sin Si 1 449573lS cos 03 o.ooooooo Substituting these values in eq. (i), Chap. VI, we have log cos # 9 9969260 log cos 6> 00 9.9999974 9-99 6 9 2 34 log Vcos # cos # 00 9.9984617 a. c. log " " 0.0015383 log n .1362153 log Vcos V 9.9482660 log 2 3 .9030900* a. c. log Vcos V a cos # 00 0.0015383 0.9891096 2.2418773 log 3 * 2.7472323 3 5 5 8 46'. 1 40 is taken equal to 3, because cos os = I. ELLIPTIC FUNCTIONS. Whence, by equations (i)* of Chap. VI, we get ^03 log = 4.495 73 1 6 sin 3 9.5075232,, sin (20, -0,) 40032548* 2 8 - 03 O'. 00346 2 = 279 23'.o6827 4o log = 7-5488952 sin 2 9.994 1484* sin (20, - 9 ) 7.5430436,, 20, S = 1 2 '.0039 0: = 139 35'-532I ^o log = 9-0739438 sin 0, 9.8117249 sin (20 0,) 8.8856687 20 0j = 4 24^467 = 71 59^9999 = 72. Ans. EXAMPLE. Given u = 2.41569, ff = 80. Find 0. ^4j. = 82' EXAMPLE. Given u = 1.62530, k = . Find 0. . = 87' CHAPTER XVI. NUMERICAL CALCULATIONS. E(k, 0). First Method, By Chap. X, eqs. (15), (16), and (17). EXAMPLE. Given k 0.9327, = 80. Find E(k, 0). By eq. (15), Chap. X, = 80 : 0, = 50 43 / .6, 0. = 27 4 8'.5, S =14 i6'7, 0A= 7 1 1 '-3, 0A = 3 36'A 3 V = 0.062831. /. 0* < O.OOOOOOI. 31 y y\ 67 44'. ; 46 4o'-4 J 26 o'.i ; XA = 6 45'-2 log sin ^ A = 8.77094 ; Whence, by eq. (17), sin sin" 2 sin 0j sin j 4 sin 0j sin 2 8 sin 0. sin* 16 sin sin" = 0.062794 =0.52116 = 0.29757 =0.10023 = 0.02728 = 0.00697 0.95321 Hence, by eq. (16), E(k, 0) = 32^, ^A) - 0.95321 = 2.0094 0.9532 = 1.0562. in 112 ELLIPTIC FUNCTIONS. Second Method. By Chap. X, eq. (26). EXAMPLE. Given k = sin 75, tan (f> = \ / ~^. Find ', 0). From eqs. (i4i), Chap. IV, we have k sin sin 75 o' o" log = k' cos B = cos 75 9.4129962 tan 2 = tan 2 37 30' sin0 -sin 36 4 'i6". 47 f kj = cos 9.9075648 ( tan 2 tan 2 18 2' 8".235 ) &<,. = > /, >-o / // i 9.0253880 ( sm 00 = sin 6 5' 9 .38 ) k^ cos ^ 00 9-99754S 2 1 i~ o n -1- rr 4"Ofi ^ O "J A *\f~\ 1 , \ Ldll ^"/i ft Ldll \ Z, S4- .^JvJ / - ^os= 1 a t n 745U 6 72 ( sm M = sin 9' 42 '.90 j ^/ = cos 03 9.9999982 ^ 04 = (i>^ 03 ) 2 4.3002761 /&/ = o.ooooooo From eqs. (14.,), Chap. IV, we have = 47 3' 30".94 ; ,= 62*36' 3 ".io; 3 240 o' o /x .i9. NUMERICAL CALCULATION'S. E(k, (p). 1 13 Applying eq. (26), Chap. X, we have /P log = 9.9698876 a. c. 2 9.6989700 9.6688576 .4665064 k a 9.7699610 a. c. 2 9.6989700 9.1377886 .1373373 00 9.0253880 a. c. 2 9.6989700 7.8621466 .0072802 ^os 7-45ii672 a. c. 2 9.6989700 5.0132838 .0000103 .6111342 i .61 1 1342 = 0.3888658. From eq. (23)*, Chap. IV, we find F[k, (j>) = 0.9226874. Hence = 0.3588016 sin 0, = 0.3290186 . sm 4 = 0.0522872 K sin 03 = _ 0.0013888 . sin 4 = o.oooooio 0.3799180 114 ELLIPTIC FUNCTIONS. Whence E(k, 0) = 0.3588016 + 0.3799180 = 0.7387196. Ans. EXAMPLE. Given k = sin 75. Find E\k, -\ From Example 2, Chap. XIII, we find =0.4421761 log 0.3888658 = 1.5897998 \ogE\k, j) =0.0319759 E\k, ^) = 1.076405. Ans. \ ^ ^ EXAMPLE. Given = sin 30, = 81. j. E(k, 0) = i. 33 1 2 4- EXAMPLE. Find (sin 80, 55). ^^. 0.82417. EXAMPLE. Find ^(sin 27, H. ^w^. 1.48642. \ 2 / EXAMPLE. Find (sin 19, 27). Ans. 0.46946. CHAPTER XVII. APPLICATIONS. RECTIFICATION OF THE LEMNISCATE. THE polar equation of the Lemniscate is r = a Vcos 26, referred to the centre as the origin. From this we get dr a sin 20 dO ~ Vcos 20 ' whence the length of the arc measured from the vertex to any point whose co-ordinates are r and s = (dr\ \ 7/1 / r ' I * v " I \dOI \ I /cos l ., r 7/3 I < _ I oflv / I I / J /^/-vo 0/3 I ( = a I = - = a Vcos 20 J Vi 2 sin a Let cos 20 = cos* 0, whence s / r I cos 4 cos is the complement of the eccentric angle. Hence s J Vdx* + dy* aj*d Vi e* sirT^ = a(e, 0), in which e, the eccentricity of the ellipse, is the modulus of the Elliptic Integral. The length of the Elliptic Quadrant is x y EXAMPLE. The equation of an ellipse is -^-3 \-~-^ = i\ lO.oI IO required the length of an arc whose abscissas are 1.061162 and 4.100000 : of the quadrantal arc. Ans. 5.18912; 6.36189. RECTIFICATION OF THE HYPERBOLA. On the curve of the hyperbola, construct a straight line perpendicular to the axis x, and at a distance from the centre equal to the projection of b, the transverse axis, upon the tf asymptote, i.e. equal to _- Join the projection of the r a 3 -j- d* A P PLICA TIONS. 1 1 7 given point of the hyperbola on this line with the centre. The angle which this joining line makes with the axis of x we will call 0. If y is the ordinate of the point on the hyperbola, then evidently P tan : 7?T^' and c? sin 2 a whence a I I . c^s~0\/ J ~? s = / Vt _t>> r* ~ ^ e* sin 2 i -& cos 2 v I r sin 2 Consequently ,/, cos 2 V i ^ 2 sin 2 L'i = F(k, 0) cE(k, 0) + ^ tan ^(^> 0) tan Il8 ELLIPTIC FUNCTIONS. EXAMPLE. Find the length of the arc of the hyperbola * y = i from the vertex to the point whose ordinate 20.25 4OO 4 is tan 15. Ans. 5.231184. EXAMPLE. Find the length of the arc of the hyperbola 5 = 100 from the vertex to the point whose ordinate 144 si is 0.6. Ans. 0.6582. 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