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 1991.Cornell nnittetsttg Snbtary Atijara, îinti Çork FROM J. I, Hwtckiixsow MATHEMATICSAN ELEMENTARY TREATISE ON ELLIPTIC FUNCTIONS<£amfcrtUge: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.AN ELEMENTARY TREATISE ON *9* °<*n s4>* ELLIPTIC FUNCTIONS BY THE LATE ARTHUR CAYLEY SADLERIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE SECOND EDITION LONDON: GEORGE BELL AND SONS. CAMBRIDGE: DEIGHTON, BELL AND CO. 1895PREFACE. The present treatise is founded upon Legendre’s Traite des Fonctions EUiptiques, and upon Jacobi’s Fundamenta Nova, and Memoirs by him in Crelles Journal: comparatively very little use is made of the investigations of Abel or of those of later authors. I show how the transition is made from Legendre’s Elliptic Integrals of the three kinds to Jacobi’s Amplitude, which is the argument of the Elliptic Functions (the sine, co- sine, and delta of the amplitude, or as with Gudermann I write them, sn, cn, dn), and also of Jacobi’s functions Z, II, which replace the integrals of the second and third kinds, and of the functions ©, H, which he was thence led to. It may be re- marked as regards the Fundamenta Nova, that in the first part Jacobi (so to speak) hurries on to the problem of transformation without any sufficient development of the theory of the elliptic functions themselves; and that in the concluding part, start- ing with the developments furnished by the transformation- formulae, he connects with these, introducing them as the occasion arises, his new functions Z, II, ®, H: there are thus various points which require to be more fully discussed. Not included in the Fundamenta Nova we have the important theory of the partial differential equation satisfied by the functions ©, H, and, deduced therefrom, the partial differentialVI PREFACE. equations satisfied by the numerators and denominator in the theories of the multiplication and transformation of the elliptic functions: these I regard as essential parts of Jacobis theory, and they are here considered accordingly. For further explanation of the range and plan of the present treatise the table of contents, and the first chapter entitled “ General Outline,” may be consulted. I am greatly indebted to Mr J. W. L. Glaisher of Trinity College for his kind assistance in the revision of the proof-sheets, and for many valuable suggestions. Cambridge, 1876. NOTE TO THE SECOND EDITION. The publishers regret that owing to the death of Professor Cayley whilst this edition was passing through the press, the latter portion has not had the benefit of his revision.CONTENTS. CHAPTER I. General Outline. page Origin of the elliptic integrals, 1 to 11.........................1 The addition-theory, 12 to 14.....................................5 The addition of the three kinds of elliptic integrals, 15 to 17 . . 6 The elliptic functions am ; sin am, cos am, A am ; or sn, cn, dn, 18 to 27.....................................................8 Further theory in regard to the third kind of elliptic integrals : addition of parameters, and interchange of amplitude and para- meter, 28 to 31.................................................13 The second and third kinds of elliptic integrals expressed in terms of the argument u ; new notations, 32 to 34.....................14 The functions Qu, Hu ; the ^-formulae, 35 to 42...................16 The numerators and denominator in the multiplication and trans- formation of elliptic functions, 43.........................19 CHAPTER II. The Addition-Equation. Landen’s Theorem. Introductory, 45.................................................21 First proof of addition-equation (Walton), 46....................22 Second proof (Jacobi), 47........................................23 Forms of addition-equation, 48 and 49............................24 Third proof (a verification), 50.................................26 Fourth proof (Legendre), 51......................................27 Fifth proof (Jacobi by two fixed circles), 52.....................28 Landen’s theorem from foregoing geometrical figure, 53 to 56 . . 30 Sixth proof, 57..................................................33Vili CONTENTS, CHAPTER III. Miscellaneous Investigations. page Introductory, 58...............................................35 Arcs of curves representing or represented by the elliptic integrals F(k, φ), E(ky φ), 59 to 64................................ib. March of the functions F(k, φ), E(ky φ), 65 to 71 . . . . 41 Properties of the functions F(ky φ), E (£, φ), but chiefly the com- plete functions Fxky EJcy 72 to 79............................46 Expansions of sn uy cn uy dn u, in powers of u, 80, 81. . . . 56 The Gudermannian, 82 to 90.....................................58 CHAPTER IV. On the Elliptic Functions sn, cn, dn. Introductory, 91.....................................................63 Addition and Subtraction-formulae, 92 to 97.........................ib. The periods 4Ky 4&K'y 98.............................................66 Property arising from transformation in preceding article, 99 . .68 Jacobi’s imaginary transformation, 100..............................ib. Functions of w+(0, 1, 2, 3) A+(0, 1, 2, Z)iK'y 101 to 103 ... 69 Duplication, 104.....................................................71 Dimidiation, 105 to 110..............................................72 Triplication, 111....................................................77 Multiplication, 112 to 119...........................................78 Factorial formulae, 120 to 125.......................................92 New form of factorial formulae, 126 to 129...........................97 Anticipation of the doubly-infinite-product forms of the elliptic functions, 130.................................................101 Derivatives of sn uy cn uy dn u in regard to ky 131 .... ib. CHAPTER V. The three kinds of Elliptic Integrals. Introductory, 132...................................................103 The addition-theory, 133 to 137.....................................ib. New notations for the integrals of the second and third kinds, 138 to 140.....................................................107 The third kind of elliptic integral: outline of the further theory, 141 to 150.........................................................108 Reduction of a given imaginary quantity to the form sn (α+βί), 151 to 154.....................................................114 Addition of parameters, and reduction to standard forms, 155 to 175 117 Interchange of amplitude and parameter, 176 to 188 . . . . 133CONTENTS. IX CHAPTER VI. The functions II (u, a), Zu, ©w, Hm. page Introductory, 190....................................................142 Values of n (« +a, a) in the three cases a=\iK', a—\K’, a=\K+\iK' respectively, 191 to 193........................................143 The function Zu, 194 to 199.......................................145 The function Qu, 200 to 202 ................................... 149 Expression of n iu, a) in terms of Qu, 203 ....................... 150 The function Qu resumed, 204 to 209................................. 151 Recapitulation, 210..................................................154 The function Hu, 211 and 212.........................................155 The function n (u, a) resumed, 213 to 218.........................156 Multiplication of the functions Qu, Hu, 219 and 220 .... 159 Tables for the functions Zu, Qu, Hu, 221..........................160 CHAPTER VII. Transformation. General Outline. Introductory, 222 .............................................. 164 Case of a general quartic radical Vjf, 223 to 226 .... ib. The standard form dx-r-sll-x2. 1 -k2x2, 227 ..................... 167 Distinction of cases according to the form of n, 228 and 229 . . ib. n an odd number : further development of the theory, 230 to 235 . 169 Application to the elliptic functions, 236 ........................ 172 n an odd-prime, the ulterior theory, 237 to 239 ................... 173 Connexion with multiplication, 240 to 245 ......................... 175 CHAPTER VIII. The quadric transformation n = 2; and the odd-prime trans- formations n - 3, 5, 7. Properties of the modular equation and the multiplier. Introductory, 246 ............................................. 179 The quadric transformation, 247 to 263 .........................ib. The cubic transformation, 264 to 266............................... 188 The quintic transformation, 267 to 271 . . .... 191 The septic transformation, 272 and 273 ......................... 194 Forms of the modular equation in the cubic and quintic transforma- tions, 274 to 277 .............................................. 195X CONTENTS. PAGE Properties of the modular equation for n an odd-prime, 278 to 281 . Two transformations leading to multiplication, 282 and 283 The multiplier M, 284 to 288 ................................... Further theory of the cubic transformation, 289 to 298 A general form of the cubic transformation, 299 and 300 . XX'2 dk Proof of the equation nM2 = , 301 to 303 Differential equation satisfied by the multiplier if, 304 Differential equation of the third order satisfied by the modulus X, 305 to 308 . A relation involving JZ, K, A, A, G, 309 200 201 203 206 216 218 220 222 224 CHAPTER IX. Jacobi’s partial differential equations for the functions H, ©, and for the numerators and denominators in the MULTIPLICATION AND TRANSFORMATION OF THE ELLIPTIC FUNC- TIONS sn, cn, dn. Outline of the results, 310 to 313..............................226 Differential equation satisfied by 0w, 314 to 316 .... 229 Same equation satisfied by Hw, 0 (w-f A), H (u+K), 317 and 318 . 231 Differential equation satisfied by 0 X^, &c., 319 and 320 . . 232 New form of the two differential equations, 321 and 322 . . . 233 Equations satisfied by the numerators and denominator, 323 to 331 . 235 Verification for the cubic transformation, 332 to 339 .... 245 CHAPTER X. Transformation for an odd and in particular an odd-prime ORDER : DEVELOPMENT OF THE THEORY BY MEANS OF THE ^-DIVISION OF THE COMPLETE FUNCTIONS. The general theory, 340 to 345 ............................... 251 Additional formulae, 346 to 351 .............................. 256 The 2a>-formulae, 352 to 355 260 n an odd-prime; the real transformations, first and second, 356 to 363 .................................................... 263 Two relations of the complete functions, 364 and 365 . . . 274 The complementary and supplementary transformations, 366 to 371. 275 The multiplication-formulae, 372 280CONTENTS. xi CHAPTER XL The ^-functions : further theory of the functions H, ©. PAGE Introductory, 373 ............................................. 282 Derivation of the ^-formulae, 374 to 382 ........................ib. 0, H expressed as ^-functions, 383 to 387 ....................... 291 New developments of the functions H, 0, 388 to 392 . . . 297 Double factorial expressions of H, 0; general theory, 393 to 399 . 300 Transformation of the functions H, © (only the first transformation is considered), 400 and 400* . ..................... 306 Numerator and denominator functions, 401 and 402 .... 309 CHAPTER XII. _ Rdx Reduction of a differential expression —=. Introductory, 403 ............................................ 311 Reduction to the form Rdx-h- ±(1 ±mx*)( 1 ±na?), 404 to 411 . . ib. Reduction to the standard form Rdoc-- J\-xA. l—k2x2, 412 to 415 . 315 Further investigations, 416 to 422 ............................. 319 CHAPTER XIII. Quadric transformation of the elliptic integrals of the first AND SECOND KINDS : THE ARITHMETICO-GEOMETRICAL MEAN. Introductory, 423 ............................................. 326 Geometrical investigation of the formulae of transformation, 424 to 427 327 Reduction to standard form of radical, 428 and 429 .... 330 Continued repetition of the transformation, 430 to 433 . . . 331 Reduction to standard form of radical, 434 and 435 .... 334 Application to integrals of the third kind, 436 .................. 336 Numerical instance for complete functions Eu Fu and for an incom- plete F, 437 ......................................... 337Xll CONTENTS. CHAPTER XIV. dx dy The general differential equation = Jx Jy PAGE Integration of the differential equation, 438 to 440 .... 339 Further development of the theory, 441 to 450 ......... 342 CHAPTER XV. On the determination of certain curves the arc of which is represented by an elliptic integral of the first kind. Outline of the solution, 451 to 453 ..................... 352 General theorem of integration, 454 to 462 .............. 354 CHAPTER XVI. On two integrals reducible to elliptic integrals. Introductory, 463 ............................................. 360 Investigation of the formulae, 464 to 467 .........................ib. Further developments, 468 to 472 .............................. 364 ADDITION. \ Further theory of the linear and quadric transformations. The linear transformation, 473 to 481 ........................... 370 The quadric transformation, 482 to 486 ......................... 376 Combined transformations : irrational transformations, 487 to 491 . 379 Index........................................................384 A student to whom the subject is new should peruse Chapter I., not dwelling upon it, but returning to it as he finds occasion; and he may after- wards in the first instance confine himself to Chapters II., III., IV., XII. and XIII.ERRATA. p. 71, line 12, for 2K' read 2iK', and line 17, for 4JT' read 4iA". p. 107, line 3 from bottom, for ,—du0— read / -—. r ' l + wsn2w / 0 1 + wsn2w 1 p. 118, line 12, for 4 read- 2 s/ ~P p. 228, line 2, dele of the three functions of nu. p. 283, No. 375. It might have been proper to state explicitly that the square brackets denote infinite products obtained by giving to m the values 0, 1, 2...to infinity. p. 287, line 8 from bottom, for No. 376 read No. 377. p. 290, bottom line, before p. 93, insert 1.1. p. 310, bottom line, for No. 310 read No. 311. Legendre’s earliest systematic work on Elliptic Integrals is the Exercices de Calcul Integral sur divers ordres de Transcendantes, et sur les Quadratures, Paris, t. I., 1811 ; t. m., 1816, and t. n., 1817 : the later work is the Traité des Fonctions Elliptiques et des Intégrales Euleriennes, Paris, t. i., 1825; t. il., 1826, and t. m., 1828—32; the greater part of Legendre’s own results on the theory of Elliptic Integrals, contained in the first volume of the Fonctions Elliptiques, had been already given in the first volume of the Exercices. Jacobi’s work is the Fundamenta Nova Theoriœ Functionum Ellipticarum, Königsberg, 1829 : the Memoirs in Crelie’s Journal extend from 1828 to 1858, some of them, in connexion with the Fundamenta Nova, being published shortly after the date of that work. The Memoirs by Abel, published for the most part in the earlier volumes of Crelle’s Journal, 1826 to 1829, are collected in the Œuvres Complètes de N. H. Abel, par B. Holmboe, Christiania, t. i. and il., 1839, except the great memoir on Transcendent Functions, presented to the French Academy, and published, Mémoires des Savans Etrangers, t. vu., 1841.I.] ] CHAPTER I. GENERAL OUTLINE. Origin of the Elliptic Integrals. Art. Nos. 1 to 11. 1. We consider the integration of a differential expression Rdx 71’ where R is a rational function of x; X a rational and integral quartic function of x, with real coefficients*: the values of the variable x are real, and such that X is positive, or f X real. 2. This can be by a real substitution in place of x (that is a substitution where p and q are real) reduced to the form Rdx V ± (1 ± mx2) (1 + nx2) ’ where R is a rational function of the new x; m and n are real and positive; and the signs are such that the function under the square root is not — (1 + mx2) (1 -f nx2). * The references here and elsewhere to reality, and any references to sign or numerical limits, are regarded as in general holding good: it will be under- stood, however, that imaginary values might be admitted throughout, and the various theorems presented in a more general but less definite form: and there will be occasion to refer to and employ such extensions of the original real theory. C. 12 GENERAL OUTLINE. [I* 3. The rational function R is the sum of an even function and an odd function of x: the differential expression is thus separated into two parts; that depending on the odd function may be integrated by circular and logarithmic functions (as appears by making therein the substitution \/x in place of x); and there remains for consideration only the part depending on the even function of x: or, what is the same thing, we may take R to be an even function of x (that is a rational function of x2). a ~f" i)x~ 4. This being so, we can by a real transformation - , - - in C -j- cLx~ place of x2 transform the differential expression into the form Rdx V1 — ar . 1 — k2x2 * where R is a rational function of the new x2; k2 is real, positive and less than 1 (and therefore also k, assumed to be the positive root of k2, is real, positive and less than 1). 5. In the last-mentioned expression a? may be included between the limits 0, 1, or it may be >1 /A2; but in the latter case, we can by the substitution 1 ¡kx in place of x transform the expression into one of the like form in which the new x2 lies between the limits 0 and 1 : we therefore assume that ¿c2 lies within these limits. 6. By decomposing into an integral and fractional part, and the fractional part into simple fractions, and by integrating by parts, the integration is made to depend upon that of the three terms dx a? dx dx VI — x2.1 — k~x2 ’ V1 — x2. 1 — kV1 (1 + nx2) Vl — x2.1 — k2x2 9 where n is real or imaginary. Or, w'hat is the same thing, the three terms may be taken to beI.] GENERAL OUTLINE. 3 dx (1 — k?a?) dx dx Vl — a?. 1 — Ar5#2 * Vl — x?. 1 — k2af ’ (1 + no?) Vl — a?. 1 — A2#23 that is dx Vl — kVdx dx Vl — ¿r2.1 — A2#2 ' Vl — #2 (1 + n#2) Vl — op . 1 — k2x2 7. Writing herein x = sin , and putting for shortness Vl — A2 sin2 (f> = A (A;, <£), these are A /r* ,t , A (A;, d>) d<6, and ^r—--. A /y—77 , A (A;,) r (1 + w sm2 ) A (A, <£) and we have thus the three kinds of Elliptic Integrals: viz. these are first kind F(k, ) = second kind E(k,) d> third kind II(n,k,d>) = f7T—t------. A /7 ,v, - J(l + nsm2) A(A, cf>) the integral being in each case taken from = 0 up to the arbitrary value . It would of course be allowable under the integral sign to write for any other letter 0, taking the integral from 0 = 0 to 0=. 8. is the amplitude, k the modulus, n the parameter. The amplitude is a real angle ; as already mentioned, the modulus k is positive and less than 1; whence also A', = Vl — A;2, called the complementary modulus, is real, positive and less than 1. Moreover A (A, <£) = V1 — k2 sin2 , does not become = 0, nor consequently change its sign, and it is taken to be always positive. The parameter n, as already mentioned, may be real or imaginary: it is in the first instance taken to be real; and it will appear that the case wrhere it is imaginary can be made to depend upon that in which it is real. Supposing it to be real, there is a distinction according as it is negative and greater 1—24 GENERAL OUTLINE. [i. than 1 (viz. in this case the denominator 1 +n sin2 becomes = 0 for a real value of ); or else as it is negative and less than 1, or positive. 9. Instead of the complete notation A (k, ), we frequently express only the amplitude, and write simply A; and simi- larly Fj>, E(j>, II for F(Jc, (j.>), &c. respectively: viz. in these cases it is assumed to be understood what the unexpressed letters k, or k and n, are. We may in like manner express only the modulus, or the parameter, and write Ak, II (n, k), Tin, or IIk &c., but there is less frequent occasion for this, and the nota- tions when used will be explained. 10. The integrals, taken up to the value \tt of the amplitude, are said to have their complete values, and these are frequently denoted by means of a subscript unity; thus F (k, r) = FJc, or simply FY; and so EJc, Ely &c. 11. The three elliptic integrals are not on a par with each other; but they depend, the second and third kinds upon the first kind; or we may say* that they all three depend on the differential expression ^ fk ^) ' ^us there f°r each °f them an addition-theory depending on the integration of the differ- ential equation d4> . A (k, J A (k, f) V’ not for the first kind a theory depending on this equation and for the other two kinds like theories depending on the equations A (k, cj>) d(j> + A (k, yfr) dyfr = 0, _________#__________ I dÿ__________=0 (1+n sin2 $>) A (k, (f>) (1+77 sin2 ^r) A (k, yfr) 9 respectively: these last are equations not admitting of algebraic integration, and which do not present themselves in the theory. And the like as regards multiplication and transformation. * The statement is made provisionally: the three kinds, as will appear, depend each of them on the functions sn u, cn u, dn u.I.] GENERAL OUTLINE. 5 The Addition-Theory. Art. Nos. 12 to 14. 12. The differential equation dx dy .. VI VT ’ where Y is the same quartic function of y that X is of xy admits of algebraic integration: and in particular this is the case with the equation dx _ t Vl — ¿r2.1 — hPa? Vl — y2.1 — &2y2 and in this last equation we may take the constant of integra- tion, say m, to be the value of either of the variables x, yy when the other of them is put = 0. Writing x — sin <£, y = sin yfry m = sin we obtain for the differential equation d dyfr A Ayfr ’ an algebraic integral such that the constant of integration //, is the value of either of the variables , yfr when the other of them is = 0 ; viz. this is an integral involving the sines and cosines of , yfr (and fi), but which (as being algebraic in regard to these sines and cosines) is spoken of as an algebraic integral. 13. The integral in question, say the addition-equation, may be expressed in (among others) the various forms cos fi — cos cos yfr — sin sin yfrAfiy cos (f> = cos yfr cos fM + sin yfr sin yuAy cos yfr = cos cos fi + sin sin fiAyfr} 1 — cos2 — cos2 yfr — cos2 A* + 2 cos cos yfr cos yu — k2 sin2 cf> sin2 yfr sin2 //, = 0, sin fi = sin cos yfrAyfr + sin yfr cos (j>A(f> (—)*, cos (i = cos cos yfr — sin sin yfrAAyfr (-l·), A/jl = AtpAyfr — k2 sin sin yfr cos cos yfr (-l·), * The notation hardly requires explanation; (rf·) shows that the function is a fraction the numerator of which is written down, and the denominator of6 GENERAL OUTLINE. [i. where in each case there is a denominator = 1 — Jc? sin2 sin2 yfr, and sin = cos ft cos yfr + sin ft sin A/iAyfr (+), A<£ = ApAyfr +sin ft sin yfr cos ft cosyfr (h-), where there is a denominator = 1 — h? sin2 ft sin2 yfr, and in these last formulae we may interchange , yfr. 14. It is to be remarked that, considering ft as variable, we have d(f> dyfr _ dp A (f> A yfr A ft ’ viz. the addition-equation is (not the general, but) a particular integral of this differential equation. Writing this equation under the forms d __ dfx dyfr dyfr __ dp d(f> A A p A yfr ’ A yfr Aft A (f> ’ we naturally regard the integral equation, any form of it which gives ft in terms of , yfr as an addition-equation: and any form of it which gives <£ or yfr in terms of ft, and yfr or , as a sub- traction-equation. The resulting notion of subtraction may be regarded as included in that of addition, and it will hardly be necessary again to refer to it. The Addition of the three kinds of Elliptic Integrals. Art. Nos. 15 to 17. 15. We assume throughout , yfr, ft to be connected by the foregoing addition-equation: recollecting that this is an integral (taken with the constant determined as above) of the differ- ential equation ^ = 0, and reverting to the definition of which is afterwards stated: it is, I think, a very useful one generally, but there is in Elliptic Functions an especial need of it, from the frequent occurrence therein of groups of complicated algebraical fractions having the same de- nominator.I.] GENERAL OUTLINE. 7 the function F, it at once appears that for the first kind of elliptic function we have F -f Fyjr — Fjt = 0, (viz. (f), yjr, fji being connected as above, the integrals F, F\fr, Fft satisfy this relation) : this is the addition-theorem for the first kind of elliptic integrals. 16. It can be shown that for the second kind E + Fyfr — Eft — k2 sin sin yfr sin ft; and that for the third kind TT . , TT I TT 1 x n Vasin ft sin sin \lr II + Hyfr — IIfi = -p tan 1 —--------------- , - -j , T va I —n cos ft cos q> cos yfr .-log 1 +n — n cos ft cos <6 cos \fr+n V— a sin ft sin sin yfr 2 V— a 1 + n — n cos ft cos cos yfr—n —a sin ^ sin sin yfr9 (k2\ 1 + —J, and n being real, the first or second form is real according as a is positive or negative. 17. The mode of verification is obvious; in fact, repre- senting either of the last-mentioned equations by U=0, and considering U as a function of the variables <£, yjr, we have ITT dU 7 , d Ü 7 . dU= d$d*+d^d* d A5 so that to sustain the assumed equation U = 0, we must in virtue of the addition-equation have identically dU d(j> ¿Pp.o, viz. this equation, if true at all, can be nothing else than a form of the addition-equation: or, what is the same thing, the addition-equation will be reducible into the last-mentioned form: which being so, it gives dU =0, and thence by integra- tion U = const., and then determining the constant by the con-8 .GENERAL OUTLINE. [I- dition that for yfr = 0 the value of is = the value of the constant must come =0; and in this manner we must from the addition-equation arrive at the required equation U = 0. The Elliptic Functions am ; sinam, cosam, A am ; or sn, cn, dn. Art. Nos. 18 to 27. 18. We have spoken of as the amplitude of F; or writ- ing F = u, then is the amplitude of u; say = am u, and then sin , cos , A are the sine, cosine, and delta of am ut say these are sin. am u, cos. am ut A . am it, which may also be written sinam u, cosam u, A am u, or, as an abbreviation, sn u, cn u, dn u. 19. But in adopting the last-mentioned forms we introduce a new mode of looking at the functions; viz. sn u is a sort of sine-function, and cn u, dn u are sorts of cosine-functions of u ; these are called Elliptic Functions; and we may develope the theory from this point of view. Observe that the fundamental equation is u = F or d = A0 du : this may be written d sin = cos A = sn u, this is d sn u = cn u dn u du : say sn' u = cn u dn u, moreover cn2 u = 1 — sn2 u, dn2 u = 1 — k2 sn2 u, and differentiating and substituting for sn' u its value, we find cn' u = — sn u dn uy dn' u = — Jc2 sn u cn u , and as above sn' u = cnwdnw, which five equations constitute a foundation of the theory. Observe also that sn 0 = 0, cn 0 = 1, dn 0 = 1, sn (— u) = — sn uy cn (— u) = cn u, dn (— u) = dn u.I·] GENERAL OUTLINE. 9 20. But this theory is already furnished by the addition- equation ; viz. starting from the equation F 4- Fyfr = Ffi, then writing F = u, Fyfr = v (and therefore = am u, ifr = am v) we have F/jl = u + v or fi = am (u 4- v): the equations which deter- mine sin /a, cos fi, A/a in terms of the sin, cos, and A of and yfr give the sn, cn and dn of u + v in terms of those of u and v: viz. these equations are sn (u 4- v) = sn u en v dn v + sn v en u dn u cn (u 4- v) = en u cn v — sn u dn u sn v dn v (-), dn (u 4- v) = dn u dn v — h2 sn u cnwsnvciu; (+), where the denominator is = 1 — Jc2 sn2 u sn2 v, and we may on the left-hand sides write u — v instead of u 4- v, changing in each of the three numerators the sign of the second term. 21. These equations may be obtained independently: viz. in any one of them differentiating the right-hand side in regard to u and substituting for sn' u, cn' u, dn' u, their values, we obtain a symmetrical function of u, v; hence the same result as would have been obtained by differentiating in regard to v: the ex- pression in question is thus a function of u + v; and writing therein v = 0, we find it to be the sn, cn or dn (as the case may be) of u + v; which proves the equations. 22. We thus see that F is an inverse function, the direct function being sn; and that cn, dn are connected therewith as the cosine with the sine. It may be remarked that there are six quotients, sncn, sn + dn; cn-f- sn, dn-f-sn; dn-r-cn, cn-r-dn, which are in some sort analogous to the functions tan, cot: if all these functions had to be considered, appropriate notations would be —, &c. ( viz. = — u. &c ). These are not required: cn V cn u cn J it is however in some of the formulae convenient to have a symbol for the single quotient sn-i-cn: and considering thisGENERAL OUTLINE. 10 [I. as standing for sin. am -r- cos. am, it is=tan. am, and we accord- ingly write it as tn: viz. we have , = tan . am u. = tn u. A more complete notation employed by Dr Glaisher consists in writing sc, sd, cs, cd, ds, dc for the six quotients sn -f- dn &c. and he further writes ns, nc, nd, for the reciprocal functions 1 -r* sn, etc. 23. In further illustration suppose that the theory of the circular functions sine, cosine, was unknown, and that we had defined Fx to be the function f dx Jo Vi -a2' Then taking the variables x, y to be connected by the differ- ential equation + dv_ = 0 VI— a? VI — y* and supposing that 2 is the value of y answering to x = 0, we have Fx + Fy — Fz. But the differential equation admits of algebraic integration : and determining in each case the constant by the condition that for x=0, y shall be = z, the algebraic integral may be expressed in the two forms #Vl — y2+ yVl— a? =z, xy —Vl—— 2/2=VT — £2, so that either of these equations represents the above-mentioned transcendental integral; and we have thus a circular theory precisely analogous to the elliptic theory in its original form. But here the function Fx is the inverse function sin-1 a?, and the last-mentioned two equations are the equivalents of the equation sin-1 x 4- sin-1 y = sin ~lz, whence writing sin-1 x = uy sin-1 y = v, and therefore x — sin u, y = sin v, z = sin (u -f v): also assuming Vl — sin2 u = cos u, andI.] GENERAL OUTLINE. 11 therefore Vl — sin2 v = cos v, and Vl — sin2 (w + v) = cos (u + v), the equations in question become sin (u + v) = sin u cos v + sin v cos u, cos (u 4- v) = cos u cos v — sin m sin v, and it is clearly convenient to use these functions sin, cos in place of F, denoting as above sin-1. 24. In the theory of the circular functions we have an addition-theory, which gives rise to and may be considered as in- cluding a subtraction-theory: and this leads to a multiplication- and division-theory: viz. we find from sin u, cos u, the functions sin or cos nut sin or cos — u; we have similarly for the elliptic functions sn, cn, dn a multiplication- and division-theory. These will be considered in detail; they are referred to here only for the sake of the remark that there is for the elliptic functions a “transformation-theory” which has no analogue in the circular functions, viz. we determine in terms of the functions of u the like functions with an argument u/M, and a new modulus A in place of the origiual k : the transformation is of any integer order n} and there is, for each value of n, a relation called the modular equation between k and the new modulus A. And it is convenient to notice that in the multiplication-theory the sn, cn and dn of nu, and in the transformation-theory the same functions of (u/M, A), are fractions having a common denom- inator, so that in each case there are three numerators and a denominator which come into consideration. 25. The circular theory gives rise to a numerical transcend- ant 7r, viz. ^7r = £3T4159... is a quantity such that sin^7r=l, cos \nt = 0, t being the smallest positive value of the argument for which the two functions have these values : and in develop- f dec ing the theory from the integral J —- ^7r would be the complete function defined from the equation dx Vl — Æ212 GENERAL OUTLINE. [I- Moreover the circular functions are periodic, having for their common period four times this quantity, = r: viz. we have sin cos (u 4- 27r) = sin cos Vu 26. Corresponding to ¿7r we have in elliptic functions in the first instance the complete function FJc, also denoted by K, viz. K is a real positive quantity defined by the equation ___, J o v 1 — &2 sin2 or, what is the same thing, fe2 ’ where K is of course not a mere numerical transcendant, but a function of k: K is such that we have snlf=l, cnif=0, dn K = k\ Writing v = K, we obtain simple expressions for the sn, cn, dn of u + K, and thence for those of u -f 2K and u -f 4K ; viz. it ultimately appears that the sn, cn and dn of u + 4ÜT are the same as the sn, cn and dn of u respectively : or the functions have a real period 4jST. 27. But the form of the integral suggests the consideration of another quantity j** dx Jo Vl —a?.l — ’ this is a complex quantity transformable into the form [1 dx ^ j p dx io k2x2 Jo Vl — ¿r2.1 ~ k'W ’ viz. K' being the same function of the complementary modulus Jcf that K is of k, the value is = K + %K\ We have 1 ik' Bn(JSr + tJT)=if cn(K + iK’) = -^y dn(K + iK') = 0, and then forming the sn, cn and dn of u -f K + iK', &c. it ulti- mately appears that the functions of m+4 (K + iK') are equalI·] GENERAL OUTLINE. 13 to those of u respectively: viz. there is a second period 4(K+iK'). But as above seen 4K was a period, and thus the periods may be taken to be 4Kf &iK' respectively—only it must be borne in mind that K, K + iK' have, K} %K' have not, analogous relations to the elliptic functions. This is the theorem of the double periodicity of the elliptic functions. Further theory in regard to the third kind of Elliptic Integrals: Addition of Parameters, and Interchange of Amplitude and Parameter. Art. Nos. 28 to 31. 28. We may differentiate an algebraic function sin <£ cos R(sin2 ) A (k} ), where R (sin2 ) denotes a rational function of sin2 ; and thereby obtain an expression involving two or more terms .It of the form —---------; ■ A fl—¡-r with different values of n. (1 -f-7isin2<£) A(k, ) Conversely, integrating such expression we obtain an equation containing two or more terms of the form II (n, k> ), that is elliptic integrals of the third kind with different parameters. In particular there may be two parameters only; viz. these being n, n, then we have either nn' = k2 or (1 + n)(l + n) = k'1: the resulting formulae are useful for the reduction of an integral of the third kind to a like integral where the parameter is of one of the standard forms cot2 0, — 1 + k'2 sin2 0, — k2 sin2 9. 29. There may be three parameters ; the theorem is in this case a theorem for the “ addition of the parameters/’ To explain this, suppose that two of the parameters are — &2sin2^, —i2sin2^ (this if p, q are taken to be real, is a particular assumption, limiting the generality of the result; but allowing them to be imaginary, it is no restriction): then the third parameter is — h2 sin2 r, where the angles p, q, r are connected together by that very relation which is the addition-equation for the integrals of the first kind, Fp+Fq — Fr = 0 (rather it is, in the first in- stance, Fp 4- Fq — jPV = const., reducible to the last-mentioned particular form): the theorem then gives II (— k2 sin2 r, h, ) in terms of II (— k2 sin2 p, k, ) and II (— k2 sin2 q} kf ) ; and it is in this sense a theorem for the addition of parameters.14 GENERAL OUTLINE. [I- 30. The theorem leads to an expression for an integral of given imaginary parameter in terms of two integrals of real parameter, one of them of the form - k2 sin2 9, the other of the form cot2 \ or — 1 + k'2 sin2 X. 31. There is a further theory of the “ interchange of amplitude and parameter ”: differentiating the two sides of the equation in regard to n, and, after multiplication by a factor, conversely integrating in regard to this variable, we obtain expressed as a sum of certain integrals in respect to n. Express- ing this parameter in one of the standard forms, for instance n = — k2 sin2 0, the integrals in regard to n become integrals in regard to 9, viz. these are the elliptic integrals F (k, 0), E (k, 0) and an integral of the third kind II (n, k, 0), where the para- meter n' is = — 1c2· sin2 : that is n = — k? sin2 0, n = — k? sin2 . We have a relation between the integrals II (n, k} ), II (n\ k, 0) : this relation [involving also F(k, ), E(kf )} F(k, 0), E(k, 0)] is a form of the so-called theorem for the interchange of ampli- tude and parameter: those belonging to the other two forms of parameter n = cot2 0 and n = — 1 + k'2 sin2 0 are less elegant, inasmuch as in the 9 functions the modulus is k' instead of k. The second and third kinds of Elliptic Integrals expressed in terms of the argument u ; new Notations. Art. Nos. 32 to 34. 32. The introduction of u, = F, as the argument in place of u, in fact supersedes the consideration of the elliptic integral of the first kind: by introducing u as the argument in the integrals of the second and third kinds, we obtain d (1+w sin2 ) A (k, ) ’ E {k} )— I (1 — k2 sn2 u) du} II (w, k, <£) = du 1 + n sn2u ’I·] GENERAL OUTLINE. 15 which functions changing the notation might be called E (k, u) and II (n, k, u) respectively. But it is found convenient to con- sider somewhat different functions; viz. in place of the integral of the second kind Jacobi considers Zu = u ^1 — — k2 ƒ sn2w du, where E, K denote the complete functions E^k, FYk respec- tively : Zu is of course a function of k, so that this complete E expression is Z(k, u): it is = — E{k, u), differing from E(k} u) by a multiple of u. We have it is clear Z(K) — 0, and it was in fact in order to the existence of this relation that the change was made from Eu to Zu. 33. As regards the third kind, the parameter is taken to be = — k2 sn2 a (to meet every case a must not be restricted to real values) and the function considered is tt / \ _ f k? sn a cn a dn a sn2u du 11 {u, a)- |o i_k2 > [being of course a function also of kf so that its complete ex- pression would be II (u, a, k)] : viz. writing n = — &2sn2a, this is in fact a multiple of in------nyJtvT: , =-[F(k, if,)- n (n, k, )}. J(1 -H n sin2(f>) A(k, ) n 1 v r/ v r/i ’ 34. The advantage of the new forms is very great: thus the addition-theorem for the second kind of integral is Zu + Zv — Z (u + v) = k2 sn u sn v sn (u + v) and that for the third kind gives in like manner the value of II (u} a) 4- II (v, a) — II (u + v, a) in terms of the functions of u, v, u + v: the theorem for the addition of the parameters gives a very similar expression for II (u} a) + II (u, b) — II (u, a + 6), and the theorem for the interchange of amplitude and para- meter is in fact a relation between II (u, a) and II (a, u).16 GENERAL OUTLINE. [I. The Functions ®u, Hu. The q-formulce. Art. Nos. 35 to 42. 35. From the function sn u we derive a new function ©z^ by the equation 0^ _ ^ ehu*(l-^-k*f0dufQdusn*u (K, E denoting as before the complete functions FYk, Exk)\ this may be regarded as one of a system of four functions, ®u, © (u + K), © (it + %K'), © (u + K + iK')\ or writing Hu = — ie2K © (u + iK), the functions may be taken to be ®u, ®(w + K), Hu, H (u + K). 36. The function Zu is at once expressed in terms of ©w, and its derived function ©'w; viz. we have Zu = . ©w The function II (u, a) has a simple expression in terms of ©, viz. we have II (u, a) u &a ©a + ^log © (u — a) © (u + a) ‘ 37. Writing herein u + a for u, we have U(u + a, a) = (u + a)m-\log y; and for the values a = \K, \iK', \K + \%K' the function II (u + a, a) is expressible in finite terms by means of the func- tions log sn u, log cn u, log dn u respectively: the resulting equa- tions give, after all reductions, the formulae next referred to*. 38. The functions sn u, cn u, dn u are found to be fractional functions, the three numerators and the denominator being the four functions above spoken of; viz. we have i /y — snu = -j-Hu cnu=\/^H(u+K)+, dnw = V&'©(tt+iT)-7-, * This is not Jacobi’s method nor perhaps the most direct or natural way of obtaining the formulae in question; but the connexion of the formulae with the expression for II (u, a) is very noticeable: Note to Ed. 1. The method is at any rate a very natural way of obtaining the formulae.I] GENERAL OUTLINE. 17 where denom. = ©it. It may be remarked here that the functions H,e are not doubly periodic, the change u into u+4- n{1 + (5^}' where (except that in the first product the simultaneous values m = 0, m' = 0 are to be omitted) m, m have all positive or negative integer values, including zero, but under the following condition, viz. taking fi, ¡j! to denote each of them an in- definitely large positive integer, p being also indefinitely large in comparison with so that p! + fi = 0, then m the limits are m = — fM to 4- fi, m' « a m —— fj/ » + P, m » » 3 II 1 1 I—» » +fi> m a t t 1 m = — ¡j> — 1 ,, + /*'· = n jl+ =HlU, U (m, m) = H 1 + u (m, m')j ’ c. 218 GENERAL OUTLINE. [I. 40. Giving to m all its values, and reducing by means of the factorial expressions of sin x, cos x, the expressions become singly infinite products of circular functions such as 7T sin (w + Zm'iK'); or writing = x, these are expressible as products or series involving cos 2x, or the multiple sines or cosines of x, with co- J*K efficients which are functions of the quantity q, = e K ; viz. we have thus the ^-formulae which Jacobi obtains in quite a dif- ferent manner (from a transformation formula, by writing therein ujn for u, and taking n infinite), and which in fact led him to the functions H and ©. The formulae are very remarkable as well in themselves as from their origin, and the connexion which they establish between Elliptic Functions and the Theory of Numbers: as a specimen take here the identity {1 + 22 + 2^+22» + ...^ = 1 + 8{14^ + ^ + t^+...}> which not only shows that every number is the sum of four squares, but affords the means of finding the number of decom- positions. 41. The four functions @w, © (u + K), Hu, H (u + K) con- sidered as functions of a>, = ttK'¡K, and v = iTuj2Ky each of them satisfy the partial differential equation d2a da dv2 da) = 0. This equation, not given in the Fundamenta Nova, but obtained by Jacobi (Grelle, t. ill. (1828), p. 306), is, in fact, an immediate consequence from the expressions of the functions as series in terms of q (= e~u) and u; but it is also obtainable from the finite expressions of these functions. It may be right to remark that in some places in the present work, to is used in a different sense, viz. it is such that q = eiwto. 42. There is no proper addition-equation for the functions Hy ©: the nearest analogue is the system of equationsI.] GENERAL OUTLINE. 19 © (u + v) © (u — v) = H(u + v)H (u — v) = ©%©2w — H2uH2v ©*0 H2u®2v-®2uH2v ©20 involving, it will be observed, the H, © as well of u — v as of u + v. But these formulae show, what follows also from the double-product expressions, that these functions have a multi- plication-theory ; and the double-product expressions also show that they have a transformation theory. The functions ©u, Hu, Hxn, ©iW- differ only by constant and exponential factors from the Weierstrassian functions A1 u, A1 A12u, Al3w. The Numerators and Denominator in the multiplication and transformation of the Elliptic Functions. Art. No. 43. 43. We are thus, in the multiplication and in the trans- formation of the elliptic functions, led to expressions for the three numerators and the denominator of the functions of nu, or of (u/M, X) (ante, No. 24), in terms of the functions H, © ; and by the aid of the above-mentioned partial differential equation we obtain partial differential equations satisfied by the nume- rator- and denominator-functions in question: thus, considering the denominator only, and writing for convenience x = Jk sn u, a = & + ^, v — n in the case of the transformation of the nth order sn (u/M, X), but =n2 in the case of multiplication sn nu; then the denominator, considered as a function of x and a, satisfies the partial differential equation (1 -arf + xl)^ + {v-\){axc- 2a;8) ~ dr + v(v-l)x2z-2v (a2-4)^. (Jacobi, Crelle, t. iv. (1829), p. 185.) As regards the transforma- tion formula, it is to be observed that X, quit function of k, must consequently be considered as a function of a, and the expression of z as a function of x, and of a directly and through X, is so 2—220 GENERAL OUTLINE. [I- complex, that not only the equation is practically useless, but it is difficult to verify it even in the simple case of a cubic transformation : but as regards the multiplication formula, the equation is very convenient for the determination of the actual expression of z as a function of x and a, or, what is the same thing, of sn u and k. The equation requires some change of form to adapt it to the three numerators respectively: and the resulting equations are in like manner practically useless for transformation, but very convenient for multiplication. Concluding Remarks. Art. No. 44. 44. The foregoing outline is purposely very brief as to the theory of transformation, and as to the ulterior theory of the third kind of Elliptic Integrals; as to these it is completed by the outlines prefixed to the chapters on these subjects respec- tively, and generally the outlines or introductory paragraphs to the several chapters may be consulted: as thus extended, the outline is intended to cover the whole of the present treatise up to the end of chapter xi., and also chapter XII., which contains the reduction of the differential expression Rdx -r- JX to the like expression with the radical in the standard form Jl—xW—k^x*. The subsequent chapters may be regarded as supplementary; the outlines or introductory paragraphs will explain what the contents of these are; I only remark here that chapter XIII., relating in fact to Landen’s transformation, belongs to the elementary part of the subject, and might have been brought in at an earlier stage; the only reason for deferring it was the convenience of using the form of radical Ja? cos2 + 62 sin2 , instead of the standard form Jl — k* sin* ; generally whatever relates to the non-standard form of radical is given in these supplementary chapters. But the foregoing outline does not extend to the new chapter xyiii. and following chapters which form the conclusion of the present treatise.II.] 21 CHAPTER II. THE ADDITION-EQUATION : LANDEN’s THEOREM. 45. As already mentioned the addition-equation is the integral of the differential equation d i d^-o A Ayfr ’ (A<^ = yi — A^sin2^), &c.) the constant of integration ¡jl being the value of either variable when the other is put = 0. Of the proofs which are here given several are only verifications of the theorem assumed to be known: but the first one is a direct investigation. The fifth proof (Jacobi’s by means of two fixed circles) leads so naturally to Landens theorem, that, although belonging to a different part of the subject, I have given it in the present chapter. It may be remarked that taking , ^ each of them small we have A = Ayfr = 1, and the differential equation thus be- comes dcf> + dyjr = 0, or integrating we have + yfr = fi, viz. ft is here the value of either variable when the other variable = 0. In the proofs which follow jjl has throughout this signification; there are it will be seen cases in which the sign of a radical has to be determined and this must be done consistently with the foregoing relation, -f = /x.22 THE ADDITION-EQUATION. [II. First Proof (Walton, Quarterly Math. Journ. t. xi. (1870), pp. 177—178). Art. No. 46. 46. Rationalising the differential equation, we have d2 — dyfr2 = — A2 (sin2 dyfr2 — sin2 yfr eZ<£2), or, as this may be written, (d — cos2 yfr) = — k2 (sin2 dyfr2 — sin2 yfr d2) (cos2 — cos2 yfr). The left-hand side is = — sin ( + yfr) {d -l· dyfr) sin ( — yfr) (id

cos yfr, y = sin 0 sin and therefore COS ( + yfr) = x — y, cos ( — ^) = x + y, this is = dy2 — dx2. The right-hand side, omitting the factor — k2, is (cos cf> + cos yfr) (sin dyfr + sin yfr d(f>) x (cos — cos yfr) (sin — sin ^ d), where the first factor is = cos sin yfrdcf) + cos yfr sin dyfr + sin cos dyfr + sin yfr cos yfrd sin i/r d(j) + cos yfr sin dyfr -f sin cos (cos2 yfr + sin2i/r) dyfr -f sin yfr cos yfr (cos2 + sin2) d, it is = dy-bxdy — yeZ#; and similarly the second factor is = — dy + xdy — ydx. Hence, restoring the factor — k2, the right-hand side is = ¿2 [dy2 - (*dy - 3/<*»)2], or the differential equation is dy2 — d*2 = &2 [dy2 - ( sin yfr = y cos cos yjr + ^ J{k2- l)y2 — 1. Let fx be the value of yfr corresponding to the value = 0, then writing = 0, yfr = /*, we have 7 cos /* 4 ^ J(k2 — 1) y2 4 1 = 0, giving y2k2 (1 — sin2/*) = y2k2 — y2 41, that is y2 (1 — k2 sin2/*) = 1, or y = ^, whence ^ J(k2 — 1) y2 4 1 = ■^>S—; and substituting we obtain cos /* = cos cos y/r — sin cos yfr 4 c sin sin yfr = 0, then differentiating we have (— b sin cos yfr 4 c cos sin yfr) d(f> 4· (— b cos sin yfr 4 c sin (f> cos i/r) dyjr = 0; say this is Md 4 Ndyfr = 0. But we have M2 4 (6 cos cos yfr 4 c sin sin yfr)2 = b2 cos2 yfr 4 c2 sin2 yfr, N2 4 (b cos cos yfr 4 c sin sin ^)2 = b2 cos2 4c2 sin2 <£, that is iU2 = — a2 4 52 cos2^/r 4 c2 sin2^ = 62 — a2 — (62 — c2) sin2^, .A2 = — a2 4 b2 cos2 4 c2 sin2 = b2 — a2 — (ft2 — c2) sin2 <£,24 THE ADDITION-EQUATION. [II. and the differential equation thus is d dyfr = = o; y&2 - a? - (¥- c2) sin2 Jb2 - a2 - (&2 - c2) sin2 f viz. an integral of this equation is a + b cos (f> cos yfr + c sin sin yjr = 0. But observe that the differential equation contains the single 52 __ ^2 be constant 70------, the integral equation the two constants - , -, b2 — a2 a a b2 — c2 which of course cannot be expressed in terms of y------------but b2 — a2 4 only in terms of this and an arbitrary constant, say /z. Hence the assumed equation is the general integral of the differential equation. b2 — c2 To complete the investigation write p—— = k2, and assume cl b2___c2 - = — cos /z, then the equation p—— = i2, or c2 = b2 — (b2 — a2) k2 becomes c2 = b2{\—k2sin2¿z), or say c = — &A/z: substituting these values of a and c, the equation becomes — cos /z 4- cos cos yfr — sin sin ^A^z = 0; viz. we have cos /z = cos <£ cos yfr — sin sin A/z, as the integral of the differential equation = 0. It is clear that /z is the value of either variable correspond- ing to the value 0 of the other variable. Forms of the Addition-Equation. Art. Nos. 48 and 49. 48. We have (cos /z — cos cos yfr)2 — sin2 (f> sin2 yfr A2/z = 0; or, expanding and reducing, 1 — cos2 — cos2 yfr — cos2 fi + 2 cos cos yfr cos /z — k2 sin2 sin2 sin2 /z = 0,II.] THE ADDITION-EQUATION. 25 which is symmetrical in regard to the three quantities: hence we have also (cos — cos /x cos yfr)2 = sin2 fju sin2 yfr A2 , (cos yfr — cos fj, cos )2 = sin2 /x sin2 A2yfr, and extracting the square roots, it appears that the signs on the right-hand side must be H-: we thus have cos — cos /x cos yfr = sin ¿x sin yfr A, cos yfr — cos fju cos (f> = sin /x sin (f> Ayfr, to which join the original equation cos fju — cos cos yfr = — sin sin yfr A/x. 49. From the rationalised equation, writing sin2/x=l—cos2/*, we obtain (1 — A2 sin2 (f) sin2 yfr) cos2 /x — 2 cos <£ cos yfr cos p = 1 — cos2 — cos2 yfr — k2 sin2 <ƒ> sin2^, that is [(1 — k2 sin2 sin2 yfr) cos /x — cos cos yfr]2 = (1 — k2 sin2 cos yfr — sin sin yfr A Ayfr, which gives the value of /x in terms of and yfr. Combining with this the equation cos /Jb — cos cos yfr = — sin sin yfr A/x, we have the value of A/x : and if from cos /x we proceed to find the value of sin2 /x, we have (1 — k2 sin2 sin2 yfr)2 — (cos (j> cos yfr — sin<£ sin yfr AAyfr)2, which is readily found to be = (sin 0 cos yfr A yfr + sin yfr cos ¿a ¿)2;26 THE ADDITION-EQUATION. [iL and extracting the square roots, the sign on the right-hand side is +: we have thus the formulae sin ¡l = sin <£> cos ^ A^ + sin yfr cos A , (--) cos fjL= cos cos yfr — sin sin yfr A Ayfr, (~) Afi = A<£ Ayfr — k2 sin sin ^ cos <£ cos (-*-) where the denominator is = 1 — Jc2 sin2 sin2 yfr. We have in like manner sin = sin fi cos yfr Ayfr — sin yfr cos fi Afi, (-=-) COS , yfr. Third Proof of the Addition-Equation (a verification). Art. No. 50. 50. Writing the equation in the form cos ft, cosec cosec yfr — cot cot yfr = — A/*> then, differentiating the left-hand side, the coefficient of d is — cos fi cosec cot cf> cosec yfr + cosec2 cot yfr, = —} .—. (cos yfr — cos ft, cos d>), sin2 smyjrK T ^ ^/y which in virtue of the form cos yfr — cos fi cos = sin fi sin A^r, sin a IS = —T—·------r sin 9 sm yfr and similarly the coefficient of dyfr is = ~~=—~r~v~ T Afr sin 9 sm yfrII.] THE ADDITION-EQUATION. 27 so that, omitting the common factor, the differential equation becomes dtpAyfr + dyjrAff) = 0, which is right. Fourth Proof (Legendre, Traité des Fonctions Elliptiques, t. I. (1825), p. 20, by a spherical triangle). Art. No. 51. 51. Consider a spherical triangle ABC, obtuse-angled at C, such that the sides CB, CA are =, yfr respectively, and C that the cosine of the angle C is = — A/*. This being so, the equation cos p — cos cos yfr = — sin , yfr which satisfy the relation in question). And the other two equations cos — cos p cos yfr = sin p sin yfr A, and cos yfr — cos p cos = sin p sin A(f>, show that cos-4 = A(f>, and cos B = A yfr: so that the sides a, b, c of the spherical tri- angle are , yfr, p respectively, and the cosines of the opposite angles A, B,C are A , A yfr, — Ap respectively. Now considering the consecutive position A'B' of the side AB, and letting fall on AB the perpendiculars A'p and B'q, the equation A'B' = AB gives Ap = Bq, that is A A' cos A = BB' cosB, or db cos A+ dacosB = 0] viz. this is the differential equation d(f)Ayfr + dyjrA = 0.28 THE ADDITION-EQUATION. [II. Fifth Proof (Jacobi, Crelle, t. ill. (1828), p. 376, by two fixed circles). Art. No. 52. 52. Consider two fixed circles as shown in the figure, and suppose that we have L Radius of larger circle = R, „ smaller „ =r, Distance OQ of centres = D. Write moreover R — D r ^ = Cm whence easily 4>DRII.] THE ADDITION-EQUATION. 29 viz. k and /x are given functions of R, r, D: noticed that (iZ + D)2-^ 4 J?2 sin2 /x (1 + A fif and it may be Imagine now a variable tangent AB, and assume zAOL = 2, ZB0L = 2yfr, then letting fall on AB the perpendicular OG, we have Z AOG = 7r — (<ƒ> + y/r), Z QOG = + + D cos (<£ — t/t) = r; that is, (R + D) cos cos yfr - (R — D) sin sin yfr = r, or what is the same thing cos cos yfr — sin 0 sin ijr A/x = cos /x, which is the integral equation. Also AM2 = AQ2- MQ2, = J?2 + D2 + 2Dii cos 2 — r2, = (i2 + D)2 — r2 — 4Dii sin2 <£, = {(ii + i))2 - r·2} A20 ; and similarly j?Ji2 = {(R + D)2 - r2} A2^|r. Now, varying the tangent, let the new position be A'B!; then clearly A A': BB' = A if: BM; that is or d(f> : — == AM : BM, d , d± _ n. AM+BM~ ’ viz. substituting for AM, BM their values, we have the required differential equation d dyfr = A * A> ’ corresponding to the above integral equation.30 THE ADDITION-EQUATION. [n Landeds Theorem, from the foregoing geometrical figure. Art. Nos. 53 to 56. 53. Suppose that the large circle and also k remaining constant, the small circle is varied; that is, let r, D vary subject to the foregoing condition M- iDR ~ (R + D)2 — r2 ’ it is readily shown that the radical axis of the two circles re- mains unaltered. In fact, taking the centre of the larger circle as origin and the axis of x vertically downwards, the equations of the two circles are a2 + f - R2 = o, (x — D)2 4- y2 — r2 = 0, and thence for the radical axis 2Dx — jR2 — D2 4- r2 = 0, or x ii2 + D2 - r2 2D J2R ~ k2 > which is constant. In particular the smaller circle may reduce itself to the point F (one of the limit-circles of the original two circles, or what is the same thing an antipoint of their points of intersection, viz. that antipoint which lies within the smaller circle) : and then, taking the distance OF =8, we have 4&jR 4 DR (R + Sj2 = (R + D)2 - r2 ’ or, what is the same thing, S(R2 + D2-r2) = D (R2 + B2). 54. Reverting now to the original two circles, if in the figure Z AOG = <*>(= ir — — yfr) and Z QOG = x(= — ^), then obviously AA' cos MAO = AMdx, that is R. 2d sin a> = AMdx; or what is the same thing 2AG d = AM dx; hence the equa- i<}> _ dty d _ dyfr _ dx AM ~BM~ 2AG ’II·] THE ADDITION-EQUATION. 31 and observing that AG2 = AO2 — OG2 = P2 — (P cos x — r)2, the equation is d — dyjr _ dx V(P + P)2 — r2 A (*, £)”A (fe, ~2 ViF-(i) cos x~~r)2' V 7* We have OP =------------. and thence OP = P---------, whence from cos% cos^ the triangle OAPt in which the angles A, P are = + yfr — ^7r (that is 2 — x — ^7r) and ^7r + % respectively, we have v D-------: — cos (26 — v) = R : cos v; COSX \ 'T A/ that is D cos x — r = — P cos (2 — ^), which is an integral equation corresponding to the above differ- ential equation dcf> _ ¿x V(P + P)2 — r2 A (&> ) 2 VP2 — (P cos ^ — r)2 Writing now ZAPO = 0, then X=d—\ir, 2—x=2 — 0+^7r, and the integral and differential equations become respectively P sin 6 — r = R sin (2<£> — 6). , d$ __ dd V(P + P)2 - r2 A (k, 0) 2 VP2 — (P sin 0 — r)2 55. Suppose now that the smaller circle reduces itself to the point P, then retaining 0 to denote the angle in this state of the figure, we must in place of P, r write 8, 0; and the equations become 8 sin 0 = JR sin (2 — 0), d d0 (P + 8) A (k, ) 2 VP2 — 82 sin2 0 ’32 THE ADDITION-EQUATION. [II. or writing herein \ = B/R, these are \ sin 0 = sin (2 — 0), d(f> , ,, _ ^ d0 and 4(1 + *) A (*,<#») where in virtue of the relations k- = 4\ have k2 = (i+\y , and therefore also k' = A(X, 0)’ 4&R , S (STif*"1*·-!·” 1-X , , 1-k' ---— and X = ^. 1 + X 1+k 56. The result would have come out more simply by con- sidering ab initio the smaller circle as replaced by the point F: viz. the chord AB would then pass through the point F, and the points M, Q each coincide with F: but it was interesting to consider the theory in connexion with the original figure of the two circles. The theorem gives, it will be observed, a transformation of the differential expression m^° an exPress^on a (X 6) 9 involving a new modulus X: viz. considering X as derived from \ —1c' k by the equation X = -----j-,, then we have between the two i T •I' variable angles , 0 an integral equation X sin 0 = sin (2 — 0) answering: to the differential relation A ^ ^ · or 6 A(&,0) A (k, 0) since <£, 0 vanish together this last is equivalent to F(k,4>) = %(l+\)F(\0). The integral equation gives X tan 0 = sin 2 — cos 2 tan 0, that is whence sin 0 — X H- cos 2 sin 2 sin 2 Vl + 2X cos 2 4* X2 9 V(1 + X)2 — 4A. sin2 4X 2 observing that -^=i2 and 1 + X = ^—77, this is + X/ 1 + fc sin 2 = , or sin 0 = ^ (1 + k‘) Vl — ^sin2^»II] THE ADDITION-EQUATION. 33 Sixth Proof of the Addition-Equation,. Art. No. 57. 57. The rationalised equation in , yfr, fA may be written sin2 /a — cos2 (j> — cos2 yfr + 2 cos cos y/r cos fA — k2 sin2 fA (1 — cos2 ) (1 — cos2 = 0 ; viz. this is k1'2 sin2 fA — A2/!- (cos2 + cos2 \jr) -f 2 cos fA cos cos yfr — k2 sin2 p cos2 cos2 ^ = 0, or as it may also be written A;'2 sin2 fx · — A2/* cos2 + 2 { · cos yu cos <£ ■ } cos ^ + {— A2/*, · — k2 sin2 fj, cos2 ^>} cos2 y[r = 0; viz. the left-hand side is a quadriquadric function of cos <£, cos yfr: say this is u, and represent it successively under the forms A' + 2B' cos + (7 cos2 <£, and A + 21? cos yfr + C cos2 where of course A', B\ (J are given functions of cos and A, B, C are the like given functions of cos : we have -r——7 = 2 ((7 cos + B'), d cos + jB')2 = (S'2 — A'C), whence -=——7 = 2 slB'2 — A'G\ or what is the same thing d cos 0 ^ = — 2 sin (j> VB'2—A'C', and similarly ~ = — 2 sin yfr Vl?2 — A 0: wherefore the differential equation is sin (f>d + fB2 — AO sin ^ cty = 0; we have B*—AC=cos2/a cos2 (j> + (A;'2 sin2/a—A2/4COS2<£) (A^-f-A^sin^cos2 0) = &'2 sin2 ¡a A2/x 4- (cos2 + A^A?'2 sin4 fA — A4//,) cos2 — k2 sin2 fA A2fA cos4 , c. 334 THE ADDITION-EQUATION. [II. and the coefficient of cos2 is 1 — sin2 ¡x + k?k'2 sin4 fi — 1 + 2k2 sin2 fM — k* sin4 fi, = (k2 — k'2) sin2 fi A2fi; hence the value of B2 — AC is sin2 /¿. A2fL [k'2 + (k2 — k'2) cos2 — A;2 cos4 }, = sin2 fju A2fL. (1 — cos2 <£) (¿'2 + i2 cos2 ), = sin2 fi A2fi sin2 A2 \ that is we have *JB2-AC = sin sin AfiA; and similarly Vi?'2 — J/C" = sin fjL sin ^ Afi Ayfr; hence the foregoing result is d(f> Ayfr + dyfr A = 0, the required differential equation. It may be remarked that this, like the third proof, ante, No. 50, is a verification, the difference being that we use the rationalised integral equation instead of the original irrational equation: and that they are each of them closely connected with the second proof, ante, No. 47, although this is less in the form of a verification.CHAPTER III. MISCELLANEOUS INVESTIGATIONS. 58. The present chapter contains, in relation to the first and second kinds of elliptic integrals, various matters not very closely connected which it was convenient to give here before going on in the following Chapter IV. with the main theory: the contents will be seen from the headings of the several articles. Arcs of curves representing or represented by the elliptic integrals E (k, <£), F(k, ). Art. Nos. 59 to 64. 59. The elliptic integral of the second kind occurs naturally as representing the arc of an ellipse: viz. taking the /jj2 equation of the ellipse to be — +1-2 = 1, this is satisfied on writing therein x = a sin, y = b cos (observe that is the complement of the eccentric anomaly, or say of the parametric angle).: we then have dx = a cos d, dy = — 6 sin d: and thence ds2 = (a2 cos2 cf> + b2 sin2 ) d2 = [a2 — (a2 — b2) sin2 <ƒ>] d2, (P — ¿2 so that taking k =----------(= eccentricity) we have ds = a A (k, ) d), the arc being measured from the extremity of the major axis: the length of the quadrant is = aEJc. In the case of the circle 8—236 MISCELLANEOUS INVESTIGATIONS. [III. the length is aE10, = a. ; and, as the minor axis diminishes, k increases and EJc diminishes, until ultimately for the indefi- nitely thin ellipse k becomes = 1, and .9 = aE11, = a. 60. We may also represent the arc of the hyperbola: taking the equation to be -2 — |-a = 1, and expressing x, y in terms of the parametric angle u, that is writing x = a sec u, y=b tan u, we have dx = a sec u tan u du, dy — b sec2 u du, and thence ds = ^ - Vi>2 + a2 sin2 u, cos2 u which does not immediately express the arc s by means of an elliptic integral: to obtain an expression of the required form assume k = ■ - and therefore k' = ——— va2 + 62 va2 + 62 (k = reciprocal of the eccentricity); and consider an angle connected with u by the equation tan u = k' tan : the expres- sions of x, y in terms of are *=cgiving , „ dy = ak'2 sin d<\> cos? bk' d cos2 {A is written for shortness to denote A (k, ) and so presently F(j>, E to denote F(k, ), E (k, ) respectively}: and thence , _ bk' d ~~ cos2 A ’ a value which of course may also be obtained from the fore- going expression of ds in terms of du. We obtain by differentiation d·^ tan<*> = (cos^TA^>“ A* + A*)d’ and conversely, integrating from zero, we haveIII.] MISCELLANEOUS INVESTIGATIONS. 37 whence, substituting for the integral and observing that b/k' = a/Jc} we have s = ^ {tan A + — Efy), where (see figure) s denotes the arc AM measured from the vertex A of the hyperbola. 61. As regards the geometric signification, observe that for the point M on the hyperbola, the construction of the angles u, . Moreover the perpendicular GZ on38 MISCELLANEOUS INVESTIGATIONS. [III. X lKj2 'll2 the tangent at M is given by = ~4 + > and substituting for x, y their values in terms of cj> we find GZ = a cos ; hence if Y be the intersection of this tangent with the circle, we have also Z YGZ = <£. Further MZ2 = x2 + y2 — a2 cos2 ; or since x2 4 y2 = a2 + b2 tan2 <£, this is MZ2 = a2 sin2 + b2 tan2 , = a2 tan2<£ ^cos2 + ^, = a2 tan2<£ ^ — sin2<^ , a2 tto2 or finally MZ — ~ tan A. Hence the formula is or, what is the same thing, JfZ - JO = | {E+ - k'*F), the quantity on the right-hand side being it is clear positive, viz. it is in fact = ah f COS2 = 90°, we obtain IO-IA=^(E1-F*Fl) (where I represents the point at infinity on the curve or the asymptote) as the expression for the excess of the length of the asymptote over the arc of the curve. 62. It is less obvious how to find a curve the arc of which shall express the elliptic function of the first kind. LegendreIII.] MISCELLANEOUS INVESTIGATIONS. 39 remarked that in the particular case k = 1/V2, the solution was afforded by the lemniscate (a?2 -f y2)2 = a2 (a2 — y2). Observe that the curve is a horizontal figure-of-eight, the extremities being given by y = 0, x = ± a, and the branches at the origin being inclined to the axis of x at angles = + 45°. The equation is satisfied on writing therein x = a cos Vl — 4 sin2 , a . , y = sin cos

, x2 — y2 = a2cos4<£): and hence determining the element of arc ds, = Vdx2 + dy2, we have dx = - ° S^n -—(— f + sin2 <£) d(f>, dy = ~L(1 — 2 sin2 0) ci^, V1 — £ sin2 V 2 whence attending to the identity sin2 <ƒ> (— § + sin2 )2 + £ (1 — \ sin2 )(l — 2 sin2 <£)2 = d2 we have or finally whence ds2 = | d 1 — £ sin2 ’ a d V2 Vl — ^ sin2 ’ *)■ s denoting the arc measured from the extremity x = a, y = 0 (<£ = 0) to the point belonging to the value of the parametric angle. The same result may be obtained by means of the polar equation r2 = a2 cos 20, introducing instead of 6 the variable connected with it by the equation sin = V2 sin 6. At the origin we have = 90°, and the length of the quadrant of the curve is thus = F, V2 VV2/ It thus appears that the lemniscate serves to express the function F of modulus 1/V2.40 MISCELLANEOUS INVESTIGATIONS. [III. 63. For the general representation of the function F(Jc, ) Legendre used the sextic curve where observe it is not the arc s, but the difference of this arc and an algebraic function, which is equal to the function F (k3 ) : and the solution is not an elegant one. 64. A very beautiful solution was obtained by Serret (improved ' upon by Liouville), Liouv. t. x. (1845), pp. 257 and 351 : and I have found that the theory admits of further development: I reserve the whole investigation for a sub- sequent chapter, remarking here that Serret’s solution was suggested to him by a different treatment of the lemniscate ; viz. the equation of the curve is satisfied by x = h sin *(1 + sin2 ). y = bh cos (1 + m — cos2 (f>), where, h being the modulus, the values of h, m, 6 are and it is then easily found that k2 s =F(ky (f>)— -pt sin cos A(ky <£), values which lead to so that the arc is expressed as a multiple of which is an expression in the nature of an elliptic integral. To compare with the former solution observe that we haveIII.] MISCELLANEOUS INVESTIGATIONS. 41 Vi — J sin 1 _i±*2 and thence V2 (1 + z2) dz d _ 2dz -----— ana . . · ■==· = -, =. 1 + z4 V1 — ^ sin2 Vl + z4 As remarked by Dr Fiedler, if in connexion with the lem- niscate we consider the circle x2 -f y2 = az (x + y) (z & variable parameter) this touches one of the branches at the origin, so that the origin counts for three intersections and there is besides a single intersection the coordinates of which are the foregoing values of x and y in terms of the parameter z. March of the Functions F(k, ), E (kt ). Art. Nos. 65 to 71. 65. To gain some idea of the march of the functions F, E, we may, taking as abscissa, trace the curves y=l/A, y = A<£>: the areas of these curves included between the axis of y and the ordinate corresponding to the abscissa <£ will of course represent the values of the integrals F, E. 66. If k = 0, then A = 1, and the curves y=A) y = 1/A<£, each reduce themselves to the line y = 1. Here of course If k > 0, <1, which is the standard case, then the curve y = A is an undulating curve lying wholly below the line y = 1, and the curve y=l/A<£ an undulating curve lying wholly above this line. As increases from zero the functions F, E each continually increase from zero, the function F being always the larger; and it is moreover clear that for a given value of , as k increases the function F increases and F(f> = E = (f).42 MISCELLANEOUS INVESTIGATIONS. [III. E diminishes, and conversely as k decreases then Ftp dimi- nishes and Ecj> increases. In particular k = 0, Fl} = K, = i'rr, and also E1 = ^tt, so that as k increases from zero, F1 or K increases from r, and E1 diminishes from f 7r.III.] MISCELLANEOUS INVESTIGATIONS. 43 67. We see moreover that for each of the functions F$,E<$> (k having a given value) it is sufficient to know the values of the functions for values of from 0 to \ir: in fact we have F(—a) = — Fa, Fir = 2Flf and then Fa = Fir — F (it — a), = 2FX — F (it — a), giving the values from a = it to \ir, and Fa = Fir + F(a — it), = 2F1 + F(a — it), giving the values from a = it to 27r; and so on. Or what is the same thing we have in general F (mir ± a) = 2mF1 ± Fa, and similarly E(rmr ± a) = 2mE1 ± Ea. 68. If k = 1 there is an entire change in the form of the curves, viz. the curve y = becomes y = cos , which is a curve lying as before wholly below the line y — 1, but which, instead of being included between this and the line y = 0, passes below the last-mentioned line, and is in fact included between the lines y = 4-1, y = —1. And the curve y=l/A<£ becomes y — l/cos, where the ordinate becomes infinite for <£ = ^-7r: we have then between the yalues \ir, f7r a branch lying wholly below the line y = — 1, the ordinates at the limits being = — x , then from f7r to f7r a like branch lying wholly above the line y — + 1, the ordinates at the limits being each = -t- qo ; and so on. Observe that in this case E = ƒ cos = sin (f>, so that 2?! = 1, and, completing a former statement, we may say that as k increases from 0 to 1, Ex decreases from ^7r to 1. We have also F = viz. we have F = log tan (^7r -f (observe that log tan is here the hyperbolic logarithm of the tangent,) and in particular F1 = co (a value agreeing with the form of the curve), so that, completing a former statement, we may say that as k increases from 0 to 1, Fx increases from ^7r to x . d ;os , which admits of finite integration,44 MISCELLANEOUS INVESTIGATIONS. [ill. 69. This case (corresponding to the extreme value & = 1 of the modulus) is one of great interest: writing u = Fcj> = log tan +|<£), we have = gud u (read Gudermannian of u, after Gudermann, by whom the form was specially considered), and then sin <£ = sin gud u, cos = A (f> = cos gud u, or as we may for shortness write them sin (/> = sg u, cos (j> = A = eg u ; viz. we have here the two new functions sg, eg, replacing the sn, cn, dn of the general case. 70. We have in a subsequent part of the subject to con- sider the expressions K'¡K, and q = e~nK’IK; and it is convenient to notice here that 1 x = kIII.] MISCELLANEOUS INVESTIGATIONS. 45 o' II k' = 1, 8 II h HN II II 8 0 II C3H V 2 || 11 *1 II II K' = l K ’ q = e~7r fc=l; o' II II 8 II 3 ^1^ II 0 2 = 1: viz. as k increases from 0 to 1, K'/K diminishes from oo to 0 and q increases from 0 to 1. The annexed figure shows the curve x = k} y = K'jK. It shows also a construction which will present itself in the sequel: viz. considering an abscissa x — ky and the abscissae x = \, x = y, which belong to the double ordinate and the half-ordinate respectively; then if T, T' be the complete functions to the modulus 7, and A, A' the complete functions to the modulus A, we have it is clear 2^-A' r ~K’ K~ A‘ 71. Conversely if A be such that A!IA. = ZK,IKy then A is less than k ; and similarly if 7 be such that %K'[K = T'/T, then 7 is greater than k: and not only so, but if, starting from k, we repeat this process of the double ordinate so as to obtain a series of moduli A, A1? A2... then we approximate very rapidly to a modulus = 0: and similarly if, starting from k> we repeat the process of the half-ordinate so as to obtain a series of moduli 7, 7i> 72· · · then we approximate very rapidly to the modulus = 1. And the like conclusions follow if n denoting any number greater than 1 (say n a positive integer = or > 2), we have A' K' K' r A ~nK’ K~n T*46 MISCELLANEOUS INVESTIGATIONS. [III. Properties of the Functions F {k, ), E(k, <ƒ>), but chiefly the complete functions FJc, EJc. Art. Nos. 72 to 79. 72. Starting from the expressions when k is small we may under the integral sign expand in ascending powers of k, and then, integrating from 0 to \ir by the formula 1.3...2ti— 1 7r 2.4...2?i 2 obtain the formulae / Is l2 32 F1k = ^7r \1 +2+ 2^74.& + EJc = iTr (l - ^ Jc2 - ' J, k* - 12.32.52 22.42.62 12.32.5 22.42.62 or what is the same thing, introducing the notation of hyper- geometric series ?(., A 7. *) -1 + these are FJc = . F ( J, 1, A;2), EJc^fr.Fi-b l 1, k2). 73. Suppose & is very nearly 1, k' is small and we have & = 1 — ; to find the value of Fj k we may write rbr J ijT—< d(f> jtt-c V cos2 -1- A;'2 sin2 rh*—« 'Jo ; d Vcos2 <ƒ> h- A;'2 sin2 ? where 6 may be taken an indefinitely small quantity which is nevertheless indefinitely large as regards A/. This being so, writing in the first integral \ir — u in place of , since through- re fai out the integral u is small, the integral becomes I — S S Jo VAP + ^H2’HI.] MISCELLANEOUS INVESTIGATIONS. 47 _ . . . 1. ke + VA,5i + &V i *· Jf · i , which is = ^ log------y--------, or neglecting k in regard to ke, this is = | log ^ , or say = log^. In the second integral k' sin (¡> is throughout small as regards cos <£, and the integral is __ fi*-* d_ Jo cos <£’ 2 which is = log tan it — £e), or what is the same thing = log -. Hence we have 2e 2 4 Fik=* log -jjj + log -, = log p, as an approximate value of FY kf k being nearly equal to unity. 74. The functions F{k, ), E{k, ), considered as functions of k, satisfy certain differential equations. Write for shortness E, F to denote the functions E(kt ), F(k, ), and A to denote A (k} ). Then dE _ f k sin2 d dF _ f k sin2 d dk~~} A * dk~J- ~ A3 ’ 1 v / and writing herein sin2 <£ = — (1 — A), the two expressions de- pend on the integrals ƒ, ƒAc£<£, ƒ : the first two of these are F, E respectively: as regards the third of them, we have d sin cos + k2 sin4 d A A3 or what is the same thing l9 d sin ¿cos 6 A 4 —&'2 A A:'2 A =~A~’ =A-A»· and thence by integration #-l P k- sin cos (f> JfiE '48 MISCELLANEOUS INVESTIGATIONS. [III. The foregoing expressions of thus become dE 1 (F F\ dk’ k^-V· dF_ 1 IV k sin cos dk~mp ’ whence also F= E-k dE dk’ and in particular if = J7r, and E, F now denote the complete functions E1 k, F1 k, then f - \ <*-*>■ fk=w^E-k'^· Let E\ F denote the complementary complete functions d k d E1 k\ F1 k'; then observing that = ~ jp ¿p » we ^ave S=- p dik=-w^E'-^· 75. If we now consider the expression EF' + E'F — FF\ and form its differential coefficient in regard to k, this (sub- dE stituting therein for , &c. their values) is found to be = 0: the expression in question is therefore = a constant; and if to find its value we take k to be indefinitely small, then writing it under the form {E — F)F' + EFt and observing that F' is equal 4 to the indefinitely large quantity log^, but that this is multi- plied by the indefinitely small quantity E — F, — — \nrlc2, andm.] MISCELLANEOUS INVESTIGATIONS. 49 consequently that the product is =0, there remains only the second term E'F, which is = r (viz. for k = 0, we have E1 = 1, F = r); we have therefore EF’ + E'F-FF'^^ tt; or writing this at full length, EJc. FJc' + EJc'. FJc — FJc. r, a relation between the original and complementary complete functions EJc, FJc, ) Later on, instead of these quan- tities we write K, K', E, E\ and the equation is EK' + E'K-KK’ = \ tt.} 76. The equation in question has been proved in a very elegant manner by Dr Glaisher, Messenger of Mathematics, t. IV. (1874), p. 95. Writing for convenience A2 = c, &'2 = c, and ■u = EF' + E'F- FF\ then from the definitions of the functions, (1 — cx2) + (1 — c'y2) — 1 Ji Vi-«2.i-( -dxdy, • c#2.1 — y2.1 — c'i/2 where, and in what follows, the integrals in regard to x, y re- spectively are taken from 0 to 1. Differentiating with regard to c, observing that dc' = — dc, and reducing, we have 2 —= [[ r - + car4 - cV + cVy - ctff dc — — y-)-. (1 — cx’-. 1 — c'y^f where the numerator is = (1 - rf)(l - cV) - (1 - 2/2)(l - ca*); ’ Vl — ¿i^d# hence 2 . f Y* ~ ^ f. C-W* dc J (X—caPv J (\ — —ev- il (1 -cx?f J (l-y-)2(l-cy2) _ f\/l-y2dy f (1 — car1)dx J (1 — c'y2)^ J (1 — x2)2 (1 — cx2)^ where = pq —p q suppose^ f \/l—a?dx P = --------F, ? In — _ i (1 — c^r4) d# j (l-«a)4(l- (l—c#2)* * J (1 — #2)^(1 — c#2) and p\ qf are the like functions with c' in place of c. c. J 450 MISCELLANEOUS INVESTIGATIONS. [III. We have __ i {(1 — x2) + (x2 — cal·)] dx q J (1 - serf (1 - c*2)1 f x2 dx — P P I ~t~ -----~ ·' v 1 — x2.1 — cx2 (— x Vl — x-\ f Vl — x2 dx ■p+r?rr5rJ+J (ir^· ’ where the second term, taken between the limits, vanishes; and we have therefore q =p +p, = 2p. And similarly q = 2p' : hence 2^c=P'2p'~P'2p’ =0; hence u is independent of c, and putting c = 0, we find that its value is = \ir, and the theorem is thus proved. 77. Reverting to the equations *■*-*»· and from these eliminating successively E and F, we find /I , 1-3&2 dF sin cos <£ _ n ( k) dk* + k dk + A3 n -Pi — + — — dE + E-a™AC0^t>-n. (1 + * ctt + A ~0, and in particular if <£ = ^7r, and E, now again denote the complete functions thenIII.] MISCELLANEOUS INVESTIGATIONS. 51 We have consequently a particular solution of each of the differential equations a ■ 1 -sfr<*y V_Q {l fcW+ k dk y~°’ and we can in terms of the foregoing expressions obtain the complete integrals of these equations; for this purpose, trans- forming to a new variable k\ connected with k by the equation k2 + k'- = 1, it is easily found that the transformed equations are (1 7d2y 1 — %k'2 dy ^ -—A·-»-0· «•«ff-fT)»**·1 where the new equation in y is as regards k' of the same form as the original equation in regard to k: hence, Fxk being a particular solution, another particular solution is FJc’; and we have the general solution y = olFJc + a FJc'. And moreover, observing that the equation in E is satisfied by the value E = k it appears that the equation in z must be satisfied by the value 2 = k’2 ^y + k , viz. this is = i'2 {aFJc + dFJc) + k’% (a Jr F,k - d | ~ Fjty : reducing by the formulae dFJc dk 1_ k’°-k (EJ- - k'-FF), dFJc' dk’ 1 k'k* (.EJc'-tfFJc'), this is z = olEJc + d {FJc' — EJc’): where, instead of a, a, we may of course write ¡3, ¡3we have thus y = dFJc + dFJc', z = f3EJc+13' (FJc — EJc'), 4—2MISCELLANEOUS INVESTIGATIONS. 52 [III. as the complete integrals of the differential equations in y, z. And more generally the equations being n /.n d'y 11~dv „ isip 4»cos ft _ o ^ ^ k dk y A3 ’ n /-n, 1~^ I - sineos_ q , (1 fc d/fc+ A then, to obtain the complete solutions, we must to the expres- sion for y add the term F(k, <£), and to that for z the term E (Jc, ). 78. To obtain developments for FJc, EJc when k is nearly = 1, or k' is small, observe that FJc is a solution of (1 — k'2) — + * — y = 0 (1 * }dk2 + k’ dk' J U’ having, when k' is small, the value logp: and conversely, that a solution of the differential equation satisfying the fore- going condition will be the required value of FJc. Such a solution is y = P log ~ + Qf where P — 1 and Q are each a func- tion of the form Bk'2 + Gku +_____ Substituting in the differen- tial equation, we have first (1 — k'2)~— + ( > dk'2 + 1-3 k'2dP V dk' -P=0, and then (1 - k'2) + 1 ^ ^ ( L ' dk'2 * k' dk’ ^ 1 k’ dk' + A10, and the first equation then gives p = 1+|>+^| e* +..., = F(i, h 1, k'2). Represent this for a moment by 1 + mjc'2 + mjc'4, -f &c.,III.] MISCELLANEOUS INVESTIGATIONS. 53 and assume for convenience Q — — rn^AJc2 — m2A2k'A — m3A3k'6 — ... Substituting these values, the equation to be satisfied is found to be k"' ** kf* k's k's s 1 ^ 1 II o - 8m2 — 12m3 — 16m4 - 20 m5 + 4?^ + 8 ra2 + 12m3 4- 16m4 + 2 + 2/?^ + 2w2 + 2 m3 + 2m4 — 2 Ax- L2m2 J-2 — 30m3^l3 — 56m4J.4 - 907?i5ilg ... + 2^!^! + 12to2.42 + 30/713^.3 4- 56m4il4... — 2 mj ¿1- 4m2^L2 - 6m3i3- 8m4.A4 - 10m5^45... + 6^! 4" 12^2^2 -f 18m3.43 + 24ra4.A4 ... 4- Wjij -f- 711‘2 A .> + m3A3 4- ??l4J4 ... viz. this gives 2 — 4m2 — 4m1-d1 = 0, 6 m1 — 8m2 — 16m2J.2 4- 9 m1A1 = o, 10 m2 — 12m3 — 36m3A3 4- 25m2A2 = o, 14ra3 — 16m4 — 64m4 A4 + 49^3^4 3 = o, &c., &c. ; or, observing that 4m4 = = 1, 16m2 = 9ml, 36m3 = = 25m2} &c., we have 2 — 4 mx = 6m1 — 8m2 = 9mj(il2- -A,), 10 ra2 — 12m3 = 25w2 (As ■ -A2), 14wi3 — 16m4 = 49m3 (A4 ■ -At), &c., &c. ; 1 ’ “1 ’ that is h = 1^ 6-2 = 9(A-A),=|, 10-^ = 25(^3-^2),=|, 4Q 7 14-*| = 49 (Aa-A9),=ì9 &c., &c. ;54 MISCELLANEOUS INVESTIGATIONS. [III. finally A' 1.2’ a _ JL JL ^2~1.2+3.4’ 2 2 2 As = 172 + 374 + 576 ’ __2 2 2 2 ^4“rT2+3.4 + 5.6 + 7.8’ &c„ &c., and thence ^=lr’g # l3 12.33 /. 4 2 2 \ + r-.¥k V 0g k' 1.2 3.4/ 13.33.53,,, + 2-. 4-. 6- C + &c., , 4 2 2 2 \ (logA/ 1.2 3.4 5.6/ where the limit of the subtracted series is = log4, or 1’38629... From this we obtain EJe by the formula leading to EJi = k'2FJc -k’( 1 - k'2) FJe: E,k = 1 + ^W-A-r W-iW 1-^5 + 23.43.6 + &c., 1 4 5.6III.] MISCELLANEOUS INVESTIGATIONS. 55 where in the several subtracted series, the numerator of the last fraction is 1, but the other numerators are each 2: the limit of the subtracted series is as in the former case = log 4, or 138629... : hence in the two cases respectively the successive 4 partial series converge to log —log 4, = — log A;'. We have thus the values of FJc, EJc for k nearly = 1, corresponding in a remarkable manner to those previously given for the case of k small. 79. Kummer has given, Crelle, t. xv. (1836) p. 83, the following general formulae in relation to hypergeometric series, 2cF(\a, £ft q2) = F{a, ft J («■+ ft+1), i(l + q)}+F {a, ft *(a+/8 +1), * (1 - 5)); 2cWH« + 1)i|(/3+1),!,<72} = F{a, ft £(<* + £ +1), + ft i(a+£+l), where c, d are constants to be determined: as regards c, writing q = 0, we have at once c = F{a, ft, J (a + + 1), : as regards d, imagining the series on the right-hand side expanded, taking their difference and dividing by q, and then writing q = 0, we find 2d = F fa, ¡3, \ (a + ft + 1), £}, where in general F (a, ft,y, m) denotes ^ F (a, ft, 7, x), writing therein x = m. Taking now a — ft = \\ and q = 1 — 2k'2, whence 4 (1 + ?) = &'3> 4 (! - ?)= we find 2cF(i, i, i h 1, *'*)+*U 4,1, n ={FJJ+FJc)+frr, 2dqF(l f, f, q2)=F{\, 1, **)-.?(*, 1, k% ^k'-FJc)^, in virtue of the expression for FJc, FJc obtained ante, No. 72. Hence, conversely FJc = \tt {ci’ii, i, q2) ~ dqF{l, f, f, q% FJc' = {cFO. i. - —, w2/ Jo Vl—-|-sm and with a little more difficulty \ird = &T + K and we have thus the expressions for FJc, FJc' given by Jacobi, Fund. Nova, pp. 67 and 68. Expansions of sn u, cn u, dn u, in powers of u. Art. Nos. 80, 81. 80. Writing x = sn u we have f dx u = -- J V1 — ¿c2. 1 — fc2x- = jdx |l + (1 + k?) ~ + (3 + 2k- + at*) I + Ac. qp3 rf*> * + (l+^)i- + (3 + ^+3^)^r + &c. O And hence reverting the series we obtain the expression for sn u in terms of u; the first three terms may be calculated in this way without difficulty, but a larger number of terms has been found by other means, and I give the final result as follows*: sn u — u - (! + ¿2) f! + (l + 14fc3 + fc*)|j - (1 +135&2 + 135^ + ¥) ^ * These expressions for sn u, en u, dnu are taken from Hermite’s “Note sur le calcul différentiel et le calcul intégral,” 8vo. Paris, 1862 ; published also in the 6th Edition of the Differential and Integral Calculus of Lacroix.III.] MISCELLANEOUS INVESTIGATIONS. 57 + (1 + 1228*2 + 5478/fc4 + 1228&6 + hf>) - (1 + 11069&2 +165826&4 + 165826/fc6 + 11069P + k + &c., where as usual nl is written to denote the factorial 1.2.3...n. In the particular case k = 1, the last term is = 353792.j1 , = 691.512 . d. 81. The corresponding expressions for cn u, and dn u are cn u = u v- ~2! + (l + 4fr)Jj - (1 + 44&2 + 16&4) g" + (1 + 408/:- + 912/·4 + 644·)** , O: + &c. dn u = n +v(i + k>)l£ - P (16 + 44&2 + fr) ^ +1- (64 + 912/.·- + 408^ + t) , O : + &C. where observe that as far as the fifth order we have sinwVd+&2 , , sn u = -- , ·.·: — , cn a = cos u, dn u = cos tea. Vl + k2 It is to be remarked that the series are not (as are the series for sine u and cosine u) convergent for all values of the variable.58 MISCELLANEOUS INVESTIGATIONS. [III. The Gudermannian. Art. Nos. 82 to 90. 82. It has been already remarked that, for k = 1, the function F (k> ) becomes = log tan + ^<£), and that instead of the general function am u, we have the gudermannian gd u, giving rise to the two functions sin gd u and cos gd % or say sg u and cg^. We have in regard to these a theory correspond- ing to that of the functions of am u (sn u, cn u, dn u), discussed in the following Chapter: and it is convenient to consider in the first instance the special case in question, k = 1. 83. Starting from F = log tan (¿7r + = u, where as a definition = gd ?£, or what is the same thing, u = log tan (^7r + \ gd u); we have = tan (¿7T + i gd u) = 1 + tan £ gd u _ cos £ gd u + sin £ gd u 1 — tan ^ gd u cos \ gd u — sin \ gd u and thence sin gd u or sg u ^ and cos gd u or eg u — 1 + sin gd u cos gd u cos gd u 1 — sin gd u ’ eu — e~u — i sin iu sinh u = tanh u, eu + e~“ ’ — cos iu ’ cosh u 9 H _ 1 1 = sech u: eu 4_ e-u ’ cos {u ’ cosh u9 (where sinh u, = \(eu—e~u)> and coshw, = £(eu + e~ll)> denote the hyperbolic sine and cosine of u; and similarly tanh u and sech u denote the hyperbolic tangent and secant of u). It may be added that eg2 u + sg2 u = 1, and further gd' u = eg u, sg' u — eg2 u, eg' u = — sg u eg w, also sg iu = i tan u, eg iu = sec u.in.] MISCELLANEOUS INVESTIGATIONS. 59 84. The equations may also be written sg u — — i tan iu, sin iu = i tg u, cgu = 1 cos iu3 tg u = - i sin iu, 1 cos iu =---, cgu tan in = i sg u; (tg w denoting tan gd u) which may also be arrived at as follows, viz. considering the angles 0, connected by the equation cos 0 cos 0 = 1, or as it may in various forms be written, sin 0 = i tan , 1 COS , cos 0 = - sin <£ = cos = tan <ƒ) = — i tan 0, 1 cos 0 ’ — i sin 0, then cos 0d0 = i sec2 , that is or ^ cos 0 0 = i log tan (¿7T + \); whence assuming = gd u we have 0 = iu, and thence the fore- going relations. 85. We easily obtain the addition-equations sg (u + v) = Sg U + Sg v, (-T-) eg (u + v) = cgucg V, (-^) where denom. = 1 + sg u sg v; viz. if for a moment eu = a, ev = ft, then a2 - 1 2a ft'2 — 1 2/3 Sgw=ai+^T’ cgM=^in ’ sg®=^TT’ cgi,=^n’ and substituting these values, the expressions for sg (u + u), / v * a2£2 — 1 , 2aft . . , eg (u + v) come out = j ana ^ respectively: which proves the formulae. 86. To deduce the equations from the general formulae for sn (u + v), cn {u + v), dn {u + v), (see next Chapter,) observe that putting k = 1, and consequently sn = sg, cn = dn = eg, these be- come60 MISCELLANEOUS INVESTIGATIONS. [III. Sg (u + v) = SgUCg2V + SgV eg2 U (-T-), eg (u + v) = eg u eg v — eg u sg u cgvsgv (^-), where denom. = 1 — sg2 u sg2 v. Here in sg(w-f v) the numerator is sgi/(l — sg2v)+sg'y(l—sg2?/), which is = (sg u + sg v)(l — sg u sgv), and in cg(w4-v) the nume- rator is = eg u eg v (1 — sg u sg v), and the denominator is = (1 + sg u sg v) (1 — sg u sg v); whence, throwing out the factor (1 — sgw sg v), we have the formulae in question. 87. It is easy to derive the formulae for the sg and eg of the sum of any number of functions. Writing for convenience sgu = x} egu = Vl — = x', sgv = yy egv = Vi — y2, = y, the foregoing formulae may be written sg (u + v) = x + y (-5-)» eg (w+t;)=0y (-0. where denom. = 1 + xy · and then introducing a new angle w, and writing sgiv = z, eg w = z\ we find sg (u + v 4- w) = x 4- y + z + xyz (4-), eg (u + v + w) = x'y z (4-), where denom. = 14-xy + xz + yz\ and so when there is a fourth angle sgw=t, cgco=t', we have sg (u + v + w + <») = x + y + z 4- 14- ®yz + xyt + xzt + yzt (4-), eg (u+v + w + ®) = dyz’i (4-)? where denom. = 1 4- xy 4- xz 4- yz + ^ + xyz^ 5 and so on, the law being obvious. 88. If the angles of all of them — retaining x to denote sg u, and putting for x its value = V(1 + x) (1 — x), we have sg mi = -J- {(1 -f- x)n — (1 x)n) (■=■), eg nu = (1 4- ocfn (1 ““ (4-), denom. = \ {(1 4- x)n 4- (1 “ x)n} 5 whereIII.] MISCELLANEOUS INVESTIGATIONS. 61 and observe that, n being even, the expressions are rational, but n being odd, the numerator of eg nu contains the factor Vl — ¿c2. The formulae are valuable for their own sake; and they afford very convenient verifications of formulae relating to the general functions sn, cn, dn: viz. putting in these k = 1, they must of course reduce themselves to the far more simple formulae for sg, eg. 89. We have as above e?*-l and the right-hand side is expansible in terms of the Bernoulli Numbers viz. we thus have + &c. Thus the term in u11 is this agrees with the result ante No. 80, for we have And moreover 2eu 1 cs -£(e« + c-u)> u62 MISCELLANEOUS INVESTIGATIONS. [m. where the coefficients 1, 5, 61, ... are those of the secant-series, 90. The foregoing values of sgw , cgu, give _ 2e“ + » (<>m ~ !) i (eu — i)3 e2™ + 1 eM + 1 that is ¿«a» _»(**-»). Î ’ onn q rra i 2 + ^^-e-’1) _ 1+sin ui 6 ' eu + e~u 9 cos ui 1 + tan 1 — tan \ui 5 that is e*gd« _ ^an Qt- or what is the same thing, * gd u = log tan (^7r + £ui) ; with which compare the original equation u = log tan (^7t + £ gd u). If in the first of these for u we write 1 gd uf it becomes that is ¿gd (\ gd it) = log tan (Jtt + \ gd u), \l / ¿gd (7 gd «)=«■> a remarkable property of the function gd^; there is no ana- logue to this as regards the general function am u.IV.J 63 CHAPTER IV. ON THE ELLIPTIC FUNCTIONS sn, Cll, dll. 91. We now commence a systematic development of the theory of the elliptic functions properly so called, the functions sn, cn, dn. Addition and Subtraction Formulae. Art. Nos. 92 to 97. 92. The addition formulae are sn (u + v) = sn u cn v dn v + sn v cn u dn u (-^), cn (¿4 + v) = cn u cn v -snwdnwsni/dnv (-h), dn (u + v) = dn u dn v — k2 sn u cn u sn v cn v (-^), where denqm. = 1 — k2 sn2 u sn2 v. And we thence deduce the subtraction formulae, by writing — v for r, and therefore — sn v for sn v but without altering cn v and dnv, viz. these are sn (u — v) = sn u cn v dn v — sn v cn u dn u (-l·), cn (u -v) = cn w cnv + sn u dn u sn v dn v dn (u — v) = dn u dn v + k2 sn u cn n sn v cn v (-^), with same denom. = 1 — L·2 sn2 u sn2 v. As remarked in Chapter I. these are given by the addition equation or they may be at once deduced from cn2 u = l— sn2 u, dn2 u = l — k2 sn2 w, sn' u — cn u dn u, cn' u = — sn u dn w, dn' u = — k2 sn u cn u.64 ON THE ELLIPTIC functions sn, cn, dn. [iy. 93. Writing sly c1} dj_ for the sn, cn and dn of u and s2, c2, d2 for the sn, cn and dn of v, we have / x 4- sxYdY SIl(it + ")= ’ cn (u 4- v) = c1Co — s^s.d., l-Ws*s2 y j / v d\d-> Jc“S\G\S2c-i dn <» + .)-—rrRTsT ■ 94. But I found that each of these expressions is one of a set of four expressions, viz. that we have ■ sYc2d2 4" sX\d\ s-r sn (u 4- v) ^ — k2s^s22 sYc,cL· 4- sx^dj S\C\d2 4~ s^c-dii Sjd]C‘2 4 s2d2c-± cn (u + v) = CiC2 4- s^Ssdo dYd, 4- tes&sx, 3 CjCo “■* S]d]S>dj2 S]Cj0^2SdCodj 1 S\‘S2‘ SjC2d2 s-jC\d\ 1 — s2 — s22 4* kPs^s.? CjdxCzdz — k'%s2 dn («+«)= 4 C1C2 + Sjd-±s2d2 d\d2 k“S\C\S2fi2 SjdjC2 s2d2Cj d\d2 4“ lc2S\C\S2c2 k-Q 2C: A/ dj 02 sYc2d2 s2C\d\ c^c.dc, 4- k'%s2 1 — 1d*s? — k2s22 4- k2S!%2 CiC2 4" dvs2d, dld2 4 kP'S-^C\SX2 with the like formulae for sn (u — v), cn (u — v), dn (u — v). 95. These are mere algebraical transformations of the original formulae considered as depending on the radicals c2 = V1 — Sx2 and Vl — k%2, &c. Thus we have {SiC2d2 4~ s&di) ^SjC2d2““ s2C\di^ —— s 2c2'd·2 s>2c-i,d^y =**. i-^2.1 -tesf.-sf. 1 -*1a.1 -¿v, =S^ — S2 .1 — kPs^Sn, which proves the identity of the first and second of the two ex- pressions for sn (44 4- u2); and so in other cases.IV.] ON THE ELLIPTIC functions sn, cn, dn. 65 96» We may add or subtract the expressions for sn(u + u), sn (u — v), &c. Thus we have sn (w + t>) + sn (u-v) = i sn (u + v) - sn (u - v) = 1 ; but we may also multiply such expressions, and by selecting the suitable forms the final results are obtained at once without any reduction ; thus we have , \ S\C 2d2 "f* S? - S22 whence sn (u + v) sn (u-v)= 1 ^ 97. Although the formulae are so numerous that they can- not be remembered, and in the manner just explained any one of them may be obtained with extreme facility, yet for con- venience of reference I reproduce the whole series of 33 equa- tions given Fund. Nov. pp. 32—34. We have throughout (1) to (9). Denom. = 1 — k%s2. , sn (u + v) + sn (u — v) cn (u + v) + cn (u — v) dn (u + v) + dn (u — v) sn (u + v) — sn (u — v) cn (u — v) — cn (u + v) dn (u — v) — dn (u + v) sn (u + v) sn (u — v) 1 + k2 sn (u + v) sn (u — v) 1 -h sn + v) sn (it — v) c. — 2s\C2d2 (+). = 2CA (+). = 2dxd2 (+), —— 2s2C\d\ <+). = 2s1d1s2d2 (-). ~= 2k^SiC\S2c2 (+). = s?-s2* (+>. = dx2 + k%%2 (-). = c2 + S\d2 (->. 566 ON THE ELLIPTIC functions sn, cn, dn. [IV. (10) to (33). 1 4- cn (it 4- v) cn (it — v) = Cj2 + c22 1 4- dn (it 4- v) dn (it — y) = d,2 + d22 (-). 1 — k2 sn (it 4- y) sn (it — v) = di + Ps,2^2 (-). 1 — sn (it 4- y) sn (it — v) = Cj2 + s22(Z,2 (-)» 1 — cn (u + v) cn (u — v) = «!2ii22 + s./d^ (+)> 1 — dn (it 4- y) dn (it — v) II $. + ·* (-). {1 + sn (u 4- y)} {1 + sn {u — y)} — (^2 i ^1^2)^ (-). {1 ± sn (it + y)} {1 + sn (it — y)] =(cx+s^y (-*■), {1 + k sn (tt 4- y)| {1 + k sn (tt — y)} =(d2 ± ks&y (-),· {1 ± k sn (it + y)] {1 + & sn (tt — y)j = (dx ± ks2ciy (-). {1 + cn (it 4- y)} {1 + cn (u — y)} = (Cx + c2)2 (-), {1 ± cn (it + y)} {1 + cn (it — y)j = («9jC?2 4“ 52t?x)“ (+). {1 ± dn (it + y)j {1 + dn (it — y)j = (d1 ± d2)2 (-). {1 + dn (it 4- y)j {1 + dn (it — y)} = Ic2 ($xC2 4" ^2^1 )2 (-). sn (it + y) cn (it — y) = «?xCxti2 4- 52C2dx (-)> sn (it — y) cn (it 4- v) == SjCjd2 s^c2di (-). sn (it 4- y) dn (it — y) — S\d\C2 4~ s2d2C\ (-). sn (it - y) dn (it 4- v) — SxC?xC2 52C?2Cx (-). cn (it 4- y) dn (it — y) —— C\d]C2d2 ~~~ k 2SjS2 (-). cn (it — y) dn (it 4- v) — CjdiC2d2 4" ^ 2$xS2 (-). sin {am (it 4- v) 4- am (it — y)} —— 25xCxd2 (->, sin [am (it 4- v) — am (it — y)} = %s2c2dx (-). cos {am (it 4- v) 4- am (it — y)} = Cx2 — s2d2 (-). cos {am (it 4- v) — am (it — y)} The Periods 4K, 4dK'. = c22 - s22c£x2 Art. No. 98. (-)· 98. The theory of the periods depends on the equations 1 sn 0 = 0, cn 0 = 1, dn 0 = 1, sn K = 1 , cn K — 0 , dnK=k', sn (K + iK') = ^ cn (K + iKr) = , dn (K + iK') = 0;IV.] ON THE ELLIPTIC functions sn, cn, dn. where K, K' are the complete functions Fk, Fk'. To prove these observe that writing dx 67 u __ p dx Jo Vl — x2.1 — k2x2 9 we have sn u = f, cn u — Vl — |2, dn m = V1 — whence writing J=0 we have the first triad of formulae, and writing f=l the second triad. For the third triad, writing £ = 1 jk, we have dx .-rT______________ Jo Vl — x2. 1 — k2x2 1 _ / f1 f*\ dx \io J i / V1 — x2.1 — k?x2 = K+ [* ■■ dx. J i vl — ¿c2 . 1 — k2a? and to transform the integral we write (x=ly z=0; x=l/k> z= 1), 1 wrhence # = dx = Vl - fcV’ k'2zdz (1 — k'2z2)^ ’ i Vl — A/2^2 Vl — x2 1 Vl - Vi - A^2 F Vi - J or multiplying dx idz Vl — ¿a?2 - 1 — k^x1 1 — z2. 1 — k2z so that the integral is . p_________ lJoVl -2*. 1 ■ k'2z2 5—268 ON THE ELLIPTIC functions sn, cn, dn. [IV. and the value of u is = K + %K'. Hence writing u = K + iK', £= 1/k, and observing that the value of Vl — f2 is viz. that it is = — ik'/k, we have the required formulas "I __ n JrS sn(K + iK') = jc, cn(ÆT+iZ') = -^, dn (K + iK') = 0. Property arising from the transformation. Art. No. 99. 99. In the foregoing relation between x and zy write for a moment x = sin , z = sin % the differential equation is d __ id% Vl — Ic2 sin2 cp Vl — k'2 sin2^ ’ whence, assuming sin <\> = sn (v, k), sin % = sn (w, &'), this is dv = idu> or we have ii = m+ const. But we have simultane- ously x = l, z — 0; and for x = l, vis=K, and for z — 0, u is = 0: hence the constant is = K, or we have v = iu + K: con- sequently x = sn (m + K, k), z = sn (u, &'). Substituting in the integral equations between xf z, we have sn (iu + K, k) = 1 dn (w, &') ’ en (iu -f K, k) — — sn (w, k') dn (u, k') 9 dn {%u + K, k) — dn^u ’kf which are equivalent to the equations obtained in the next article. Jacobis imaginary transformation. Art. No. 100. 100. Write sin = i tan yfr, whence also cos = sec yfr, and sin ^ = — i tan cf>; consequently d(j> = idyfr sec y}r, and dd> idyjrIV.] ON THE ELLIPTIC functions sn, cn, dn. Hence, putting sin = sn (v> k), sn yfr = sn (u, &'), we have dv = idw, or since v, u vanish together v = in; that is sin <£ = sn (iw, A;), sin yfr = sn (^, ¿'). The integral relations between , ^ give isn(w, k*) sn (in, k)= ■/—, 7 cn (w, k) cn (in, k) = dn(m, &) = 1____ cn (u, k') 9 dn (u, k') cn (u, k') ’ It may be observed that in this transformation writing = in we have = gd u. It is to be further observed that writing sin yjr = y, and as before sin = w} we have r_ ty_________L_ Vl -y* Vl-AV’ that is 1 ly = k'z, which exhibits the relation between this and the transformation in the preceding article. Functions of u + (0, 1, 2, 3)K + (0,1, 2. S)iK'. Art. Nos. 101 to 103. 101. It is easy from the foregoing values of the sn, cn, dn of K and K + iK' to obtain the values given in the following table: for instance we have sn(u + K) = snKcnudnu + l—k2sn2Ksn2 u, = cn u dn u -s- dn2 u} = cn u -r- dn u; sn(u — K) = -cnw-rdnw, &c. Similarly finding sn (u + K + iK'), and in the resulting formulae substituting u — K for u and reducing, we have sn + ¿A'): and so in the other cases.70 ON THE ELLIPTIC functions sn, cn, dn. [IY. + 0A' + iK' + 2 iK' + 3 iK' Functions of u + (0, 1, 2, 3)iT + (0, 1, 2, 3)iK'. + 0£· + K + 2 K + 3 K sn w cn« (-H) - snw - cnw (-r) cn w - ft' sn u (-i-) - cnw ft'snw {—) dn w V {-) dn w ft' <-) denom. = dnw denom. = dnw 1 <-) dnw -1 (H-) - dnw(-f) - i dn u (-f-) -iW (-=-) i dnw (-r-) ¿ft' {—) -¿ft cnw (~r~) ¿ftft'sn u (-5-) -ikcnu (-i-) ¿ftft'sn w (-i-) denom. = ftsnw denom. = ft cn u denom. = ft sn w denom. = ft cn u sn w cnw (-~) - sn u - cnw (-f-) - cn w ft'snii (-i-) cnw -ft'snw (-?-) -dn w -k' (H-) - dn u -k' (+) denom. = dn u denom. = dn w 1 (*> dnw (-^-) -1 (-5-) - dnw (~) i dnw (+) ik' (-=-) -i dnw(-i-) -ik' (+) ikcuu (+) -¿ftft' snw (-i-) ikcnu (-J-) -¿ftft'sn w (H-) denom. = ft sn u denom. = ft cnw denom. = ft snw denom. = ft cn w where the arrangement hardly requires explanation: the table shows for instance that sn (u + iK') = 1 cn (u + iK’) = — i dn u (■*■>. dn (u + %Kf) = — ik cn u (-X where denom. = k sn u; it sometimes, as here for dn (u+iK'), happens that there is in the numerator and denominator a common factor k, this is of course to be omitted. 102. The table, writing therein u — 0, gives the values of the functions of mK 4- miK'. In particular, where there is a denominator k sn u, the functions become infinite: it is necessary to attend to the ratios of these infinite values, and the convenient course is to write 1/k sn u = I, where I is regarded as a definite infinite value. The table thus givesIV.] ON THE ELLIPTIC functions sn, cn, dn. 71 sn iK' = I, cn iK' = — ¿7, dn iK' = — iA;7, sn 3iAT' = 7, cn 3iK' = il, dn 3i7T = i&7, sn (2AT + iK') = - 7, cn (2if + ¿Z0= dn (27l 4- iK') = -ikI) sn (27T + 3^7T) = — 7, cn (2K + 3iK') = ~ il, dn (2 + 3i7T) = ¿*7. We may from these reproduce the original formulae which involve u; thus sn u (— IcI2) 1 cnudnu sn (u + iK') = - 1 — k212 sn2w _ — A72 sn u _ 1 — k2I2 sn2 u 9 k sn u 9 and so in the other cases. 103. The table shows that the functions have 2K, 2K' as half-periods : we in fact deduce sn (u + 2 mK + 2m'iK ) = (—)m sn u, cn ( „ „ ) = (-)»+«' cn w, dn( „ „ ) = (-)"*' dn w; whence taking m, rrt each even it appears that 4Kt 4K' are whole periods; viz. that increasing the argument by 4emK -f 4 rn'iK'y the functions are severally unaltered. Observe however that sn {u -f 2iK') = sn u. Duplication. Art. No. 104. 104. Writing v = u, we deduce the functions of 2u, or say the duplication-formulae. We have sn 2u = 2 sn u cn u dn u (-5-), cn 2w = cn2 u — sn2 w dn2 u = 1 — 2 sn2 u + k2 sn4 u (h-), dn 2u = dn2w — k? sn2 u cn2 u = 1 — 2A2 sn2 -w + k2 sn4 u (-r), where denom. = 1 — k2 sn4 u ;72 ON THE ELLIPTIC functions sn, cn, dn. [IV. or if for convenience we write sn u = x, cnii = \/l — x2, dn u = V1 — k2x2; then the formulae are sn 2u = 2% Vl — a? Vl — k2x2 (-^)> cn 2u = 1 — 2x2 + k2a? (-r), dn 2u = 1 — 2k2x2 + k2a? (-l·), where denom. = 1 — k2a It may be added that 1 — cn 2u = 2^ (1 — k2x2) = 2 sn2dn2 u (~), 1 + cn 2u= 2 (1 — x2) =2 cn2 u (-l·), 1 — dn 2u = 2k2x2 (1 — x2) = 2&2 sn2 u cn2 w (-h), 1 + dn 2u = 2(1 — &2#2) = 2 dn2 ^ (-r), the denominator being as above 1 — k2x$t or 1 — k2 sn4 u. And we thence deduce sn2 u = 1 — cn 2u (V), cn2 u = dn 2u 4- cn 2u (-4-), dn2 u = k'2 + dn 2w. + &2 cn 2u (-l·), where denom. = 1 + dn 2u. Dimidiation. Art. Nos. 105 to 110. 105. In the expressions for the functions of 2u, writing instead of % we have the functions of u expressed in terms of those of and from these equations can obtain the ex-IV.] ON THE ELLIPTIC functions sn, cn, dn. 73 pressions of the functions of \u in terms of those of u. Thus, writing for a moment x = sn \u, we have sn u = 2x Vl — a? Vl — } cn2 \u — Jc'2 (1 -f cn u) (-0. dn2\u = k'2 (1 + dn u) (-), where denom. = k’2 + dn v — k2 cn u. 106. But, ante No. 104, it appears that we expressions have also the sn2 = 1 — cn u (-)> cn2 \u — dn u 4- cn u O). dn 2^u = k'2 + dn u + k2 cn u 0-). where denom. = 1 + dn u. In passing to the expressions of sn^, cn-^, dn^M, the radicals must of course be taken with the proper sign. We deduce the following special formulae:74 ON THE ELLIPTIC functions sn, cn, dn. [IV. sn = $n = dn = \K 1 Vi+AT' ./ft' \/l + A/ sjk' %K 1 isjk' sjk' Jl + k’ ~ Jl + k' ±K + iK' 1 Vl -ft' i Jk' ~ -i«/fc7 IK+ iK' 1 Jl-k’ t Vft' \iK· i Ijk \/l + & ~^jk~ Vl + ft K+&K' 1 i Jl-k Vl-ft sjk vs ||JT i «yi+ft s/ft — \/l + ft K+fiK 1 ijl-ft - Vi^ft v* sjk iJr+JiJT ^ i /“ — / -» 1 -¿7* ^'{s/iTF-iVi^ft'i ^/2 W1 + & + »V1- V2 v'ft iJST+ifJT 1 N/2Vfc'"v 1 + fc-Wi-i:} 1 + i j^/ ft' Vft ^{VTTft'+tVi-ft'i V2 pr+fzlT 1 1 + i «/ft' *J% ijk -^|Wiift'+»vr-ft1 pr+fzJT s/2 ^Wl + fc + tV/l-*} l-ijk' si^ \Jk -^Vrrft'-iVi~ft'} where for the last set of formulae we may substitute: sn2= cn2 = dn2 = \K+\iK' \ (&+**0 ikf ~lk /s' (kf-ik) \K+\iK' ik' ~k A;' (Aj' + zA) iK+%iK' | (fc-i&O ikr k A/ (A/ + iA;) %K+%iK' | (*+»*') ikf k A;'(A:'-? A;) iIV.] ON THE elliptic functions sn, cn, dn. 75 107. We find sn (u 4- 1 Vi + fc' k' sn u 4- cn u dn u 1 — (1 - k') sn2 u * 1 dn u 4- (1 4- k') sn Mcnt^ vr+*' cn ii4-snw dnw ’ sn (u 4- \ ) '1^ Ml> II (1 4- k) sn u 4- i cn u dn u 1 4- k sn2 u II a- j(1 4- k) sn u 4- i cn u dn u V (1 4- k) sn u — icnudnu3 sn (^4* \K 4- %iK') Ik + ik’ — ik' sn u 4- cn u dn u ~ V k 1 — k (k 4- ik') sn2 u 3 _ Jk + iK V k cn u 4- (k — ik') sn u dn u dn 4- k sn wcnw 5 where the first expressions are those given at once by substi- tution in the general formula for sn (u + v). 108. To identify the two expressions of sn (u 4- |A), writing for convenience sn u = x, observe that in the first expression the denominator is 1 — (1 — k')x2, and multiplying this by 1 4- (1 — k') x2, the product is 1 — 2x2 + k2xl·. And in the second expression the denominator is V1 — x2 4- x Vl — k2x2, which multiplied by Vl — x2 — x Vl — k2x2 gives 1—x2 — x2 (1 — k2x2), = same value, 1 — 2x2 4- k2x4: reducing in this manner the two expressions to a common denominator, the numerators would be found to be equal. Similarly as regards the two expressions of sn (u + \K 4- %iK'), we have {1 — A; (& + ik') x2} {1 — k (k — ik') x2} = 1 — 2k2x2 4- &V, and {V1 — k2x2 4- kx Vl — x2] {Vl — k2x2 — kx Vl — x2) = 1 — k2x2 — k2x2 (1 — x2), = same value.[IV. 76 on the elliptic functions sn, cn, dn. As regards the two values of sn (u 4- ^iK'), we have {(1 4- k) x 4- i Vl — x2 Vl — k2x2) {(1 4- &) # — i Vl — ¿r2 Vl — te2} = (1 + ¿)2 a2 + (1 - #2) (1 - k2x% = (1 4- kx2)\ and the identity is at once established. 109. We deduce without difficulty from the second formulae: _ 1 dnu + ^ + ^snwcnii sn2 (u 4- \K) sn2 (u 4- £iK') 1 4- Jc' dn u 4- (1 — k') sn u cn u 9 1 (1 4- k) sn u 4- i cn u dn u · “ k (1 4- k) sn u — i cn u dn u 9 k + ik' cnii + (i- ik') sn u dn u cn u 4- {k 4- ik') sn u dn u 5 to these may be joined the formulae obtained by considering u + | Ky &c. as the halves of 2u + K, &c., see No. 106, viz. we thus have sn2 (u 4- %K) sn2 (u 4- \iK') dn 2u 4- k' sn 2u k' + dn 2u 9 1 k sn 2u 4- i dn 2u Tc sn 2u — i cn 2u 9 sn2(u + iK + &K') = j 1 k cn 2 u + iti k cn 2u + ik' sn 2u ’ 110. Observe that in the first expression the denominator multiplied by 1 — k2a4 is k' (1 — k2af) 4-1 — 2k2x2 4- k2xA, = 1 4- V — 2k2a9 4- (1 — k') k2xt, = (1 4- k') {1 — (1 -k')x2}2. In the second expression, multiplying the numerator and denominator by sn 2u 4- i cn 2u, the expression becomes an integral function (sn 2u, cn 2u, dn 2uf; having therefore a de- nominator (1 — fc4)2, = (1 4- kx2)2 (1 — kx2)1.IV.] ON THE ELLIPTIC functions sn, cn, dn. 77 In the third expression the denominator multiplied by 1 — k2af is 1 — 2a?2 + k2o& + 2ilex Vl — a?2 Vl — k2x2, = [ik'x + \f l — x2 Vl — k2x2}2, = (ik' sn u + cn u dn u)2; by aid of these remarks the identifications can be easily effected. Triplication. Art. No. 111. 111. Writing v=2u, and using the duplication-formulae, we obtain the functions of Sn, These are easily found to be sn 3u = Sx — (4 + 4&2) a?3 + 6&2a?5 — Me9 (^), cn Su — (1 — 4a?2 + 6k2af — 4A4#6 + k*x?) *Jl — x2 (-i-), dn Su = (1 — 4k2a? + 6Z?2a:4 — 4&2a?6 -f kta?) Vl — ¿2a?2 (-h), where denom. = 1 — 6^'^ + (4&2 + W) a?6 — S/^a?8. And we may add 1 — sn Su = (1 + a?) {1 — 2a? 4- 2&2a?3 — fe4}2 (-h), 1 + sn 3w = (1 — a?) {1 + 2a? — 2^2a?3 — A^a?4}2 (-i-), 1 — k sn Su = (1 + kx) {1 — 2kx -+■ 2&a?3 — k2xf\2 1 + k sn Su = (1 — kx) {1 + 2kx — 2ka? — irva?4}2 (-r), the denominator as above. The duplication and triplication formulae possess various properties which are in fact particular cases of those for the multiplication by any even or odd integer n: and it will be convenient to defer the consideration of them until other in- stances of the formulae are obtained.78 ON THE ELLIPTIC FUNCTIONS sn, cn, dll. [IV. Multiplication. Art. Nos. 112 to 120. 112. It has been seen how the functions of 2u and 3u are obtained: to consider the general question of determining the functions of nu, suppose n =p 4 q, and imagine that the functions of puy qu are known. We may write sn pu = Ap (-)> sn qu = Aq cn pn = Bp (->, cn qu = Bq dn^M = Gp dn qu = Gq where denom. = Dp, denom. = Dq. The addition-formulae give sn (p + q)u = ApDpBqGq 4 BpGpAqDq (-p), cn (p 4 q) u — BpDpBqDq ApGpAqGq (-p), dn (p + q)u = CpDpGqDq - k2 ApBpAqBq (-p), where denom. = Dp2Dq2 — k2Ap2Aq2; and the functions on the right-hand side are consequently pro- portional to Ap+qy Bp+qy Gp+qy Dp+q respectively. We have Ax=x, Bx = Vl — a?y Cj = V1 — k2x2y Dl = 1; and hence writing p = q =1, we find four valued which have no common divisor, and which may therefore be taken for the values of A2, B2, G2, D2 re- spectively: viz. we thus obtain A2 = 2# Vi — #2 Vi — k2x2, B2= 1-2x2 4 k2afy C2 = 1 — 2 k2x2 4 k2x*y k2a*y the foregoing duplication-formulae. And similarly, writing p = 2, q = 1, we obtain the triplication-formulae. But at the next step, if we write p =q = 2 we obtain four values, and if we write p = 3, q = 1 we obtain four other values of higherIV.] ON THE ELLIPTIC FUNCTIONS Sn, cn, dn. 79 degrees; these are of course proportional to the former ones, and they contain a common factor, throwing which out they would coincide with them. And so in general, for a given value of p + q the degrees are lowest when p, q are as nearly as possible equal: that is p + q even, when p = q> and p + q odd, when p~ q = 1: or what is the same thing, the proper partitionments are 4 = 2 + 2, 5 = 3 + 2, 6 = 3+3, &c. Taking the functions thus obtained for the values of Ap+qy Bp+qy Cp+qy Dp+q, we may write p + q odd ; p ~ q = 1. •Ap+q= ApDpBqGq + BpCpAqDqy Bp+q — BpT)pBqT)q ApGpAqGqy Cp+q = CPDP CqDq k“ApBpAqBqy Dp+q = A2A2 - ¥Ap2A p + q even \ p — q. A-2p CM II OpDp, B-ip = BP2DP2 - AP*Cp2, Gw = cyzy - k2Ap2Bp\ Ap II $ - tfAp*. 113. The calculations for the cases 4 and 5 may be per- formed without difficulty: but for 6 and 7 they become very laborious: the results have however been calculated by Baehr, Grunert’s Archiv xxxvi. (1861), pp. 125—176, and for con- venience of reference I reproduce them here, partially verifying them as afterwards mentioned. The whole series of formulae for the cases n = 2, 3, 4, 5, 6, 7 are as follows:80 ON THE ELLIPTIC functions sn, cn, dn. [IV sn2 u= Xsjl -x2jjl- k2x2 into cn2 u = dn 2 u — denom. = 1 - cn 2u = l-k2x2 into 1 + cn 2u — 1 — x2 into 1 - dn 2u = 1 -X2 into 1 + dn 2u = 1 -k2x2 into 2 1 1 1 2 2 - 2x2 - 2 k2x2 2x2 2k2X2 + k2x4 + k2x4 -k2x4 (+) (-*-> (+) (-h (+) (-H sn 3 u — cn 3 u= dn 3m= denom. = 1 - sn 3m = 1 - k sn 3 u = X J\-x- Vl - kV il + x) (1 + kx) into into into into sq. of into sq. of 3 1 1 1 1 1 - (4 + 47c2) X2 — 4æ2 -M2x2 0 - 2x - 2 kx + 6 k2x4 + 6k2x4 + 6 k2x4 - 6 k2x4 0 0 0 - 4 text - ák2x6 + (ík2+àk4)x6 + 2k*a? + 2 kx3 -k4x8 + k4x8 + ft4x8 - 3 k4x8 - k2x4 - k2x4 <+> (-) (+> (+) sn 4m = xsjl-x2,jl-k2x2 into cn4w= I : 1 dn áu — ! denom. = 4 : 1 1 1 - (8 + 87c2) x2 : -8 a;2 - 8k2 x2 0 + 20 it2 x4 i + (8 + 207c2) cc4 + (20 k2+ 8k4) x4 -20 ft2 X4 0 - (247c2 + 327c4)a^ ~(S2k2 + 2àk4)x6 ' + (32ft2 + 32&4) X6 - 20 k4 x« + (547c4 + 167c6)z8 + (16λ;2 + 54Λ;4)*8 — (16&2+58&4+16fc6) x8 + (8fc4 + 8fc6)z10 - (24£* + 32&6) a;10 - (327c4 + 247c6) æ10 + (327c4 + 327c6) a:10 - 47c6 x12 + ( 8kt + 207c6) x12 + (20 A:6 + 8Æ8) x12 - 207b6 X12 - 8k6 x14 - 8k8 x14 0 + k8 λ;16 + k8 x16 + k8 .τ16IV.] ON THE ELLIPTIC functions sn, cn, dn. 81 sn 5u=x into cn5u— fjl-x2 into dn 5u = yv/1 - k2x2 into x° 5 1 1 X2 - (20 + 20k2) - 12 - 12ft2 x4 +16 + 94ft2 + 16ft4 + 16 + 50k2 + 50ft2 + 16ft4 xs -( 80ft2 + 80ft4) - 80ft2-140ft4 -140ft2-80ft4 X8 -105 k4 + 335ft4 + 160ft6 + 160ft2+335ft4 X10 + 360ft4+360ft6 - 264ft4 - 464ft6 - 64ft8 - 64ft2 -464ft4 -264ft6 X12 - (240ft4 + 780ft6 + 240 ft8) + 208ft4 + 508ft6 + 208ft8 + 208ft4 + 508ft6 + 208ft8 X14 + 64 k4 + 560ft6 + 500 k8 + 64ft10 - 64ft4-464ft6-264ft8 -264ft6-464ft8-64ft10 x16 - (160ft6 + 445ft8 +160ft10) + 160ft6 + 335ft8 + 335ft8 + 160ft10 #18 + 140ft8 + 140ft10 -140ft8 -80ft10 - 80ft8-140ft16 a·20 - 50ft10 + 50ft16 + 16ft12 + 16ft8+50 ft10 X22 0 - 12ft12 - 12ft10 X24 + k12 + ft12 + ft12 (+) (+) (■*·) 1 - ft sn 5u = (1 - x) into 1 - ft sn 5u = (1 - kx) into square of square of Denom. = #° 1 1 1 xl - 2 - 2ft 0 X2 - 4 - 4ft2 - 50ft2 x:i + 10ft2 + 10ft + 140ft2 + 140ft4 X4 + 5ft2 + 5ft2 - (160ft2 + 445ft4 + 160ft6) X5 -12ft2- 8ft4 - 8ft -12ft3 + 64ft2 + 560ft4 + 560ft6 + 64ft8 X6 + 4ft2- 4ft4 : - 4ft2 + 4ft4 - (240ft4 + 780ft6 + 240ft8) X7 + 8ft2+12ft4 i + 12ft3 + 8ft5 + 360ft6 + 360ft8 X8 - 5ft4 - 5ft4 - 105ft8 ! ^r»9 - 10ft4 -10ft5 - (80ft8 + 80ft10) ! a?10 + 4ft6 + 4ft4 + 16ft8 + 94ft10 + 16ft12 a*11 + 2ft6 + 2ft5 - (20ft10 + 20ft12) .r12 - ft6 - ft6 + 5ft12 (+> (-5-) In the Tables which follow, some obvious abbreviations are made use of. Thus we must read in the table for sn 6u \ 6 + (- 32 - 32&2) #2 + (32 + 208£2 + 32^) ¿c4 - &c., and in that for sn 7u, 7 + (— 56 — 56k2) x2 + (112 + 532&2 + 112&4) x4, — &c., the numerical coefficients in this last case being printed to the middle term only: — 56 for (— 56 — 56), and + 112 (+532) for (+112+ 532 + 112), the expressions being symmetrical as here shown. The numerical coefficients of denom. 7u are in a reverse order the same as for sn7ut and those of dn7ti the same as for cn 7a, but in a reverse order, as is sufficiently indi- cated in the tables. c. 682 ON THE ELLIPTIC functions sn, cn, dn. [IV. 43 ta - ce 10 rH 00 co o Q0 co + © τΗ CN io o i-H rH + 1 + 1 05 05 05 O ^ ^ 00 ~ H H H H S «cí^pceoooooooorH rü rii -ϋ 00 O Ο (Μ CO CO ^ O 00 O ri O oo T* cq -Φ o 00 l> 00 T(l CO © co io + © h* © IN 00 il CO CO CN © co + I co co il CO io H H -il il o + 1 + 1+7 00 IO CO il co co co ri o il CO CO IO CO H il 00 N «Í I + I + I + I + I co CO CO 00 H io CO il Ü 05 CO + I + I + h* © h* © © t> (M » CO IN 10 (N io IÍ5 CO © © © © h* ¿é rió rig rii »·'»·' o" Hi H· «O 00 i-t ώί ώί ^ -Já ^ H>** Tji*“ 5, ^ ^ ^ rC¿ -Ci >ÍÍ lií ^ co co cm o cq co co IO CO CO CO CO CO IO CO N co io io OH io io co Η* o © CO © © Htl h N cq cq h + I + I + I + © ao oo oo ao oo © cq o i< cq ii o cq CD i 00 t> 00 i CO © IO CO © © IO © + I co © H* H* + 1 + I H* (M © ri + I H* © + I + I + © Hi co © © H* co H* 1 + 1 © © © ri IO co ríl © co H* ri 1 + 1 H* © cq CO H* © co cq co 1 + 1 + I + 6—2 denom. ut suprà. (—) denom. ut suprà.sn7w = Tinto Denom. 84 ON THE ELLIPTIC functions sn, cn, dn. [IV. rü OD OO-S I-H rü ü 'C co co*· < rió ü rü ü ü ü rü ^ ^ ^ r. e» « r—H tH tH I + ü •s O *. ÍO 3 3 7k Se I Si » S ® § rü ^ rü -ü rü -ü rü rü rü rü rü r- r, rt ürüüüürürürürü”^^ * ' r-r--^r-r-r--·' 0(MO HC^HTfl-HiOCiWOOOOCSlOCC ^^H^CO©CO©OHCO«(M CQ CO C0 — J J + (M ^ (M © O N W ^ ÍD co O 05 t>“ TH + I + I + (M O ^ I + I + I + ΟΟ^’ΨΦΝΟ^ΝΟΟ (Maooo5t»o©>o^© 05«Kîo©©œ©ooo5 O00H^0DO5t(IH5O00 O Tí cq N 00 © H M ^ CO O G0 (N © r- w. ÍÍ5 05 CO rl H CO ^ CO H co co o τη I + I+ +I + I + I + 1 + I + I + I + I + 3 5 9 *ï H08 + 961 + sï‘I 96ΐ +08 IV.] ON THE ELLIPTIC functions sn, cn, dn, 85 © I- Sí "S rió rió rió rii rió 2 8 % riè rió rii ~ -r 00 rió ^ ^ ^ CÖ to tc CO** to*' 00~ ' Γ^ί rió ^ S^2i2i2í2sSiSe'¿i4s¿íSr¿ oí οι*' cq*' oíΛ oí*' i·** «o** oo~ oo~ oo*' oo*' oo~ ^ ^ ^ _ .. 2 § ?! S S r¡ó rió rió rió rió rió rió ώί-^^ώί-ίέ^-ίί^ k O m ^ h* eo oo ao en 0 00 ^ ■> II s CD CD O· rH I + H< © ^fl CD 00 <© JO » 00 00 © S^^^íSiSiSiSiSí rü rió rió rió rió rió rü rÍÓ rü rió rió rió rió rü & 3e ST ST ST ST sT ST ST Si ^ ^ ^ ^ ^ ^ ® i n^r.r.r.^r.or.0 Π ~ M “ « ^ (N CD Cí O CO Ci W _ .. o ao oo io CO GO 00 I + I 00 CD CÍ LO 00 LO O Oí CD 00 CD CO + t CD Oí LO LO CO 1-1 CO 05 © © Oí o © -Ψ H LO H< 05 © co H CO -Ψ + 1 + 1 CD 00 CD O CO ^ ri 00 O lo OÍ H* Oí Oí D IO (M CO Η Oí Η· LO LO 05 rH Oí Oí ι-l “ + 1 + 1 CO 00 Η o ri Ή· CO Oí © Ή GO ri Ή © Ή O - - Ο* CO CO Ci CD Η © cd co η + © 00 Ci HI rH 00 t- © c- Oí © © © I + I + I + I + I + I + I ©oi©Hi©aocc©©aoHiHi© ©©©©©©©©ι>·αο©αοαο LOLOOOGOCO©©rlHCiLOtH Λ Λ1 ί—Ν ri ι»Λι —1 (—1 ΓΤλ rCi net <*o Oí CD I—I WM I + I + I 1 I H Oí 00 O O 00 © © I> GO Oí Ci © © © Ci rl © Η Ή © © © © © rl Oí HI © Π © Oi © © ... ^ Ci 00 © © 00 00 Π © © GO 00 © + I + I + I © Oí © HI OÍ Ή © O» © © Hl © © © GO Ci t— H· 00 © HI © rl Oí 00 © - I + I ... -.· oo ^ Ci 00 Oi © © © © © rl rl © HI © rl + I + © © HI rl I + ! + I + I + I + Oi + + I + I + ~) denom ut suprà. (-^) denom. ut suprà.86 ON THE ELLIPTIC functions sn, cn, dn. [iv. 1 - sn 7u= (1 + a?) into square of 1 - 1 sn 7u=(1 + kx) into square of 1 1 1 1 X1 - 4 1 - 4 1 X2 - 4 1 - 4 l2 iC3 + 8+ 28 1, 12 + 28+ 8 1, l3 X4 - 14 12 - 14 l2 X5 - 84- 56 l2, k4 - 56- 84 1, l3 Xs + 112+ 28 l2, k4 + 28 + 112 l2, l4 X7 + 64 + 204+ 82 12, k4, k8 f 32 + 204+ 64 1, l3, l5 X8 -144-305- 16 l2, l4, k6 - 16-305-144 l2, l4, l6 X9 - 82-200-128 l2, k4, l6 -128-200- 32 l3, l5, w X10 + 64 + 456 + 368 l2, k\ k* + 368 + 456+ 64 l4, l6, 18 X11 + 112+ 56 k\ 1« + 56 + 112 15, l7 X12 -224-644-224 l4, 1«, k8 -224-644-224 l4, l6, 18 #13 + 56 + 112 fee, k8 + 112+ 56 l5, l7 X14 + 368 + 456+ 64 16, 18, l10 + 64 + 456 + 368 l4, 16, 18 X15 -128-200- 32 16, is, iio - 32-200-128 l5, l7, 19 X16 - 16-305-144 16, is, lio -144-305- 16 l6, 18, lio a;17 + 32 + 204+ 64 I®, 1», lio + 64 + 204+ 32 l7, 19, l11 a;18 + 28 + 112 18, 110 + 112+ 28 l8, 1™ X19 - 56- 84 18, lio - 84- 56 l9, l11 a;30 1 - 14 110 - 14 l10 X21 + 28 + 8 110, 112 + 8+ 28 l9, 1U X22 - 4 l12 - 4 1!0 X23 - 4 1*12 - 4 111 X24 + 1 l12 + 1 112 (-r) denom. ut supra. (-r-) denom. ut supra. The formulae for 84 were obtained by E. H. Glaisher, Proc. R. Soc. t. xxxiii. (1882), pp. 480—489, viz. he gives the nu- merators of sn u, cn u, dn u, and the common denominator: also as subsidiary results used in the calculation, the numerators and denominators of sn2 u and sn4 u. 113. It will be observed that the forms are essentially different according as n is odd or even. When n is odd, the numerators and denominators, say A(x)y B (x), G (oo) and D (#), are of the forms ®(1, Vl — ai2, (1, «2)i("s'1) V1-&W, (1, viz., the degrees are n2, n2, n2, n2 — 1.IV.] ON THE ELLIPTIC FUNCTIONS SU, Cn, dn. 87 But, n even, the forms are *(i, a?)i(n*_4) Vl - h?x\ (1, (1, s?)in\ (i, viz., the degrees are n2 — 1, n2, n2, w2. The rational functions (1, ¿r2) presenting themselves in the foregoing forms may be called A'(x), JS' (#), (?'(#), JD' (x): the degrees in x2 are ¿(w2—1), £(n2— 1), £(ti2—1), ^(w2—1) or n2 — 4), ^w2, Jw2, ^w2 according as n is odd or even. 114. Whether n is odd or even, if we change k into 1/k and x into kxy the functions A'f D' each remain unaltered, while the functions B\ G' are interchanged : thus n = 2, A' becomes = 2 & G' D' n — 3, A* &c. ; = 1 2 k2X2 + yz frx4, k2 = 1 ~Ffe2 + F'fc4*4’ = 1 -P4*· = 3 " (4 + p) + I·**®* - ^ ^ and the same is the case with the functions A, B, C} D, except that A is changed into kA. 115. But there is another change, x into 1/kx, the effect of which is different according as n is odd or even. If n be odd, then disregarding a monomial factor Itf-aP, the change x into 1/kx interchanges A', Dr and also interchanges B\ <7: thus88 ON THE ELLIPTIC FUNCTIONS Sn, cn, dn. [IV. n=S, A' becomes 3 - (4 + 4>te) + 6fr-¿j - *· -L , .B' = — (l — fik?a? + (4i2 + ite) te — '¿tete^j ; 1 ~ Wte + Gk*tete “ +kl tea*’ If teaf C' D' ^1 — 4tea? + 6tete — 4>tete + teaP’j ; = ^1 — 4#2 + 6k2x4 — Wa? + kta^j ; = — ^3 — (4 + 4&2) x2 -f 6A;2#4 — . If passing to the functions A, D we write down the general formula, this is D (x) = (-)i(”-1) fci(B2+1) ^ (¿) , implying ¿> (i)= {~)Hn~l) k'iint~v *"*A{x)’ and we thence deduce thati» A ^ j + U I'ij - 1 + it .4 i.t) + II (), 1 + cn 2pu = 2 cn 2pu (-t-), 1 — dn 2pu = 2k2 sn2 pu cn2jpu (-i-), 1 + dn 2pu = 2 dn2^w (-h), denom. = 1 —k2 sn4pu, and substituting for the functions of pu their values we have 1 — cn 2pu = 2AP2GP2 (-h), 1 + cn 2pu = 2BP2DP2 (+), 1 — dn 2pu = 2k2Ap2Bp2 (-=-), 1 + dn 2pw = 2G2B2 (-=-), where denom. = Dw, as for the other 2p^-functions. 119. We may in the multiplication-formulae write k = 0, viz. we then have a? = sinw, and snww, cn nu, dn nu = sin nu, cos nu, 1 respectively: this however affords a verification only of the terms not multiplied by any power of k. A more complete verification is obtained by writing k = 1, we then have x — sg u, V1 — x2 and Vl — k2x2 each = eg u; and sn nu, ennu, &nnu = sgnu, cgnu, eg nu respectively. Recalling the formulae sg nu = \ {(1 + x)n — (1 — x)n} (-=-)> eg nu = (1 — x*)**1 (■*■)> denom. = £ {(1 + a?)w + (1 — x)n} (“0> the terms of the fractions require to be each multiplied by (1 — aP)^{n2~n), viz. the formulae then are sg nu = | {(1 + x)n — (1 — x)n) (1 — x2)* denom. = i {(1 + %)n + (1 — x)n} (1 — xi)^{n2^n). Thus n — 3, the formulae are sg 3u = x (3 + x2)^ — x2)3, = x (3 — 8 a?2 + 6a?4 + Oa?6 — xf) (-l·), eg ?>u = (1 — a?5)4 Vl — a?2, = (1— 4a?2 + 6a?4 — 4#® + a?8) V1 x2 ( : ), denom. = (1 + 3a^)(l — x2)3, = (1 + 0 a?2 — 6a?4 -f 8a?6 — 3a^),92 on the elliptic functions sn, cn, dn. [IV. agreeing with the foregoing values of sn Su, cn 3u, dn 3u on putting therein k= 1. 120. In the expressions for the numerators and denomi- nator of the functions of nu9 the rational functions of a? may be decomposed into their simple factors. This may be effected a priori by considering what are the values of x (that is sn u) which make these functions respectively vanish. But in, the particular case n = 2, it may be done a posteriori, by means of the duplication-formulae, and the formulae obtained for the dimidiation of the periods. Then, using { } to denote a product, as explained by the appended values of m, m', we have Factorial-formulce. Art. 120 to 125. Write (m, m') = 2mK + 2m' iKr (m, m') = (2m +1) K + 2m' iK'y (m, m) = 2mK + (2m' +1)iK'y (m, m) = (2m + 1)K + (2m' + 1)iK\ sn 2u = 2x Vl — x2 Vl — k2x2 (--), m — 0, 2 m = 0, 1 m = 0, 2 m' = 0, 1 m = 0, 1 m' = 0, 1. 121. Thus in cn 2u the product isIV.] ON THE ELLIPTIC FUNCTIONS sn, cn, dn. which is 93 "(1+sniz)(1 sn*if) sna-K + iK'^i1 X sn (\K + iKr )J\ snQK + iKy’ = {1 — (1 + kf) #2} {i — (i — k ) #2}> = 1 - 2#2 + kw. So in dn 2u the product is (1+m(iJf+*iirj) l1 + sn(| 'AK+tiK') which is -fl+ ' sn (JZ + f iK'}) i1 + sn (AK + ’ a sn QK + %iK')) (X sn {hK + \iK')} ( 1 + ItJnX1 sn (^K + UK’))' ' sn (jK + \iKr)i V sn {\K + %iK’)> = ^1 — kx2 {k — 2^1 — kx2 (k + 2, = l-2fe2+^. And in the denominator the product is (* + sn \Jk) (X + snf^jfr') 1 + .i‘&r'v) i·*· sn m'). which is = 1, ,, sn -(-/*, -M). ft where — p> —¡t! are of the form in question. One of the foregoing values is sn ^ (0, 0), = 0; and if we exclude this there remains a system of n2 — 1 values.IV.] ON THE ELLIPTIC FUNCTIONS sn, cn, dn. 95 123. Consider next the distinct values of sn-(m, m). Suppose in the first instance that m, m each extend from — (p +1), — p,... — 1, 0, 1... p (viz. that each has n -f 1 values). I call the values — (^> + 1), p extreme values and the others intermediate; so that m, m! have respectively 2 extreme and intermediate values. We have in all a system of (n +l)2 terms, viz. these are m, mf both extreme 4 m extreme, m mean 2n — 2 mf extreme, m mean 2n — 2 in, vi both mean n2 — 2n +1 n2 + 2n + 1 Now m, mf both extreme the values are sn (± iT ± %Kf), and these are excluded from consideration. If m is extreme, m! mean, the values are &n[±K + 2m—— iK'^j, say for shortness sn(±K+a), that is sn(iT+a) and sn(—K+ a), where a has \{n — 1) pairs of equal and opposite values. But sn(if+a) =— sn(—iT+a), =sin(iT-a); hence sn(K+ot) has \ (n — 1) values; similarly sn (— K+ a) has \ {n — 1) values; or si) (Hl· K. ot) has {n — 1) values. And in like manner, mf being extreme, the value is = 8x1(1^' + ^^^), =sn (±iK' + (3), which has (n — 1) values. We have thus in all (n - 1) + (n - 1) + (n -1)2, = n2 — 1 values. And as for sn - (m, m), it may be shown that these are all n K the values.96 ON THE ELLIPTIC FUNCTIONS Sn, CD, dn. [IV 124. Consider in like manner sn-(ra, m'): here m has the nx values — (p + 1), -p}... — 1, 0, 1, ...p, say - (p + l)>p are ex- treme values and the others mean; and m has the values —p, ... — 1, 0, 1, ...p, say 0 is the extreme value and the others mean. The cases are m, m both extreme 2 exclude m extreme, m! mean 2 n — 2 reduce to 71 — 1 m mean, m extreme n — 1 is 71-1 m, mr both mean n2 — 2n + l n2 — 2?? + 1 n2 + n 7l2 -1 or number in resulting system is = n2 — 1. And so sn - (m, m') has same number = n2 — 1 of values. 125. We now obtain, n odd. sn nu — nx 1 + - sn -(m, m) (-)> cn?i^ = Vl— a2 J1 + ■ sn (m, m') \ n I dn mi = Vl — k2x2 1 + sn - (m, m') [ n ') where denom. = 1+- sn - (m. mf) [ n v (+), the number of factors being in each case n2 — 1, viz. the values of m, m' are those belonging to the several systems of (n2 — 1) values as above explained.IV.] ON THE ELLIPTIC FUNCTIONS Sn, cn, dn. 97 New Form of the Factorial-Formulae. Art. Nos. 126 to 129. 126. The formulae may be presented in a different form : 00 observing that to each term 1 + - in the numerator or denomi- nator there corresponds a term 1 — ^, and combining together the pair of factors, also making an easy change of form, we have x1 | sn nu = nx 1 — ■ sn· cn nu = V1 — oc2· (1 — af | sn 2^K — ^(m)mr dn nu = Vl — k2x2 j 1 — A2 sn2 ^ (m, m)j x2 j denom. = 11 — k2 sn2 — (m, mr) x2 - , where as regards the values of m, mf observe that these are m = 0, rri = 1, 2... ^ (n — 1); m = 1, 2, ... or \ (n — 1), w! — 0, ± 1, ± 2 ... (n — 1) ; viz. there are in all \ (n — 1) + \ (n — 1) n, = | (n2 — 1) combi- nations. 127. Restoring for x its value sn a, and observing that ^ sn2 u _______sn2 a______sn (u + a) sn (u — a) 1 — k2 sn2 u sn2 a ~~ sn a sn (— a) and combining all the constant factors (that is factors indepen- dent of u) into a single factor At we find sn nu = A sn u sn c.98 ON THE ELLIPTIC functions sn, cn, dn. [IV. or as this may be written sn nu = A jsn jjw + i (m, m')J j , where m, m have now each of them all the values 0, ±1, ±2 ... ±£(n-1). Proceeding in the same manner with the other equations, we have, with the same limits for (m, m'), the system sn nu = A jsn ~ (m> j > = B |cn ~ (m, m')J j-, dn nu = C jdn j~u + ^ (m, m') |, cn nu where the coefficients A, B, (7 have to be determined. The values are /k\*{nl) /1 A = 5=(J) , C = (± 128. To show this, write in the formulae u + K in place of u. Observing that we have sn (nu + nK) = (—)* (wt1) sn (nu + K), cn (nu + nK) = (—)*(n_1) cn (nu + K), dn (nu + nK ) = dn (nu + K), and that the products on the right-hand sides contain n2 terms, n2 — 1 being evenly even, or (—)w2_1 = +, we obtain (-)Un-i)?^ = A . dn nu cn dn u + - (m, m) n 7 u + - (m, m) n j /IV.] ON THE ELLIPTIC functions sn, cn, dn. 99 ' r i, ,/i\ sn u + - (m, m ) dn u + - (m, m) V L n . agreeing with the original equations if only (—)è(»—i) ^ = 0 but these reduce themselves to the two independent equations The change of u into u + iK' gives in like manner two inde- pendent equations, one of which is and we thus have A, B> C, subject to an indétermination of the signs of A and C. 129. But it may be shown that the signs of A, By C are (—)^(n~1), +, +. For this purpose recurring to the original equations and writing therein u = 0, we find, observing that where in each case the combination 7/i = 0, m' = 0 is to be omitted, viz. the products each contain n2 — 1 terms. Grouping A2 = then sn nu = n, en u = 1, dn n — 1 7—2ON THE ELLIPTIC FUNCTIONS sn, CD, dn. 100 [IV. together the opposite terms sn a, sn (— a), &c., and recollecting that \{n2 — 1) is even, we may write n — A sn" - (m, m) ■, 1 = B jcn2 - (m, m')|, 1 = C j dn2 - (m, m')l, ^ V ^ hX J n and we may in each product consider separately the \ (n — 1) terms in which m is = 0, the \ (n — 1) terms in which m is = 0, and the \ (n — l)2 terms in which neither m nor m! is = 0. As regards these last we may consider that m has any value whatever from 1 to \ (n — 1), and ra' any value whatever from + 1, to ± J (n — 1) ; uniting together the terms which belong to the same value of m but to opposite values of m' these are conjugate imaginaries and their product is positive: hence the whole third product is positive. Taking next the terms for which ra' is =0, each term is real and positive; hence the whole first product is positive. There remains only the second pro- duct ; viz. as regards A this is jsn2^(0, m')j, where m has the values 1, 2, ... ^ (n — 1). Each term is the square of a pure imaginary, viz. it is real and negative; and the sign is thus (~)i (n~1}. But as regards B the product is jcn2 ^ (0, m')j, where each term is positive (since cn^(0, m) is real): hence the product is positive. And so as regards G the product is dn2 i (0, , which is in like manner positive. Hence in the three cases respectively the sign of the first product is 1), +. And the required quantities A, B, C have these signs accordingly; wherefore we have A = (-)»<*-« ** 5 = (f,) . c = (£) as mentioned above.IV.] ON THE ELLIPTIC FUNCTIONS sn, cn, dn. 101 Anticipation of the doubly-infinite-product Forms of the Elliptic Functions. Art. No. 130. u u 130. In the formulae No. 126 for u write -, then o& = sn - , n n = ^ when n is very large. Moreover when m, m! are finite, then in like manner sn ~ (m, m) is = ^ (m, m): and substituting these values and writing n = oo we obtain the following formulae: sn u = " f * (»l, «')} (+)> cn u = {l + “ J l (m,m) j (-)- dn u = jl + ( (m, m )) (-0, denom. = f u } 1 + (m, m')j ’ where m} mr have each of them every integer value from — x to + x , the simultaneous values m = 0, m' = 0 being excluded from the numerator of sn u. I defer the further consideration of these formulae, only remarking that not only they are not as yet proved, but that, in the absence of further definition as to the limits, they are wholly meaningless. Derivatives of sn u, cn u, dn u in regard to k. Art. No. 131. 131. We have seen, Chap. III. No. 74, that dk kk'>( > k'*A ’ where F, E, A stand for F(k, ), E (k, ), A (k, ) respectively. But we have u — F, giving sn u = sin , cn u = cos , dn u = A ; also E-^d^>=J0dn *udu, =f()du(l—k2+ki!cn2u) =fc2u +F J0cnHidu,102 ON THE ELLIPTIC FUNCTIONS sn, cn, dn. [IV. and therefore hence E — k'2F = k2 f0 cn2 u du; dF k {c 0 , snwcnw 1 o cn 2udu-;----- dk k'2 r dn u But sn n = sin , viz. considering sn u as a function of u, k, where uf =F(k, ) is a function of k and we have sn u, a function of only without k; and we hence obtain d sn u dF d sn u _ du dk^ dk ’ that is d sn u dk , dF = — cn u dn u ^ , or finally and thence and d sn u ~sr = d cn u '~dF~ ddnu ~dJT k k cn u dn u/0 cn2 u du + sn u cn2 u, k k sn u dn u f0 cn2 udu— j~r2 sn2 u cn iiy = psnwcn uj0 cn2 u du — ^ sn2 u dn u. And it will be convenient to repeat here from Nos. 74 and 75 the following formulae, in which we now write K, K\ E, E' for the complete functions FJc, FJc\ EJcy EJc'\ dE 1 / jp dk=k{-E~K)’ dE' dk dK’ dk kk'2 (E’ - k*K'), giving EK' +E’K-KK' = ^.CHAPTER V. THE THREE KINDS OF ELLIPTIC INTEGRALS. 132. In the present and following Chapters we revert to the notation of the elliptic integrals F, E, bringing up the theory to the point at which it is expedient to introduce the elliptic functions sn u, cn u, dn u: and explaining the resulting new notations. The Addition-Theory. Art. Nos. 133 to 137. 133. We have throughout <£, jl connected by the addition-equation: regarding herein /i as a constant, this gives = 0 : hence if TJ be any function of , fi, such that in virtue of the addition-equation we have dU dUAt A or (what is the same thing) if this last equation be a form of the addition-equation, we hence derive d(f>+(^~dyfr=Q, that is dU=0, or by integration U = funct. fju: and if moreover the function U is such that it vanishes for = 0, y]r = //,, then the constant of integration, funct. /£, is = 0; and we have U = 0 as a consequence of the addition-equation. For instance the function U = F + Fyjr -F/i = 0, as the addition-theorem for the first kind of elliptic integrals.104 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. 134. Again we have E + Ety — E}i — k2 sin (f> sin yfr sin fx = 0 as the addition-theorem for the second kind of integrals. In fact the equation to be verified is (A<£ — k2 sini/r cos sinjjl) A — (Ayfr—k2 sin cos yfr sinjjl) Ai|r = 0, that is A2 A) = 0 ; which in virtue of sin d> cos yjr Ayjr + sin -\lr cos (f> A S1Ufi =------ 1 — ** Bin** sin**---------’ and the identity sin2 = (sin2(f> — sin2yfr) (1 — k2 sin2<£ sin2i/r), reduces itself to A2(f> — A2\fr + k2 (sin2<£ — sin2^) = 0, which is an identity. 135. Again for the third kind of integrals, writing ,, x /, k2 \ n n sin ) A 1+a R2d] *** 1 1 dR (1 4-ft sin2^)A>/r 1 4- olR2 dyjr C, we have d.R 1 d Q2 + oP2 * 'q — - C d pdQ\ d)’ dR = 1 D.I (q¥- p dQ\ dyjr Qt + aP1' d+)’ A^=0, and the equation thus becomes {l 4- n sin2 <\> 1 4-n sin2 >/r} ^ ^ ^ But in virtue of the addition-equation, as shown in the next No, Q2 4- olP2 = (1 4- n sin2 fx) (1 + n sin2 ) (1 + n sin2 yfr), and the equation to be verified thus becomes (1 4- ft sin2 fi) n (sin2 yfr — sin2 ) 4«£-p8W-(«3rp3?K 136. In regard to the expression for Q2 + aP2, observe that this is = (1 + n - n cos p cos cos yfr)2 4- ft2 a sin2 p sin2 sin2 yfr, or putting herein cos cos yfr = cos p + sin sin yfr Ap, this is = (1 4-ft sin2 p — n cos p Ap sin sin yfr)2 4- n2a sin2 p sin2 sin2^, = (l4w sin2 p)2 — 2(l+n sin2 fx) n cos pAp sin sin yfr 4-ft2 (1—sin2^)(l — &2sin2/it) 4- fl4-ft4“ 4-i2^ sin2yi6^sin2<£sinS/r, = (1 4- ft sin2 fx) {1 4- ft sin2 p—2n cos pAp sin sin yfr 4- (n2 4- nk2 sin2 fx) sin2 cf> sin2 yfr}.106 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. But we have (1 — sin2 ) (1 — sin2 yfr) — cos2 n — sin2 sin2 yfr A2fi = 2 cos /jlA/jl sin cf> sin yfr, or what is the same thing, 2 cos fjb Apsuuf) sin yfr = sin2 ¡x — sin2 — sin2 yfr+k2 sin2 /x sin2 <£ sin2 tJt, and substituting this value within the {} we obtain the above expression for Q2 + aP2. 137. We have d-P 7~v (¿0 I I \ * J * i (¿-j- --T T7 = (1 + W“W cos fx cos

cos . n sin tx sin sin yjry = n sin pb sin yfr {(1 -{- n) cos — n cos /x cos yfr (cos2 + sin2 0)} = n sin fi sin yfr {cos + n (cos — cos fi cos tJt)}, that is o ~ p V d d and similarly O—_ P^ = n sin fi sin yfr (cos <£ + n sin /x sin yfrA), = n sin p sin (p (cos yfr + n sin /u, sin Ayfr). The equation to be verified is thus (1+ n sin2 p) (sin2yfr—sin2 + n sin p sinyfr A ) A

(cosyfr4- n sin p sin Ayfr) Ayfr, breaking up into the two equations sin2 yfr — sin2

— sin

= sin2 yfr A2 — sin2 A2yfr, the former of which is equivalent to the equation which gives the addition-theorem for the second kind of integrals, and the latter is obviously true.V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 107 New Notations for the Integrals of the Second and Third Kinds. Art. Nos. 138 to 140. 138. If in the equation E = /0 Ad we write sin = snu, and consequently d = dn u du, A<£ = dn u, the value of E be- comes =/0 dn2 u du, or what is the same thing /0 (1 — ¥ sn2 u) du. Jacobi, changing the original signification of the functional symbol E, calls this Eu, viz. he writes Eu = Jo dn2 u du7 the effect being to throw the addition-theorem into the form Eu + Ev — E (u + v) = jfc2 sn u sn v sn (u + V). He further considers in place of Eu a new function Zu9 differ- ing from it only by a multiple of u, viz. we have Zu = Eu — u, = u (l — — &2 Jo sn2 u du, where E is the complete integral of the second kind. Sub- stituting for E its expression in terms of Z, we thus have Zu 4- Zv — Z {u + v) = k2 sn u sn v sn (u + v). 139. If similarly in the equation U = I J 0 d(f> (1 + n sin2 ) A<£’ we write sind) = snu, and therefore f^- = du, the value of H ; and if, changing the notation, this were -L- du becomes , , + n sn2 u called Hu, we should have n u = du 1 + n sn2 u 5 or what is the same thing, Hu — u = ƒ. — n sn2 u du 1 + n sn2 u108 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. The effect would be to change the addition-theorem into nM+n«-n(M+t»)=|r^; where R = n sn u sn v sn (u 4- v) 1 + n — n cn u cn v cn (u + v) * * 140. Jacobi makes however a further change of notation, viz. expressing the parameter n in the form — k2 sn2 a*, he omits from IIi^ the term u and multiplies the remaining term by a constant factor; he writes in fact n k2 sn a cn a dn a sn2 u du 1 — k2 sn2 a sn2 u The full advantages of the change will appear in the sequel, but it is convenient to mention here that the addition-theorem takes the form Il (w, a)+ II (v, a) — II (u 4- v, a) __ . 1 — k2 sn u sn v sn (u 4- v — a) sn a “2 » i + ¿2 sn w sn v sn + a + a) sn a ‘ The Third Kind of Elliptic Integral. Outline of the further Theory. Art. Nos. 141 to 150. 141. We have a theory for the addition of the parameters, including in it a theory of the reduction of the parameter to the forms — 1 4- k'2 sin2 0, and — k? sin2 0 respectively. This is de- rived from the consideration of the function _ sin cos (f> (1 + £ sin2 ) A ’ where f is an arbitrary constant. Taking also p an arbitrary constant, we obtain dct _d$ 1 — (2 + £) sin2 -f (1 + 2f) k2 sin4 (f> — %k2 sin6 1 + par2 , A (1 + f sin2 ) + p sin2

) (1 + n sin2 ) (1 + m sin2 |1 A A' B ] 1 + pm2 A + 1 + n sin2 1 + n' sin2 1 + m sin2 j ’ whence integrating from = 0, AUn + A'lln' + BUm + ^F=J^ dm f J 1 + pm2 3 where the integral on the right-hand side is 1, , /— 1 i 1+,ctV—p = -j= tan-1 m Vp, or —== log--------==+ Vp r 2 V — p 1 — -ct V — p according as p is positive or negative. 142. There are two particular cases, f = — 1, that is m = and f = 0, that is m = s^n ft CQS , —k2 sin2g, — L·1 sin2#, the relation between », »', m gives a relation between p, q, 0, viz. this is found to be 1c (1 — i2 sin p sin q sin 0) = Ap Aq A0,Ill V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. being in fact equivalent to the relation Fp + Fq- F6 - F1 = 0, or (what is the same thing) p, q, 0 being connected by this equation, and writing lip, &c. in place of II (— k2 sin'J p), &c., we have between the three functions the foregoing relation AUp + Ant + BID+lr-f^^. 145. It is natural in place of 0 to introduce a new angle ff such that F0 + F1 = F0\ the relation between p, q} 0' being con- sequently the algebraical relation answering to Fp+Fq—F0*=0. The function H0 can be expressed in terms of U0', and the resulting equation is found to be (np - f)+(n, - F) - (iw - F) smp v r 7 sinj 2 7 sin# 7 = k2 sin p sin q sin &. F+ ^ log [A +k2sinpsin q sin .B]A' [Af + k2 sinp sin q sin ,Br] A where A} B, A'y B' are certain functions of 0' and . 146. This equation assumes a very simple form on writing therein sin<^=snw, sinp = sn a, sin q = sn6, and therefore (by reason of Fp + Fq — F0' = 0) sin 0’ = sn (a + b): and by intro- ducing Jacobis notation for the function II; viz. making the changes in question the three terms on the left-hand side are to common factor pres II (u, a), II (w, 6), II (w, a + b): the logarithmic term is considerably simplified and the final equa- tion is II (u, a) + II (u, b) — II {uy a + b) = i2snasn6 sn(a+6).u + \log 1—k2 sn^ sna sn&sn (a+b—u) H-A^snii sna sn&sn (a+b+n) viz. we thus see the theorem in its true point of view as a theorem for the addition of the parameters. 147. We also gain a further insight into the problem of the determination of the function Tin with an imaginary value112 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. of n : viz. any such value is expressible in the form — i2sn2(a+6t), where a and b are real; the function II??, or say II (u, a + hi), is then made to depend on the two functions II (u, a), II (u, hi), which have each of them a real parameter, viz. in the second function the parameter is ?? = — k2 sn2 bi, which is a real positive quantity. 148. There is another theory, the interchange of amplitude and parameter. Starting with the equation d n=f ft j o (1 + nsin2<£)A we have depending on the integral [ . 0 A , which dn * 6 6 J (1 + n sin2 <£)2A is expressible in terms of F, E and II. The terms involving and n combine together into a term ^ EE Vot, where as before a = (1 + w) ^ > and we have this term equal to a function of n, <£, where enters through the functions sin , cos , A, E, F, but which is algebraical in regard to n, viz. the actual equation is an n2 v a n V a + J A sin (f> cos k2\ (1 + ?isin2<£) Va ’ / fc2\ a = (l + ?i)ilH---J as just mentioned: so that integrating in /- . f dn regard to n, we have II v a depending on the integrals J ’ r dn f J nVoc * J ( dn ---------------p- ; these are really elliptic integrals as (1 + n sin2 <£) Va J r 6 at once appears by writing therein n = — kr sin2# (viz. we thus adopt for the parameter n the before-mentioned form —k2 sin2 0), reducing the integrals to the forms C dO fdO f sin2 Odd J sin2 0 AO ’ JA0’ J (1 — ^sin2# sin2<£) A0 ’V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 113 or ultimately to the forms ƒ S ’ ^6de’ and f (1 — k2 sin2 cf> sin2 0) A0y dd viz. the first two of these are F0 and E0} and the last is the in- tegral of the third kind H(n', 0) with parameter n', = — A^sin20: the final result is cot 0 A0 {n (rc, ) - F} - cot A(f> {U (n\ 0) - F0] which is the equation for the interchange of amplitude and parameter. 149. If as before = sn u, 0 = sna, then using Jacobis notation, the functions on the left-hand side are II (u, a), n (a, u): and the functions E, E0 are in the same notation Eu, Ea: the equation therefore is or, what is the same thing, II (u, a) — II (a, u) = uZa — aZu. Fund. Nova, p. 146. 150. The foregoing outline of the theory of the elliptic integral of the third kind brings up the theory to the point immediately preceding the introduction of Jacobi’s function ©: viz. his functions II (u, a), Zu are in fact each of them expressed in terms of the new transcendent ©, by the equations the second of these leads at once to the just-mentioned equa- tion II (u, a) — II (a, u) = uZa — aZu (interchange of amplitude and parameter): and by means of this theorem we can from either of the addition-theorems (for the amplitudes and the parameters respectively) at once derive the other theorem. = E0F — E > (1 + n sin2 <£) Vl — Jc2 sin2 9 or expressing only the parameter, and writing for shortness Vl — k? sin2<£= A, Tin = f <*+ J 0 Consider the function , (1 4- n sin2 <£) A ‘ sin cos m (1 + £ sin2 <£) A * where f is a constant. Taking also p a constant, we form the equation d'sr _ d(j> 1 — (2 + £) sin2 4- (1 + 2£) k2 sin4 — sin6 1 + p-cr2 A (1 + ?sin2 <£)2(1 — k2 sin2 ) + p sin2 0(1— sin2 )9118 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. where the denominator, being a cubic function of sin2 0, may be put = (1 + n sin2 ) (1 + n' sin2 ) (1 + m sin2 ). The expression risk which multiplies is then a fraction with this denominator, and breaking it up into partial fractions there is an integral part ~, and we have d'sr (1 A A' B 1 + ptsr2 A {f + 1 4- n sin2 1 4 vl sin2 1 4- m sin2 <ƒ>, whence, integrating from = 0, we have AUn + A'Un' + Mm + \ F= [ ■ d" ., g J 1 4- pvr2 where the integral expression on the right-hand side is re- tained to stand for —¡= tan-1 srVp, (p positive) v p or £ log , (p negative); 1 — 'CJ V— p and we have thus an identical relation between three functions II each with the same modulus k and amplitude , but with the parameters ny n'y m respectively. 156. The relations between these quantities and the values of the coefficients A, A', B are given by the equations nrim — — Ar’f2, (1 + n) (1 4- n ) (1 4- m) = k'2 (1 + £)2 (A?2 4* n) (k2 + n') (A? + m) = — A;2A/2p, or, what is the same thing, n + n' + m = 2£ — k2 + p, nn' + m{n + n')= f2 — 2 k2£ — p, nn'm = — A^f2.V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 119 Hence considering n, n as given, we have 1 = , ^2d + g)2 rm' (1 + n) (1 4- w') * or, what is the same thing, and then ? = m = + n 1 + n (A? + ri)(k2 + n) k'2 + k2 1 + n 1 + n n n -k2? nn p — — (k- + n) (fc* + w') , and then further = {n3 + (2 + £) n2 + (1 -f 2£) k2n + £&2} -r-n {n — n') {n — m), A' = {n'3 + (2 + f)rc'2 + (1 + 2£)k2n' + f&2} + ri (n'-n) (nf - m), 5 = {m3 + (2 + f) m2 -f (1 -f 2£) k2m + £&2} -r- m (m — n ) (m — w'). The reduction of the general formula is somewhat laborious; there are two important particular cases which it is as well to discuss separately: these are / tan \ , „ A / sin cos \ and f=°(«=—a—^j· The Case f=-l, CT = -”^. 157. We have here m = — 1, nn = k2, (k2 + n) (k2 + nf) = k2p, which last equation may be written (k*+n)(t-+^=k% or, what is the same thing, p = (l + w)fl + ^) , =a.120 THE THREE KINDS OF ELLIPTIC INTEGRALS. [V. We then have nrfi __ 1/·2 A = in3 + n2 — ten — te}-i-n(n — n') On + 1), = - - , = 1 c n(n — n) and similarly A'= 1; also B = 0; so that the parameters n, n being connected by the relation nn = k2, we have between the two functions II the relation Tin + Tin' = F + fy~^ , (/) Leg. p. 68*, viz. a being positive, the integral, substituting therein for w its value, is = 1 va A and a being negative it is = -7=10 g 2 V—a A + \/— a tan A — V — a tan The Case f = 0, = —?cos *. A 158. The general expressions are not immediately ap- plicable : they give m = 0 and then 5 ^, but the two terms 1 B -r, and ------ „ -, are together equal to a determinate constant, f l + msm2(f) ° 1 the value of which, = — te/p, can be found by writing in the first instance £ = 0 : the formula becomes cfe __ / k2 A A' \ dp 1 + p'sj2 \ p 1 An sin2 1 + n' sin2 ) A * or, what is the same thing, AUn + A'Hn--F=[T^~, p Jl+pvr1’ * (ƒ') is the formula thus designated, Legendre, Traité des Fonctions Elliptiques, 1.1. p. 68 ; and so for the other formula referred to in a similar manner in the present Chapter.V.] where THE THREE KINDS OF ELLIPTIC INTEGRALS. 121 n + n' = — k2 + p, nn = — p, . n2 + 2n + k2 n2 4- 2 n' + k2 n(n — n') ’ n (n — n) We have therefore n+n +nn' = — k2, or, what is the same thing, (1 + w) (1 4- n') = A/2, which is the relation between n, ri. And then writing , (n + l)2-k'2 A —77- , n(n —n) jnd substituting for k'2 its value we find A = n * , and similarly A' = n . n J n Moreover, writing for p its value, the formula becomes n + 1. n which is the relation between two functions II. Tin + —Tin' + —,F=f -—dvr, -, (/) Leg. p. 72, n nn J 1 — nn rn2 a ° - 159. The two formulae (ƒ') and (gf) enable us to perform the reduction of functions of real parameter. We may consider the four cases I. n positive, = cot2 0; v- = A Qc',6) sin 0 cos 0 II. III. IV. n negative and between 0, — k2y V— a = cot 0 A (k, 0). n negative and between -k2 and — 1, k'2 sin 0 cos 0 VS =' A (k\ 0) = — k2 sin2 0; = — 1 4- A/2 sin2 0; n negative and between — 1, — oo , JL sin2 05 V— a = cot 0 A (k, 0) ]122 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. where in each case the value is annexed of Va or V—a as the case may be. Observe that in the cases I. and III. a is positive, or the function is circular: in II. and IY. a is negative or the function is logarithmic. 160. It is very noticeable how the formulae (ƒ') and (g) give each of them a relation between two circular functions or two logarithmic functions, but not in any case a relation between a circular function and a logarithmic function. Treat- ing n, nf as coordinates, we shade by vertical lines the spaces for which n is circular and by horizontal lines those for which n' is circular: the two curves nn' = k2 and (l + n)(l +n') = k'2 are then hyperbolas lying wholly in the spaces which are either cross-shaded or else white, viz. the corresponding values n, n' are both circular or both logarithmic.V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 123 161. In the formula (f), taking n = — ^ ^ we have n' = — k~ sin2 6, and thence, substituting for a its value, _ 1 A + cot 0 A (Jc, 0) tan ~~ + 2 cot 0 A (Jc, 0) A — cot 0 A (Jc, 0) tan ’ or as this is better written, _ _ 1 . cot A (k, ) + cot 0 A (k, 8) + 2 cot 0 A (k, 8) l0g cot A (Jc, 0) - cot 8 A (k, 8) ‘ This equation shows that a logarithmic function of parameter which is negative and in absolute magnitude greater than 1, may be reduced to depend on a like function where the parameter is negative and in absolute magnitude less than Jc2. The first-mentioned kind of logarithmic functions presents the difficulty that the function under the integral sign becomes infinite in the course of the integration (viz. for the real value sin2 <*>---1/»): we therefore always consider the reduction as made, and attend only to the case where the parameter is of the form — Jc2 sin2 0. 162. The formula (ƒ') gives also a relation between two circular functions of positive parameter, viz. writing therein n =» cot2 6 we have ri = Jc2 tan2 0; and the relation is II (cot2 0) + II (Jc2 tan2 0) = F -f sinflcos# A (Jc, 0) tan-1 A (Jc\ 0)tan (j> A (Jc, <£)sin#cos0 ’ which in fact serves to reduce a circular function of positive parameter greater than Jc to a like function of parameter less than Jc: but the original form II n + II -ptan- va 1 V a tan A is for this purpose equally if not more* convenient.124 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. 163. The formula (g') gives in like manner a relation between two logarithmic functions, or two circular functions: as regards the first case observe that if ny n are both negative they are both in absolute magnitude greater than 1 (viz. 1 + n, 1 + n' are each negative); and we have thus a relation between two logarithmic functions with parameters of this form ; but such functions being excluded from consideration, the formula is not written down. There remains the case where the para- meters (being by supposition logarithmic) are each negative and in absolute magnitude less than k2: viz. writing n= — k2 sin2 0, n = — k2 sin2 X, the relation between the parameters is (1 — k2 sin2 6) (1 — &2sin2X) = A/2, or what is the same thing (cos2 6 4- k'2 sin2 0) (cos2 X 4- k'2 sin2 X) = k/2, or as this may be written (1 4- k'2 tan2 0)(1 + k’2 tan2 X) = k'2 (1 4- tan2 0) (1 4- tan2 X), whence finally the relation is 1 = k' tanX tan 0 (answering ic will be observed to the transcendental relation F0 + FX = F}). We then have 1 +n_ 1 +A/2tan20 _ k'2 + k'2 n ” — k2 tan2 0 ’ “ ¿2 ( 1 tan X)’ “_i^co^X, and completing the substitution, the formula becomes cos2 0 II (— k2 sin2 0) + cos2 X II (— Jc? sin2 X) r, , . ^ i /A — k2 sin 0 sin X sin \ 2 6 \A + A;2 sm ^ sm X sm

J where as above 1 = k' tan 0 tan X. The formula enables the reduction of a logarithmic function of parameter — k2 sin2 0 in absolute magnitude greater than (1 — k') (or for which tan0>l/V&') to a like function of parameter in absolute magnitude less than (1~A/) (or for which tan 0 < 1 s/k'). But it is convenient, not using the formula, and therefore without thus restricting the value of 0y to retain — k3 sin2 0 as the expression for the parameter. 164. In the same formula (g') if the parameters are both circular they may be taken to be n—cot2# and n'=—l+£,2sin20; and the formula becomesCORNELL UNIVERSITY V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 125 n (cot2 0) - k'2 sin2 0 cos2 0 1 — k'2 sin2 0 II (— 1 + k'2 sin2 0) _ k2 sin2 0 ~ sin 0 cos 0 _a sin cos A (&', 0) ~ 1 -k'2sin2 0* + A (k\ 0) n tan 6 A (k, ) ’ which is a formula for the reduction of a circular function of positive parameter cot2 0 to a circular function of negative parameter — 1 + F2 sin2 0. 165. The above formula cos2 0II (— k2 sin2 0) + cos2 X II (— k2 sin2 X) „ 1 . - . . /A — k2 sin 0 sin X sin cos \ 2 & vA<£ + k2 sin 0 sm X sin <ƒ> cos / may be written under a slightly different form : viz. expressing it first in the form cos2 *[H(- k2 sin2 0) — F] + cos2 X [Ü (- k2 sin2 X) — F] = (1 — cos2 0 — cos2 \)F + ± sin 0 sin X log il, and dividing the wdiole by sin 0 sin X ; then reducing the coefficients of the several terms by means of the relation 1 = F tan X tan 0, and finally restoring the value of ÎÎ under a slightly altered form, the equation becomes —[ü (- k2 sin2 0) - F] + —=—— [II (- k2sm2 X) - F] sm 0 L J sinX L v 7 J ¿2 = p cos X cos 0 where as before 1 = F tan X tan 0. 166. If to fix the ideas we consider herein 0, X as positive and less than \ir, then writing 0' = 7r — X, the relation between 0, 0' will be F tan 0 tan 0' = — 1 (0 and 0' each positive but 0 < ^7r, 0' > ¿7r). Substituting for X its value 7r — 0' the formula becomes cos0A0rTT/ 7. . 0/lx -J-.-. cos0A0 rTT/ 7o . „ [n (- &2 sm2 0) - J7]-. ¿j— [n (- k2 sin2 0 ) ■ ,0L\ 7 J sm 0 L 7 sin i ■F] l? = — T7 COS /! rr 1 1 (k Aé + &2 COS 0 COS 0 Sin COS \ 0 COS 0 . jP + A log 77- 7 r------71-----^ . ,------V ), 2 6 A — &2 cos 0 cos 0 sm <£ cos <£/ ........~ 12 V*' A (j) which is a form used in the sequel.126 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. The general Case resumed. 167. Returning now to the general equation AHn + A'Un' + BUm + \ F= f—, £ J 1 + pm2 write n = -&2sin2^, n' = — k2 sin2 q, m — — k2 sin2 0; then introducing these values we have £ = — k2 sin p sin q sin 0, p = —-j^ cos2 p cos2 q cos2 0, k'2 (1 + £)2 = (1 — k2 sin2 p) (1 - k2 sin2 q) (1 — k2 sin2 0) ; or writing this last under the form k' (1 + £) = Ap Aq A0, we have k' (1 —k2 sin p sin q sin 0) = Ap Aq A0 as the relation between the parametric angles pt q, 0. This is in fact equivalent to the transcendental equation Fp + Fq-F0-Fx = 0, and it suggests the introduction into the formulae in place of 0, of a new angle 0', such that F0 + F1 = F& and consequently Fp + Fq- F& = 0. 168. But let us first express B in terms of the original angles p, q> 0. We have g _ m3 + (2 + f) rr# + (1 + 2 J) k2m + £k2 m (m — n) ’ the numerator is — sin6 0 4- (2 — A2 sin 0 sin p sin q) k4, sin4 0 + (1 — 2A;2sin0sin^sin^). — A^sin2# — k* sin 0 sin p sin q,V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 127 = — A4 sin 0 [sin 0 (1 — 2 sin2 0 4- k2 sin4 0) + sinpsing(l — 2k? sin2 0 + k? sin4 0)], = — k* sin 0 [(sin 0 4- sin p sin q) cos2 0 A2 0 — A?'2 sin2 0 (sin 0 — sin p sin gr)], and the denominator is — k? sin2 0 (sin2 0 — sin2 p) (sin2 0 — sin2 q\ whence „ _ cos2 0 A20 (sin 0 4- sinp sin g) — k'2 sin2 0 (sin 0 — sin p sin q) k2 sin 0 (sin2 0 — sin2 p) (sin2 0 — sin2 g) 169. The relation beween 0, 0/ may be written under the forms sin 0 = — cos 0 = cos 0' "A07- ’ ¥ sin 0' A0' ¥ A°~ A0' ’ sin 0' = cos & — — cos 0 ~AF’ k' sin 0 A0 A0' = A0 * Hence in the last-mentioned expression of B> the nu- merator is ku sin2 0' / cos 0' . \ ¥2 cos2 0' /cos 0' \ -&r~ r + sm ^siniJ+ U^' + sin p sm V * A/2 cos2 0' /cos 0' which is k' 2 = [&'2 s^tl2 ^ (” cos & 4- sin p sin q A0') 4- cos2 0' A2 0' (cos 0' 4- sin p sin q A0/)]. But in virtue of the relation between p, g, 0' we have cos 0' = cos p cos q —sin p sin q A0\ or the numerator is k’2 = g, [k'2 sin2 0' (cos p cos q — 2 cos 0') 4- cos2 0' A2 0'. cos p cos g], _^.n say this is128 THE THREE KINDS OF ELLIPTIC INTEGRALS. [V. Then we have il cos p cos q = (cos2 0' A2 0' + k'2 sin2 6') cos2 p cos2 q — k'2 sin2 0'. 2 cos p cos q cos 0'. But 1 — cos2_p — cos2 q — cos2 0' — k2 sin2 p sin2 q sin2 0' = — 2 cos p cos q cos 0\ say R sin2 0' — cos2 p — cos2 q = — 2 cos p cos q cos 0', where R = 1 — k2 sin2 p sin2 q. Hence il cos_p cos q = (cos2 0' A20' + k'2 sin2 0') cos2 p cos2 q + k'2 sin2 0' (.R sin2 O' — cos2 p — cos2 q), = (1 — 2k2 sin2 0' + k2 sin4 0') cos2 p cos2 q + k'2 sin2 0' (R sin2 0' — cos2 p — cos2 q\ = cos2j> cos2 q + sin2 0' [— 2k2 cos2p cos2 q — k'2 (cos2 p + cos2 g)], -f sin4 0r \k2 cos2 p cos2 q + k!2 (1 — k2 sin2 p sin2 5)], which is = cos2 p cos2 q — sin2 0' (cos2 p A2 q + cos2 q A2 p) + sin4 0' A2 p A2 q, = (cos2p — sin2 ff A2 p) (cos2 q — sin2 0' A2 q), so that the numerator is k'2 1 = ^-----------(cos2 p — sin2 0' A2p) (cos2 q — sin2 & A2 q). A50 cos^) cos £x r 'i 'l/ 170. The denominator is k2 cos & ( . 0 cos2 ff\f . „ cos2 0'\ - -KT [sm p - W ) [sm'q - w) * 1*2 rtAO ff ------A^0'— (cos2 ^ s^n2-^ (cos2 & ~~ sin2 q A20')} J/*2 nna Qf ------A50‘ ^C°s2 P ~ S^n2 ^ (c°s2 i “ s^n2 ^ A2#)5 which isV.] whence THE THREE KINDS OF ELLIPTIC INTEGRALS. 129 Write B = M= then B = M -Æ'2 Æ2 cos p cos q cos 0'* -Ay Æ2 sin y cos^) cos 2 k'2 sin y (=_________________\ \ k2 cos p cos q cos 0/ * cos yAy (- M cos sin 0 / 5 and similarly a = æcos£A£ sm p A'--McoaqAq. sin q 171. The equation is AUn + A'lin +BUm+\F= [ , f J1 + pisr2 or since A + A' + B = 1-i * this is 4 (lira - i1) + (lira' - F) + B (Ilm - F) + F=ƒ - that is -r)+ cos <£ y ~ (1+ £ sin2 0) A 1 A;2 cos cos q cos 0 sin cos 1 _ (i+fsin2£)A and the formula thus becomes (lip - ff) + (% - F) - COgS.^ ^ (n<9 - F) sinj? sin# Æ2 = p: cos p cos q cos 0. F j^o ¿'(1 4- f sin2 0) A<£ — k2 cos^ cos ^ cos 5 sin cos 2 ° A/(l + f sin2 <£) A<£ -f k2 cos^) cos ^ cos 0 sin cos * 172. Representing, for convenience, the logarithmic term by ^ log fl, so that Q _ &'(1—A?sin^sin9,sin0sin2<£)A<£--A:2cospcos<7COs0sin<£cos k' (1—A;2sin sinq sin 0sin2 )A+k2cos pcos gcos 0sin 0 cos 9 P-Q = pqTQ suppose, we have, ante No. 165, writing & for \, and — 0 for 0 (thereby passing from the relation F1 = FX + F0 to the actual relation F1 = F0'-F0)> cos 0 /TTZ) T?S cos & A0' /TT/j/ ns, k2 a /)/ et --v—7i— (n^ — F)-----. — „ (110 — F) = — 77 cos 0 cos 0 . F smfl v 7 smi v 7 k &'A-f ¿2cos0' cos 0 sin cos / R + S \ + tlogt'a»-fcoaycos9«m»cos»r*1°g^,,nPl1<>sej ; and using this to introduce into the formula 5?!^ (n ) A J(?k' — A^7 s*n ^ C0S-P cos ? s^n ^ cos & or, multiplying by A0' and omitting the factor k\ say P— Q=(A0'+i2sin^sinycos0,sin20)A^)—i^sinfl'cos^cosgsin^cos^, = [A0r A — Ar2 sin 0’ cos ff sin cos <ƒ>] + A^sintf'sin^cos^Kcos#'-- cosjpcos#), =—sinpsing A0'] + A2 sin p sin ^ cos 0' sin2 A } = A0' A0 — ¿2 sin 0' cos sin <£ cos 4- k2 sinp sin ^ sin [cos 0’ sin A — cos <£ sin ff A#']; and similarly P -f Q = A0' A<£ + A:2 sin ff cos ff sin cos + k2 sin sin q sin <ƒ> (cos 0' sin <£ A<£ + cos sin ff A0'). Also ^ ~ 7 / * . A;2&' sin 0' cos ff sin cos P + £ = A;'A4> +-----------^------*-------- , or, multiplying by A07 and omitting the factor k\ say P -1- S = Aff A; 9—2132 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. and similarly, R — S = AiP A — k2 sin 0' cos 0' sin cos <£. 174. Write for shortness A#' A<£ — i2 sin 0' cos O' sin cos = A, A0' A<£ + &2 sin 0' cos 0' sin <£ cos cf> = J/, cos 0' sin A<£ — cos sin 0' A0' = B, cos 0' sin <£ A<£ + cos sin O' A0r = 5', then the logarithmic term is at once expressible in terms of these quantities, and substituting in the formula, we have which is in fact the formula connecting the three functions Up, Uq, HO', or in the original notation the angles p, q, 0' being, it will be remembered, connected by the algebraical equivalent of the equation 175. This apparently complicated formula is wonderfully simplified by introducing into it Jacobi’s notation; viz. writing sin p = sn a, sin q = sn b; and therefore sin O' = sn {a + b); also sin = sn u; then omitting a common factor 1 — k2 sin2 0' sin2 , we have and the formula becomes II (u, a) + II (u, b)—H(u, a + 6) = ft? sn a sn 6 sn (cH- b). u viz. it is in fact a formula for the addition of the parameters. II (— k2 sin2 p), II (— sin2 q), II (— k2 sin2 0'); Fp+Fq-FO' = 0. A = A'= dn (a + b + u), dn (a + b — u\ — B = sn (a + b - u) dn (a + b + u), B* = sn (a + 6 + u) dn (a + b — u), - , 1 — k2 sn a sn b sn (a + b — n) sn u * 1 + k? sn a sn b sn (a + b + u) sn u *V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 133 Interchange of Amplitude and Parameter. Art. Nos. 176 to 188. 176. Starting with n = f------^------- J0(l+nsm2)A’ and taking throughout the integrals in regard to from this inferior limit 0, we have dU j f — n sin2 d(f> n -j. dn J (1 + n sin2 )2 A * d -ƒ■ (1 + n sin2 <£)2A A2 -n. But writing a = (1 + n) ^1 + — j we have 2a f dd> A sin d> cos d> ! w J (1+ « sin2$)2A — 1+n sin2) ^ \ [______di JJ (1 + n si + (l+ 2 + 2 k2 3k2 n v? d

_ A sin cos fc2 „ _1 n J (1+n sin2 n2 n ^ •i(2«-”s)n· viz. we have 2a ( f d(j> if ____________________ n \J (l+ n sin2 )2 A _ A sin cos k2 „ J 1 + n sin2 n2 --(F-E)-^U. n dn 177. This equation may be written 9„dn _ A sin cos _ k2 1 f v ^ ^ da134 THE THREE KINDS OF ELLIPTIC INTEGRALS. [V- or, what is the same thing, 2adn + Uda = A sin 6 cos -——l—- - JfiF ^ - (F-E) — , ^ Tl+?ism2<£ n2 y n viz. multiplying each side by |a~2, this is c£. II Va = | A sin <£ cos —^— WFJ? (1 + 7isin2) va n2v a d?i Va’ where of course of 71. a, = (1 + n) (14- , is V, nJ regarded as a function Integrating each side we have II Va = 0 + £ A sin cos <£ f-—----T- - \1s?F [—- 2 r W (1 + w sin2) Va 2 ./n*Va ~i(F-E) f-^=, J va where the constant of integration may of course be a function of k, <£, but it is independent of n. The formula is simplified by representing the parameter n under any one of the foregoing forms cot2 6, — l+&'2sin20, — &2 sin2 6. The last is the most interesting case, but it is proper to consider them all three. First case, n — cot2 6, 178. Here , 2 cos 0 ™ A (&', 0) dn =-----. d0, v a = -r—^------^, sin3 6/ sin 6 cos 6 and the equation becomes A {k\ 0) sin26d6 sin {4^d)a n =C + k*F ¡-——r-g. -r 0 cos 0 J cos2 0 A (&, 0) A . , . f cos20d0 — Asm9 cos cos2 0) A (&', 0) ; where the integrals in regard to 0 may be taken from the inferior limit 0.V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 135 The integral u dO and we have A2 A(*\ 0) dd sin2 6 sin & is =F(Jc', 6) f do sin2 0 sintf. ,,, m ^ I c·* a <*',*)" 5» * 4 ^ ~ ^ Moreover, writing cot2 = v! we have cos2 0 dd A sin in <ƒ> cos ♦ƒ, f dd J A(*', ; + (sin2 0 4- sin2 <ƒ> cos2 0) A (A/, 0) A f d<9 A sin <£> cos J A (A;', 0) sin cos J (1 + n' sin2 0) A (k\ 0) ’ ƒ< = _ ^sva±F(k\ 0) + . A n (»', 0). cos v sm cos Substituting these values and for greater clearness writing IT (w, kt ), A (At, <£), F(k, <£), i7(&, ) instead of II, A, jP, E, putting also for 0 its value = ^7r, determined as presently mentioned, the formula is, (n = cot2 0, n — cot2 cj>) -^¡¡’ 6\ n (n, k, 4>) + .A^’^ n(n\ k’, 8) sin 0 cos 0 sm cos . (¿') Leg. p. 133. -i’+Sr*4**'-^· «+^4<*· +>*■<*■*> + -F(ifc, )F(k't 0)-F(k9 4>)E(V, 0)-E(k, )F(k’, 0). ,179. If in the formula, instead of \ir3 the term had been (7, then G is independent of 0} and by the symmetry of the formula it must be independent also of <ƒ>: it is thus an absolute constant: to determine its value take 0, each indefinitely small: then F (fc, ) = E(k, ) = } F(k', 0)=E(k'3 0) = 0, n (n, k, ) = f ^ = 4= tan-1 *Jn = 0 tan-1 ^, r J1 + n sin2 <}>*Jn 0 and similarly II (n', k’, 0)= tan-1 ^:136 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. we have therefore = ^7r — to, a> being indefinitely small: then n' = cot2 = tan2® = to2 is indefinitely small, and therefore dd f d8 , f sin28d8 TT / ' 7' m - f _ f d8 , f sin28d8 (n,K,tf)-j Q+n, s{n20) A ^ 0) JA(k,' 0) njA (k,' ^, de rr/if £\\ „/ fsin28dd = F(h,6)-nj—lc7( and thence ■^r^K n (»', k’, B) - F(k, 0) sm

cos \sm (j> J cos 4> sin

), = —T± and----------- , = —j: cos \sin (p J sm

cos

= <0, these £ach contain the factor co, and the function on the left-hand side is thus = 0; hence the equation becomes A MW- p. 134. + FJcF (k\ 8) - FJcE (k\ 8) - EJcF{kr, 8), viz. we have thus an expression for the complete function Ili (n, k) in terms of the functions of the first and second kinds. jfc2 181. If in this equation we write — in place of n, and k2 assume also — = cot2 X, so as to write X in place of 8, we n have AflX\ nx (-,k)- + ^Flk A (k\ A) sm A cosA \n ! cosX + FJcF(k\ \)—FJcE{k', X) -EJcFik', A).V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 137 182. A (k\ 8) sin 8 cos 8 Adding the two equations together, we have „ , A (k\ A) „ (sin 8 . ... sinX . . .) 7r + FJc ^-a A (k, 0)-\---- A (&, X)!· (cos# v ' cosX v ') + FJc{F{¥, 8)+F(K, \)-E(E, 8)-E{¥, X)} -E1k{F(¥,8)+F(k\ X)}. But in virtue of cot'-’d cot2 X = k", or k tan 8 tan X = 1, we have F(k', 6) + F (k', X) = FJc\ E (¥, 8) + E (kX) = EJc’ + k'1 sin X sin 0, and further cos0 . &sin0 . .. k sin X = A yjj m, cos X = A nj a., A (k, X) = a (k’,ey a (kf,ey A (k',8)’ A (¥, X) A (¥, 0) sin X cos X sin 8 cos 8 ,=*Ja, smXA(k,\) sin# A (A?,#) r · i ------—- = y a sm2 X, -----V - - = ^a sm2fl; cos X cos 0 hence the equation becomes Vajll^ZO + n^, *)} = 7r + FJc {Va (sin2 0 + sin2 X) — k'2 sin 0 sin X} + FJc (FJc’ - EJc’) - EJcFJc’. 183. But we have 1 — sin2# — sin2 X = cos2 0 — COS' 0 k’2 sin2 0 cos2 0 k'2 A2(k\0) A2(k\ 0) ’ ~ a’ that is sin2 0 4- sin2 X = 1 H-or Va (sin2 0 + sin2 X) = Va + — , a Va sin X sin 0 = sin 0 cos 0 _ 1 A(*', 0)'THE THREE KINDS OF ELLIPTIC INTEGRALS. 138 [V. whence on the right-hand side the second term is FJc Va, or the equation may be written Vo a ¿) + nig, k)-FJc =ir+F1kFlk'-F1kE1k'-E1kFJc', which is in fact a consequence of two former results (k2 1\ m ^7r 11,(71, k) + n, g , -FJc = , and FJcFJc—FJcEJc—EJcFJc = — J7r, viz. the equation is hereby reduced to the identity ^7r = it — \ Second case, n = — 1 + k'2 sin2 0. 10, TJ /- k'2sin 0 cos 0 dn K ri, m 184. Here Ja = ^ - ¡MO A (i, tf), and taking as before the integrals in regard to 0 from the inferior limit 0, the general equation, ante No. 177, becomes k'2 dTL=C + (F—E) I ff - k*F f-------f^- V, 0) J A(i, 0) J (\-k*&i. d0.A(k\ 0) A (k', We have 0f + A sin in (f> cos ƒ cos2 + k'2 sin2 0 sin2 ƒ; d0 =F(k', 0), and a(*', ey ( d0 _1 p/„ ^ k'2 sin 0cos 0 J (l-kt2sm20f~fr {i’ ~aW7W1 moreover, writing , k'2 tan2 = n\ we have . . f A (&', 0) <20 _ A sin 9 i A(kf, 0)d0 sm 9 cos 9 J cog2 ^ ^,2 g-n2 ^ gj^2 ^ , cos ^ J i+ nf sin2 9 > ---Acos£^ . A - — II(w', 0). sm 9 sm9cos9Y·] THE THREE KINDS OF ELLIPTIC INTEGRALS. 139 Making these substitutions, and writing also A (k, ) instead of A, &c., the formula becomes (w = -l + Psin20, nf = &'2tan2<}>,) k'2 sin 0 cos 0 rTT , , _/? [n (nf 4 F (4 <£)] (l')Leg. p. 138. = ^(i’ rn (n, if, 0) - cos3 6 F(k\ 0)] sm (f> cos L v 7 ^ + ^(4 0) W 0)-i?(4 <¿>^(4, 0)-^(4 0)^(4, 0), the value of the constant G being here = 0, since the two sides each vanish for 0 = 0. 185. Write <£ = ^7r— co, co being indefinitely small; it is to be shown that A-(k^- [n (n', Jc, 0) - cos3 F(Je, 0)] = fcr, sm cos

) Leg· p·138· = }7T + FJcF(k\ 0)-E1kF(k', 0)-F1kE(k', 0), giving the value of the complete function II x (n, k) for the form in hand n = — 1 + 42 sin2 0. 186. As regards the subsidiary proposition, observe that k'2 A (k, ) tan 0 + A (4 ) an sin cos A (4, 0) 5 but, tan g> being indefinitely small, n (tan2 co, 4, 0) = F (4, 0) + g>2 M,140 THE THREE KINDS OF ELLIPTIC INTEGRALS. [v. where M is finite: the equation thus is n *·,«> + ton- . , A (Jc, ) sm cos A(lc, 0) that is AjJ, k> 0y + = tan-1 sm 9 cos 9 sm 9 cos 9 sm cos A (A;, 0) or putting herein = |-7r — &>, then since cos 0 vanishes, and the function under the tan-1 becomes infinite, we have ■A ^7 n (»', k\ 0) = Jtt, sm cos 9 v 7 whence the required relation. Third case, n = - Jc2 sin2 ft 187. Here Va = ¿A (&, 0) cot ft , and the v a ^ W ft) general formula becomes A (4,9) n - 0 + (i· - *> ƒ - j/ar.^8) A . , , f Jc2 sin2 0d0 + &sm cos J (1_&Ssin2 0sin^)A(^ 0) * where as before the integrals in regard to 0 may be taken from the inferior limit 0. Since in the present case the modulus as regards both the and the 0 functions is k, we may instead of A (Jc, 0) write simply A0 and so F0, E0. We have and moreover writing n' = —Jc2 sin2 , then f Id2 sin2 0d0 _f Jc2 sin2 0 d0 J (1 - k2 sin2 0 sin2 ) A0 ” J (1 + n sin2 0) A0 ’ 1 sin2 [n (»', 0)-«];V.] THE THREE KINDS OF ELLIPTIC INTEGRALS. 141 whence the formula becomes (n = —Jc* sin2 0, nf— — k2 sin2 <£), cot 0 A0 [II (n, ) — F] = (nf) Leg. p. 141. cot A(f> [II (nr, 0) — F0] + E0 F — F0E, the constant in this case being evidently = 0. 188. Writing <ƒ> = we have n,7i (F.E0 -E,F6), (p') Leg. p. 141. which is the value of the complete function n^, or say of Ui (72, k), for the form n = —k2 sin2 0.142 [VI. CHAPTER VI. the functions n (u, a), Zu, ®u, Hu. 189. The functions referred to all depend on the modulus Jc, which may be expressed when necessary: as regards II (u, a) this is seldom required, but the other functions will be fre- quently written Z (u, k), © (u, k), H (u, &), so as to put the modulus in evidence. Introductory. Art. No. 190. 190. The function II (u, a) has already been defined as A^snacnadnasn^dii Jo 1 — k2 sn2a sn2u 9 and the several properties already obtained for the function n (n, k, c/>) admit of being translated into this new notation. But in the present chapter the theory is established in a different manner, by expressing this function II (u, a) in terms of the new function ®u. This function %u may be considered as originating from the function Zu, which has already been mentioned as introduced in place of the function E (k, 0), viz. writing E to denote the complete function EJk> we have Zu ~u(l k2lo sn2u du} or, what is the same thing, E — — TrliJr J0dn 2udu.TI.] THE FUNCTIONS II (u, a), Zu, ©u, Hu. 143 The new function ®u is in fact /2k'K _ /2k'K ~i^u2+f0dufodudn2w = V^Te ’~V~^re /WK y-^r where the exterior factor is fixed upon for reasons which will appear: the original function Zu is thus expressible in terms of the new function ®u and its derived function, viz. ©^ we have Zu = ^-, but the employment of the symbol Z as a separate notation is nevertheless convenient. The function ®u is one of a series of four functions, ®u, © (u + iKf)> ®(u + K)y ®(u + K + iK'); but it is found con- venient instead of ®(u + iK') to introduce a new function Hu} and write the four as ®u, Huy © (u + K\ H(u + K). The following article is in the nature of a lemma. Values of II (u + a, a) in the three cases a = \iK\ a = \K, a = ^K + ^iK' respectively. Art. Nos. 191 to 193. 191. We have d >i2snacnadnasn2(w + a)< du \uJta>a) l — Jc2 sn2 a sn2 (u + a) first if a = \iK\ we have sn\%K\ cnUK', dn|iK' = ^=, VT+k, vk yk 9/ , .T_,x 1 (l + íOsnw' + '*'CI1'wcln*¿ n + ----=-(-------.-----, ~ , (CMlfe No. 105> v 2 7 k (1 + &) snw — ^ cnttdnw The right-hand side of the foregoing equation is therefore a fraction, the numerator of which is ik(1 + k) [(1 + k) snu + icnwdnu\ its denominator being k [(1 + &) sn u — i cn u dn ti] + k [(1 + h) sn u + i cn u dn u\144 the functions II (u, a), Zu, ©w, Hu. [VI. viz. this is = 2k (1 + k) sn u, a mere multiple of sn u: and we thus have du II (« + \iK', Jti') = i*(l + *) 1 + i cn udn u 1 + k sn u f- or observing that log sn u, and integrating so that the value may vanish for u = — \iK'} we have \i{ 1 + k) (u + \iK‘)-ilogsn^ + i log 192. Secondly a = \K, snJjST, cn \K, dn^iL = •JV 7, W, sn Vl + &'’ Vl +k . i^\_ 1 dnit + (l+i:')snwcnii (w + 2 ) 1 +1c' dnw + (l — k') snw cnw5 and then d gU (u + iK, *Z) = i(l-40{l + or observing that (1 + kf) sn u cn u dn u d , , —IcPsnuciiu -j- log dn u =-=------, du ° dn u and integrating so that the value may vanish for u — — \K, we have no + pr, JiT) = i(l-F)(m + l-fiO-ilogdnw + llogVE 193. Thirdly a = + \iK’, snqK + ^iK'), cn(%K + \iK'), dn{\K+\iK') =\/~k-’ \Z~Y ’ v-^ o / , i 77 i -rx/\ k + ikr cn u + (k — ik') sn u dn u sn2 (u + \K + hiK) = —-7------77-^-------i—, k cn u + (k + %k) sn u dn uTHE FUNCTIONS II (u, a), Zu, ©M, Hu. 145 VI.] and then (n + ^K + ^iK', \K + \iK') =£(&+*')+i cn u sn u dn u cn u whence observing that d , — sn u dn u log cn w =-----------, dn ° cn u and integrating so that the value may vanish for u = -\K-\iK'9 we have n (u + jsK + \iK\ \K + \iK') = 2 “f· ^ ) {u -f- + \'iK ) — ^ log cn u 4” 2" log The Function Zu. Art. Nos. 194 to 199. 194. We now proceed to consider the function Zu: it has already been, ante No. 135, seen that we have here the addition- equation Zu + Zv — Z (u + v) = k2 sn u sn v sn (u + v). We have as before Z {K) = 0, and therefore also z(iK) = Hi-k'). 195. Starting from the equation E Zu = — u + /o dn2w dn, and writing herein iu for u and k' for k, we have Ef Z {iut k') = — -gr, iu -f* i J0 dn2 {iu, k') du} E' . ^which observing that dn {iu, k') — ^ ^ may also be written dn2 u . [ dn2w \146 the functions II (u, a), Zu, ®u, Hu. [vi. and it is to be shown that we have „ snudnu iru ltx ~ criM----2KK'+lZ{m’k)· I stop to remark that u being indefinitely small this equa- tion is which is true in virtue of EE - _ 7r K + 1C~ L~ 2KIT" 196. To prove the theorem, we verify without difficulty that d sn u dn u . dn2 u = dn2wH--------1; du cn u we have therefore d sn u dn u c TTU du cn u or integrating from u = 0, sn u dn u = dn2 u + dn2 (iu, 1c) — 1, cnw = /o dn2 u du + Jo dn2 (iu, ¥) du — u. But the integrals in this formula are E Ef = Zu + gU and — iZ {iu, 1c) + u respectively, and substituting these values and reducing by EE' _ 7r K + K7~ 1 ~ 2KK1 ’ we have the required formula ~ snudnu ttu T,x ----2KK' + lZ{lU' k )- 197. Writing in this equation u — iK’, and observing that Z(-K',k') = -Z(K',k') = 0,VI.] THE FUNCTIONS II (it, a), Zuy ®u, Hu. 147 since Z (K'y k') is what Z (K) becomes on writing therein k' for k, we have Z(iK') sniK’ dn iK' l^iK' ITT 2K ’ which is infinite, =kl, if I is the infinite value of sn iK'. Writing in the addition-equation n = v = \iK', we have 2Z (| iK') = Z (iK') + k2 sn2 \%K' sin iK', sn iK' dn iK' : ~^ÏK' + k2 sn2 J iK' sin iK' — ITT 2K or substituting for sn2£iüT' its value = 1 — cn iK' r+dn iK'f this is . n , fdn %K = SinîA <----rTrri cn iK where the first term is k? (1 — cnijfiT')) "T+dniiT' j ITT 2K’ sin iK' (dn iK' 4- k2 cn iK' + k'2) cn iK' (1 + dn iK) ' and substituting herein for sin ¿if', cn iK’, dn iK' their values, = /, — ii, — ikl respectively, and then making I infinite, the term is = i (1 + k), and we thus obtain Z(\iK') = kJ(\+k)-^K. It will be recollected that z(p:)=i(i-n and we thence by the addition-equation find Z(\K + liK’)=\(k^ik')^^K 198. Starting from Zu = E K u -f /0 dn2 u du, 10—2148 THE FUNCTIONS II (u, a), Zu, ®u, Hu. [VI. we have E Z(u + a) = - ^ (u + a) + ¡-a dn2 (u + a) du, E = ~K + + J° ^n2 (u + a) du + Jo dn2 u du, E = Jr u + Jo dn2 (u + ci) du + Za, ii that is Jo dn2 (u + a) du = ~ u + Z (u -f a) — Za. And similarly, observing that Z(—a)= — Za, we have E Jo dn2 — a)du = ^u + Z (it — a) 4- whence J0 dn2 (u + a) dn — J0 dn2 (u — a) du = Z(u + a) — Z(u — a) — 2Za,. E 199. We find without difficulty d cn u dn u cn2 u du sn u sm u — dn2 u, = dn2 (u + iK') — dn2 u, and thence cn u dn u sn u = G + ƒ dn2 (u + iKf) du— ƒ dn2 u du, = C + Z(u + iK')-Zu. To determine the constant, write u — — \iK\ we have _cnj k') _ jlKK' e (u> k) ©(0, kf) 0(0, Jfer) * © (0 k\ — 1 e(«.fc)=ejo7F)e iKK'^uQ^k^ or substituting for 0 (0, fc'), 0 (0, k) their values, this is the formula of the text.151 VI.] THE FUNCTIONS II (it, a), Zu, ©it, Hu. and thence sn2 (it + a) — sn2 (it — a) = 4 sn a cn a dn a sn w cn it dn it = 2 (1 — k2 sn2 a sn*2 it)2 ’ cZ sn a cn a dn a sn2 u du’ 1 “ k? sn2 a sn2 it ’ d as is at once verified from the relation sn u = cn u dn u. du The equation may be written — ^dn2 (it + a) + ^dn2 (it — a) _ d k2 sn a cn a dn a sn2 it _ d2 _ , du ’ 1 — k2 sn2 a sn2 it * du2 'Uj a whence multiplying by du and integrating from it = 0, d — \ ƒ o dn2 (it + ct) tZit + -J- ƒ o dn2 (it — a) du = ^ II (it, a), or, what is the same thing, ^II(it, a) = Za-\-^ Z(it — a) — % Z(ti + a). Substituting herein for Z(u — a)y Z(u + a) their values ©' (it — a) ©'(it + a) © (it — a) ’ © (it + a)’ multiplying by du and integrating from it = 0, we have n (if, a) = uZa + \ log , . ’ _ _ _ . . . ©'a where for Za we may of course substitute its value, = . The Function ©it resumed. 204. We have d tt / k2 sn a cn a dn a sn2 it cZit n (it’ a)= 1 — k2 sn^a sn2 it : Art. Nos. 204 to 209. = - ^ ~ log (1 - sn2a sn2it), that is @'a ©'(it — a) _ ©'(it + ct) ©^ + © (it - a) © (it + a) — ~ log (1 - k2 sn2 a sn2 it),152 THE FUNCTIONS II (u, a), Zu, ® u, Hu. [VI or what is the same thing, log © (it - a) + log © O + a) = 2 log ©a + log (1 — k'2 sn2 a sn2 u). Integrating in regard to a we have 0 (u — a) ® (u + a) = G ©2a (1 — k2 sn2 a sn2 u), where of course the constant of integration G may be a function of \L To determine it write a = 0, we have 9 b'K ®2u = C®20 = C^^, and then the equation is © (u — a) 0 (u + a) = 2k'K U ®2 a (* ” sn2 a sn2 w)· 205. Writing the differential formula under the form k2 sn a cn a dn a sn2 u = Za + \Z(u — a) — \Z(u + a\ 1 — k2 sn2 a sn2 u ^ if we herein interchange a, a, this becomes &2 sn u cn a dn u sn2 a „ . „ , x , „, , v l-f M·.»·.— *·-**<—«>-»*<« + »>. and adding the two together we have Zu + Za — Z(u +a) = k2 snwsna sn (u + a), viz. we thus reproduce the addition-formula for the function Z. 206. Starting with n (u, a) = uZa — ! log 0 (u + a) and writing herein n + a in place of u, we have II (u + a, a) = (u + a)Za-% log-^— ; we have in the present chapter found the values of II (u + a, a) in the several cases a = ^iK'} a = ^K, a = \K + \iKf.VI.] THE FUNCTIONS II (u, a), Zu, @w, Hu. 153 207. First a = \iK', we have %i(l + k)(u+i iK’) - ilogsn u + i log (q!) = Z$iK')(u + $iK')-lz log ®(u + iK’) ®u that is log sn u = [i (1 + k) — 2Z iK')} (u + \ iK') — A . i ® (u + iK') + og we)+ 08 R* which substituting for Z iK') its value becomes -i· iir , ·. .Trt. , (—10 (w + iK')[ -M(u + i,K) + log « or writing the first term under the form — ~^.{K' — 2iu), and taking the exponentials of each side, sn u = 2iu) — i © (u + iK') e ■ vi· 208. Secondly a — \ Ky we have l(l-k'){u + \K)-^logdn u + % log Vi' = Z{\K){n + \K)~h log that is log dn u = {1 — k' — 2Z if)} (a + \K) 4- log \/k' + log 0 (u + K) 0w ’ where the term in u + J K vanishes by reason of the value of Z (\K)y and passing to the exponentials we have dn u = Vk' ®(u + K) ©¿i 209. Thirdly a = £ K + £ iK', we have | (4 + +2^+2 — h l°g cnit + J log sj = Z(hK + i iK')(u + \K + \iK')-h log e (tt + q/- —}154 THE FUNCTIONS II (u, a), Zu, @m, Hu. [VI. that is log cn u = {* + ik' -2Z(^K + %iK’)) (u+^K+ \iK') + ®(u+K' +iKr) ©it or substituting for Z(\K+%iK') its value, the first term is ~ (u + \K + \iK'), which is = — ■— (K' — 2iu) + ^: hence passing to the exponentials and observing that we have V k ~ V k ’ ‘H. == 0 4K cn u = e '-2 iu) fk' s/k ®(u + R + iKQ Sn Recapitulation. Art. No. 210. 210. We now see that the elliptic functions sn u, cn u, dn n, that the elliptic function of the second kind considered as a function of u, and for convenience replaced by Jacobi’s Zu, and that the function of the third kind considered under Jacobi’s form II (u, a), are all of them expressed in terms of the single function © (u), and the ¿-functions K, K\ viz. that we have sn u = e -£g{K'-2iu) . ~l%(u+iK') vk cn m = e iK{K'~Uu) @ (M + K + iK') O), dn u = Vk' © (u + K) (-h), denom. = Su, viz. these are fractional functions having the common denomi- nator Su, and having also ©-functions in their numerators; and further that Zu = ©'^ ©i’ II (u, a) = ©'a ©a + ilog © (u — a) © (u + a) ’VI.] THE FUNCTIONS II (u, a), Zìi, ©w, Hu. 155 and conversely that ©w is a function derived from sn u by the equation ©M=ei (1_ i) (involving the ¿-function E, Legendres And we have also proved the formula f¥K --5^ 1 ©24 = kK cn u or as this may also be written 0 (iu, k'), © (iu) and the formula m) ~ \! kK' ITU2 e4KK’ cn (iu, k') © (iu, k') ; © (u -1- a) 0 (u — a) = ©22t ©2& (1 — ¿2 sn2 u sn2 a). The Function Hu. Art. Nos. 211, 212. 211. If introducing for convenience a new function Hu*, we write Hu = - ie~& (K’~2iu) © (« + iKr), and therefore also H (« + K) = - ie~”K {K’ 2iu)+ii" ®(u+K + iK'), —rgAK'-2iu) Trx = e 4K 0(u + A), * If instead of Jacobi’s ©, H we use the four functions 0«, ©2w, 6stt, = Qu, Hu, j^J^H (u + R), yJk'Q (u + K) respectively, then 0^/, 02w, 03w are the numerators, and On the common denominator, for the three elliptic functions sn, cn, dnw. The four functions 0lf 02, 03, 0 have been tabulated under the superintendence of Dr Glaisher, but are still unpublished.[VI. 156 the functions II (u, a), Zu, ®u, Hu. then the formulae for the elliptic functions become sn«= -i Hu v& cn u = H (u + K) (-0. dn u = Vi' 0 (u + K) (+)> where denom. = ©w. It hence appears that Hu is an odd function of uf which, for u indefinitely small becomes = /2 kk'K u. 212. Combining with @ (u + a) 0 (u — a) = 02m ©2a (1 ~ &2 sn2 u sn2 a), the equation sn (u 4- a) sn (u — a) = sn2 u — sn2 a 1 — A;2 sn2 u sn2 a 5 and attending to the expressions of sn u, sn a in terms of H, 0, we have H (u + a)H (u — a) = (if 2w©2a — H2a&2u). The Function II (m, a) resumed. Art. Nos. 213 to 218. 213. We deduce the addition-equation for the function of the third kind II (u, a), viz. we have first II («, a) + II (v, a) — II (u + v, a) = U°g © (u - a)0(i/-a)0(ii + v + a) 0 (u + a) 0 (v + a) 0 (u + v — a) (=^-log fl, suppose), where the logarithmic term containing the functions © may be in three different ways made to depend on the functions sn.VI.] THE FUNCTIONS II (a, a), Zut &u, Hu. 157 214. First we have ©(tt--a)0(tf--a) = ^©2£(>-'tO ©2(i(u + v)-a) {1 — k2 sn2 \ (u — v) sn2 (u +v) — a)}, © (u + a) © (v + a) = ^ ©2 £ (a - v) ©2 (i (u 4- v) + a) {1 — k2 sn2 ^ (w — v) sn2 (w + v) + a)}, ©a © (u + v — a) = ©21 (u + fl) ©2 ( J (u + v) — a) {1 — k? sn21 (w + t/) sn2 (u + v) — a)}, ©a © (m + t> + a) = ©2 \ (u + tf) ©2 (i (w + v) + a) {1 — k2 sn2 \ (u + v) sn2 + v) + a)}, and taking the product of the first and fourth expressions divided by that of the second and third, we have {1—&2sn2|(^—?;)sn2(|(^-H;)—o>)]{1— ^2sn2^(tt+r)sn2(^(^+?;)+a)} {1— k2sn2^(u—v)sn2(^(t64"'y)+a)j {1— &2sn2^(iJ-H;)sn2(^(w+v)--a)} ' 215. Secondly we have ©2 {u — a) ©2 (v — a) = ©20 . © (u — v) © (u + v — 2a) 4- {1 — k2 sn2(u — a) sn2 (v — a)}, ©2 (u + a) ©2 (v + a) = ©20 . © (u — v) © (u + v — 2a) -r {1 —k2 sn2 (u + a) sn2 (v + a)}, ©2a ©2 (u + v — a) = ©20 . © (u + v) © (u + v — 2a) — {1 — k'2 sn2 a sn2 (u + v — a)}, ©2a ©2 + v + a) = ©20. © (/a + v) © (u + v + 2a) 4- {1 — k2 sn2 a sn2 (i£ + v + a)}, and then in like manner we obtain n = — k2 sn2 (u 4- a) sn2 (v + a)} {1 — k1 sn2 a sn2 {u + v — a)j — k2sn2 (a — a) sn2(v — a)} {1 — &2 sn2a sn2(u + v + a)| 216. But, thirdly, from the form originally obtained for the addition-equation, the same quantity should be ^ 1 — k2 sn a sn u sn v sn (u + v — a) ~ 1 + k£ sn a sn w sn vsn (ii + v + a)* The transformation is effected as follows:158 THE FUNCTIONS IT (u, a), Zu9 ®u, Hu. [VI. We have {1 — k2 sn2 ^{a H- v) sn2\{u — #)} sn u sn v = sn2 — sn2 £ ( m — v), {1 — k2 sn2 4- a) sn2 (£ (u + v) — a)} sn a sn (u 4- v — a) = sn2 \ (u 4- v) — sn2 (£ + v) — a), and taking the products of the two sides each multiplied by — k2y and adding a common term on each side, we have {1 — ¿2sn2A(w 4- v)sn2|(w ~ v)} {1 — &2sn2|(w 4- v)sn2(-J(-it 4- v)-a)} x {1 — k2 sn a sn u sn v sn (u 4- v — a)}, = {1 -k2 sn2£(w 4- v) sn2\(u -v)} {1 —J?sn2£(tt4-«0sn2(i(tt+v)“"a)} —&2{sn2£(w 4- v) — sn2£(w — v)} {sn2J(w 4- v) — sn2(J(-u 4- v) — a)}, = 14-/4 sn4 \{u 4- v) sn2 ^(u — u)sn2(|(?£ + v) — a) — &2 sn4 + v) —k2 sn2 ^(u — v) sn2 (£(& 4- tf) — a), = {1 — k2 sn4\{u 4· v)} {1 — k2 sn2^(w — i>) sn2 (J (u 4- v) — a)}*. Changing the sign of a we have a second like equation, and dividing one by the other, we find the required equation {1—fc2sn2K^4tt)sn2(Ku4-tt)4-ft)Hl—/?sn2^(M-tt)sn2(^(^4-fl)—a)} (1—fc2sn2-!(ii4v)sn2(£(w+v)—a)}[l— &2sn2£(«—fl)sn2(^w+v)-|-a)} 1 — k2 sn a sn u sn v sn (u 4- v — a) ~~ 1 + k2 sn a sn u sn v sn (u + v 4- a) ’ 217. The conclusion is II (u, a) 4* II (v, a) — II (u + v, a) = \ log fl, where fl is expressed in the three forms just obtained. * The identity, writing therein u, a, v for \(u-v)y 4(a + t>), £(w + v)-a, becomes 1 - fc2 sn (a + u) sn (a - u) sn (a 4- r) sn (a - v) {1 - k2 sn4 a} {1 - k2 sn2 u sn2 v} {1 - k? sn2 a sn*u} {1 - k2 sn2 a sn2 v} *VI.] THE FUNCTIONS II (¿¿, d), Zu, ©U, Hu. 218. In the equation 159 interchanging u and a, we obtain (observing that © is an even function) which is the theorem for the interchange of amplitude and parameter. And we hence deduce II (u, a) + II (u, b) — II (u, a + b) = II (a, u) + H (b, w·) — II (a + 6, w) + w — Z (a +5)}. Here on the right-hand side by the addition-theorem the first term is = £ log 12', where 12' is the same function of a9 b, u that 12 is of u, v, a: we have thus 12' in three forms one of which is 1 — sn a sn b sn (a + b — it) sn u ~~ 1 k2 sn a sn b sn (a + b + u) sn it ’ and the second term is, by the addition-theorem for Z} = k2 sn a sn b sn (a + b) u; we have therefore n (u, a) + n (itt b) — n (u, a -h b) = k2 sn a sn b sn (a + b) it + \ log 12', which is the theorem for the addition of parameters. Multiplication of the Functions ©¿¿, Hu. Art. Nos. 219, 220. 219. From the equation and thence n (uf a) - n (a, u) = uZa - aZu, ©(li+ti) ©(zj — v) =160 THE FUNCTIONS II (u, a), Zu, ©It, Hu. [VI. we deduce 0 (2m) = and it is hence easy to see that ®nu 4- 0n2 (u) is a rational and integral function of sn2 u of the degree %n2 or £ (n2 — 1) (that is n2 or n2 — 1 in sn u) according as n is even or odd. More precisely we may say that Snu . ©^“^O 4- 0n2i£ is such a function, reducing itself to unity for sn u = 0; and it thus appears that considering sn nuy cn nu, dn nu as expressed in terms of sn u by the multiplication formulae, in such wise that for u = 0 the denominator is = 1, then this denominator will be respectively. It will appear in the sequel how we thence obtain the expressions of these numerators and denominator. = %nu. 0n2_10 4- %n2u. Vi' 0 {nu + K) 0w2—10 4- ®n2u} Tables for the Functions Zu, ©w, Hu. 221. I annex the following tables taken from Dr Glaisher’s paper “ Values of the Theta and Zeta functions for certain values of the Argument.” Proc. R. Soc. t. xxix. (1879),VI.] THE FUNCTIONS II (u, a), Zu, ©w, IZw. 161 pp. 351—361, of the values of Zu> Hu for the values u = 0, K, &c. u Zu Qu Hu 0 0 2 h'^K? Tri 0 K 0 2 %K% 2^k^K? TT* iK· 00 0 . _i2hW T i 7T* K+ iK' -4- *K 2hiRi _i 2^^ * —I 7T^ q 1 7T* 2 K 0 2 h'^K? 0 2 iK' * K - 3_1 .. 0 2 K+ iK' 00 0 . _i2^k'^K^ n t 1 7T^ K + 2iK' ITT ~K . ziici 9_1—i- 7TlT , 2?k?K? 9 *4 2K + 2iK' ITT ~ K 2h'lK$ ~ 9"1 —i— 7T^ 0 * By mistake given as 0 in p. 164 of the Fundamenta Nova. Wo have moreover u Zu Qu Hu w 4(1-*') 2T^‘*'i(l + *')i 7TT k'i (1 - i:')i %K -1(1-*') „ „ (1 + *1)* „ (1-*')* \K+iK' -if+4(1+*') 2 “i^^il-*')* (!+*·) 9_1-TT-*'4(1+ *')*(! + *) 2M %K+iK' -if-4(i+*') „ „ (l-fc'^a-i) „ „ (1 + k’)i (1 - {) c. 11162 the functions Π (u, a), Zu, Θμ, Hu. [VI.VI.] THE FUNCTIONS II (u, a), Zu, Su, Hu. 163 _jrX' where in these last formulae q is = e K and 0 is the angle of the modulus, k = sin 6; whence (2* + (1 + k'ff . 1fl (ti-a+k'f? cos \6 = ---—j----— , sin \Q = ---------——, 24 24 .. _ {2* + (1 — &')*}* . ' (2i-(l-A:')i)i cos i (w* — =-&--^-----J-L, smi(TT-0) = '----^----'-Lt and where we have also u e2« ii2tt \K+ \iK* q ~ 1 - kh'i {(1 + k')k + i (1 - k')h 7T q-?Kkik'l TT {(l-*')* + i. >« (k'-ik) 3 * » ,» (-k' - ik) %K+%iK' 3~* n >1 (kf-ik) 9 3 * » n (-fc'-ifc) %K+%iK' 3~* ,, ,, (fc'-M'fc) _ 9 3 ¥ If M (_*/ + ,·*) 11—2164 [VII. CHAPTER VII. TRANSFORMATION. GENERAL OUTLINE. 222. The theory of transformation is considered in the first instance in regard to the differential expression ^which, for doc \ the elliptic integrals, has the particular form ^----— ^ , and then to the elliptic functions sn, cn, dn. Case of a general quartic radical VX Art. Nos. 223 to 226. 223. Consider the differential expression where Y is a V Y given rational and integral quartic function of y. Write herein y=y- where U and V are rational and integral functions of xy one of them of the order p, the other of the order p or p — 1: such a fraction is said to be of the order p. It is to be shown that the coefficients of U, V may be so determined as to lead to an equation Mdy _ dx where X is a rational and integral quartic function of xy and if is a constant. We have iy.^VU'-VlDd*, r= y,(r, uy,VII.] TRANSFORMATION. GENERAL OUTLINE. 165 where considering Y as a homogeneous quartic function of (1, y), then (V, U)4 is what this becomes on writing therein V9 U in place of 1, y respectively: viz. (V, U)4 is a homogeneous quartic function of Z7, V, and therefore of the order 4p in x; VUr — V'U, if V, U are of the same order p, would at first sight appear to be of the order 2p — 1, but in this case the coefficient of vanishes and the order is really = 2p — 2 ; viz. whether the orders of Uy V are p, p or p, p — 1, the order of VU' — V'U is = 2p — 2. The foregoing values give dy (VU'-TlDdx vf” ^(v7uy 224. It is at once seen that if (V} U)4 has a square factor (x — a)2 then x — a divides VU' — V'U. Similarly if (V, U)4 has 2p — 2 such factors, or if it is = T2X, where T2 is of the order 4p — 4 and therefore X of the order 4, then the product T of the roots of the square factors divides VU' — V' U, and since VU'— V'U and T are each of the order 2p — 2 the quotient (VU' — V'U)+T must be an absolute constant if-1. But in this case we have Mdy dx an equation of the required form. 225. Regarding ¡7, V as being each of them of the order p, the expression contains 2p + 1 constants, and in determining Uy V so as to satisfy the condition ( Vf U)4 = T2X we determine 2p — 2 of these: there thus remain three arbitrary constants: this is as it should be, for if the required condition is satis- fied by any particular values ¡7, Vy it will also be satisfied by the new values obtained by writing in the fraction CL Qx in place of x the function ^ + three arbitrary constants. We may by such linear transformation make either U or V to be of the order p — 1, or if we please begin by assuming this166 TRANSFORMATION. GENERAL OUTLINE. [VII. to be so. But we cannot have either U or V of an order inferior to p — 1 ; for if this were the case VU’ —V'U would be of an order inferior to 2p — 2, while in fact it divides by T which is of the order 2p — 2. Considering F as a given quartic function of y, the function X is obtained as an arbitrary linear transformation of a deter- minate quartic function of x : or what is the same thing, it is a quartic function containing a single parameter which cannot be assumed at pleasure, but is a determinate function of the coeffi- cients of F, different according to the different values of the number p : which number is termed the order of the transform- ation. 226. It is to be observed that we cannot have any other really distinct transformation of the differential expression 3£~^dx . /— into the form —7=— with the same radical vl and a con- vx stant value of M: for suppose that such transformation existed; say by writing y = Function (z) we could obtain ^L· = where Z is the same quartic function of z that X is of 1 nr · j. M~xdx N~xdz . j . xy and N is a constant: then — 1=- = —=— = —=r-, that is VF vi nZ Xdx Mdz = —r=r; such an equation is integrable algebraically when v X VZ M, X are commensurable, that is proportional to integer numbers m, n; and from the form of the integral we infer that the equation is not integrable algebraically unless M, N are commensurable: hence N, M must be commensurable or the last-mentioned equation must be of the form ; and . . Vi ^ we have thus a known algebraical relation between the quanti- ties x, z such that by means of it we can pass from one to the other of the transformations y — y — Funct. (z): the two transformations would on this account be regarded as not essentially distinct the one from the other.VII.] TRANSFORMATION. GENERAL OUTLINE. 167 The standard form . 1 ■■■.-. Art. No. 227. , Vi-sM-Jfcw 227. The theory applies in particular to the case of a differential expression of the form dy Vi-yM-xy’ viz. this by a transformation of the form y = ^, of the order ft, can be converted into one of a like form in regard to %, that is we obtain a relation Mdy _ . dx Vi- y*.i -xy ~ xki -kw 3 where, h or A being given, the other of them and also the value of the multiplier M are each determined, not uniquely but by means of an equation called the modular equation, between k and X : more precisely, if k or X be given, the other of them may be taken to be any particular root of the modular equation, and then the coefficients of U, V, and the multiplier M, are determinate functions of k, X. Distinction of cases according to the form of ft. Art. Nos. 228 and 229. 228. In the case where ft is a composite number = qr, the modular equation breaks up, and the transformation in fact decomposes into distinct transformations. That this may be the case is clear a priori, viz. if we have 2 = 5 a rational * l function of % of the order q, giving rise to a relation Mftz dx Vl-22.l-i V1 — æ2.1 — k?x*3 and y = W* a rational function of z of the order r, giving rise V 2 to a relation M2dy dzTRANSFORMATION. GENERAL OUTLINE. 168 [VII. then for z substituting its value in terms of x, we have clearly y = ■? a rational function of x of the order qr} giving M^M^dy _ dx Vl-I/2. l-Xy-'Vi-a?.1 but to show that the case is of necessity so would require further investigation, and the question is not entered upon in the present work. Assuming the property in question, it appears that the transformations belonging to the several prime numbers need alone be considered; viz. the cases n = 2 and n an odd prime =p. The case n = 2 presents certain peculiarities. 229. n = 2. There are in this case two distinct rational J)X transformations, one of them of the form y = ^ ^ ^ (viz. here 1—k' y vanishes with x)} for which the new modulus is X, = —■ , cl "4~ and the other of them of the form y — , for which the c + dx2> 2s/k new modulus is 7, = : these will be considered. It is to be observed that for the case in question n = 2, X and 7 correspond respectively to the real moduli X and \x belonging to the case n, an odd prime, as presently mentioned: A' j£' p»' viz. we have the equations i ^ = = % pr precisely corre- 1 A' K' A' sponding to the equations - = n ~~ afterwards men- tioned. But in the case of n an odd prime, X, X1 are roots of one and the same irreducible equation: moreover (as afterwards appears) y, = sn X^ and y, = sn , X^ are each given in terms of x, = sn u, by a rational transformation of the form y = ^ where y vanishes with x: whereas in the present caseVII.] TRANSFORMATION. GENERAL OUTLINE. 169 n — 2, the corresponding functions V> = sn {(1 + ¥) u, X}, y, = sn {(1 + 7} are (as will be seen) given in terms of x, = sn u, the former by an irrational, the latter by a rational transformation, y in each of them vanishing with x. Instead of at once proceeding to the case of n an odd prime, we take in the first instance, n any odd number whatever. n an odd number: further development of the theory. Art. Nos. 230 to 235. 230. We have here the formula = a(l, ' V ~ (1, ’ viz. the numerator is an odd function of the order n, and the denominator an even function of the order n — 1. We may proceed somewhat further in the determination of the form: for this purpose take P, Q even functions of x, such that P + Qx is of the degree ^ (n — 1): for instance n = 3, P + Qx=-&+ fix, ord. P = 0, ord. Q = 0, n = 5, P + Qx = a + fix + ya?, ord. P = 2, ord. Q = 0, n = 7, P + Qx = a + fix + 7a? + 8a?, ord. P = 2, ord. Q = 2, n = 9, P + Qx = a + + yx? + 8a? + ea?, ord. P = 4, ord. Q = 2, and so in general; viz. n = 4*p — 1, the orders of P and Q are each = 2p — 2, but = 4p + 1, order of P is = 2p and that of Q is = 2p — 2. 231. This being so, assuming 1 — y _ (P — Q#)21 — # l + y“(P + Q#)21 + # ’ we see that a(P2 + 2PQ + Q20 y’~ P* + 2PQa? + Q‘a? ’ is a function of the above-mentioned form; and not only so, but170 TRANSFORMATION. GENERAL OUTLINE. [VII. forming the equations l-y = (P-Q*)!!(l-a:) O), 1+y = (-P+Q^)a(i + «) (+), where denom. = P2 + 2 PQx2 + Q2x2, we see that 1 — y and 1 + y have each of them the required property of having in the numerators a square factor of the proper order. 232. It is next to be observed that the functions P, Q may be so determined that the expression for y remains unaltered 1 1 when we simultaneously change x into y-, and y into — . rCX \y To see how this is, write for shortness xN (1, of) ^=p(hv Ny D being as above functions each of the order 1) in x\ We have * (L· «· and considering the coefficients, say of N (1, a?2), as given, we can at once determine those of D (1, a?) in such manner that, il being a constant, we have identically N(k2x2, l) = ilD(l, x2). In fact the coefficients of D will be those of N taken in the reverse order and multiplied each by the proper power of k. This being so, we have and this identical equation, writing for xy becomes whence identicallyVII.] TRANSFORMATION. GENERAL OUTLINE. 171 Suppose that writing ^ for x, y is changed into y, then or multiplying by y and reducing by means of the result just obtained, we have 02 viz. writing — = - we have y = —; and thus we may simul- fc A Ay taneously change xt y into -r- > > the theorem in question. lex A y 233. Or, in a somewhat different form, the theorem is at once seen to hold good provided we have * (i _ £Wi _ ^ = M\ aV\ ¥) y (1 — k2a2x2) (1 — k2b2a?) ... 9 for then, making the change in question it becomes 1 1 (1 — k2a2x2) (1 - k2b2x2)... Xy~ Mkn(ab...y ^ ^ ^ ^ which is in fact the original equation provided only X = M2kn (ab...)4* We thus in effect determine X as a function of k (viz. these are connected by an equation called the modular equation), and then the coefficients of P, Q are determined in terms of k3 X. 234. The required condition being satisfied, we may in the formulae which give 1 — y, 1 + y make the same change; and it is easy to see that the resulting formulae will be of the form 1 - Xy = (P'- Q’xf (1 - kx) (+), 1 + Xy = (P' + Q'x)2 (1 + kx) (-5-), Tcn 1 * Comparing with the former equation \=—2, we have —=M[ab...)*.172 TRANSFORMATION. GENERAL OUTLINE. [VII. the denominator being of course the same as before : hence the required condition as to the square factor is also satisfied by each of the functions 1—Xy, 1+Xy; and the integral relation between y, x leads thus to the required differential equation Mdy _ dx V1 — y2.1 — X2y2 Vl — x2. 1 — A2#2 235. Supposing that n is not a prime number it will be the product of two or more odd primes, and the transformation will break up into distinct transformations each of which may be separately considered. We therefore now assume n an odd prime: the modular equation is in this case an irreducible equation of the order n +1, so that A being given we have n 4-1 different values of X; and corresponding to each of them we have a distinct formula of transformation. This modular equation is conveniently expressed as an equation between the two quantities u = \/k, and v = viz. it is an equation of the form ('u, v) = 0 where (u, v) is a rational function of the degree w+1 as regards each of the quantities (u, v) separately. It is to be added that k2 being as usual positive and less than 1, there are two and only two real values of X2 (which values are also positive and less than 1) : and corresponding to them there are two real transformations: but this is a property which may in the first instance be disregarded. Application to the Elliptic Functions. Art. No. 236. 236. We have in what precedes a purely algebraical theory of transformation: in particular, in the case where the order n is an odd number, if in the formulae wre write y = sin %, x = sin , the differential equation becomes - ; r ^ A(\, X) MM) and further assuming sin % = sn (v, X), sin = sn (u> A), then it becomes Mdv = du, giving (since u and v vanish together) v = jj ; whence x = sn (u, A), y = sn iXJ : and the theoryyn.] TRANSFORMATION. GENERAL OUTLINE. 173 is an algebraic theory of transformation, serving to express sn x) in terms of sn (u, k). The theory may be completed algebraically without much difficulty in the cases, n = 3, 5, 7 ; but there is great difficulty in doing this generally for larger values of n: and it is in fact completed by Jacobi, not algebraically but transcendentally, by expressing X and the coefficients of the transformation by means of the sn, cn and dn of ^ (m and m' integers), or say by means of the functions dependent on the rc-division of the complete functions K, K'. n an odd-prime, the ulterior theory *. Art. Nos. 237 to 239. 237. In particular when n is an odd-prime, there are as already mentioned two real transformations; a first transforma- tion from k to a smaller modulus X, involving the functions of —; and a second transformation from k to a larger modulus X, n ° iK* involving the functions of , And in these two cases (taking K, A, Aj, Kf, A', A/ for the complete functions to the moduli k, X, Xx, k\ X', X/ respectively) the modular equation is replaced A' ]£' J£' A ' by the equations ^ = n > ~K==n7t respectively: viz. these transcendental equations contain the relations between the original modulus k and the new moduli X and X1 respectively. * Observe that X, heretofore used to denote any one whatever of the n + 1 roots of the modular equation, is in what immediately follows used to denote a particular root, and another particular root, the roots belonging to the first and second real transformations respectively. In Nos. 241 et seq. X is again used at the beginning to denote any root, and (X) a determinate root correspond- ing thereto, these are taken to be first the particular roots (X, XJ, and secondly the particular roots (\lt X). It would, abstractedly, be advantageous to reserve X as the symbol of any root whatever, using \, X2 for the particular roots : but this would have occasioned a very frequent alteration of Jacobi’s notation.174 TRANSFORMATION. GENERAL OUTLINE. [VII. 238. The equations just referred to are obtained from the following: K K' nM* if’ A _ K K* MS Al nM\' which present themselves in the theory. As regards these equations it may be observed here as follows: 239. The first transformation is a relation between sn xj , sn (u, Jc), and it leads to the equation A = . Effecting on the transformation-equation Jacobi’s imaginary substitution, we obtain from it a complementary first transform- ation, giving sn X'j in terms of sn (it, Jc'), and this leads to K' the equation A' = Similarly the second transformation is a relation between (iET’ * sn^u> an(^ ^ ^eac*s the equation A1 = §. Effecting on the transformation-equation Jacobi’s imaginary substitution, we obtain from it a complementary second trans- formation, giving sn X^ in terms of sn (u, k'), and this K' leads to the equation A/ = , or recapitulating, K first transformation gives A = , IT M> _K MS K' complementary second „ A\ = , the chief object of the complementary transformations being in fact the deduction of these second and fourth equations. sn complementary first second A' = · A, = A',-;VII.] TRANSFORMATION. GENERAL OUTLINE. 175 Connection with Multiplication. Art. Nos. 240 to 245. 240. The theory of transformation is connected in a very remarkable manner with that of multiplication. This is the case as well for an even as an odd number n, and indeed the connexion will be exhibited in the case, n = 2, of the quadric transformation, but here one of the transformations is irrational: and it is convenient to restrict the attention to the case n an odd number, where the transformations are both rational; or rather (this being the only case which has been completely developed) we may at once take n to be an odd- prime. 241. This being so, starting with the transformation-equa- tion y = of the order n, which gives Mdy __ dx V1 — y2.1 — \2y2 V1 — x1.1 — k2x? ’ we may imagine a new variable ^ connected with y by a P transformation-equation z = of the same order n (P, Q rational and integral functions of y) giving Ndz _ dy V1 - Jg*. 1 - (A)2 z2 ~ V1 - y2.1 - xy’ where (X) is not of necessity the same function of X that X is of k, but a like function; viz. X, k are connected by the modular equation, and changing herein k into X and X into (X) we have the relation between X, (X). And we have then z a fractional function of x such that MNdz dx \fl-z2.\-{\)2z2^ Vl — ¿c2.1 — k2a?* 242. It is a property of the modular equation that we may have (X) = k. and further that when this is so MN = -: the n last-mentioned equation then is dz _ ndx Vl — z2. 1 — k2z2 Vl — x2. 1 — kPa? *176 TRANSFORMATION. GENERAL OUTLINE. [VII. viz. x being as before taken = sn (u, k), we have z = sn (nu, k) ; and the relation between z, x then gives sn (nu, k) as a function of sn (u, k), viz. the expression is a fraction, the numerator being an odd function of the order n2 and the denominator an even function of the order n2 — 1 ; this is in fact the expression of sn (nu, k) in terms of sn (u, k) given by the multiplication- equation. Observe that for obtaining in this manner the trans- formation x to z (or sn (u, k) to sn (nu, k)), the transformation x to y may be any one at pleasure of the different trans- formations, but that (regarding it as given) we must combine with it a determinate transformation y to 2, the resulting transformation x to z being of course independent of the selected x to y transformation : there are thus as many ways of obtaining the final x to z transformation as there are trans- formations x to y. In the case n an odd-prime, this may be considered more in detail. 243. Selecting the root X of the modular equation we have a real transformation (Jacobi's first transformation) y = ^ giving (M real) Mdy _ dx Vl - f. 1 - \y ~ V1 - a?. 1 - tea?' and selecting the root Xx of the modular equation we have a real transformation (Jacobi's second transformation) y = 5 * 1 giving (Mx real) Mxdy_______________dx -X,y- VI-as*, Now X is in fact the same function of k that k is of Xx: this at once appears from the before-mentioned relations A' Kf K' A/ A ~~n K’ K~n A,’ UX Hence taking z such a function of y, X as is of x, k, the *1 differential relation between z, y is Ndz dyVII.] TRANSFORMATION. GENERAL OUTLINE. 177 and consequently, MN being = -, we have dz ndx 244. Or again, taking z such a function of t/, Aq as ~ is of x, A:, the differential equation between z, y is Ü Nxdz dy Vl - s2.1 - fe2 Vl - 3/2.1 - xxy ’ and consequently, M1N1 being = ^, we have in this case also dz ndx Vl-sM-te2 Vi-#2. 1 -tw’ so that in each case, x being = sn(?£, k), we obtain the same value 2 = sn (nu, k): viz. in the first case we pass by a first and then a second transformation from k through X to k; and in the second case by a second and then a first transformation from k through X1 to k. 245. As regards the equations JOT=i, these follow from the before-mentioned equations ^ K ljr K M~nA’ Ml Aj’ viz. N being what M1 becomes on changing therein k, X1 into X, k, and N what M becomes on changing k, X into X1} k, we derive from these N-— N — — iV“K> iVl — nK’ and thence the equations in question. A/ K' Jacobi in connexion with the equations = n -gr and A / 1 K remarks, Fundamenta Nova, p. 59, that if n be a Ai n K composite number =wV', then, in the transformation of the order n, there is corresponding to each real root of the modular c. 12178 TRANSFORMATION. GENERAL OUTLINE. [VII. equation a relation of the form ^ ^ ^: whence in particular A' ]£' if n be a square number, the equation is = -gr, viz. we then have \ = k, showing that in the case where n is a square number there is among the transformations of the order n one wThich gives the multiplication by V71. He further remarks, p. 75, that \ being any root whatever of the modular equation there exist equations of the form a A + if3 A' aK+ibK' nM ’ a'A' + ij8'A = a'K' + ib'K nM where a, a', a, a' are odd numbers, 6, £>', y3, /S' even numbers, such that aa' + bb' = 1, aa' -f fi/3' = 1: and (same page in a foot- note) as follows: “ Accuratior numerorum a, a', 6,6', &c. determi- nate pro singulis ejusdem ordinis transformationibus gravibus laborare difficultatibus videtur. Immo haec determinate, nisi egregie fallimur, maxime a limitibus pendet, inter quos modulus k versatur, ita ut pro limitibus diversis plane alia evadat. Id quod quam intricatam reddat qusestionem, expertus cognoscet. Ante omnia autem accuratius in naturam modulorum imagina- riorum inquirendum esse videtur, quae adhuc tota jacet quaestio.” That some such equations exist may be inferred without difficulty from the general formulae of transformation, but the strict proof, and certainly the determination in question, would depend upon investigations out of the field of the Fundamenta Nova. The property is used by Jacobi to show that the proof which he gives of the equation M2 = ^ ~, where \ denotes in the first instance the real root, applies to the case of any root what- ever.Vili.] 179 CHAPTER VIII. THE QUADRIC TRANSFORMATION, n = 2; AND THE ODD-PRIME transformations n = 3, 5, 7. properties of the MODULAR EQUATION AND THE MULTIPLIER. 246. The case n = 2, although very analogous to the case n an odd prime, presents, as remarked in the preceding Chapter, some essential differences; there are analytically dis- tinct transformations relating to the two new moduli X and y respectively, viz. these are not roots of one and the same irreducible modular equation: and it is an irrational trans- formation which in some sort corresponds to one of the real transformations in the other case. There is an a priori necessity for this: viz. as sn 2u is not a rational function of sn u, we cannot have here two rational transformations leading to the duplication: the duplication must arise from the com- bination of a rational and an irrational transformation. It should be noticed that the case may be studied quite inde- pendently of, and in fact previous to, the general theory ex- plained in the preceding Chapter. The Quadric Transformation. Art. Nos. 247 to 262. 247. It has been shown geometrically that, considering 1 __ a new modulus X connected with k by the equation X = 9 and establishing between — 0), or, what is the same thing, Vl — A3 sin2 12—2180 QUADRIC TRANSFORMATION. [VIII. we have between , 0 the differential equation (1 4- k') d d6 Writing herein sin = x, sin 0 = y, the relation between y, x is ^ __ (1 + k') x Vl — ¿r2 ^ V1 — k2a? this is in fact the first form of quadric transformation, and (as is about to be shown) it is connected with a second (1 + k) x form2' = TT^· Modular relations. 248. From the original modulus k we derive two moduli 7, A; these form a decreasing series 7, A, A, the relations between them being 2 vT k 2 Vx 7 = ‘1 + *’ _1+X’ 1-* 1 -X 7 = ' 1 + * ’ k' -1 + X’ Jfc = 1 -7' 1+7" A 1-ff 1 + *'1 and the corresponding complete functions T, T', K, K', A, A', are connected by the equations (l + XJA.K-jljf, ta + XjA'-K'-jljI·, whence also ^ ^ = 2 ^ . jFVrsi and Second Transformations. 249. We pass by a quadric transformation from the , . dx dy differential expression . - - to - - or r Vl-aM-te2 Vl - y2.1 - A2y2VIII.] QUADRIC TRANSFORMATION. 181 “--- — , viz. in the former case the transformation is v 1 — y'2.1 — , dx idX and . - = — — ; Vl — a?. 1 — k2a? Vl — X2.1 — &'2X2 and the differential relation is therefore changed into (1 +X)dY _ 2 dX Vl-FM—V2F2" Vl-X2.l-jfc'2X2> the integral equation is changed into viz. this is F (1 + AQX Vl- F2 Vl -X2.1-£'2X2' (i + aqx 1 + ¿'X2 ' which integral form gives therefore the last-mentioned differ- ential relation: observe that this integral form is what the second form becomes on writing therein X, F for x, y, and for k the complementary modulus k'. Moreover since X, F increase simultaneously from 0 to 1, the differential equation leads to (1 + A) A' = 2K', which is another of the above-mentioned integral relations.VIII.] QUADRIC TRANSFORMATION. 183 252. Similarly, if in the second form we effect Jacobi’s imaginary transformation, then the differential equation is changed into dY _ (1 + k)dX Vl — F2.1 - y'2Y2 Vf^XM -k'*X* ’ the integral relation between x, y is changed into leading to Y (l+ifc)XVl-X2 Vl - F2 _ 1 — (1 +k)X2 ’ vr^pz2 ’ which integral form gives rise therefore to the last-mentioned differential relation: observe that this integral form is what the first form becomes on writing therein X, F for x, y, and for k the complementary modulus k'. Moreover as X passes from 0 to _ — , F passes from 0 to 1, and as X, continuing vl +k to increase, passes to 1, F passes from 1 to 0: the differential equation gives therefore 2r' = (1 + k) K', which completes the set of integral relations. The Duplication Theory. 253. We may in two different ways combine the two transformations, and thus in two different ways obtain a “Duplication by two quadric transformations.” First duplication (through A). Writing (l+\)y (l+&')Wl-a? 1 + X#2 ’ V1 — l&a? we have by what precedes dy _ 2 dx _ 1 dz Vl-2,2.l-\y“·1+x Vl - tf2.1 - k2x? ~ l + x Vl-s2. l-jfci ’ and therefore dz 2 dx184 QUADRIC TRANSFORMATION. [vill. where, from the assumed integral equations, _ 2# Vl — ¿r2. Vl — k2x2 1 — A 254. Second duplication (through y). Writing ^=0* + v')y^i-y2 __ (l + k) x Vl - 7y ’ y 1 +  ’ we have by what precedes (1 + k)dz __ 2dy V1 — -S'2.1 — k2z2 V1 — y1.1 — ry2t/2 * dy __ (l4-&)d# Vl-ƒ .l-7¥~ VI^#2. 1 -A^c2’ and therefore cfo _ 2dx Vl — z2l — A^2 V1 — ¿c2.1 — Ar*#2 * and the two integral equations give, as in the first duplication, _ 2# V1 — #2 V1 — k2a? 1 — A; V 255. In the first duplication, assuming x = sn (w, jfc), y = sn (v, X), ^ = sn (w, k\ and observing that u, v, w vanish together, we obtain = (1 + k') u, w = 2u, and the formulae are a? = sn(w, k)t —-r~fi (1 + K) sn (u, k) cn (uf k) * -sn =A) the value is v = A — (1 + k') u, and the integral equation is / a i---77 1 + X — 2 sn2 (u, k) sn(A— 1 +* u, *-)-1 + x_2xm*(Uik) ’ or, what is the same thing, __ 1 — (1 4- k') sn2 (u, k) 1 — (1 — k') sn2 {u, k) ’VIII.] QUADRIC TRANSFORMATION. 187 but the left-hand side is = en (1 4- k'u, X) + dn (1 + k'u, \), and substituting herein the values of the two terms from the table No. 257 the formula is verified. 260. The Fourth form is __ 1 4- ka? ^ 2 Vi. # Here 1 rH 1 II 1 rH 1+ y— (1 4-a? Vi)2 (-). 1 — 73/ = — 7 (1 — a?) (1 — for) (-), 1 + 72/= 7 (1 4- #) (1 4- kx) (-). where denom.= 2 Vfor, and hence Vi — y2.1 — 72y2= — 7 (1 — for2) Vl — ¿r2. 1-kW dy = — 2 Vi (1 — for2) cir where denom. = 4for2. Consequently dy _ (1 + k) dx V1 — y2.1 — 72y2 V1 — op. 1 — k2x2 261. To connect with the standard form, putting x = sn (u, k), y = sn (a, 7), we find v = C 4- (1 4- k) u, and then, since x = 1 gives y = - + ^, = - , we have T 4- ¿P = G 4- (1 4- k) K, or since 2 Vi 7 (1 4- k) K = F, this gives C = %T\ and therefore v =iT/+ 1 4- ku and y = sn (¿17' 4-1 4- ku, 7): wherefore the equation is sn (ir'+ 1 4- ku, 7) = 1 4- & sn2 (u, i) 2 Vi sn (u, i) The left-hand side is 1 __ 1 4- k sn2 {u, k) 7 sn (1 + ku, 7) * 7 (1 4- i) sn (u> fy ’188 CUBIC TRANSFORMATION. [yin. or. what is the same thine:, = ^ ~^-^ SD ^U’ ^ which is right. 6 2\/*sn (u,k) 6 262. Making in the third form Jacobi’s imaginary trans- „ iX iY ,A, %Y 1+lc'X2 Vl-X2 ^ Vl-F2 vT^F2 1-VX* 1 +k'X2 giving F = 2 fjpx ’ v*z' ^is *s four^1 f°rm> writing therein X, Y for x, y, and for h the complementary modulus Jc'. And similarly making in the fourth form Jacobi’s imaginary - F 1-1 +TX2 Vl - 72_ 2 Vyl-Z VF^X2 ’ glVmg transformation, it becomes 1 4- V — 2Z2 F = ^ _|l2\'X2 9 vlZm ^is *s third form becomes on substituting therein X, F for y> and for X the comple- mentary modulus X'. 263. The cases n = 3, 5, 7 are worked out in accordance with the general algebraical theory explained in the preceding Chapter. In the case n = 3, it is to be observed, that the process introduces a single indeterminate quantity a, in terms of which the moduli 1c, X are expressed; the resulting form, containing only this parameter, is an interesting and valuable one, but it is nevertheless proper to obtain the modular equa- tion, and express the formula in terms of the two quantities u, v connected by this modular equation. I have in regard to this same case n = 3 gone into some details to connect the formulae with the transcendental ones depending on the trisec- tion of the complete functions, as obtained from the general theory for the case of an odd-prime. The Cubic Transformation. Art. Nos. 264 to 266. 264. We write 1 — y /I — aa?y 1 — x 1 + y \1 4 ax) 1 4 x ’ _ x \2a 4 1 4 a2#2} ^ 1 4 a (a 4 2) ¿r2 * givingVIII.] CUBIC TRANSFORMATION. 189 and kx> then the conditions in order to the change x, y into 1 \y> are a2 = n, fc2 (2a +1) = fla (a + 2), It is moreover clear that X = H2’ 1^ M = 2a+l. 265. We have at once everything expressed in terms of a, viz. we have first il = a2, and thence and then a3 (*2 + a 2a+1 ’ X2 = a 1 — 3/ = (1 — ax)2 (1 — x) (+), i + y — (i + a#)2 (i + x) t k \2 l-\y=[l--x) (1 -kx) / k \ 2 1 + Xy = 1 1 + - xj (1 + kx) where denom. = 1 + a (a + 2) x2, and thence dy _ (2ot + l)dx Vl-yM-xy Vl — x2. 1 — &2#2’ the factor 2a + 1 being obtained directly from the consideration that, x and y being small, y = (2a 4-1) x. The modular equa- tion is here replaced by the two equations k2 = a3 (2 + a) 2a+ 1 X2 = a which in fact determine X in terms of k. We obtain (1 — a) (1 + a)3 (1 +0)(l-aY K &+1 (2a +1)3 ’190 CUBIC TRANSFORMATION. [VIII. and thence *Jk\ = a (2 + a) 2a +1 s/k’\' = 1 — a2 2a + l ’ hence Vk\ + Vi'X' = 1, which is a form of the modular equation. We have — = a1, that is writing \/k = u, *J\ = v, we have A* us ,, /7— a (a + 2) - , . . 9 a (a + 2) a = —. Moreover \/A?X = ^, that is v?v2 — ^~ , or v 2a + 1 2a + 1 (substituting herein for a its value), v (2uz + v) * or u ('u? + 2v) = v3 (2u3 + v), that is it4, - v4 + 2uv (1 — u2v2) = 0, which is the modular equation, expressed as an equation be- tween u = \Jk, and v = 266. Introducing into the equations u, t; in place of a we have 2/ = {(v + 2uz) vx + w¥) (-r-), i +yz=(v + ^3#)2 (i + x) (-?-), 1 — y = (-y _ w3a?)2 (l — #) (—), 1 + tfy = v2 (1 4- wwr)2 (1 + u4x) (h-), 1 — = a2 (1 — wy#)2 (1 — w4#) (-i-), where the denominator is in the first instance obtained in the form if + uz (u3 + 2v) of ; or, altering this by means of the modular equation, we have denom. = v2 {1 + vvf (v + 2u2) of]; and then vdy Vl — y2. 1 — ify2 (v + 2uz) dx Vl — of. 1 — u8ofVIII.] QUINTIC TRANSFORMATION. 191 The Quintic Transformation. Art. Nos. 267 to 271. 267. We write 1 — y __ (1 — x) (1 — ax + fix2)2 1+y (1 + x) (1 -f olx -f fix2)2 * . . x {(2a +1) + (2afi + 2/3 + a2) a? 4* fi2x?} giving y- 1+(2/9 + 2a + a2)is2 + (/32+ 2a/S)ie4 ' And then the conditions in order to the change x into p-Cl, k2 (2a/3 + 2/3 + a2) = ft (2a + 2/3 + a3), ^(20 + 1) =il(^+2a/3), Jc5 1 where il2 = —. It is moreover clear that 2a -f 1. u20 268. Assuming k = u4, \ = v4, we have il3 = —, and thence %(/ fi = fil = —. Substituting these values the last equation be- comes (2a +1) uv4 = us 4- 2olv, that is 2av (1 — uv3) = u (v4 — u4), or 2a = The second equation becomes (v2 — u2) (2fi + a2) = u2 (1 — u3v) 2a, u (v4 — u4) v (1 — uv3) * that is 2fi 4- a2 = u3 (v4 — u4) (1 — u3v) V 1 — uv3 u3 (v3 + O (1 - U3V) V 1 — uv3 i u3 f(fl3 + w3)(l - - u3v) V ( 1 — uv3 u3 (v2-u2)(\ + u3v) V 1 — uv3 whence192 QUINTIC TRANSFORMATION. [vm. And dividing this value by the value first obtained for 2a, we have _4ii2 (1 + uhi) __u(tf — u4) a v2-\-u2 ’ v (1 — uv3) ' whence — (v2 + u2) (v4 — u4) + 4mv (1 — uv*) (1 + u3v) = 0, or, what is the same thing, u6 — v6 + 5u2v2 (u2 — v2) + 4*uv (1 — wV) = 0, the modular equation between u = \/k and v = v^X. 269. We then have v — u5 2a + 1 = u(l — uv3) ’ v2 (1 — 1 — tw3 ’ /32 = —, M V2 ’ 1 — uv3 and hence y = _ v(v — u5)x + u3 (v2 + u2) (v - U5) a? + u10 (1 — uv3) x5 v2 (1 — uv3) + uv2 (v2 + u2) (v — u5) Qp -f tfu6 (v — u5) xA ’ or if we please 1 — y __ 1 —x 1+y 1+x' , - u(v4 — u4) u5 1 — wx + — x2 2(1 — uv3) v 1 + u (v4 — u4) x~\— op { 2(1 — uv3) v leading to v (1 - uv3) dy _ (v — ?is) ¿¿e Vl — y2.1 — a8?/2 Vl — #2.1 — ^8a?2 * 270. If from the original equations we eliminate k, i2, we obtain (a2 + 2a£ + 2£)2 (2a + ¿8) - (a2 + 2a -f 2j8)2 (2a + 1) /3 = 0, viz. this is 2a2 (1-/3) {a3 — 2y8 (l + * + /3)} = 0.VIII.] QUINTIC TRANSFORMATION. 193 But o = 0 gives simply y = x: 1 — /3 = 0 corresponds to k = X — 1, and does not give a transformation : rejecting these factors, we have K3 — 2/3(1 +a + @) = 0, viz. if a, /8 are connected by this equation, and 1 — y _ 1 -a?/I — ax + f3x-\- 1 +y l+«\l + oa: + jfcteV ’ then there exist values of M, k, A, such that Mdy ^_______dx Vl — y2.1 — \2y2 Vl — m?. 1 — tyx* ’ viz. we have ¿= 2o+ 1, , or, what is the same 82(32 + 2 i1+<2 -a2) *2+^i2· Moreover dy = -jj2 (1, ^)4 dx, but the numerator (1, a?y con- tains, not the square, but only the first power of 1 + (2 — a2) x2 + x*; we in fact find dy = ^{2a+I+(-CL2-4}OL+2)x2+(2QL+l)xt}{l+(2--OL2)a?+xi}dx, and consequently dy __ 2a+1 + (-«2— 4a+ 2)^ + (2a+ 1)#4 dx 1 — y2~ 1-f (2 — a2) x2 + ¥ ' 1 —a?’ c. 13194 SEPTIC TRANSFORMATION. [VIII. dso viz. the factor which multiplies -—— is not a mere constant; J- “* %Xj and we have thus no quintic transformation. The Septic Transformation. Art. Nos. 272 and 273. 272. We write \ — y _ \ — x f\ — ax + fix2 — ya?\2 1 + y 1 + # \1 + a# + /S#2 + 7a?) 9 and thence the conditions in order to the change x, y into 1 1 kx 9 \y are 72 = fl, k2 (/32 + 2/87 + 2a7) = (2/8 + 27 + a2), & (2/8 + 2a/3 + 27 + a2) = fl (/S2 + 2a/3 + 27 4- 2«7), k (1 + 2a) = il (t2 + 2/87), k7 where O2 = . Writing as before k = u4, \ = v*, we have A. ^14 __ fl = — ; and thence 7, = v i2, = —. Moreover, by taking 5? and y each indefinitely small we obtain at once 1 + 2a = , and substituting these results in the last of the four equations we find 2/8 = w¥ (— — J ·. and the second and third equations become v2 (/32 + 2/87 + 2«7) = w6 (2/8 + 27 + a2), n2v2 (2/3 + 2a/8 + 27 -f a2) = /82 4- 2a/8 -f* 27 + 2«7, in which equations a, /8, 7 are to be considered as given functions of u, v, M: the equations therefore determine the relation between u and v (the modular equation) : and they also determine the multiplier M as a function of u, v. 273. The final results are simple: but it is by no means easy to deduce them from the equations, or even to verify them, when known : we have (1-m8)(1-v8) = (1-mv)p,Vili.] THE MODULAR EQUATION. 195 or, as this may also be written, (v — u7)(u — v7) + 7 uv (1 — uv)2 (1 — uv + u2v2f — 0, for the modular equation: and then M is given in either of the two forms 1 _ 7u (1 — uv) (l — uv+ u2tf) M _ v (1 — uv) (1 — uv 4- u2v2) M u — v7 9 v — u7 9 values which are identical in virtue of the last-mentioned form of the modular equation. And then as above 2a = i-i, = 7 which are the values of the coefficients a, j8,7. if v 9 Forms of the Modular Equation in the Cubic and Quintic Transformations. Art. Nos. 274 to 277. 274. In the cubic transformation, the modular equation is originally given as an equation of the fourth order between (w, v): but we thence easily derive equations of the same order, 4, between (V2, v2) (u4, v*), and (w8, v8): the forms are 1 u u2 u3 u4 1 j +1 1 V + 2 i v2. . V3Ì -2 V4 -1 1 u2 u4 w6 us 1 1 + 1 V2 - 4 V4 46 V6 ! -4 Vs + 1 1 13—2196 THE MODULAR EQUATION. [VIII. 1 u4 w8 u12 1 + 1 V4 -16 + 12 v8 + 6 v12 + 12 -16 V16 + 1 1 u8 w16 tt24 w32 1 + 1 Vs -256 + 384 -132 V16 + 384 -762 + 384 ^24 -132 + 384 -256 VS2 + 1 ! 1 275. Here I. is the original form u4 — v4 + 2uv (1 — urv2) = 0. II. may be written (1 — ^8) (l — v8) = (l — vrv2)4. Jacobi ob- tains this, Fund. Nova, p. 68, as follows: we have (1 — u4) (1 + v4) = 1 — + 2uv (1 — u?v2) = (1 — u2v2) (1 + uv)2, = (1 — uv) (1 + uv)3, (1 4- u4) (1 — v4) = 1 — u4v4 — 2uv (1 — u2v2) = (1 — u^v2) (1 — uv)2, = (1 + uv) (1 — uvf, whence the form. Writing k2 = u8, k'2 = 1 — us, X2 = v8, A/- = 1 — v8, the equation is k'2\'2 = (1 — Vk\)4, or, what is the same thing, V&X + V&v = 1, the irrational form obtained ante, No. 265. III. may be written (u4 — v4)4 — 1 (hi4*)4 (1 — u8) (1 — v8) = 0 : which form can be at once derived from II. under the form (1 — u8) (1 - Vs) = (1 — u2v2)4, by writing therein 1 — u2vr = — (u4 — v4) +- 2uv.VIII.] THE MODULAR EQUATION. 197 IV. may be written (u8 — v8)4 = 128a*vP (1 — u8) (1 — Vs) (2 — u8 — v8 + 2u8tf): or say (k2 - A2)4 = 128 A2A2 (1 - ¥) (1 - A2) (2 - k2 - A2 + 2k2X2), Fund. Nova, p. 67, viz. this is the modular equation expressed rationally in terms of k2, A2. Writing, with Jacobi, q — 1 — 2k2, l = 1 — 2A2, it becomes (q-l)4=64 (1 — q~) (1 — l'2) (3 + ql). 276. In the quintic transformation the modular equation is originally given as an equation of the order 6 between u, v: this may be expressed as an equation of the same order 6 between (u2, v2), (u4, v4), (u8, v8), viz. the four forms are 1 u u2 u3 ti4 u5 w6 1 1 l +1 V + 4 v2 + 0 V3 1 L . .. V4 - 5 ! V5 -4 v6 -1 ! 1 u2 u4 u6 w8 U10 It12 1 | + 1 t>2 -16 + 10 v4 + 15 -20 Vs + 15 V10 + 10 -16 t;12 + 1 198 THE MODULAR EQUATION. [VIII. III. IV. 1 u4 u8 u12 u16 w20 u24 1 ‘ ! 1 + 1 V4 , -256 i + 320 - 70 v8 -640 i +655 V12 + 320 -660 + 320 V16 | + 655 -640 V20 - 70 + 320 -256 V24 + 1 j 1 1 U8 U16 U24 U32 M40 u48 ! ! + 1 - 65536 + 16384C -138240 + 43520 - 3590 + 163840 -133120 -207360 + 133135 + 43520 i -138240 - 207360 + 691180 -207360 -138240 j + 43520 + 133135 - 207360 -133120 + 163840 - 3590 + 43520 -138240 + 163840 - 65536 i 1 + 1 i 277. Here I. is the original form uG — vG 4- 5u'2v2 (u2 — v2) + 4 uv (1 — u4if) = 0. II. may be written (vr — v2)G — 16u2v2 (1 — u8) (1 — v8) = 0. This Jacobi obtains, Fund. Nova, p. 69, directly as follows: writing the modular equation in the form (u2 — tf) (u4 + 6u2v2 4- v4) = — 4uv (1 — w¥), from this we deduce (it2 — v2) (u + v)4y = (u — v) (u 4 v)5, = — 4uv (1 — m4) (1 4* v4), (tt2 — v2) (u — v)4, = (it — v)5 (u 4· v) , — — 4*«; (1 4- w4) (1 — v4), and thence the form in question.VIII.] THE MODULAR EQUATION. 199 The form IV. may be transformed into: (u8 — y8)6 = 512u8v8 into 1 us u16 w* uà- 1 + 128 \ -320 + 270 - 85 - 7 ! v3 -320 + 260 + 405 -260 -M 1 v16 + 270 + 405 -1350 + 405 + 270 1 | v24 - 85 -260 + 405 + 260 -320 | v32 + 7 - 85 + 270 -320 + 128 and thence into: (u8 — v8)6 = 512 u8v8 (1 — u8) (1 — v8) into 1 tt8 u16 u™ | +128 -192 + 78 - 7 -192 -252 + 423 + 78 + 78 + 423 -252 -192 - 7 + 78 -192 + 128 which is the modular equation expressed rationally in terms of u8, v8, — k2, If we herein write q = 1 — 2k\ l = 1 — 2\2, this becomes : (? 1 l I2 1* — If — 256 (1 - g2) (1 — P) into 1 q q* q* + 405 + 486 - 9 + 405 -270 - 9 + 16 which is equivalent to the form given Fund. Nova, p. 67. The equation may also be written (q - If = 256 (1 - q2) (1 - P) {16ql (9 — qlf + 9 (45 — ql) (q — Z)2}.200 THE MODULAR EQUATION. [VIII. Properties of the Modular Equation for n an odd prime. Art. Nos. 278 to 281. 278. The cubic, quintic and septic transformations supply illustrations of certain properties of the modular equation for any odd prime value of n. It may be convenient to mention here that the equation has been further calculated for the odd prime values 11, 13, 17 and 19, by Sohnke, in the Memoir, Equationes modulares pro transformatione functionum ellipti- carum, Crelle, t. xvi (1836), pp. 97—130; the results are given in a tabular form in my Memoir on the transformatioq of elliptic functions, Phil. T?'ans. t. 164 (1874), pp. 397—456. The degree in u, v respectively is = n + 1. 279. The equation remains unaltered if for u, v we write therein — it, — v respectively. Connected herewith we have an important property not explicitly noticed by Jacobi. In general an equation F (it, v) — 0 of the order v in u and v respectively can be transformed into an equation of the order 2v, in u2, v2 respectively; viz. the transformed equation is F {it, v) F (— u, v) F {it, —v)F{— u, —v) = 0, where the left-hand side is a rational and integral function of it2, v2 of the order 2v in these quantities respectively. But as regards the modular equation, since F (— u, — v) =F (u, v), and therefore also F (— u, v) = F (it, — v), the transformed equation may be written F {u, v)F(u, —v) = 0, and it is thus an equa- tion in it2, v2 of the order v, =n+l, only. It has just been seen how in the cases n = 3 and n = 5, we obtain equations not only in {u2, v2), but also in (it4, v4) and in (us, v8), of the same order, 4, 6, in these quantities respectively: and the same thing might easily be shown in the case n = 7. 280. The modular equation remains unaltered when for u, v we write therein v, (—Y^^u; viz. n = 3 or 5, (v, — it), but n = 7, (v, it) in place of (u, v). Taking the equation inVIII.] TRANSFORMATIONS LEADING TO MULTIPLICATION. 201 (u?, v2) (u4, v4·) or (u8, -y8) this merely means that the equation is symmetrical as regards the two variables, but as regards the original form as an equation between (u, v), we have, as just stated, n = 3 or 5 (mod. 8) a skew symmetry, but n = 1 or 7 (mod. 8) a complete symmetry. The above change it, v into [v, (—)^(n2-1) changes the / Y2 multiplier M into > and ^ thus appears that, given the expression of the multiplier in terms of (u, v), we can deduce the modular equation: thus, n = 3, v 4- 2m3 ’ 3M — u + 2v* ’ whence (2u* 4- v) (2v* — u) — 3 uv = 0, the modular equation. And so also, n = 5, v (1 — uv*) 1 _ — u (1 4- it*v) v — u5 ’ oM —it — v5 ’ whence ouv (1 — uv*) (1 4- usv) —(v — it5) (v5 + u) = 0, the modular equation. 281. The modular equation remains unaltered on changing therein u, v into i ^ respectively. The modular equation also remains unaltered on changing therein k, X into k', V respectively, that is u8, v8 into 1 — it8, 1 —v2; this appears from the equations expressed in terms of q = 1 — 2k2 and 1=1 — 2X2; viz. by the change in question q, l are changed into — q,—l\ and the equation remains unaltered. Two Transformations leading to Multiplication. Art. No. 282. 282. It appears from the property stated in No. 280 that we can by a twice-repeated transformation obtain a multiplica- tion, thus, n = 3,202 TRANSFORMATIONS LEADING TO MULTIPLICATION. [VIII. v (v 4- 2it3) x 4- v2 4- vht2 (v 4- 2u3) x2 gives dy _ v 4- 2 us dx V1 — y2. 1 — ^ V1 — x1. 1 — usx2 and writing (v, — it) for (it, v), and (z, i/) for (3/, #), gives _ u (u — 2v3) y 4- v6ys u2 4- uH2 (u — 2v3) y2 dz _ it — ^ dy Vl — z2. 1 — u8z2 u Vl — y2.1 — i?8?/2 ’ = -a , —------------------ v 1 — a;2.1 — it8#2 283. Similarly, ?t = 5, _ v (v — it5) x 4- it3 (it2 4- v2) (v — u5)x? 4- it10 (1 — uv3) ^ v2 (1 — iti;3) 4- wy2 (it2 4- v2) (v — u5) a? 4- u^v3 (v — it5) x* gives dy _ v —u5 dx Vl —ya.l—t*y ~^(1 “ tw8) Vl -a?.l--it8tr2 and _ u (u 4- v5) y — fl3 (it2 4- v2) (w 4- v5) y3 4- i;10(l 4- it3v)y5 ?.t2 (1 4- usv) — ?t2v (it24· ir)(it 4· v5) i/24- it3v6 (it 4- v5) y* gives cfo __ it 4- v5 dy Vl -s2. l-it8^2”W(1 4-it3v) Vl-2/2.1 ’ whence cfe __ da? V1 — 22.1 — u8z2 V1 — a?2.1 — it8a?2VIII.] THE MULTIPLIER. 203 The Multiplier M. Art. Nos. 284 to 288. 284. The above-mentioned values of M} lead to con- venient expressions of nM2; thus »-3, 3 u (2 us + v) oM2 = v u + v5 1 — UV’ u v — u5 1 + usv ’ K Hr T| m- V (u — v7) , = 7? 7M2 =-----^1 u{v — u7) It will be shown that we have in general /if > _ ^2 dk _ v (1 — -y8) du ~~ u (I — u8)dv’ or, what is the same thing, if = 0 be the modular equation, then -nM2 -w8) d(f> di' a formula which is here to be verified in the three cases n = 3, n = 5 and n = 7. 285. In the case n = 3, we have if _ v __ 2 Vs —u 1 ” v + 2uz ~~ 3 u 9 also du _ W — u-\- 3 uzv2 dv 2us + v — 3u2v3 9 and the equation becomes 2v* — u _ 1 — if 2if — u 4- 3uzv2 2^¿3 + v 1 —us‘ 2uz + v — SuV ’ But writing 3 = —u^(^u then in the last fraction the numerator becomes = (2v3 — u) (1 + 4- 2uH), and the204 THE MULTIPLIER. [VIII. denominator = (2u3 4- v) (1 4- u2v2 — 2uv5): and the equation thus is j _ 1 — y8 1 + u2v2 4- 2it5v 1 — it8' 1 + u2v2 — 2 uv5 * But we have 1 — u8 = (1 4- it4) {1 — v4 4- 2^y (1 — &2y2)}, = 1 — u4v4 4- it4 — y4 4- 2uv (1 4- it4·) (1 — u2v2)y = 1 — it^v4, + 2u5v (1 — u2v% = (1 — u2v2) (1 4- u2v2 4- 2usv), and similarly 1 — v8 = (1 — u2v2) (1 + it2v2 — 2wy5), which proves the theorem. 286. In the case n = 5 we have y (1 — wy3) _ ¿6 4- y5 y — u* hit (1 4- usv) ’ and the equation becomes (1 — uv3) {it 4- y5) __ 1 — y8 du (y — it5) (1 4- u3v) 1 — it8 dy * The modular equation may be written (by No. 277) (it2 - y2)6 = 16 u2v2 (1 - it8) (1 - y8), whence differentiating and multiplying by it2 — y2, and reducing, we have 6uv (1 — u8) (1 — y8) (itdu — vdv) = it (it2 — v2) (1 — u8) (1 — dv8) dv 4- v (it2 — v2) (1 — v8) (1 — du8) du, or, as this may be written, v (1 — y8) (5^2 — it10 4- y2 — 5u8y2) dw = (1 — w8) (5y2 — y10 4- w2 — ou2v8) dv, y dw 1 — y8 _ oy2 — y10 4- it2 — 5i^2y8 # u dv 1 — it8 5u2 — i^10 4- y2 — 5ii8y2 5 that isVili.] THE MULTIPLIER. 205 or, observing that from the modular equation we obtain 5u2 — u1Q 4 v2 — 5u8v2 = (1 — (v2 4 on2 4 4w5u), 5v2 — v10 4 — 5u2v8 = (1 — uW) (it2 4 bv2 — 4w,=(! — u8) (1 — v8) — (1 — uv)8, = 0, we have *dv=~V7(1 ~ + u O ~ uv)7 ’ and thence £ (1 - ve) ^ = - v7 (1 - O (1 - if) + (1 - if) u (1 - uv)7, = — v7 (1 — uv)8 -f (1 — v8) u (1 — uv)7, = (1 — uv)7 (u — v7). And similarly il1 - «8)^= (1 - UV7) (v - u7); whence 1 — v8 du 1 — u8 dv ’ = -(1 -«*) fv^(1"M8) d dû’ u — v7 V — u?y the formula in question. Further theory of the Cubic Transformation. Art. Nos. 289 to 298. 289. The cubic transformation may be considered from a converse point of view. Writing x = sn (u, k), z = sn (3u, k), we have 3*(i-g)(i-g)(i-g)(i-g) Z (1 — Jâo?a?){ \ — lc2fi-af)(l — k‘-'Y:af)(l — k-S-af) ’ where a = sn 7 = sn 4K 3 ’ 4K + 4 iK' 3 /3 = sn , S = sn UK 3 ’ - 4K+ UKVIII.] FURTHER THEORY OF THE CUBIC TRANSFORMATION. 207 these being the roots of 3-4(1 +k2)a? + 6kW 0; and it is to be shown that this relation between z3 x may be decomposed into two transformation equations between {yf x) and (z, y) respectively. 290. We take these to be y 1 - k?a?o? ’ 2 1 - \20y ’ giving respectively and Mdy _ dx Vl-yK 1 - ~ ’ dz _ SMdy Vi -z2.i- kn* ~ Vf-y. T- \y ’ where observe that a, which enters into the relation between 4 k y, xf being as above the real root sn —^ , the equation between o y, x is a first transformation, and consequently that the relation between zt y ought to come out a second transformation. 291. Writing -----5 4jBT n----------j— , 4 K v 1 — a2 = cn , v 1 — ]c2ol2 = dn , o o we have SK ( 4>K\ 4fK sn 8“ = sn V~ 3 J = “ sn "3" ’ that is 2 Vl — a2 Vl - ^a2 = - (1 - ¿“a*), and similarly 2 Vl —/S2 Vl — A2/S2 = —(1 — W).208 FURTHER THEORY OF THE CUBIC TRANSFORMATION. [VlII. Also , 4 iK' + 4 K 4 iK' -4 K ft — a- yS = sn----s---- sn-----s— - 1 - k-a?ft2 ’ 7 + S = _ 2/3 Jl -a?J 1 - (1 - fcW) 1 — k2cL2/32 1 - k2a2(32 9 and thence that is ¡3 + y + 8 = — k20L2/3y8, y8 + 8/3 + fiy = — a2, or, what is the same thing, 111 ,oo y8* 8/3*/3y~~ ^ 1 1 1_ a2 . /3 y 8 /3y8 ' and, moreover, since 3(1-3(1-8(1-?)(1-l) = 3 — 4(1 + k2) x~ + 6k2xA — A+z8, we have ¿ = or = M\ a2) 292. Determination of y — ^ % ■- , leading to X KTOL~X~ Mdy dx Jl - y2.1 - \y Jl-oc2 .l-tesc2'VIII.] FURTHER THEORY OF THE CUBIC TRANSFORMATION. 209 We determine M so that x = l, y = 1 shall be correspond- ing values, viz. we have 1 = iM) 1 -¿V , or M = — 1 — a2 * a2 (1 — k2ar) ’ and then (denom. = 1 — kWa?), writing 1 -y = (l (-S) (-)> , v (. /I x1 ) (+); 1 ~ Ma2\ the term in { } is taken to be a perfect square, II 1 ^¿5 suppose, viz. this being so we have 2 1 .— k2a4 / 1 Vl — k2a2\ /=" Vor 7 ” J > 11 _ 1 - ¿2a2 ƒ*“ Jfa2’ ” 1 - a2 ’ which agree; and then 1—2/ = (l _a:)(1-/) O)- whence also l+y=(l+ *)(!+ƒ) O)· We next determine \, so that x being changed into ~ y shall be changed into ^ : we thus have \y 1 1 1 - k2a2x- \y “ Mk?cx.4x ' _ a? * a2 c. 14210 FURTHER THEORY OF THE CUBIC TRANSFORMATION. [VIII. or, multiplying by y, 1 _ 1 X “ M*k? a4' that is X = M2k?a4, *»( 1 - a2)2 (1 - A?a2)2 ‘ Observe that, a being real, we have 1 — a2 < 1 — Arta2, and hence X < A;3, viz. we pass from a modulus i to a smaller modulus X. And then the expressions for 1 — y and 1 + y lead to l-\y=(l- kx) (1 - kfxf (-5-), 1 + \y = (1 + kx) (1 + kfxf <», so that we have the required equation Mdy _ dx Vl — y2.1 — X2y2 V1 — of. 1 — krx2 293. Modular equation. Next, for finding the modular equation, we have or J7dc : X = X'2 = ¿3 ^ ^ a8) (1 - A^a2)2 ’ 1 (1 — &2a2)4 1 — k2a2 ’ {(1 -¿2a2)4-^(l-a2)4}, where the term in { } is 1 - 4A?a2 + 6*V - 4W + A;8a8 — Ic6 + 4fk?a2 — 6AAZ4 + 4&6a6 — A^a8, = (1 - A;2) {1 + Ar2 + A4 - 4 (A? -ffc4) a2 + 6&V - ¿«a8}, = (1 - A?) {(1 - A?)2 + &2 [3 - 4 (1+ A?) a2 + 6^0* - A**8]}, = (i-*2)3; y s '^(l-^a2)2’ ¿'2_. - ¿2a2 ’ that isVIII.] FURTHER THEORY OF THE CUBIC TRANSFORMATION. 211 and hence that is i—f /r-;T7 k'1 + &2 (1 — a2) Vxife + = l, the required equation. 294. We have next (0 being arbitrary) (denom. as before = 1 — k2a.2x2). And taking /, 1 fiyS ,, , · 1 a2 0 = — irp —7-, that is — n/rA = -pz—s , M a2 ’ M0 £78’ then 0 and similarly 1 + _1+/3y8 “ ¡3yS H1+DK)K) (->. (->, (+x also, changing x, y into ^~, 1 — \0y = (1 — kfix) (1 — kyx) (1 — kSx) (-r·), 1 X0y = (1 + kfix) (1 + kyx) (1 + khx) (-l·) ; consequently 1 — X^&y- (1 — Pa2ic2)( 1 — k2fihj-)(\ — ki'fxi)(l — k282x'2) ’ We have 0 = hence(^_J_ W2 _ -3 . if a2 ’ nence a - Mi „« > ^/U2> J/2 ¿‘««if2 14—2212 FURTHER THEORY OF THE CUBIC TRANSFORMATION. [VIII. but A = M2Jc?a4, hence X02 = — ^ and A2#2 = — 3M2Jc2a2; whence, 1 — a2 A putting for shortness If, = —, =—o77, we have A ° a2 (1 — k2 a2) ol2B ^ _ _ 3 - 3fc2^2 " — 7, «> 12, a-c/2 — \ ^ — làa2A2 oPB2- 295. It is to be shown that 0 is connected with A as a is with k ; viz. that we have 3 - 4 (1 + X2) fr + 6\20* - A4#8 = 0. Substituting for 62, A2#2 and X#2 their values, the expression on the left-hand side is = ~ kcL·® {(27 "l8k2ai ~ ***> A2Bi ~ 4“6 (A = l — a‘,B=l — k-Tr), viz. the term in {} is a function (1, a2)8 the coefficients of which are ' 27, — 54 — 54k, 27 + 90k + 27k4, - 4 - 18k - 18k4 - 4k, — 2k — 46k — 2Ar®, 14*4 + 14**, — ** + 10*6 — k, -2k- 2k, - k, and this is equal to the product of 3 — 4(1 + k) or + 6ka.4 — ka? by a function (1, a2)4, the coefficients of which are 9, — 6 — 6*2, 1 - 4*2 + k, 2k + 2k, k,vm.] FURTHER THEORY OF THE CUBIC TRANSFORMATION. 213 viz. this other factor is = {3 — (1 + k~) a2 — Ba*}-, The first factor vanishes and we have thus the required relation 3 — 4 (1 + \2) fr + 6V04 - V08 = 0. We have 0 = — where M, a, y8 are all real but j3 is a pure imaginary, hence also 0 is a pure imaginary. Now the equation in z, y corresponds with the differential relation dz SMdy Vl — z2.1 — k2z2 Vl - yM - X2y2 ’ and we thence see that 0 must denote one of the quantities 4A 4iAf 4A + 4iA' — 4A + 4iAx sn — , sn—, sn-------g----, sn-----------; and, being as just shown, a pure imaginary, it clearly denotes 4^ A.^ sn — - , viz. the transformation from z to y is a second trans- formation. Writing now JL N z i - x2&y ’ we may determine N so that corresponding values shall be z = 1, y — — 1 (or z = — lf y = 1), viz. this will be the case if 1 = l - xy '■ at i-0* or say and the value of N thus determined will be = . To verify this we have to prove the equation 02(l-\20a)=3Jf(l-08). Substituting for O2, \02 and M their values, the equation is {- * -k'A*) + SAB (B - BA )} = 0, ( A = 1 - a2, B = 1 — Jc2a\ as before).214 FURTHER THEORY OF THE CUBIC TRANSFORMATION. [VIII. We have B - k2A = l — ]c2, and the term B3 — k*A3 contains this same factor. Omitting the factor in question, 1 — k2, the term in {} is = — a2 (1 + ¥ - 3^a2 + *W) + 3 {1 -(1 +**) a2 + £2a4}, viz. this is = 3 — 4 (1 + k2) a2 + 6&2a4 — i^a8, which is = 0, and the theorem is thus proved. 296. Starting from the equation z i - \a0y ’ where 1,-1 are corresponding values of y, and 3 - 4 (1 + A2) d2 + GX2#4- \*0* = 0, we have l+Z=(l-y)(l-Zj (*), i-^ = (i + y)(i+|)2 (-5-). 1 + = (1 - Xy) (1 - Xgy)- O), 1 — = (1 + \y) (1 + \gy)· (+), where denom. = 1 — \2&2y2, viz. in obtaining the above we have l + z = l +SMy (l-^) (+)> that is -{i+(sir+i)y+^i »■}(+), - = 3M+1, 9 1 _aM & ’ 1 - V04 1-02 ’ 1 - \20·- 1 - 6>r ’Vili.] FURTHER THEORY OF THE CUBIC TRANSFORMATION. 215 which agree. We have therefore dz __ 3 Mdy Vl — z2. 1 — k2z2 Vl — y2.1 — \2i/2 * and the proof is thus completed. 297. The investigation would have been very similar if, in the formula MV d) y 1 — A^a2#2 a had denoted any other root of the modular equation, or, what is the same thing, if a were replaced by any other root j3,7 or 8: there would have been in each case a corresponding equation in (z, y) giving by its combination with the assumed equation the triplication. In particular if the root had been /3, then the equation in x, y would have been a second transformation and the corresponding equation in (z, y) a first transformation. But if the root had been 7 or 8, then in either case the equation in (x, y) and the corresponding equation in (y, z) would have been each an imaginary transformation. 298. Returning to the quantities a, ft, 7, 8, which denote 4 K ±iK' 4*K 4- 4dK' -4>K+4iK' sn ^ f sn ^ , sn ^ > sn ^ > respectively the two equations obtained in No. 291 belong to a system which may be written a2 = . . . — fiy — /3S — 78, jS2 = . — ay + a8 . . + 78, 72 = a/3 . — a8 . + £8 . , 8? = -afi + «7 . +@y . . , k?d/3y8 = . — — 7 — 8, / or if for shortness s = % V3, then we may Hj write a/3y8 = — -jp or k2oifiy8 = — $, and the last set of equations becomes216 FURTHER THEORY OF THE CUBIC TRANSFORMATION. [VIII. SOL - /?— 7— 8=0, a + sy8 + 7— 8=0, a — fi + sy+ 8=0, a + /3 — 7 + 58 = 0, which must be equivalent to two equations only: in fact the equations may also be written 2a . +(5 —1)7+ (5+ 1)8 = 0, 2/3 -- (s + 1) 7 + (s — 1) 8 = 0, -(s- l)a + (s +1)/3+ 27 . =0, -0 + l)a-(s-l)/3 . +28 = 0, which linearly determine any two of the quantities in terms of the remaining two, for instance a and /3 in terms of 7 and 8: but then, substituting for a and £ their values, the third and fourth equations are satisfied identically. A General Form of the Cubic Transformation. Art. Nos. 299, 300. 299. Consider the two quartic functions X = (VIII.] DIFFERENTIAL EQUATION FOR MODULUS X. 223 and after all reductions we arrive at Jacobi’s form 3 \(dkf (d2X)- - (dXf (d?ky) - 2dk dX (dk daX - dX d3k) +<*)’ w {(¿4-*)’ m - (£*)’ w} - o. 307. It may be remarked that if til· l· Wa = dp'thatis p = logF’ and therefore k2 = e2p „ V2 = ~- ,, kk' = eP and therefore X2 = we have 1 + e2? ’ 1 + e2* = dq, that is q = log ^ 1 1 + e^’ e2q 1 4- e2q , V2 = 1 + < , w = gg 1 + 6 and the equation then is n = A/^ V dq ’ which is readily converted into 2p Y (g>w ~ />Y") ~ 3 W2 - j> V*) *wy _ / ep y / _ /_e^ y £ \1 + eW ’ \1 + e2v * pr ’ where p, p", p" and q , <7", g'", are the derived functions of p} q with respect to the independent variable. 308. The equation in No. 305 is easily verified in the case 1 - k' of the quadric transformation: we have here X = ^ ^ ; and we thence find il = , 37 = ———L> and the equation takes 1 + * ^ ¿'(l + £')f the form J2 A A Tm-' A F-^2 + 20 - *T_0 1+U·' ¿ft- L <*# vi+AJ 1+A+k'(i+vf ~ ’224 RELATION BETWEEN M, K, A, E, 0. [VIII. viz. dividing by 2, and reducing, this is i a (-v+&\ _ _k_\ (i - k'f 1 + k' 1c dk' \ 1 + ¥ ) 1 + k') // ^ | _j_ But the first term is 1 \k -1+2 k'+k'* k \ 1+Ic \k' ‘ (1 + kj 1 + k’\ ’ = k’(\ + ky(~1 + A:)’ and the equation is verified. -y-kj k’il + k’f ’ In the case of the cubic transformation, the equation in Jacobi’s form No. 306 might be verified (although not without some difficulty) by means of the expressions, No. 265, P„«3(2 + q) * 1 +2a \2 = a ( 2 + ay 1 + 2a/ ’ of the moduli ]c, \ in terms of a parameter a: but the verifica- tion in the next following case of the quintic equation would apparently be very difficult. Jacobi remarks that if a method existed for finding the algebraical solutions of a differential equation, then, by means of the foregoing differential equation alone, it would be possible to obtain the modular equation in the transformation of any order n whatever: but, the mere verifications being so difficult, it does not appear that anything can be done in this manner in regard to the modular equations. A relation involving M, K, A, E, G. Art. No. 309. 309. Immediately connected with what precedes we have a result which will be useful in the sequel: we have dK dk 1^ kkf* (E—k'2K), dk k E dK kk'*Kdk+ K = 0, that isVIII.] RELATION BETWEEN M, Ky A, E} G. 225 and similarly if A, G are the complete functions to the modulus X (A = FiX, as before, G = E^X), then dX ~X G XX'2A dX -4- dA ~A = 0. Hence establishing the equation dX GdX dA _ dk Edk dK X ~ XX'rA + X =~k~ W*K + K ’ and observing that M = - ^ and therefore = ~ , we ° n A M K A obtain dX GdX dM _ dk Edk ~X ~ XX'2A ” M ~Jc~ kk'2K ’ viz. this is dX f /g (? XX'2 dit/) (¿A; /,/o A’x xxi2\ “A~Jd lA]=W2{k~~~Kj> or, eliminating dX by the relation = ^j^2 , this is ?l/Y2 | A M dx\~ K’ which is the result in question. Observe that j— is the total aX differential coefficient, viz. if M is taken to be a function of k, X, then, in the differentiation, k must be treated as a function of X. The equation, as involving not only Ky A but also E, Gy is in its actual form only true for the first transformation, and it does not readily appear how it should be modified in the case where X is any root whatever. c. 15226 [IX. CHAPTER IX. JACOBI’S PARTIAL DIFFERENTIAL EQUATIONS FOR THE FUNCTIONS H, ©, AND FOR THE NUMERATORS AND DENOMINATORS IN THE MULTIPLICATION AND TRANSFORMATION OF THE ELLIP- TIC FUNCTIONS sn u, cn u, dn u. Outline of the Results. Art. Nos. 310 to 313. 310. The functions ©w, Hu have an important application to the theory of multiplication, and theoretically a like one to the theory of transformation. To explain this, recalling the formulae in terms of sn uy the three numerators and the denominator of these functions are respectively = Hnu ©"^H), H (nu + K) 0^0, © (nu + K) ©^O, x)0 (f+A’x) 0lW_1°’ and ® , x) each divided by ©nit; where for shortness ©jO is written in- stead of its value = sj· the proof need not be at present considered. Observe that for u = 0 the denominator is Now the functions ©w, Hu, ©(w-f K\ H(u-\-K), each satisfy as will be shown a certain partial differential equation which in its most simple form is — 4^ = 0, where the dv2 dw 7T K' ITU variables are to, = , and V> = 2K> Jacobi, Grelle, t. ill. (1828) p. 306. And we hence deduce a partial differential equation satisfied by the foregoing numerator- and denominator-functions, as well in the case of transformation as in that of multiplica- tion : viz. if, in the case of multiplication by n, we write v = n2, but in the case of the transformation of the nth order v = ny then (in one of several forms) this equation is (Jacobi, Grelle, t. iv. (1829) p. 185) (1 - aa? + + (p - 1) (cue - 2**)^ dz + z/ (z/ — l)x2z — 2v(a2 — 4)^ = 0, in which equation the variables are #, = V& sn u, and a, = h + 15—2228 PARTIAL DIFFERENTIAL EQUATIONS FOR H, @, &C. [iX. 312. The form is specially applicable to the denominator of the three functions of nu, for this is a rational and integral function of k and sn2 u, which when we introduce therein x, = V&sn u, becomes a function of x and k, which is unaltered when k is changed into and is therefore a rational and integral function of x and a: and it is for the like reason specially applicable to the numerator of s/k sn nu when n is an odd number. But the form is not in other cases the most convenient one; for instance as regards the numerators of /k 1 A/^cnu, dn u, these do not thus become rational in V k 's/k' regard to a, and it would be better to have & as a variable in place of a; and in the case where the numerator contains as a factor an irrational function cn u, dn u or cn u dn u of sn^, it is proper instead of £ to consider 2 divided by such irrational factor, that is the other factor, rational in regard to sn u. But making the suitable modifications the formula is for multiplication a very convenient one: viz. we can by means of it actually determine the numerator- and denominator- functions. 313. But for transformation the formula is practically use- less ; for observe that A is therein regarded as a function of k, that is of a; viz. the modular equation must be taken to be known. Supposing that it is known, we cannot even then determine by means of it the numerator- and denominator-functions; for in seeking a solution by the method of indeterminate coefficients the coefficients of the several powers of x would be functions of (u, v) not only unknown, but in form indeterminate (as admit- ting of modification by means of the modular equation):—and even when the actual expression of ^ as a function of (x, u, v) is known, as of course it is for the cubic, quintic, &c. transforma- tions, it is, from the complexity of the modular equations, by no means easy to verify the formula: the process is in fact one of difficulty even in the case of the cubic transformation n = 3. This of course in no wise diminishes the interest of the result;IX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, 0, &C. 229 and the investigation of it being substantially identical in the two cases of transformation and multiplication, it is proper not to separate them. Partial Differential Equation satisfied by Su. Art. Nos. 314 to 316. 314. It is to be shown that the function o-, =©^, - ^ du^ du dn2 * satisfies the differential equation We have da du = (/o du dn2 u — ^ uj a, = \u (k' o2 — -f- k2 f0du cn2 u| a, dS dii dn2 u — | Jc1’2 — + k2 J0 du cn2 w j | cr, d, = — ^ jsn2 w dn2 u + ^k2 cn2^ /0 du cn2 ,230 PARTIAL DIFFERENTIAL EQUATIONS FOR H, 0, &C. [iX. and thence d 2k ( Jo du Jo du ^ dn2 u — — ^ -¡/0 du JQdu sn2 u dn2 E2 + \ k2 f0 du (cn2 u Jo du en2 u — J0 du cn4 w)j , k = — T^ljodujodu (2 sn2 u dn2 u — k2 cn4 u) + \k2 (Jo du cn2 u)2J . But we have sn2 u = 2 (en2 u dn2 u — sn2 u dn2 u — k2 sn2 u cn2 u), = 2 (k'2 — 2 sn2 u dn2 u + &2 cn4 u), or multiplying by du* and integrating twice sn2 u = k'2u2 — 2 Jo du /0 du (2 sn2 u dn2 u — k2 cn4 u), whence at length d k Jo du Jo du dn2 u = — \ku2 4- \ ^ sn2 u — ^-2 (f0 du cn2 u)2, the required value of the integral. 310. Resuming the investigation, we have ^ Kk' — ^ d E _ 1 ( /2E \ E-\ dk kk' ’ dkK~kk'2[c \k V K2r and hence s={f - ^ m+- a ^cn2 m)2} *· Substituting the foregoing values of ^ in the ° ® ° du* du dk differential equation, the several terms destroy each other, and we thus have the equation in question.IX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, 0, &C. 231 Same Equation satisfied by Hu, ®(u + K), H (u + K). Art. Nos. 317, 318. 317. The equation d2a du2 2 d2234 PARTIAL DIFFERENTIAL EQUATIONS FOR H, ®, &C. [iX. may be established in a different manner thus: writing in the first equation 7tK' TTU IT’ V~2K9 then observing that we have da _ 7r da d2a _ nr2 d2a du 2K dv ’ du2 4if2 du2 ’ 7r 2K2kk'2> da _ a /,,2 E\da nr2 da dh~ W2\ KJdv~ 2K2kk'2 day ' and the equation becomes dV do* dv2 do) = 0, satisfied by this is in like manner transformed into the same equation — 4 ~ = 0. Hence whatever function of ^ and ~ satisfies du2 day K K » Till Yli. the first equation, the same function of and -y satisfies the second equation. Let \ be the modulus in the first trans- formation of the nth order, and A, A' the complete functions, a; A nK , . K jLl ^ . nK A , nu u , —fr~ and A = - ,,, that is -T=- = -r- and = -¿t ; or the K nM K A KM’ A' second equation is satisfied by the same function of , Hence the first equation being satisfied by a = ©w, &c., theIX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, ®, &C. 235 second equation is satisfied by ♦ x)>&c- Partial Differential Equations satisfied by the Numerators and Denominator. Art. Nos. 323 to 331. 323. Start now with the equation satisfied by 2 = ® , Xj , &c. And assume 2 = (£?r)Hn-1\Kk,yHn"1)®nu.z> say for shortness this is = Cilan . z, (where a denotes ®u and consequently satisfies the equation We find /rfcrV) — 2nu [k'2 — + 2nick'2 — (iîcrn) +g [n · 2’“'“ g - 2m‘ (*■ - £ +Sn"- d? + jt 2nkk'2£l-1) l-ddl + Wo*.Ig} + 2nkk''2-uw]’ 1 fJC) or in the term ^ substituting for ÎÎ its value = {Kkr)~^n~^ this is = ii*» .n(n-l) jl (g)‘ - 2« (k'> ~~)ld£ Hence dividing the whole equation by ilan it becomes d2z du2 E\ —----u[lc* — [a < E\ 1 d a dk K \ &/ kk' d ™, -, E -HB=1TIX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, 0, &C. 237 Adding these several quantities, the coefficient of n (n— l)z dz is 1 — dn2 u, = k2 sn2 u ; the coefficient of -j- has also been du found ; and the equation thus becomes + 2nk2 (f0cn2 u du) ^ + n (n — 1) k2 sn2 u. ^ + 2nkk'2 = 0, which equation is consequently satisfied by /2 \ 7T / 0” (w) Z 1 ) e*(u)®(iT X) ’ &C' 325. It is to be further remarked that if we had started with #2 du2 — 2 n2 which equation is obviously satisfied by 2 = © (nu), &c., and had then assumed 2 = (|tt) it»2-!) (m -i(n2-l) Sn2 ( u) . z9 we should at every step of the investigation have had n2 in place of n, and should finally have arrived at the equation d^z d 7 7—h 2n2 k2 ( f0 en2 u dit) 7- + n2 in2 — 1) k2 sn2 u . z du2 au + 2n*kk'^ = 0, dk which equation is consequently satisfied by © {nu\ &c. \ 7T J 0n2 (u) 326. It will be convenient to include the two equations in the common form ^ 2 + 2vk2 (J0cn2 u du) ^ + v (v — 1) k2 sn2 u . z + 2vkk'2 ~ = 0, du238 PARTIAL DIFFERENTIAL EQUATIONS FOR H, ®, &C. [iX. where for the transformation equation v = n, and for the multi- plication equation v — n2. 327. Write in this equation x = Jk sn u. We have dx = Jlc cn u dn u du + 12 dk, if for a moment O = ^ (Jk snw), = sn u 4- 4k \k sn u en2u — k en u dn u Lcn2u du\, 2 \!k k'2l jQ 1 /. 2k2 \ k*JTc , , 2 , = —ïcSUU \ + cn p cnuan uj0cn2udu, = —sn^(l + k2 — 2 k2 sn 2u) — cn^dn u f0cn2u du, and hence V& en w dn u ^, dz_ _dz^ ndz dJc~dk+ dx> where on the right-hand side -yr is the new’ value of this CUC differential coefficient, viz. that belonging to the assumption z = & function of x, k, or (as we may express this) z = z (x, k). And thence also d2z 7 , d2z /Tdz d , , x -j--9 = k cn2m dn2w -j— + v& -j- -r- (cn w dn w) du2 cte2 dx dux = kcvL2udn2ui^- dx2 dz — V& sn (1 + i2 — 2k2 sn2 w) ^.IX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, 0, &C. 239 Substituting, the equation becomes d2z , o I o . k cn2 u dn2 u dx2 dz - + ^sn ^ (1 + A2 — 2&2 sn2 u) (Jg _ + ^ . 2vk2 VA; cn w dn u J0 cn2 udu + z. v (v — 1) k2 sn2 u + ^ {i/ VA; snw (1 + A;2 — dz + ^.2,M;'2 = 0, 2A2 sn2 u) — 2vk2 *Jk cn u dn u f0 cn2 u du} where the term involving the integral disappears, and two other terms combine together; viz. the result is d?z . k cn2 u dn2 u dx2 + ^ (v — 1) \fk sn u (1 + k2 — 2k2 sn2 u) + z . v(v— l)A;2sn2w + ^.2vkk'*=0 in which equation sn u should be replaced by its value -j=. Introducing at the same time in place of k the quantity a, = fc + i, the equation becomes (1 — aa? + a*) ^ + (v - 1) (ax - 2«3)^ dz + v(v — l) x2z — 2v (a2 — 4) ^ = 0, where I recall that the variables are x = VA? sn u and a = k -f ^.240 PARTIAL DIFFERENTIAL EQUATIONS FOR H, 0, &C. [iX. The equation is satisfied by the numerators and the denomi- nator of VX sn (J, X), V^Cn(i’ X)’ X) in the transformation of the order n, or (v = n2) by the nume- rators and denominator of \/k sn nu, a /~7 cn nu, dn nu. V k v&' 328. As already remarked, the formula is not practically useful in the transformation-theory, but it is so for multipli- cation. As regards this last theory it has been observed that although with respect to the denominator-function, and the numerator of sn nu when n is an odd number, there is great elegance in taking as above the variable to be \fk sn u, and in introducing a in place of k, yet that for the other functions, this is not the case, and it seems better to have as the variables = sn u, and k. The transformation is of course easily effected, viz. writing we find dz _ 1 dz dx ~ *Jkd%’ dz _dz f dz dk~dk~2kd£’ dz where on the right-hand side ^ is the value belonging to d2z 1 d2z and the assumption z = z (f, k). Hence also j ^ , the equation, finally restoring therein x in place of £, becomes ^ (1 - .1 - tea?) + ^[(2wfc2-1 -&2) X - 2 (v- 1) fc2*3] dz + z. v (v — 1) k2af + 2vk (1 — k2) ^ = 0 ; viz. x is here = sn u, and the equation is satisfied by the ? cn nu, dn nu. vk' numerators and denominator of s/k sn nu’ \JrIX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, 0, &C. 241 We may of course get rid of the exterior factors, and thus obtain a system of four equations, viz. fj 2« ft & 2 (1 — x2.1 — ¿V) + ^ [(2vk2 — 1 — fc2) x — 2 (v — 1) fe3] rfz + z{{v2— j/) Arte2+.¿4} + 2i/fe(l — ¿2) = 0; where A = z; (1 — Ar2), equation satisfied by numerator of sn ww, = v , „ „ cn rrn, = vl and thence dz __ 1 — 2 (1 4- k2) x2 + 3Arte* dx VZ ’ £ = Y^{(-S-3/ci)x+(2+lW+2t)^+(-d!if-9kt)af+Qkix!}, 37 =“7= (— ^ + fa5)· d* VZ c. 16242 PARTIAL DIFFERENTIAL EQUATIONS FOR H, ©, &C. [IX. P 73 P eg P o > s S-t © © .P © ,P © eg P O P © © ¿P .S *o3 11$ 73 1^3 a ip © rP P © W 1 (4 A 3i N | co X . 1$ - 1 + 1 CO rH 1 «3 00 cr* © © rP eg P ‘a o P © 73 J ll -Cg r4g a $-1 © -ea CO + + ci cl 1$ 2g 2g + + rH rH '·«—^ d I I rH 5$ 5$ 8 2g -¿g gp d i? ” ^ + £ eg © P .a eg © cp © > © rP P .a "eg P C7 © Jh © rP o •w - [3i d 1 ^ -eg rj< d ^g d rH I (+ 1 + ii' d rH 2g 00 d 2g CO rH o Tii 1 -la CO rH + 2a o d Cl -¿a o d | 1 Cl -ea CO rH 11 1 V far CO rH 2g 00 d Cl ~ -ea d rH + 1 + rft 00 M7 + o P o © rP • r—* +=> eg p CO a- o © 1 "P w o P o o 00 CO © CO m % -eg d rH 2* -eg 1 1 d + + + 1 5$ Tf I d I 1 1 rH 1$ 1$ 3* -2a 2g CO ¿a d rH -2a 1 + 1 + 1$ rf< 00 5s ci -2g I— + + rH rH 1 cpThird equation. IX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, &C. 243 5s 5a 5s j f 3a Tf 3a (N ''f (N d d rH d rH o 33 ■< d rH 1 + 1 1· + 1 3a~ 3a 3a" 3a /3a" 3a" CO 53 33 CO 55 33 3a d rH 3a ■48 ria d 3a d rH 5a 3a + + + S3 3$ ^ ^ ^ ^ tJi d I I I > 3* 1 ^ © d + I + % ST + + 1>- £3 S3 I I I i* I I Tf la 33 1 ■4a 3a CO + 1 ST la S3 1 d 3a 3a to oc ’■+= c8 + + 2 CT1 rH rH — 3)^^} + 2z> (& — k?) ^ = 0, satisfied by numerator of sn nu omitting the factor Vl — x2.1 — : for example, n = 2 (v = 4) the equation is dr ^ (1 — ic2.1 — k?x2) + ^ {(5&2 — 3) x — 2&2#3} dz + z{( 3 — 5A2) + 2fo2} + 8 (& — A8) ^ = 0, satisfied by z — x: and, w odd, //2» /7» ^ (1 - a?. 1 - kV) +{(- 3 + (2v - 1) A2) x + (- 2i> + 4) AV} dz -f -0 (v — 1) {1 -f (y — 2) &2#2} + 2i/ (A — A3) ^ = 0, satisfied by numerator of cn nu omitting the factor Vl — oo2; for example, n — 1 (v = 1) the equation is satisfied by z = 1; g(l -ic2.1 - AW) {(- 1 + (2i> - 3) A2) * + (- 2* + 4) A2*3} dz + z(v-l)l<*{l+(i,-2)x>} + 2v(k-J+**) 4·*} we have y = ^ 7 . 1 + —9 (v + 2w3) #2 u2 v 7 Hence multiplying numerator and denominator by a factor A, the denominator is = A {1+5^+2«3)^}; writing x = 0, and observing that in this case the denominator should be = > or what is the same thing = 'j^gjyr > we find A =^3^, or say A 333. We have V2 _l-v*(v + 2u3)2 k/2M2 1-w8 ^ or observing that the modular equation may be written (v3 — u)(v + 2u3) = u (v — u% this is X'2 1 — v8 u2 (t; — u3)2 _ (1 + w¥ — 2uv*) u2 (y — u3)2 k'2M2 1 — u* * v2 (y3 — u)2 ’ (1 ++ 2u5v) v2 (v3 — u)2 y but from the same equation we have (1 + vN1 — 2uv5) u2 = (v3 — u)2, (1 + u2v2 + 2u5v) v2 = (v + u3)2, whence the fraction is =246 PARTIAL DIFFERENTIAL EQUATIONS FOR H, ©, &C. [IX. or we have V Ar'Jf V — w3 -1- u3 ’ and therefore = v — 16s V + u3 ’ It thus appears that we have the function satisfying the equation 6x2z + 2 (ax — 2#3) ^ + (1 — ax2 + ¿r4) — 6 (a2 — 4)^ = 0, or, what is the same thing, the equation 6^ + 2 {(* + ^-2^f J + {l- (* +|) ^+4S+ 6F2S=0· Writing the foregoing value of £ in the form A + Bar, the equations to be satisfied by the coefficients A, B are BA + (k + ^jB + Bk'^ = 0. 334. We have k = u4, &'2 = 1 — u8, and in general, for any function il of (u, v) ,2 dfl __ 1 — u3 id£l dil 1 — v8 2u3 4- v] dk ~ 4u3 \du + dv 1—u8 2v* — u) ’ - + “V + 2“’')(2’' -“> £ + (1 + uV - 2uvs)(2u3 + v) drl,IX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, @, &C. 247 or if, as will be convenient, we write uA = a, v4 = /3, = 0, then kf2dil_ 1-02 dk 4a (2/3 + (1- 2/30 + 02)(2a + 0)y d£l dv 1 — ir 335. In particular if il, = A, = > then log A = \ log (v — u3) — \ log (v + u*\ and thence 1 dA _-|«2 fw3 — 3w2y A du V — u3 + M3’ v2 — 146 1 dA i 4 u3 A dv V — it3 V + it3 3 v2 — it6 Hence k'2dA= 1-02 u*v 6{-3(l + 2a0 + 02)(2/3-0) dk 4a (2 f3—0)v2 — uo + (1 — 2/30 4- ff2) (2a + 0)}A. u3v 03 03 But ------ = ----— = ^-----sr, and the modular equation v2 — -y4 — mV ¡3 —a02 ^ is a — /3 + 20 — 20s = 0; whence /3 — a02 = (a + 20) (1 — 02): hence 1 — 02 u3v d3 4a (2)8 - (9) v2 - i*6 4a (2)8 - 0) (a + 20) ’ and consequently -3(1+ 230 + 00(2 ¡3-0) + (1 - 2,80 + 0*) (2a + 0)} A. k'*d4 = dk 4a (2)8 — 0) (a -f 20) Also B=^(v + 2u*)A,=?(2a + 0)A. 336. Hence the first equation to be verified is 4 <2° + *> + (if, - + 2»)l~3(1'1' ** + '^ M '9) + (1 — 2j80 + 0s) (2a + 0)} = 0.248 PARTIAL DIFFERENTIAL EQUATIONS FOR H, &C. [iX. We have (2a+ 0)(2/3- 0) = 404-(2a-2/3)0-02, =3(9% by the modular equation; hence the equation is 4 (a + 2(9) - 3(1 + 2a0 + 02)(2/3 - 0) + (1 - 2/30+ 02) (2a + 0), viz. this is 6a— 6y8 +(12 - 16a£)0 + (8a - 8/8) 0s + 403 = 0. But from the modular equation = a+ 20 — 20s, on substi- tuting for f3 this value the equation becomes - 16a20 - 32a02 + 32a0* +1605 = 0, viz. this is — a2 — 2a0 + 2a03 + 04 = 0, which is in fact the equation 04 = a/3, = a (a + 20 — 203). 337. For the second equation, writing for convenience B = QA, this is 34 + (*+i)^+3r.e4{*§+*^}=0, dA or if for the term 3&'2. Q -=j- we substitute its value from the ate first equation, = — QB, that is = — Q*A, then throwing out the factor A, the equation becomes 3%*%)«-^+3l:'‘a-0· ^2 which should therefore be satisfied by Q = —2 + 2uv: viz. this is 'W* 3+(«•+5i)«-e,+s§^{(1 + *·«+«w- *>«g + (1 — 2/S0 + 02) (2a + 0)v ^9 j = 0, or, what is the same thing, it is + (l-2£0+02)(2a + 0)(| + 0)j = O.IX.] PARTIAL DIFFERENTIAL EQUATIONS FOR H, ©, &C. 249 o We have Q =—·+ 20, and then 3+(w4+Q ~ &=b(1 _ 02)2 (2a+&T' viz. this will be the case if 3a2 + (a3 + «)(§+ 20) - a2 (| + = (1 -03)2 (2a + 0)2, or since a ^ + 2$j = 0- + 2a0 it is 3a5 + (a2 +1) (0s + 2a0) - (02 + 2a0)2 = (1 - 02)- (2a + 0)2, which is to be verified. 338. The equation ff4 + 2a03 — 2a0 — a2 = 0 gives 3a2 = (— 0s -I- 2a) (0 + 2a), thereby reducing the identity to — 0s + 2a + (a2 +1) 0 — 0s (2a + 0)=(1 — 02)2 (2a -f 0), that is 0 (a2 - 0s) + (1 - 0s) (2a + 0) = (1 - 02)2 (2a + 0), or a2 - 0s = (- 0 + 03) (2a + 0), viz. this is a2 = — 2a0 + 220s + 04, the equation in question. The equation is thus (1 - 02) (2a + 0)2 + 2 (T|a_ e) j(l + 2a0 + 0*) (2/3 - 0) (-1 + 0) + (l-2/80 + 02)(2a+0)(£ + 0)| = O, or multiplying by 2 (2/3 — 0) and observing as before that (2a 4- 0) (2)8 — 0) = 302, this is 202(1 -0s) (2a + 0) + a (1 + 2a0 + 0s) (2/9 - 0) (-1 + 0) + (1 - 2/80 + 0*) (2a + 0) + 0) J = 0, or, what is the same thing, it is 20 (1 - 0s) (2a + 0) + {(1 + 2a0 + 0s) (2/8 - 0) (a - 0) + (1 - 2/80 + 0*) (2a + 0) (a + 0)} = 0.250 PARTIAL DIFFERENTIAL EQUATIONS FOR H, ®, &C. [iX. Multiplying out, this is (2a2 + 2a£) + 0 (6a - 2j3) + 0* (4 - 8a/3) - 4/303 = 0, or, what is the same thing, a2 + 3a0 + 2^ + fi (a - 6 - 4a#2 - 203) = 0, viz. substituting for /3 its value, this is a2 + Sad + 20s + (a - 6 - 4a02 - 203) (a + 20 - 20s) = 0, or working out it is 2a2 + 4a0 - 4a202 - 12a03 - 204 + 8a<95 + 406 = 0, viz. this is (a2 + 2a0 - 2a03 - 04) (2 - 40s) = 0, which is right. 339. The foregoing differential equation, written in the form 3 + (*+i)Q-Q2 + 3(1-^)^ = °, is further considered in my two papers “ On a differential equa- tion in the theory of Elliptic Functions/’ Messenger of Mathe- matics. vol. iv. (1874) pp. 69 and 110, and in the last of them it is shown that the equation can be integrated generally: the process is, by the assumption e~8(i-*>)!*, to transform the equation into a linear equation of the second order 3 (1 - — 4-1 ~ ^ — . ( k dk 1 - A2 z=0: we have a particular solution of the original equation in Q, and therefore a particular solution of this equation in z\ whence by a known method, the general solution can be obtained. The result is expressed in terms of a variable 7, — ”7= \/2 + a . 1 4- 2a, JCL where a is given in terms of k by the equation, ante No. 265, a3 (2 + a 1 + 2aX.] TRANSFORMATION FOR AN ODD ORDER. 251 CHAPTER X. TRANSFORMATION FOR AN ODD AND IN PARTICULAR AN ODD- PRIME ORDER: DEVELOPMENT OF THE THEORY BY MEANS OF THE ^-DIVISION OF THE COMPLETE FUNCTIONS. The algebraical theory of the transformation has been explained: in the present Chapter it is shown how, by means of formulae depending on the n- division of the complete functions, the prescribed algebraical conditions are satisfied; and that we thus obtain the actual expressions of the trans- formed functions sn , X j, &c. The general Theory. Art. Nos. 340 to 345. 340. We have n an odd number; m, ml any positive integers having no common divisor which also divides n; mK + miK' g> =-----------; n s a positive integer extending from 1 to 1); and when any expression depending on s is enclosed within [ ], this signifies that the product of the ^ (n — 1) terms is to be taken. The formulae for the new modulus X and multiplier M are assumed to be X = kn [sn (K — 4sg>)]4, M = [sn (K - 4m*)]1 - [sn 4s®]2. and we then assume between y and x a relation expressed in the several forms:252 TRANSFORMATION FOR AN ODD ORDER. [X. ^ 1^ II "l sn2 450) J 1 — 2/ = (1 — ®)| "l - T sn (.K — 45o>) J (-). i + y = (i+ ®) " x l2 sn (K — 450) )J (-). 1 — \y = (1 — kx) [1 — kx sn (.K — 4sw)]2 1 + \y = (1 + kx) [1 + kx sn (K — 4sco)]2 where denom. = [1 — k2 sn2 45o>. x1\ It has of course to be shown that the different expressions of y as a function of x are consistent with each other: but as- suming that this is so, it at once follows that dy dx Vl - y2. 1 - xy M Vl - a?. 1 - ’ and consequently that, writing x = sn (u, k), we have ^=sn(i’x)· 341. We start from the equation (1 — y) * (1 + y) = (1 — w) [l - gn (/_ 4g0>J - (1 + «) [l + sn (X — 4sto J ’ and show that, \ and M being assumed as above, this value of y leads to the other equations of the system. In the first place, it is clear that the assumed expression of \ j^y giyes f°r V a value of the form y~ (1, oiite a; = 1, we find y = 1: hence the last-mentioned value of y must for x = 1 reduce itself to y = 1; and we thus find viz. G = (—)^[cn. 45ft)]2 — [sn 45ft). dn 45ft)]2; or, what is the same thing, C = (-)* (n~1} [sn (K - 45ft))]2 - [sn 45ft)]2; viz. G = ilf; and the required expression of y is thus shown to 1 — y be true. Combining it with the assumed expression of ^ we at once obtain the required expressions of 1 — y and 1 + y. 342. It then appears that the change of x into changes y into : viz. writing — for x the expression for y becomes nJU rCX viz. this is — l· Mkx 1 1-fi- krx? sn2 456) J [_ 1 ~l — k2 sn2 45®. x?l I" Mkx k2x2 sn2 45ft) J [sh or, what is the same thing, _ 1 [1 — k2 sn2 45ft). xf\ Mkx [k2 sn2 45ft)] [sn2 45ro] x2 ~1 sn2 45© J254 TRANSFORMATION FOR AN ODD ORDER. [X. or finally it is M2kn [sn 4^0)]' . [1 — &2sn2 450). x2] -r- M x2 "I sn2450)J ’ viz. observing that A = M'2Jcn [sn 450)]4; this is = — . Xy Lastly, in the expressions for 1 — y and 1 + y, making the above changes x into and y into and combining with the value of y, we obtain the required expressions for 1 — Xy, 1 + Xy; and the system of formulae is thus completed. 343. We have to prove the subsidiary theorem, viz. that, 1 — w starting with the assumed value of y-^, the values of x for which y becomes = 0 and = oo respectively are as stated above. And for this purpose it is to be shown that, x being taken = sn u, the formula may be written 1 — y___[1 — sn (ti -f- 4s'g))] 1 + y [1 + sn (u -f 45ft))] ’ s' being any positive integer from 0 to n — 1, and the [ ]’s de- noting the product of the n — 1 terms accordingly. For suppose this proved, then changing u into u + 4cd, each factor is changed into that which immediately follows it; except only the last factor 1 + sn (u + 4 (n — 1) co), which is changed into 1 + sn(w+4na)); but, co being as above, we have sn('W+4nfi))=sn^; or the last factor becomes 1 + sn u, viz. this is the first factor: hence the value of the product is unaltered. 344. Now for u = 0 we have x = 0, and therefore (from the 1 — y original assumed value of ^ ), y = 0: hence also y = 0 for u = 4ft), 8od ... 4 (n — 1) ft), that is for x = sn 4ft), sn 8ft), ... sn 4 (n — 1) g) : or since in general sn 4 (n - t) co = — sn 4tco, we have y = 0 for x = ± sn 4co, ± sn 8co,. . ± sn 2 (n — 1) co.X.] TRANSFORMATION FOR AN ODD ORDER. 255 Similarly for u = iK' we have x = oo , and therefore y = oo : hence also y = oo for ^ = iK' + 4o>, iK' + 8o>... iK' + 4 (w — 1) o>, that is # = sn (¿K7 + 4o>), ... sn + 4 (n — 1) o>); or, what is the same thing, x = sn (iK' ± 4o>) ... sn (iK ± 2 (n — 1) o>) say for x = sn (¿if7 + 45o>), which is = ^—: hence y — 0, and a? sn tS© y = oo , respectively for the required series of values. 345. To prove the formula l-y — sn (u + 4/go)] 1 + y [1 + sn (u + 4s'oo)] ’ we have in general {1 + sn (« + «)} {1 + sn (u - a)} -s- cn2 a = {l + sn ^ ^| (+)> {1 - sn (u + a)} {1 - sn (w - a)} - cn2 a = |l - (^)> where denom. = 1 — k2 sn2 u sn2 a. Hence {1 — sn (u + a)} {1 — sn (u - a)} {1 + sn (u + a)} (1 + sn (u + a)} sn u sn (K — a) \ snw 1 + sn (K - a)j · Write herein successively a = 4a>, 8co,... 2 (n — 1) oo: take on each side the product of all the terms, and multiply each side of the resulting equation by : then observing that sn (u — 45o>) = sn (u + 4 (n — s) oo), and supposing as before that s' has every integer value from 0 to 7i — 1, the equation becomes [1 — sn (u + 45'g>)] -r- [1 + sn (u + 45'o))] \[ -i snu T /-i \ fi , snu l2 =(1 -sn*)[l - sn(g_4w>)J - (1 + sn «) |_1 + s^- 4s^)\ ’ viz. writing sn u = x, the right-hand side is (1 — y) (1 + y): and the equation in question is thus proved.256 TRANSFORMATION FOR AN ODD ORDER. [X. Additional Formulas. Art. Nos. 346 to 351. 346. We may in addition to the foregoing formulae write 1 - sn2{K of 1 i — 4$&>)J (+), — Vy2 = Jl — tea? [1 — tea? sn2 (K — 4s)] -5- [sn 4sg>]2, and similarly Jl — y2 = [cn (u + 4s'g))] ~ [cn 4sco]2, Jl— \2y2 = [dn (u + 4s'ft))] -f- [dn 4$g)]2. 348. From the former expression of vl —\2y2 pu therein y = 1, we deduce a value of X', which (observing dn (K — 4sco) k' x , ... ——---------*- = t-——) may be written dn 4sco dn2 4sg> ' J that + [dn 45ft)]4,X.] TRANSFORMATION FOR AN ODD ORDER. 257 and combining herewith the values of M we obtain various formulae in regard to the new modulus and the multiplier: (_)i(»-l) M A_ V kn~ 'xF* , ^ = [cn4sft>]2, /Un = [dn 4«o]2. \/f=^sn ~4sft))]2’ /xK‘k'^ W V kn = [cn “4sw)]·’ \/XJc'n~2 = [dn (iT — 4sg>)]-. 349. We may now write down the system of formulae X — kn [sn (K — 45ft))]4, X — ]cn-7· [dn45ft)]4, M = (-)i<’*~1)[sn (K - 4sa))]2 -i- [sn 4sft>]2, in \ snwL sn2% "1 , v sn \Ir L1" sn2 4s® J ( ■ Ik* =V x tsn + '(i'x)=CM[i-sid'-Vft,)] <+>> =\/^[cn(M+4s'“)]’ dn Qg., X^ = dn m [1 - i2 sn3 (K - 4sa>) sn2 w] (+), =/\/jk[dn(u + 4>s'm)l cnh-s. c. 17258 TRANSFORMATION FOR AN ODD ORDER. [X. 1 Sn {m’ X) ~ (1 Sn U) i1 sn (K - 4s®)] ( : ^ sn (K — 4sa) Denom. = [1 — 1& sn2 4sg> sn2 ^¿]. 350. To obtain a different group of formulae, observe that the equation between y, x may be written which is of the form x (x2,1)^1}— (a?2, l)^(n 1) = 0, where the co- efficient of the highest power xn is = 1 ; and that the roots of this equation are # = sn u, sn (u + 4eo)..., sn (u + 4 (n — 1) œ); whence we have the identity a?- k2 sn2 4^6) 1 k2 sn2 4fSco J = [x—sn(u + 4 s'o>)] ; ] and comparing the terms in xn 1 we have and similarlyX.] TRANSFORMATION FOR AN ODD ORDER. 259 in all which formulae s' extends from 0 to n— 1, or, what is the same thing, from — i (n “* 1) to + \{n — 1). In the first equation the left-hand side may be written = sn u -f- 2 {sn (u + 4sg>) + sn (u — 4so>)}; s = 1 to \ {n — 1), viz. this is = sn u + 2 2 cn 4sg> dn 4sg> . sn u 1 — k2 sn2 4s&). sn2 u 9 and making the like changes in the other equations we find cn 4sg) dn 4so) efs“(»’x)“"”{1 + '2Srr X""(5‘x)“c““{1+2Sr^ kM M V WM dn = dn u|l + 22 j— *■)=*“« {* + 22 k2 sn2 4sw sn2 cn4so> k2 sn2 4sg> sn2 u\ ’ dn 4s&> fc2 sn2 4sgo sn2 u dn 4sg> cn2 u .}· cn2 4sg> — dn2 4sa> sn2 u }■ 351. The last formula, which is of a different form from the others, depends on tn (u -f a)+ tn (u — a), = sn(u 4- a)cn(w — a)+ sn (u — a) cn(it + a) cn (u + a) cn (u — a) where the numerator, = sin {am (u + a) + am (u — a)}, is = 2 sn u cn u dn a, (^-) and the denominator is = cn2 a — dn2 a sn2 w, (^-) the common denominator, = 1 — k2 sn2 a sn2 u, disappearing. 17—2260 TRANSFORMATION FOR AN ODD ORDER. [X. The 2), sn (u + 8a>) ... sn (u + 2n — 1) g> is in a different order =(—)msn(u — 2(o), sn(w+4&>), (— )”*sn (u — 6g>)... ± sn(u±n—1®), where the last term is sn(u + n— lo>) or (—)m sn (u — n — lo>), according as n — 1 is evenly even or oddly even. To prove this, write 4£ + 21' = 2n, then a + 4 too — (u — 2t'co) = 2 no), = 2 mK + 2 m!iK\ whence sn {u + 4fo») = (—)m sn (u — 2t'a>). If n — 1 be evenly even, = 4^, then giving t every value from 1 to \(n — 1), 4>t is less than n, and the term is retained in its original form; but giving t the remaining values from £(n + 3) to \{n — 1), the corresponding values of tf are from 1 to ^ (n. — 3), and the term sn (u + 4too) is changed into (—)m sn (u — 2t'co). So if n — 1 be oddly even, = 4v — 2, then giving t every value from 1 to \(n — 3), 41 is less than n, and the term is retained in its original form; but giving t the remaining values from \ {n + 1) to \{n—1) the corresponding values of tr are from 1 to ^ (n — 1), and the term sn (u + 4io>) is changed into (—)m sn (u — 2t'oo). We have thus the theorem. 353. Repeating the result, and writing down the analogous results for cn and dn, series sn (u + 4g>), sn (u + 8o>)... sn {u + 2n — la>) is in a different order ► = (—)w sn (u — 2co), sn (u -f 4a>), (—)m sn (u — 6); ^X.] TRANSFORMATION FOR AN ODD ORDER. 261 series cn (u + 4©), cn (u -f 8©)... cn (u -f 2n — 1©) is in a different order = (-)«+»'cn(tt — 2©), cn (u + 4©), (—)m+m'cn(u — 6ft>)... ± cn(w + n — lft>); series dn(u + 4©), dn(^ + 8©)... dn(u + 2?i — lew) 1 is in a different order | (—)m' dn (u — 2to), dn (u + 4©), (—)m' dn (w. — 6g)). .. ± dn (u ± n — 1 ©). 354. It will be at once seen that these formulae, on writing therein u = 0, give for the series of sn, cn, dn of 4g), 8g), &c. the several values (—)w+i sn 2a), sn 4©, (—)m+1 sn 6g). .. + sn (n — 1) ©, (—)m+m' cn 2o) cn 4ft), (—)m+m' cn 6ft)... ± cn (w — 1) g), (—)m' dn 2w, dn 4ft), (—)m' dn 6g). .. ± dn (w — 1) ©. The results are also required for u = K: as to this, observe that in general sn (K + a) = — sn (— K + a) = sn (K — a) ; cn (K + a) = — cn (— K + a) = — cn (K — a); dn (K + a) = dn (— K -f a) = dn (K — a). Hence we see that series sn (.K + 4©), sn (K + 8©)... sn (K + 2n — 1©) is in a different order (—)m sn (K + 2©), sn {K + 4©), (—)m sn (K + 6©)... + sn {K + n — 1©); series cn (K + 4©), cn (K + 8©)... cn (K + 2n — 1©) is in a different order (-)m+m'+1cn(K + 2©), cn{K+4©), (-)7n+m'+1 cn (K + 6©)... + cn {K + n — 1©); ^262 TRANSFORMATION FOR AN ODD ORDER. [*- series dn {K + 4g>), dn (K + 8g>)... dn (K + 2n — lw) is in a different order (—)m' dn (K -f 2(o), dn (K 4- 4w), (—)m' dn (K + 6g>). .. in each of which formulae we may for <0 write — &). 355. It will be observed that in the formulae which con- tain only sn u, cn w, dn u (i.e. which do not contain sn (u + 4s w) &c.) and squared functions such as sn245ft>, &c., the change of form is effected simply by writing 2eo instead of 4&>: in the other formulae there are signs to be changed, and it is safer to retain the 4ft)-formulae, making the change of form only if and when it is required. We have thus: but I do not write down the other formulae in their 2ft>-form. The change from the 4cw- to the 2ft)-formuIae is, as will appear, a very essential one, and it is important to take notice ± dn (.K + n — lft>); X =kn [sn (K — 2sg))]4, X' = k'n -r [dn 25ft>]4, M = [sn (K — 2sg))]2 -f- [sn 2sft>]2, denom. = [1 — k% sn2 2sco sn2w] ; of it.X.] TRANSFORMATION FOR AN ODD ORDER. 263 n an odd-prime; the Real Transformations, First and Second. Art. Nos. 356 to 363. 356. We have co = mK + m!iKf n where m and m' are positive and negative integers having no common divisor which also divides n. It is convenient to take n an odd-prime: there are here n + 1 distinct transformations corresponding to w+1 values of to which may be taken to be K iK' K+iK' K + 2iK' K + (n-\)iK' n ’ n ’ n ’ n * n ’ or to be K iK K + iK’ 2K+iK' (n — 1) K 4- iK n ’ n n ’ n n or again K n ’ to be iK n * K ± iK' n * K±\(n-\)iK' n Two of these transformations are real: the former of them corresponding to the value w = —, and called the first trans- formation, is a transformation to a modulus X which is less than k; the latter of them corresponding to the value iK’ to = — , and called the second transformation, is a transfor- n mation to a modulus X1 which is greater than k. First Transformation, = — (to a smaller modulus X). 357. The general formulae apply at once to this case, but it is convenient to slightly alter them by omitting the factor (—f - into 2co-formulae, observe that the series / 4 K\ ( 8 K\ ( 2(n-l)K\ 1 sn fu + -^-1, sn Iu + \... sn IwH—1^—- ) is in a different order / 2 K\ ( 4Hl\ ( 6K\ = _sn^__j; 8n(«+—j.-sn . the series / 4 K\ ( 8 K\ l 2(n-l)K\ 1 cniw+-^— 1, cn(w+ — cniwH--------—-^-L—1 is in a different order / 2K\ f ^ 4K\ ( 6K\ = _cn^__rj) cn(*+—J... + en (« ± - ); and the series , ( 4>K\ , / 8K\ , ( 2(n-l)K\ ] dn \ u H — J > dn(«+—)... dn(«+--------) is in a different order In all the formulae s has the different integer values from 1 to J (n — 1), and s' the different integer values from — J (n — 1) to + \ (w — 1); or as regards the 4 + 03 a H3 N I----------------1 I £ a 03 ■ i a 03 1+ +1 03 1+ a 03 r< +1 03 SF a* tc .S *5 © o £ P* p o c3 GO s o P a>X.] TRANSFORMATION FOR AN ODD ORDER. 267 fegl Co CM ! G T? g CM I G o w w w w CM CM CM CM + + + + rH t-H rH !—1 § S3 S3 S3 G G G tc o '"O +=> II II II II G o ^1 Co ^ I G T3 G co ^ I l!s Cl G I ^ i ^ Co ?* tP I G o W s 1^268 TRANSFORMATION FOR AN ODD ORDER. [X. iK' Second Transformation, co = — (to a larger modulus \,). iK 360. Write in the general formulae co = — : the formulae n in the first instance present themselves in an imaginary form : these are given as well in the 4co- as in the 2a>-form. For the conversion observe that the series f 4 %K\ sn \ u + A l ), sn i u + 8 %K\ ... sn m + (* 2(w — l)iK'\ n /' \ nj\ n ) is in a different order 2iK\ l . 4iK\ { 6%K\ = sn \u — the series '\ / 4 iK\ ( 6iK\ -)’sn{u+ir)’ ( (n— l)iK\ sn(M± cn \u + UK' . cn ^ 8iK \ f 2 (n — 1) iK \ u H-----)... cn (u H— -----------) n \ n } is in a different order 2 iK\ ( 2iK\ ( , 4%K'\ ( 6iK\ = _cn^__j) <*(«+— , / (n-l)iK'· + cn [u +----- V “ n \ y and the series , / 4 iK\ , / 8 %K\ , / 2(n — Y)iK\ dn(“+—)· d°(”+ n ) is in a different order / (n — 1) ± dn l u + ^^---------J . 361. There is a further change of form to be made in some of the formulae. We have k snv= * — isn 2 siK n sn . , and thence sn (v + iK’Y J._______ 1 r, 2siK\ (n — 2s) iK * r— ’ sn-1-------- n X.] TRANSFORMATION FOR AN ODD ORDER. 269 Putting for a moment n — 2s = 2t — 1, we see that s having the positive integer values 1 to \(n — 1), t has the same series of values in a reverse order; whence finally writing s instead 2siK' 1 of t, — k sn —— has the same values as (2s — l)iK* ’ °r sa^ the series — h sn 2 iK' — h sn 4 iK' sn — k sn (n — 1) iK' is in the reverse order 1 “ZF’ sn — n sn 1 SiK' * ’ ’ n sn (n - 2) iK' ’ and similarly k sn ^ has tbe same values as sn (K- (2s - 1) iK' y We have moreover 7 2siK' (n- 2s) iK' Cn - 2s) iK' n n n , 2siK' (n — 2s) iK' (n — 2s) iK' n ii n which may be similarly transformed by putting therein n — 2s = 2t — 1, and finally s instead of t, as above. In all the formulae s has the different integer values from 1 to | (n — 1), and s' the different integer values from — \{n — 1) to +4 (w— 1): or we may in the 4©-formulae consider s' as having the different integer values from 0 to (n — 1).270 TRANSFORMATION FOR AN ODD ORDER. [X. O © $ 0 e 1 t δ è o o © 5Q (M ÇO co s fe s fe" s fe tí co tí Π3 I------------------------1 tí CG =c : ^ I tí co I___________________________________I g o tí 0) '■σX.] TRANSFORMATION FOR AN ODD ORDER. 271 "I « Co CM I a •s! CO CM I p CO p CO I £ CO I 1 § ,r H I =o CM ‘So CM J L P CO ss P CO 4· p CO I § P CO *5ô I S —I I— ^ Í g -3 + + £ ^ co CM ss P CO + 3 Ö CO *5ô 4 + ss^ p o IS I $ *5â k + ^ SS ^ P , ^ 1_______I *5* ^ \f< "> ss P ss P •3 s co I lí /<ί g I ·«* co P so I 4 ss p CO 4- g ·£ g Tη I I ΊΓ *5S P co + SS P CO *5£ r< iÿ P CO 4< 'SS s ^ I I ΊΓ *5è _________I P CO f< + “I Γ fg ^co Sí * VCo £ ^ Co 4- + 4 ^ss^ P CO P o ¿■H "3 SS P CO i^ri*5g .v ,. r ί *5i 4* II II II II << ^ «Ν' P o P Π3 denom. as above.272 TRANSFORMATION FOR AN ODD ORDER. 5* So £ C* s CO CO ’ n sn *·*)- sn w 1- snJX] TRANSFORMATION FOR AN ODD ORDER. 275 On the left-hand side the least real and positive value for which sn(jp ^ vanishes is ^. = 2A, and on the right-hand side it is u = — : hence we have MA = — or = A. n n nM 365. In the second transformation SB tn'2- sn / u \ snw I" sn2 u 1 , x If, >Xi)= L ttctJ (+)· On the left-hand side the least real and positive value for which sn vanishes is = 2AX, and on the right-hand side (since here the only factor which can vanish is sn u) it is u = 2K: hence ifc^Ax — K or lrr = A1. Mx K K Observe these equations, = A and y =^i· The Complementary and Supplementary Transformations. Art. Nos. 366 to 371. The first complementary transformation. 366. Start from the first transformation: this may be pre- sented in the form , , 2 s'K\ tn(w+_j s' = -$(n-l) to + ^ (n — 1). Writing herein iu instead of u, and recollecting that tn(iu, k) = isn(u, 1c), the equation becomes sn M v\ /k'n[ ( 2s'iK ,Al •x)~v v Lsnr—r · /k'n , 7/. |" ( 2siK j,\ ( 2 siK 7,\1 = ^7-sn(w, k) sn f u+ —— , k Isn (u-—, A? J s = 1 to ^ (n — 1) 18—2276 TRANSFORMATION FOR AN ODD ORDER. [X. , . , . / T (2siK ,A1* /k'n I”, sn2 {u, k ) "] which is =(-)*<” "W—.*)J V Y^l1----------------------j2sik TaJ sn I , ic j \ n ) — 1 — k'2 sn2 sn2 (u, ; or, since clearly the outside multiplier must be = -jjp* this is sn (u ^A sn(w} ¿Of-, . sn2(u, k') ~| j^l + k'2 tn2 ^ sn2 (u> ^oj · This is the first complementary transformation giving sn(-^,V^ in terms of sn (u, k'). Observe that its form is analogous to the second transformation. The second complementary transformation. 367. Start from the second transformation; this is to + 2" (p — 1 )· to(s,'x')"V/ir[tn(“+?£?)]· * The formula is ^ = ( - )*XJ M~~ 1 + tn (-) and considering the least real positive value of u for which the two sides respectively vanish: these are on the left-hand side278 TRANSFORMATION FOR AN ODD ORDER. [X. if = 2A', and on the right-hand side u = IK': hence we have K' MA' = K' or A' = ~. Similarly from the second complementary transformation, sn (it, k') sn2 (u, k') sn ( u ^ ,\ sn (u, k’) f, v^>XiJ-T^L sm (+) the least real positive values for which the two sides vanish are u a., , 2JT , K' . , K' ■ = 2A, and u =----, whence if,A/ = — or Aj = · if, 1 n n nMl 369. We have thus obtained the equations K' A' = if’ A'-^ 1 nM1 ’ to be taken along with the foregoing equations, No. 365, . if , . _K Eliminating M and M1} we obtain A' K' K A/ A=wif’ r=Mx;: the first of which is an equation between X and ky and the second is the same equation between k and \: and it thus appears that X is the same function of k that k is of The equations show that X is less than k, and Xj greater than k. The first supplementary transformation. 370. In the second transformation u _ \ sn u sn [ irr, Xi = MS Mx , sn2 u sn2 u 1 ,2 siK' L sn2 -J n L1 ,(2s-l)iK L sn2 - J n change k into X, and therefore \ into k: writing for a moment ^ as the new value of M1 the formula becomes sn (u, X) sn2 (u, X) X.] TRANSFORMATION FOR AN ODD ORDER. 279 1 - sn2 ( u, X) sn-|(25-1> Aj K nM MNX the equation becomes sn (nu, k) = nM sn i-g-, \ sn sn2 f2siA \ n ■D -H 1- sn2l M’x sn2l(^zl)^,XH n which is the first supplementary transformation. Combining herewith the first transformation, sn 2u ( u x\ mu m[Tl,\) = M snJ .2 sK - 7oo 2 sK 1 - k'2 sn2-sn2 u n we see that the two together lead to an expression of sn (nu, k) in terms of sn (u, k). The second supplementary transformation. ,371. In the first transformation sn M’ snit ^ sn2 u H 22 sK L- sn2 J - 2 sK - sn2 u change k into \lt and therefore \ into k: writing for a moment N as the new value of M, the formula becomes -,XiY ~n(W 7-V Sn(M,Xl)ri Sn^-“’ -i- j^l — \i \ n , Xjisn2^ > j280 TRANSFORMATION FOR AN ODD ORDER. Ex- change also u into ; then observing that K' a ' l 1 M = , and therefore N = rl, , = -lrr, that is n = , A IT* nMf MXN9 the equation becomes sn {nu, k) = nM1 sn I , X* sn2(i’Xi sm 2sAx ,Xi which is the second supplementary transformation. Combining herewith the second transformation, / u ^ \ sn {u, k) "tar.'N—w, - sn2 u 1 1-----^TFJ sn2 1 - n sn2u sm (2s— l)iK' we see that the two together lead to an expression of sn {nu, k) in terms of sn {u, k). The Multiplication-formulae. Art. No. 372. 372. For the actual determination of the multiplication- formulae, observe that the first supplementary transformation may be written in the form sn {nu, k) = -j- u 2s'iAf , sn [ ir>H-------, A, M or, what is the same thing, sn(m£, k) = JX V k . u =■>(»+ 2m i A.' ,X sn But the first transformation gives to + i(w- 1); rri = -\(n- 1) to + £ (w-1). /---X.] TRANSFORMATION FOR AN ODD ORDER. 281 or say “ (m- x) - [“ (“+^r) to 4- \ (w “ 1 ) 5 and writing herein 2m, iK « u , u H--------for u. Tir becomes ii 9 M u 2miK _ u 2m ih! M+ nM ’ =M+~n and the formula is / \i,„ (u Zm'ih' \ /knV ( 2mK 2m iK' » u+”■x) - V x r (“++“¡r- where on the right-hand side m has the last-mentioned values. Giving herein to m' the different values from — \ (n — 1) to + \ (n — 1) and multiplying the results together, observing that (n-i) — (n-1)? we obtain (n-i) u 2m! ih! i>A' /in*’ j l , ■, the left-hand side being = a is sn nu = (—)*(w-1) (n2_1> |sn [u -f- 2mK 2 miK' n or, formula is — sn (nu9 k)y the 2mK t 2m!iK' where on the right-hand side the { } denote the double product obtained by giving to m, m! respectively the values — \{n — 1) to +| (rc-1), or say the values 0, ± 1, ± 2, .. . ± \ (n - 1). And in the same wav ·/ cn nu = lyi»8-!) w) cn u + 2mK ^ 2m'iK\^ n n }j and dn nu = ir>( 2mK 2miK'\) u +----+-------- [ ; n n J J which are the formulae obtained Chap. IV.( 282 ) CHAPTER XI. THE (¿-FUNCTIONS: FURTHER THEORY OF THE FUNCTIONS H, ®. 373. In the present Chapter we start with the transforma- tion of the order n in the form of the first supplementary transformation, whereby the functions sn (nu, k), &c. are given in terms of sn : writing first ^ for n, we make n = oo , and (as will appear) we thus obtain the elliptic functions sn (u, k), &c., as fractions, the numerators and denominators being re- spectively obtained in terms of the circular functions of viz. as products depending on these functions, and involving 7T K' also the quantity e K , which is put = q: the elliptic func- tions have been already in Chapter VI. expressed as fractions by means of the functions H, ®: and identifying the two ex- pressions, we obtain the expressions of these functions as series involving powers of q, or say as (¿-series. Derivation of the q-formulae. Art. Nos. 374 to 382. 374. The first supplementary transformation is sn (nu, k) = nM sn V nXL] FURTHER THEORY OF THE FUNCTIONS H, ©. 283 u we write herein - for n, and make n infinite. This gives X = 0, sn (0, X) = sin 0, A = r; whence (in virtue of A = K_ nM and A' •=E\ M)’ ,, 2 K A' 7t2T niH = —, — = : 7T 71 2iT and the equation becomes 2 A' . iru snw= — sm rp. 7r -¿A sim , 7TW 2A sim , SITT- K 1 - • Sin- 2A" . (2s — \)iirK' sm2 -----■■— 2 K This is one of a group of formulae obtained in the same manner. 375. The formulae are 2 K . 7ru sn u = — sm 7T ZJ\. sin , 7T^ 1 - sm2 cn u = dn 74 = cos TTU M sin- 2K miirK' ~TT~ , 7T& 1 -■ 2K COS'5 , miirK' K sim 1-· ITU 2K cos' (2m — 1) iirK' 2K 1 - sn w = (1 — sn ^0 sn 1 - ITU 2K cos - 1 + sn u = (1 + sn ITU 2K 1 + miirK' TT~m 7m 2Z sn cos miirK' ~K~ 284 THE ^-FUNCTIONS: [XI. 1 — k sn u = 1 + k sn u = denom. = sn 1- 7TU 2K cos 1+· (2m — 1) iirK* M ITU 2K sn cos 1 - (2m — X)iirK' 2 K ,7TU 2K sn** sn (2m — X)iirK' 2K 376. We obtain in like manner another group of formulae, in which also m has the values 1, 2, 3... to infinity, cos (2m — 1) intK' 7T . 7TU j snm kg'sin 2if 21 . ,(2m-l)i7r^' “7™ 1’ Sm' 2K SU1 2KJ . (2m — l)inrK' 8ln-----2Z------- CUU~ kKC°S 2A” 21 ( 2(2m-l)t7riT sim 2# — sm-5 , ITU 2K) (_)m-icot(^ - 1)^' dn u = 1 + ^ sin2^^. £ * K 2K \ . (2m —l)i7riT . 9nru sm -----Z-------sm 2K/ The deduction of this last formula presents some peculiarity: writing + instead of (—the formula originally presents itself in the form ±*”-£±{r*\ ■ (_)»* sin (2m1) WK ^ (2m -r>i„K'\ sin2(2w~^^-Sin2||XI.] FURTHER THEORY OF THE FUNCTIONS H, @. 285 viz. the upper or lower sign must here be taken according as the number of terms in the series is even or odd. To get rid of this variable sign, write in the equation u = 0, the equation becomes - 7T 7T ^ | 11 =2^±S'2* . (2m — 1) irrK' (2m — 1) iirK’S * sin ----~~------cos sm K___________ (2m — 1) VTrK! K K and subtracting this from the general formula, each side of the equation is affected with the same sign ±, which sign may therefore be omitted: whence, observing that in general sin a cos a sin a cos a sin2 a — sin2 x sin2 a sin2 x cot a sin2 x = sin a cos a -r——= -t—-----------z—- , sin2 a (sm2 a — sm2 x) sm2 a — sm2 x it is at once seen that we thus obtain the result first written down. All the formulae assume a more convenient form by writing therein u = , viz. we have thus sin = sin x, and conse- 7r quently the elliptic functions sn , &c. expressed in terms of sin#. 377. Introducing now the quantity q,=e 7rK! " K we have . miirK' 1 / . i (1 — o,2w) —a(s--r-)—4^. COS miirK’ ~TT 1 2 (qm + q~m) = 1 +qm 2 qm ’ sin2 x , 4gamsin2a; _ 1 — 2q*m cos 2x +q*m . . „miirK1 ~ +(1 - q™J~ (1 - q^f ’ C‘286 THE ^-FUNCTIONS : [XI. and in the resulting formulae the right-hand sides contain as factors certain functions of q, which functions are afterwards determined as will presently appear. Supposing this done, the formulae of the first group are 2 Kx IT 2Kx sn- cn - dn 1 — sn 1 -f sn 1—¿sn l + &sn where 7r 2 Kx 7T 2 Kx 7T 2 Kx 7T 2 Kx 7T = 2 i- q sin # vk [1 - 2q2m cos 2# + ^m], (+) 0 f% V- =2v *v? cosa: [1 + 2q2m cos 2# + g4™], (-) = VF [1 + 2q2m~1 cos 2x + ^4m_2 ].(-) sin #) [1 — 2qm sin x + g2w]2, (-) _2\/f^3(1 + sin x) [1 + 2qm sin x + q™1]2, (-) != VF [1 — 2qm~* sin x + which will be proved 2 Kx 7T I — ^ further on, we obtain the foregoing expression for 1 — dn and conversely by the integration of this we obtain the last- / 2Kx\ mentioned formula for sin-1 isn^^J . 378. In completion of the investigation of the formulae of No. 376, observe that writing ¿=[i - (-o I=[l + ^»p, (-5-) I=[l +q™-'Y, (+) where denom. = [1 — ^2m~1]2; the formulae obtained in the first instance are 2Kx 2AK r-. « x sn------=-----sm x [1 — 2q2m cos 2x + q*m\ (-?-) 7T 7T 2Kx cn —— = B cos«[1+ 2q2mcos 2x+ j4”1], (-?-) 7T288 THE 5'FUNCTIONS : [XI. dn = C [1 + 2qMl~1 cos 2x + q1™--], (h-) 7T where denom. = [1 — 2q2m~1 cos 2x + q4m~2]. Writing in the first and third of these x = \tt> we find 2AK C 1 = k'=C.O; 7T i> whence C = V*', and 5 = 2's/k'AK 379. To determine A, write for a moment U in plaae of eix, then we have sn 2Kx AK U- U-1 [1 - q»nU2] [1 - q^U~2] 7r 7r i [1 - ^2W-1 ?72] [1 - ^2m-1 C7~2]5 which, observing that U-U-'= U(1 - U-% =-¿(1-V*). may be written 2Kx __ AK U[ 1 - ?2mi72] [1 - ^m-2[7-2] “ 7n [1 - ^2T"“1i72] [1 - ^-li/-2] ’ 1 [1 - g2m-2^72j _ g2m^/-2] sn or m t7[l - q™1-1 U2] [1 - g2™"1 i/~2] ^ . iirK' 2Kx, x1 or x write x + — ^ , sn--becomes 2 K ’ f2Kx 7T snt^r+iir)’ = W A?sn--- TT U is changed into q*U, and taking the second formula we have 1 AK 1 [1 - q2™-1 U2} [1 - q™-1 U~2] 2Kx ~m (fU [1 - q2mU2] [1 - q^U~2] ' K sn Hence multiplying, we find 1 /AK\! 1 , AK s/q l-[ir)-Tq· wl"mceir_VFXI.] FURTHER THEORY OF THE FUNCTIONS H, ®. 289 and therefore O-VF; 'KVk' VF and substituting we have the foregoing formulae for sn, cn, and dnof^. 380. We also obtain various other ^-formulae. Multiplying the expressions for jB, G we have 2 \!q. Tc _ [1 — “vF“ [i + ?m]2 ; and observing that [I _ Q2ml i n 4- t1------j — ------------ L +i [1 -22”*-1]’ we find and thence, using the foregoing value of .d, 2kk'Ks [1 - ?m]6 = u^t/q which two formulae give [1 - qm]e and to these may be joined [1 + [l-9—1]2[l-?-] = y2^, _ 4 'Jlck^K3 _ 2 \/q ~7W’ [1 + g2™]6 [1 + 9m]6 k Mk's/q ’ _ *Jlc c. 19290 THE 9-FUNCTIONS: [XI. 381. If for shortness we write: a = [1 + 52m~1], whence a/3 = [1 + qm], yS = [1 — qm], /S = [1 + 9m], and afiy = 1*; 7 = [i - g2”*-1]. 8 = [l-g"*]; then the foregoing formulae give k = k' = 2K 7r 2kK 7T 2k'K TT 2^/kK IT 2\/k'K 7T The equation &2 + &'2 = 1 gives 7s + 16g/38 = a8, or written at length {(i - 9) (i - 9s) (i - 9s).. ,}8+169 {(i + 92) (1 + g4) (1 + g6).. .}8 = {(i + g)(i + g3)(i + g5)···)8; a remarkable identity. * This is in fact the formula [1 + qm]=^ r~g2m-ij Prove^ No. 380: it occurs in Euler’s Memoir, Be Partitione Namerorum (1750), Op. Min. Coll. p. 93.XI.] FURTHER THEORY OF THE FUNCTIONS H, ©. 291 382. It will be noticed that we have obtained expressions e 2 Kx 2 Kx , 2 Kx . , _ . , . tor sn -----, cn----, dn-----, as rational tractions having a 7T 7T 7T common denominator, the three numerators and the denomi- nK' nator being each of them a ^-function, (q = e K ) , involving circular functions of x respectively. Imagining x replaced by its value we have sn«, cn uy dn-w expressed as rational fractions, the three numerators and the denominator being tjru each of them a ^-function involving circular functions of 2 K. We have already obtained for snw, cn u, dn u fractional ex- pressions having a common denominator ©m, and in their numerators Hu, H(u + K), © (u + K) respectively; and it thus appears that these functions must be, to proper factors pres, multiples of the (2m — 1) (1 V 1 + k sn- ' ' v 4 gm_* sin (2m — 1) x im—’ or, writing the series at full length, 4 Jq sin x 4 Jq3 sin 3a; 4 Jq6 sin 5# = + ~ 5(1-3») Differentiating each side in regard to x, we find without difficulty 2Kx 2JcK 7r 4 cos a; 4 cos 3a; ( 4 Vg5 cos 5x i = 1 -q 1 -q* + 7T , 2 Kx' dn------ 1-q5 *&c., or, observing that the left hand is 2 kK ----sn 7r v (*- 2Kx\ 7T / 5 2kK 2K ( 7r :---sn— - 7T 7T \2 » and writing ^7r — x in place of x, this is 2kK 2Kx 4 Jq sin x 4 Vo3 sin 3x 4 Vo5 sin ox ----sn----= — — H---------f--------1-----^— 7r 7r 1 — 5 1 — ^ 1 — ^ + &c. 384. It is this formula which leads to the identification just spoken of; viz. squaring the two sides we obtain after all reductions 2kKy 2Kx 4 K ir)“— 7T , (2q co\ - *{■1= 2q cos 2a; 4*q2 cos 4a; 6q3 cos 6x q■ + 1-g4 + I-58 +"' or, multiplying by dar and integrating from x = 0, /2MY/· .22T# , 4STV. ^ x -4 <7 sin 2a; t q2 sin 4a; ( q3 sin 6x H"" n 71 I t « t" ··* l-£2 1 -£6293 XI.] FURTHER THEORY OF THE FUNCTIONS H, ©. whence, from the definition of Zu, ante, No. 138, IK „ /2Kx\ , {q sin 2a; , q2 sin 4x _ 5* sin 6a? , v z kw)=4 ++"w"+-· and if we again multiply by dx and integrate from x = 0, IK f r, l'lKx\ j_, (1 — 2q cos 2a; + q2) (1 - 2y! cos 2a: + }; and moreover294 THE ^-FUNCTIONS: [XI. where ffn) = ryJl+l nn-l· 3 (1 - g) (1 - g2n+1) 1 - g3.1 - qm+s l-q5.l-q , 2?l+5 + ... _ qn 1 - g81 1 — q 1 — g3 1 — q5 + ... n2n+i ^ 1 — ^2n+1 1 — g2»+3 2 __ qtn+5 qn j' g g3 ^ g2n_1 = 1 -g“ jl-? +1 ~(f" + 1 -q^1 + ... and o>= qn 1 - q . 1 - q™-1 1 - g3. 1 - q2n~3 1 1 - g3.1 _ g2™-5 qn ... + 1 - g2”-1. 1 - q _ 9' ,?l f 1 -g® 1 — q 1 — g3 , + ■ ...+ 1 - g2"-1 9 1 _ q2n-i l _ qZn—3 · · · 1 _ 2 + 1 +1 ... + 1 7ign 2gn ( q q3 qm~1 ] = 1_?2»+ 1 -q^\l-q + 1 _ gs ·” + j whence 1 - g2» ’ viz. each coefficient (except A, which is an infinite series) has this finite expression, and we have /2kKV 02Kx A . (2gcos2# 4g2 cos 4# 6g cos 6# ) (it) ■",ir-A-4l-w + W^+T^+4 386. To find the value of A’ =s {(li/)2 + (W)2+&c'} ’XI.] FURTHER THEORY OF THE FUNCTIONS H, 295 multiply by dx and integrate from 0 to we have or, what is the same thing, rK A =—- ¥ I sn2 udu; 7T2 Jo viz. from the equation ZK= 0 = ^ — k2 J sn 2udu, we have A=~(K~E), 7T and the proof is thus completed. 387. Write for shortness /2 k'K y y c [1 - q*m-1]2 where, emte, No. 380, î=2^#. V& 5 [1 - g2m—1J6 = J then the relation obtained is © = £?. [1 — 252”1-1 cos 2x + g4”1”2], which is the required expression for 0, leading as mentioned above to the corresponding expressions for the other functions.In fact, comparing the forms 296 THE ^-functions : [XI. s %s 55 CM o © Os + 55 CM v. + 55 ... and thence ri+o·—*i=i+-S*_ +______2*___| g9g$ | In a very similar manner it is shown that 1 . q z q4 £2___ [1 — qmz] 1 — # 1 — 1 — ?. 1 — 221 — . 1 — j2# ________5®________ 2s 1— q·!— q2.1 — 5s 1 — <£2.1 — <£22.1 — + .· 390. Starting with the equation *- +_____2 -5* I-5* n iqa—1.1-! , g* I g**1__ I ?** + we have similarly n+o·"-1*-1 i=i+-2£l + g4^2_+_________ifl_____+ [1+5 Z J 1 + 1_22+1_92>1_?4 + 1_32 !_24,1_36+· and these two are to be multiplied together; the product will be an infinite series of the form B0 + B1 (z + z-1) + B2 (z2 + 2T2) + ..., where B0, Bly B2... are functions of q given in the first instance as infinite series, which however admit of summation by means of the last formula in No. 389, viz. 1 _1, g * | ^_____________^ [1 — 5"**] T1 — 5I — qz 1— 5.I — 5*1— 5*. 1 — q*z 59 zs + 1 - 5.1 - 52.1 - 5s 1 - qz . 1 - L K J (s = 1 to oo ), so that Hu, ©w, are constant multiples of these expressions ¡2k,K respectively ; and since ©0 = , it follows that the plete values are com- Su- V 7T 7T . 7TU . _ 7TU , Sm 2K Sm2K . siirK ' [ sm K J /2 k'K • , 7TU i Sm 2K r V 7T It is important to examine the meaning of these formulae. Consider the function which enters into the expression of Hu; or writing for greater convenience m in place of s, sayXL] FURTHER THEORY OF THE FUNCTIONS H, ©. 301 sm ITU 2K sim smJ iru 2K , m'iirK ~K~ {m = 1 to oo ). 394. Observing that in general sin2 u — sin2 a = sin (u + a) sin (u — a), this (disregarding for the moment a constant factor) is f . / iru m'iirK'V] = |sinf^ + __jJ A2 K [· = j^sin ~(u + 2ffiAK'')] , where ml has every positive or negative integer value (zero in- cluded) from — oo to + oo ; say from —jj,' to 4- fil, // = oo. Now we have sin x — x |^1 — ^^2 j > (s = 1 to oo ), which writing for x, becomes 2K . ITU IT ^m2K=2KU L 4’ or disregarding a constant factor, iru sin =[u+ 2mK], where m has every positive or negative integer value, zero included, from — oo to + oo ; say from — /jl to ft, fi = oo . Assuming for a moment that it is allowable to write herein u + 2m iK in place of u, we have sin (u + 2m'iK') = [u -f 2mK + 2m iK'], m as above, and consequently the numerator is = \u + 2mK -f 2m'iK']t302 THE ^-FUNCTIONS: [XI. m, m each extending from — oo to 4- oo as above. As regards the omitted constant factor, it is clear that sn u -i- u reduces itself to unity for u indefinitely small, and the formula thus becomes 2jK . 7TU _:_ cm ___ m, m' as before, excepting that the set of values m = 0, m = 0 (having been taken account of in the factor u) is to be omitted. sim 1 -■ sin-8 ITU 2K , siirK' K 1 + 2mK H- 2miK' 395. But when in the sine-formula we write u -f 2miK' in place of u, we assume that u + 2m'iK' is indefinitely small in regard to the extreme values ±/jl of m (viz. the infinite product x2 \ 4}7T2) to sin x only on the assumption that — is indefinitely small) : of course this is so when m' is finite, but m' acquires the values in order to sustain the assumption we must suppose that ¡M is indefinitely small as regards ¡jl ; or say that p -r /a = 0. Hence in the last-mentioned equation the limits of the doubly-infinite product are m = — fi to m — + fi; m' = — /jl to m'= + p p each infinite; but /j!+(i = Q. Putting for shortness 2mK -f 2m'iK' — (m, m') the equation is 2 K . 7TU ~tt Sin 2K 1 - . 7TU Sm 2K sin2 . S17T — K = w|l + { (m, m)) * which is one of a group of four formulae. 396. Writing for shortness as in Nos. 39 and 120, (m, m) = 2mK -h 2m'iK\ (w, m') = (2m -f 1)K + 2miK', (m, m') = 2mA" + (2m' -I-1) iK\ (m, m) = (2m + 1)K + (2m' + 1)iK',XL] FURTHER THEORY OF THE FUNCTIONS JET, ©. 303 these are 2 K . 7TU VSm2Z sirr 1- , 7TU w sin'5 siirK' sin- 1 - COS'5 K , 7Hi 2 _ 2K TmF1 = u -11 + (m —___1 , m')| ’ = 1 + (m, m')j ’ sin'5 v 7TU 1 - COS' 1 - 2 K (25 — 1) iirK' K . , 7TU 2K 1+7= (my m')j ’ sim . (2s — 1) ittK' sin2 v--------------- K = 1 + (ra, m) where on the right-hand side the limits are to be taken so that (m, m'), &c. may have equal positive and negative values, or say, as regards ra, from ra = — /x to 4-fiy m m ra' a f) w' = — fi—1 yy + fly -V » + /> -/-I „ +/*'; viz. ra, ra' have all positive and negative integer values between these limits (both inclusive) respectively: but as regards (m, m') the combination (0, 0), (which is separately taken account of in the exterior factor u), is to be omitted: fi, p are each infinite, but fJL + fl= 0. 397. The values of the Jacobian functions H, © thus are, as mentioned No. 39, Hu = VI-v/— U jl + , v 7r ( (ra, ra ) I ii+—¡wi, v 7r l (ra, ra)) ©w =304 THE ^-FUNCTIONS: [XI. limits as just mentioned. It is on account of the unsymmetrical condition p + fi = 0 in regard to the limits that H, © have a perfect periodicity as regards 4if, but only an imperfect periodicity as regards 4iK'; viz. as regards this quantity the functions are only periodic to an exponential factor pres. The resulting expressions of the elliptic functions are, as mentioned No. 130, sn“-“{i+sWj}· on“- {1+(srW)}· <+) d" “· {1+ denom. = \l + - ^=^1; { (m, m)j where as regards 4iK\ although the numerators and denomi- nator are not separately periodic, they acquire by the change equal factors, and thus the quotients are periodic as well in re- gard to 4iK' as to 4K. 398. We may state the general theory thus: consider the doubly infinite product { 1 + _______a____r-l a + mil -I- m'il'J ’ where m, m have within infinite limits every positive or negative integer value whatever. To avoid difficulties, it is assumed first that il, fT are in- commensurable, (for if they had a greatest common measure A the function would be an infinite power of the single product ) t a positive or negative integer; secondly, that the ratio il : O' is imaginary, for if it were real there would be an infinity of factors for which mil = m'il' is indefinitely near to any given real value whatever. The function a + mil + m'il' can at most vanish for a single set of values of m, m ; viz. it 1 + u a + tAXI.] FURTHER THEORY OF THE FUNCTIONS H, ©. 305 will do this if cl = — Xil — X'il', X, X' being positive or negative integers; and we must in this case replace the product by and exclude from the product the combination of values m = X, m = X'; but this makes no real difference in the theory, and we need only attend to the other case, that in which a does not vanish for any integer values of ra, m'. Thirdly, that the limits are such that to each given value of a + mi2 + ra'f2' there corresponds an equal and opposite value; or what is the same thing, regarding ra, m as rectangular co-ordinates, then that the product is extended to all integer values of m, m' lying within a closed curve having a centre at the real point given by the equation a + (viz. if a= — Xil — X'fT, then the co-ordinates of the centre are m = X, m' — X'). Say this is the “ bounding curve,” we may regard the linear magnitude of this curve as proportional to a parameter C, in such wise that C being indefinitely large, each radius vector of the curve (mea- sured from the centre) is indefinitely large. Upon the foregoing suppositions, regarding the bounding curve as given in its form (for instance, if it be a circle, or a square, or again a rectangle with its sides in a given ratio, &c.), then as G increases and ultimately becomes infinite, the product in question tends to and ultimately attains a certain definite value; but this value is dependent on the form of the bounding curve. 399. There is, however, a relation between the values of the product for different forms of the bounding curve; viz. this is U^e-H*-*** II2, where A1} A2 denote the values of the integral taken for the two forms of the bounding curve respectively. u c. 20306 THE g-FUNCTIONS: [XI. In particular let the bounding curve be the rectangle m = \ ± ¿I, m = V ± y!> and let the product, when y! /a = 0, be called no, and when fi' -r- fi — oo be called l!*,; then if II be the product for any other given form of the bounding curve, we have n = n0, =ei£.“2n„, where B0y are constants depending on the form of the bounding curve; and observing that II0 has the period 2i2, or say II0 (u + 212) = now, while II« has the period 2il', or say IIoo (u + 212') = IIo0u> we obtain II (u + 20)= eiB° (“+20>! IT, = eB*a (tt+n) II u, n (u +2fl')= e^u+m,yt noo = e2B”n'(“+n')nM,) which shows that the function IIu is not perfectly, but to an exponential factor pres, periodic in regard to the two quantities 2i2 and 212' respectively. See as to this theory my papers, Camb. and Dub. Math. Jour. t. iv. 1845, pp. 257—277, and Liouville, t. x. 1845, pp. 385—420. Transformation of the function H, ©. (Only the first trans- formation is here considered.) Art. No. 400. 400. The equation iru ©w Jwk sim 1 - 2 K . (2m — 1) intK* sm w (m = 1 to oo ), putting therein X for u, k, and attending to the relations A = , A' = ^ , becomes nM M © 1/’ sur nnru 1 SHT 2 K (2m — 1) ninrK' 2 K , (m = 1 to oo ), and we may hence deduce an equation of the form @ (£, x) = A [®(u + 2s'K)]. (s'= - i (n -1) to +%(n -1).)307 XI.] FURTHER THEORY OF THE FUNCTIONS H, 0. In fact, disregarding constant factors we have or, what is the same thing, ® S ’x)=[sin u (u+2m -1 iK>)\ > if m has now all positive or negative integer values from — oo to oo : and similarly and if in this last equation, we write for u, u + 2s'K, giving to s' all the values from — £ (n — 1) to + — 1), and multiply the resulting expressions; then by aid of a known trigonometri- question. Writing u = 0, we obtain at once the value of A, and the equation becomes which are formulae for the transformation of the functions Hy0. 400*. The theory of the transformation of the functions Hu, %u might be derived from the double factorial expressions given in No. 397: it is, however, somewhat difficult to carry out the process, and I propose only to give a general idea of it. Disregarding a constant factor we have cal formula, we see that © ^as a value of the form in and similarly 1 + 2mMA + (2m! +1) iMA!) ’ 20—2308 THE ^-FUNCTIONS: [XI. K which, substituting for MX and MX' their values — and K\ and writing for convenience l in place of m, becomes where on the right-hand side l, m have every integer value whatever from — oo to + oo: grouping these according to the remainder of the division l by n, we separate the right-hand side into n factors, each of which is a ©-function with the original periods Ky K'; thus writing l = mn + s', where s' has any one of the values 0, 1, ...n —1, and m has any integer value whatever from — oo to + oo, the factor corresponding to a given value of s' is 2 s'K 2s K viz. disregarding the constant divisor, the factor is and we thus have © (jg-, Xj expressed as a constant multiple of the product of the n factors © manner is a constant multiple of the product of the n factors H\u-{-------j ; s' having in each case the values 0, 1, 2,... n — 1, as above.XI.] FURTHER THEORY OF THE FUNCTIONS H, ©. 309 Numerator and Denominator functions. Art. Nos. 401 and 402. 401. The results obtained No. 400 may be written @hrx’=@w © («+¥) 2sK\ - / 2sK ®[u- n into constant factor as above, ’ TT ( 2sK\ ( 2sK\ We then have 2sK\ ^ ( 2sK\ © / 2sK\ ~ / 2sK\ 2sl (u+--- ©u------W-©2— V n ) \ n ) n into constant factor as above. 2 sK H2 2 sK ®2u / H2u n \ = ©20 If "" Wu~2sKJ (H)2- n ®Ht 7 0 0 o 2«1T\ = #0v1 “^sn Msn ~~n~)’ and H [ii + - (- H* ~ _ ©2m H2u 2 sK n ©20 V* ©2« ' jyj 2sK ©% /, sn2 u \ n ©20 (l- — ' sn2 .2 sKJ'· and we thence obtain 1 — k2 sn2 u sn2 @n-iQH M ,-7- ©nM = Vx. © (0, \) M ©0 sn u sn2 u 1 sn2-----310 THE ^-FUNCTIONS, &C. 402. Now in the function, see No. 359, [XI. /- / u . \ VXsnwT sn2 m 1 f .. , 2sAT| ^ 80 U ’x) ·------r L1----------------+ [1 - 4· sn-« sn· — J, sn --- n multiplying the original numerator and denominator each by ® J-A— , so that the denominator shall for u = 0 reduce itself ©n0 to — , the numerator and denominator are 0nO (A? K) 0 , H , x] , each multiplied by ©n_1 (0, X) and divided by 0n^; and in like manner the numerators of Vx sn , X^j, cn , and the common denominator (constant factor as just mentioned) are H{» ' *(£ + A’x) ■e (1+ A·x) -e (i’x) ’ each multiplied by ©n_1 (0, X) and divided by ®nu; viz. writing 0jO in place of 0 (0, X) we have thus the theorem stated Chap. ix. No. 310.( 311 ) CHAPTER XII. REDUCTION OF A DIFFERENTIAL EXPRESSION It doc 403. In the present Chapter, working out the steps of the processes referred to, Art. No». 1 to 11, we show how the dif- Jld% 'n _L QQQ ferential expression —j= is by the substitutions ^—— for x, and a + ba? c-b da? for x2, reduced successively to the forms Rdx Rdx V + (1 + mx2) (1 + nx2) ’ Vl — x2.1 — k2a? but for greater clearness we consider the substitutions under the forms x = 2_zLi^ and #2 = a—^ . l+y c + dy2 Reduction to the form Rdx V + (1 ± ma?) (1 + no?) Art. Nos. 404 to 411. 404. We start with an expression Rdx where R is a rational function of x, X a quartic function with real coefficients, and which is therefore the product of two factors f + 2rjx + 0a?, \ + 2fix + va?, with real coefficients: the values of x are real, and such that X is positive or VX real.312 REDUCTION OF A DIFFERENTIAL EXPRESSION. [XII. 405. Writing x P we have i+y ch.Jq-p)dy (1+y)2 ’ and the two factors of X become respectively (! + yf + 2i? (1 + y) (p + qy)+0(p + qyf], (1 * y {*· (1 + yY + (1 + y) (p + qy) + v(p + qyf], so that representing for a moment the functions in { } by M, N respectively, the differential expression becomes = (q-p)Rdy JMN ’ where MN is a quartic function of y. 406. To make the odd powers of y disappear in the func- tion MN, we write t+v(p + q) + 0pq = o, X + fi (p + q) + vpq = 0, for p, q being thus determined, then M, = f + 2Vp + 0p2 + (£ + 2vq + 6q2) y\ N, = X 4- 2)Ltp + vp2 + (X 4- 2pq + vq2) y2, will be functions of y2 only. The two equations give p + q and pq rationally, but in order that the resulting values of p, q may be real, the values of p + q and pq must be such that (p + q)2 — 4pq, = (p — q)2, is positive. 407. If the roots of the equation X = 0 are not all real, that is, if they are either all imaginary or else two real and two imaginary, we may take the equation X + 2px + vx2 = 0 to have its two roots imaginary, and write therefore \v > p?. But thisXII.] REDUCTION OF A DIFFERENTIAL EXPRESSION. 313 being so, the second of the two equations in p, q written in the form y + ~ (P + 9) + (p + ?)2 - (p - if = °> gives , 2ftA2 4 (Xv — u?) (p-qr={p + q+—) + v, so that p + q being real, (p — q)2 is positive, or p and q are real. 408. If the roots of the equation X = 0 are all real, let their values be a, 0, 7, 8; then assuming £4-2?7#+&r2=0(#--a)(# — y8), and \ + 2/i# + z/a^=x/(#— 7)(# — 8), the equations in p, ç become «£ - i (a + 0) (p + q) + pq = 0, 7$ - i (7 + 8) (P + ï) + pq = 0 ; whence p + q = 2 (a/3 - 78) a + /3 — 7 — S ’ g/3 (7 + 8) — 7S (& + 0) (a + 0 - 7 - 8) and thence 1 /n ^2_(a-7)(a-g)(/3-7)(^- 8) - (a + y3 — 7 — S)2 which is positive if we take for a, y8 the greatest two roots or the least two roots, or the two extreme roots, or the two mean roots; viz. we thus have (p — q)2 positive, and therefore p and q real. 409. The rational function R is the sum of an even func- tion and an odd function of y: the differential expression is thus divided into two parts; that containing the odd function may be integrated by circular and logarithmic functions (as at once appears by making therein the substitution Jy in place of y)7 and there remains for consideration only the part depending314 REDUCTION OF A DIFFERENTIAL EXPRESSION. [XII. on the even function of y, or, what is the same thing, we may take R to be an even function of y, that is, a rational function of ¿r2. 410. [It may be remarked that in the case where the function X has four real factors, say that the value is X = (x — a) (x — /3) (oc — y) (x — S), then writing y2 = -—*, or, X yO what is the same thing, x = ~, we have #_a = (a-/3)2/2(V), a? - £ = (a - £) (-*-), #_7 = a-y-(/3-y)y2 (V), x-8 = a-8-(@-8)y2 (-l·), where denominator is = 1 — y2; also dx = 2 (a — fi) y dy+(l — î/2)2, and we thence find dx __ 2 dy Jx — a. x — f3. x — y .x — 8 Jet — y—()8—y)y2. a—^8—(¡3— 8)y2* where the radical on the right-hand is in the required form; Edx but in the case where the expression is —, we have thus in V X place of 12 a function of y\ so that no part of the integral is directly reducible to circular or logarithmic functions, and the form of the result would appear to be more complicated than if we had begun by the linear substitution upon #.] 411. Restoring x in place of y, the conclusion is that the original differential expression may be replaced by one of the form Rdx Ten* where I? is a rational function of x2, and M and N are each of them a real function of the form A + Ba?*. * The above is the investigation given in Legendre’s Chap, ii., and the result is as stated: but Legendre in his following Chap. in. only assumes that the radical is reduced to the form a + px2 + yx4, and he considers (as his first case) that in which the equation a + (3x2+yx*=0 gives imaginary values of #2, that is whereXII.] REDUCTION OF A DIFFERENTIAL EXPRESSION. 315 The function MN may have the several forms ± (1 ± map) (1 + naP), where m and n are positive; but we may assume that the signs are not such as to make the function = — (1 4- map) (1 -f nap); in fact X assumed to be positive for at least some real value of the original x, cannot be by a real substitution transformed into an essentially negative function. Reduction to the standard form Rdx Vl — aP. 1 — kPaP Art. Nos. 412 to 415. 412. Retaining for convenience MN to signify + (1 + map) (1 + nx2), we have to show that -^-= can by the substitution x2 = a */MN j c + dy~ be transformed into —----t , where 8 is a rational Vl - y2.1 - kPy2 function of y2, and k2 is positive and < 1 : and since by the substitution in question R is changed into a rational function of y2, the theorem will it is clear hold good if only we have \dx _________dy \'MN Vl — y2.1 — k2y2 ’ where \ is a constant. On account of the definite form of the expression on the right-hand side, it is rather more a + fix2 + yx4 is of the form X2 + 2\fix2 cos 6 + n2xA, a case which he further con- siders in Chap. xi. The idea seems to be, that since in the case in question j) qy there are no odd powers of x, the transformation of Chap. n. is un- necessary; if, however, we do make this substitution, we obtain under the radical sign a new quartic function without odd powers: the substitution is found to be x = ^Pf \/~ (menti°ne^ *n Chap, xi.), and the radical is thereby reduced to the form m2(l+p2y2) (1 + q2y2), which is the fourth case of Chap. hi.316 REDUCTION OF A DIFFERENTIAL EXPRESSION. [XII. convenient, writing the relation between x2, y2 in the form ^ b — dx2 r, to transform this into the form dx I u — ax- *JMN left-hand side. The last-mentioned equation gives on the ydy = 1 y (be — ad)xdx (b — dx2)2 ’ V& — doc2, V— a 4- co? 9 Vfc — dx2 Vl — y2 *Jb + a — (d + c)x2i 1 V& — doc1 Vl — k2y2 V& 4- J?a — (d 4- fc2c) af ’ and we thence have dy Vi — i/2. i — (tc — ad) xdx V& — da?. — a 4- cx2. 6 4- a — (d 4- c) ¿c2.6 4- A^a — (d 4- A^c) ¿c2 Here in the denominator one of the four factors must reduce itself to a constant, and another of them to a multiple of a?, in "\*dx order that the second side may be of the required form · 413. For instance, if b 4- a = 0, d 4- k2c = 0, that is, b = — a, d a ~* co? k2=-----, then the relation between y2, a? is y2 =------=~ , and c J a + da? the differential formula, after some easy reductions, becomes /a dy _ dx V C vi - ƒ. i - ky - L ’ V a a or writing for greater convenience a = 1, then we haveXII.] REDUCTION OF A DIFFERENTIAL EXPRESSION. 317 1___cod y2 = -—-j—2, leading to the differential formula 1 -f- ax 1 dy _ dx Vc Vl — 2/2 -1 — Vl — cod. 1 + dod where Id = — -; this implies c positive, d negative and in c absolute magnitude less than c; and we have thus a formula applicable to the case MN = ( 1 — mx2)( 1 — nod); viz. assuming m>n, we may write c = m, d = — n, and the relation then is 1 7Tiod . -X ■” 7J2 y2 = --------, or, what is the same thing, x2 =---, giving 1 dy dx Vm Vl — y2. 1 — k2y2 Vl — mx2.1 — n#2 ’ where &2 = —. A more simple formula giving this same rela- t/2 tion is ¿r2 = —. m 414. We thus obtain transformations applicable to the several forms of MN, viz. numbering the cases as in Legendre’s Chap, in., but for the reason appearing in the foot-note p. 314, omitting his first case, we have 2°. MN = (1 + mx2) (1 — nod), 1 x < -, n 3°. MN = — (1 4- mx2) (1 — nx2), 1 x > -, n 4°. MN = (1 4- mx2) (1 + nod), m> n, 5°. MN = (1 — mx2) (1 — nx2), m>n, x from 0 to , or Vm from to oo, Nn 6°. MN = - (1 — mod) (1 — nod), m>n, #‘from -L to Vm vn318 REDUCTION OF A DIFFERENTIAL EXPRESSION. [XII. and writing for shortness Y = 1 — y2.1 — k2y2f the formulae are ¿•2 = 771 1 dx k dy 2°. m + n3 x2 = ■-a-A Vm V?’ or else x> = y2 dx _ A % _ «7 m *4- /1 — my2 3 *JMN~ Vm vr Q° l·2 - n /yiS — 1 dx k O . tv — m + n3 JU — "w(l-tf)’ *IMN~ < £ 1 VF’ 4° 7,2 _ m — n 1 IO dx _ 1 dy tv — n 3 1b - ml + y2 3 Jmn~ V m Vf ’ 5°. k2 = n m3 X2 = = f m ’ dx _ */MN~ 1 Vm VF’ 6°. A _ m — 7i /g2 — 1 dx 1 dy tv — n m — (m — n) y2 ' \ZMN~ Vm VF' 415. It is to be added that if in the expression we have y > ^, in which case writing F= (y2 - 1) (k2y2 — 1) the radical is still real, then assuming y = , we have = — —*L, where ** kz3 VF F = (1 — ^2) (1 — k2z2), and as y passes from ^ to oo, z passes from 1 to 0. Hence, replacing y or £ by the original letter x, the conclusion is that in every case the differential expression dx , , , . A + Ba? . r— can by a real substitution r ' V±(l ±mx?)(l ±na?) J C + Do? place of x1 be reduced to the form dx m where the variable x extends between the limits 0 and 1.XII.] REDUCTION OF A DIFFERENTIAL EXPRESSION. 319 Further investigations. Art. Nos. 416 to 421. 416. Reverting to the investigation, Art. Nos. 404—411, but abandoning the condition that the transformation shall be real, it is clear that we can by such a transformation reduce the differential expression to the form Rdx Vl - afT\ — k2a? ’ where, however, k2 is not of necessity real, or if real and positive not of necessity less than 1: it is interesting to inquire further into this question, and to show how the modulus k of this in general abnormal form is determined. The process in fact was by a substitution in place of xf or say by a linear transformation performed upon x, to transform the quartic function X into a quartic function Y containing only the even powers of the variable; the solution of this problem depends on a cubic equation which is solved rationally when we know any decomposition of the function Xy Y into quadric factors; and it was in order to have such rational solution of the cubic equation, and with a view to obtain real transformations, that we commenced by assuming the function X to be decom- posed into factors of the form (f + 2rjx + 0a?) (\ 4- 2fix + va?) or 0 (x — a) (x — /3) (x — 7) (x — S). But analytically it is more elegant to deal with the undecomposed quartic function, as was done by me in a paper in the Camb. and Dub. Math. Journal, t. 1. (1846), pp. 70—73, and I here reproduce the investigation. 417. Let the two quartic functions P = (a, by Cy d, e) (x, y )4, P' = (a\ b\ c'y d\ e')(x, yf)\ be linear transformations one of the other, say the second is derived from the first by the substitution Xyy = \x'+ fiy\ \x' + m'·320 REDUCTION OF A DIFFERENTIAL EXPRESSION. [XII. Then writing m = — Ai/U, for the determinant of substitution, we have xdy — ydx _ m {xdy — ydx) VP VF ' y , y or writing u = -, u = —,, X X and therefore xdy —ydx du xdy—ydx du ~“v?~=7F vf vF’ where Î7 = (a, 6, c, ci, e ) (1, u )4, U' = (a', b\ c\ d\ e') (1, uj, the differential equation becomes du du' W=mWf> viz. we have this from the transformation u = X + fJLU Xi + fJ'iU * The functions P, P' are obtained the one from the other by the foregoing linear substitution ; viz. if I, J are the invariants of P, viz. I = ae- 4bd + 3c2, J= ace — ad2 — b2e -4- 2 bed — c3, and by I\ J' the corresponding invariants of P'; then we have between the coefficients of the functions and the coefficients of transformation the relations T = m4/, J'2 J' = m*J, whence = P P* 418. Supposing now U' = a (1 + pu2) (1 + qu'% b' = 0, d' = 0, 6c = a (p + q), e' = a'pq, orXII.] REDUCTION OF A DIFFERENTIAL EXPRESSION. 321 we have T = ^a,'2 (p2 4- q2 4-14pq), J' = ^«3 + ?) (34pï - - g2) ; and thence (p + q)2 (34pq — p2 — q2)2 _ 27J2 (p2 4* g2 4- 14pg)3 /3 or, as this may also be written, lQ$pq{p-qY 27/2 (p2 4 q2 4 Upqf Is 9 which determines the relation between p and g. Also m _ ip2 4 g2 4 14pg\^ V<7v 127 ) 9 so that the differential equation is du __ ip2 4 g2 4 14pq\^ du' V U \ 12/ / Vl +pu2. 1 4 gw'2 419. If in particular p = — 1, then writing also —q in place of <7, this is du __ (q2 4 14g + 1^ du' Vi7 \ 12/ / Vl — u2. 1 — qu2 ’ where g is determined by the equation 108g (1 - qY _ 27 J2 (g24l4g + l)3_ 73 * Writing for shortness _ 27 J2 _ 27 1 /3 ~4il/’ the equation in q becomes (q2 4 14g 41)3 - 16i¥g (q - l)4 = 0, c. 21322 REDUCTION OF A DIFFERENTIAL EXPRESSION. [XII. or, as this may be written, q + - +14 )3 — 16M (q* — q~*)4 — 0 ; viz. writing — q~* = Vtf-l , this is 0s — M(0 — 1) = 0, which determines 0, and then 9 = *7 + $-|-4\/$4'3 ~0^1 ‘ 420. Suppose q = a is one of the values of q, the equation becomes ( — fa), then x t€ or what is the same thing, F(k, <£) = £(1 + k1)F(k1, fa). But there is as regards this transformation a peculiar con- venience in adopting, instead of the standard form of radical E (a, by ) = Jdfa\/a2 cos2 -{- b2 sin2 , where the integrals are taken from zero. Obviously Va2 cos2 + b2 sin2 = V«2 — (a2 — b2) sin2 , = a Vl — k2 sin2 fa Vl-&2 sin2 fa a new form Va2 cos2 + b2 sin2 fa (where a is taken to be > b); and I write in the present Chapter if k2 = 1 — ~ f whence also k' = ; and the two functions are a2 \ a / ’ thus = a-1 F (Jc, fa) and aE (k, fa) respectively.XIII.] QUADRIC TRANSFORMATION. 327 Geometrical Investigation of the formulae of Transformation. Art. Nos. 424 to 427. 424. I reproduce the original geometrical investigation of Landen’s transformation in the new notation, as follows: Taking in the figure P a point on the circle, 0 the centre, Q any other point on the diameter ABy QA = a, QB = b, zAQP = , and therefore Z A OP = 2, we write a1 = %(a + b), bi = s/ab, cx-\{a-b)\ we have then OA = OB = OP = a1] OQ = a1 — b, =%(a — b), =c1; QP sin = ax sin 2; QP2, = Ci2 + 2(^0! cos 2<£ + a?, and328 QUADRIC TRANSFORMATION [XIII. = b (a? + 62) + £ (d2 - 62) cos 20, = £ (a2 + 62) (cos2 0 + sin2 0) + ^ (d2 — 62) (cos2 0 — sin2 0), = a2 cos2 0 + 62 sin2 0; Oj sin 20 that is, and thence sin 0X = cos 02 = V a2 cos2 0 + 62 sin2 0 ’ cx + «! cos 20 Va2 cos2 0 4- 62 sin2 0 ’ a22 cos2 ! + &x2 sin2 0X = 425. We hence find _ aj2 (a cos2 0 + 6 sin2 0)2 a2 cos2 0 + 62 sin2 0 • /« . , x A (a — 6) sm 20 sin (20 — 02) = 2 -------7 . = ; V a2 cos2 0 + 62 sin2 0 f , . a cos2 0 + 6 sin2 0 cos (20 — 00 = . — .. =; Va2 cos2 0 + 62 sin2 0 and then further cos (**-*.) = — Vdj2 cos2 0 + 6X2 sin2 0. Oj Considering the point P' consecutive to P, we have PQdfa = PP' sin P'PQ, = 2djd0 cos (20 — 00; viz. substituting for PQ its value, we have 2d0 Vd!2 cos2 0 + &!2 sin2 0 = \f a2 cos2 0 + 62 sin2 0 dx; that is, 2 ct(f) ¿0, Vd2 cos2 0 + 62 sin2 0 Vd22 cos2 0j + 6X2 sin2 0X* the required differential relation: and by integration F (d, 6,0) = (d1? 61} 0!).XIII.] OF THE ELLIPTIC INTEGRALS. 329 426. Moreover 4a·? sin2 cos2 Write for a moment then X = a2 cos2 + b2 sin2 fa X — a2 = (b2 — a2) sin2 , X — b2 = (a2 — &2) cos2 fa (X — a2) (X — b2) = — 4 (a — 6)2 a^2 sin2 <£ cos2 <£>; and therefore (X — a2) (X — b2) + X(a — b)2 sin2 fa = 0, that is X2 4- X [— (a2 4- b2) (sin2 fa + cos2 fa) + (a — b)2 sin2 fa] + a2b2 = 0, or, what is the same thing, (X — {a2 -f b2) cos2 fa + ab sin2 fa})2 = l(a2 + b2)2 cos4 fa + (a2 -f b2) ab cos2 fa sin2 fa + a2b2 sin4 fa = | (a2 — b2)2 cos4 fa + ab (a — b)2 cos2 fa sin2 fa = 4cj2 cos2 fa (ai2 cos2 fa + b? sin2 <£x); viz. restoring for X its value, we have a2 cos2 + b2 sin2 = i (a2 + &2) cos2 + ab sin2 + 2Cj cos fa 4a^ cos2 fa^b?$m2fa = 2 (ax2 cos2<^! 4- &i2 sin2^) — 6^ + 2c2 cos ^ Vaf cos2 fa +6^ sin2^!, which is another form of the integral equation. 427. Write this in the form (a2 cos2 + b2 sin2 ) 4- Vax2 cos2 j + 6X2 sin2 = — a2b2 cos4 — 2a2b2 cos2 ^ sin2 <£j — a2b2 sin4 2 ■! Vc*!2 cos2 ^>! + 6X2 sin2 330 QUADRIC TRANSFORMATION [XIII. then combining with 2d(j> dfa a2 cos2 + b2 sin2 V cos2 Va2 cos2 + b2 sin2 = d<£i i Va22 cos2 ! + 6i2 sin2 fa-. ---_ + cx cos d>\; l r i n Va^cos^ + ^sin2^ rJ and thence by integration .#(a, 6, 0) = E(alf fa, fa) - 6^ <£>) 4- cx sin fa,' and ante, No. 425, we have ^(a, 6, ) = iF(a1} bl9 fa), which are the required transformation equations corresponding to the relation Oj sin 2 sin fa = Va2 cos2 + b2 sin2 <£ Reduction to Standard Form of Radical. Art. Nos. 428, 429. 428. The two angles correspond to each other as follows, =0, V a <£i = 0, <}> = tan' -1 1 = 7r» viz. passing from 0 to r, fa passes from 0 to tt ; and for = |7r the functions of fa are consequently the doubles of the complete functions ; we thus obtain E (a, b) =2E(a1) fa) - fa2F{a1) fa), F(a, 6)= F{al, fa), where E (a, b), &c., denote the complete functions.xm.] OF THE ELLIPTIC INTEGRALS. 331 Js , b 429. Recollecting that k2 = 1 —„, whence k = -; Cb“ a f)2 assuming also L·2 = 1-, we have ° a? fcl "a/®1 l) (a + by (1 + kj ’ 1 __ that is Jci = --r, as before, and the formulae become 1 -f- k E(h, ) = - E(k,4>i) + £ sin1; a ti/\Us U/ and F{k,) = \^F(h,$)·, «i or, what is the same thing, E (ky ) = i (1 + k') E {h, &) - F{kly,) + %(1- V) sin £, ■(i+*r where . £ (1 4- A;) sm 2<£ sm ! = * ) — —^—r; Vl — i2sm2 ^ but it is convenient, in the first instance at any rate, to retain the formulse in their original form. Continued Repetition of the Transformation. Art. Nos. 430 to 433. 430. In the same manner as c^, bly were derived from a, 6, we may from aly bx derive a2y b2y c2y and so on inde- finitely: viz. a1 = £ (a + b)> i>i = Va6, cy = | (a - 6), a2 = | (ax 4- i>x), 62 = VaA, c2 = i («! - &!>, ce3 = -^ () — dn 1, E (dn> bn> ) = cin, and in particular E (dn> bn) = dn 1 . 2^7T, E (dn, Òjj) = dn . ■^■7T. 431. Considering first the complete function F (a, 6), we have F(d, b) = F(dlt &0 ... = F(dn, bn) = %Tr + M(d, 6), viz. the complete function is given as \ir into the reciprocal of the arithmetico-geometrical mean of a, b. 432. Considering next the incomplete function F (a, b, ), the equations sin = dj sin 2 Va? cos2 + 62 sin2 ’ sm 0o = a2 sin 2cj> d? sin2 ! + òj2 sin2 0! ,&c. show without difficulty that as n increases, n continually approaches a value, = 2W into a determinate magnitude, say M (a, 6, 0): in fact n being large and therefore dn_lf bn-X) dn approximately equal, we have very nearly sin<£n = sin 2<£w_1? that is n — 2n_1 : the limit in question M () is of course to be calculated from a, 6, by means of the equation itself n = 2nM(d, 6, ): and it is to be remarked that for <£ = ^7r, the value of n is = 2n. \ir, so that M(a, b, \ir) = ^7r. The equations F(df 6, ) = \F(dlf bly fa) = ... then give F(a, b, <£) 1^ 2n F(dn, 6n, <£>n) 2n * a — Jlf (a, 6, <£) if (a, 6) ’XIII.] OF THE ELLIPTIC INTEGRALS. 333 Or if we choose to combine this with F(a, b)-^ir-r-M {a, b), then F(a, b, d>) = — M (a, b, <ƒ>) F (a, b). 7T 433. Considering next the ^-formula, this may be written [E (a, b, fa- a2 F (a, b} fa] = [E (a^, bu fa) - of F (a1} b1} fa)] + F(au blt fa) (c*!2 — - \b?) + cx sin fa, where in the second line the coefficient of F(a1,b1,fa) is — i (a2 — b2), = — ajCi, or the equation is [E (a, b, fa — a2F (a, b, fa] = [E (a1} b1} fa) - a2F(alt bly fa)] — a^F (alt blf fa) + cx sin fa. And hence observing that as n increases E (&n, bn, n) continually approaches to zero, we obtain E (a, 6, <ƒ>) — a2F (a, b, fa) = - + 4a2c2 4- 8a3c3...} F (a, b} fa) + Ca sin fa + c2 sin fa + c3 sin $3 + ... Or substituting for F (a, b, ) its value E (a, b, = {a? - 2o,c, - 4a2c2 - ...} + (Cisin fa + c2sin fa + c3sin fa+ ..,), and in particular if (f> — ^ 7r, then ! = 7r, fa = 2ir, &c., if (a, 6, 2^7t) = 17T as before, and the equation becomes E (a, b) = {a2 — 2 — 4a2c2 — .Jw -5- if (a, 6).334 QUADRIC TRANSFORMATION [XIII. Reduction to Standard Form of Radical. Art. Nos. 434, 435. 434. Introducing the modulus k, viz. writing b = ak\ and ultimately a = 1, the formula for F(a, b) becomes F1 (k) = -T- M (1, k'); viz. the complete function is given as = into the reciprocal of the arithmetico-geometrical mean of 1 and the comple- mentary modulus. Gauss has given the formula 1 -1 , 18 g,, 1°· 3^1 l-.4?kl +"· we at once connect it with the formula last obtained: viz. the right-hand side is _2 2 1 1 ~7r^lfCl~7rl +h ^l/C~(l+k1)M(ltk'Y Or the equation is M (1 + ku 1 - h) = (1 + h) M (1, *0 = (1+ h) M (l, ; which is obviously true, since in general M(a,b) = 0M(?, The formula for F(ay b, ) gives in like manner M (1, k\ ) F(l’ t^ MnTkY’ which is a formula for the numerical calculation of the function ).XIII.] OF THE ELLIPTIC INTEGRALS. 335 435. Proceeding next to the function E, we have jp /?„ i-i SojC! 4osC2 | M(l, k', ) E(k, — - — - * f C J . . Co m . Co » . \ 4- — sin + - sm 2 4- - sm 1+ (1 +¿0(1 + ¿0 sm2+(i + ¿0(1 +¿0(1 + ¿0sin<#>3 + &C., or observing that 1 _ k l+&i 2^/k^ that is 1 h 1 + ¿2 2sJk2 1 k% 1 4“ A?3 2*/%’ &c. 1 1 4- 1 1 ~1~ ki. 1 4* _______________1____________ 1 4· k\. 1 4~ k2 · 1 4· &3 k 2\/k1 ’ 4 V k2 k\! kjc2 ¥vf3 ’ the last line may be written 4- fc Vsin ! 4- J Vk-Jk2sin2 + ¿V&iAp3sin 3+ ...}. In particular, if = p, the equation becomes 2?i& = [1 — P2 (1 4- Pi + PA + \kjcjcz..........)) \ir -T- M (1, k'); or if we please, EJc = {1 - p2 (1 4- pi 4- \kjc2 4- pA&3 + ...)} ^P·336 QUADRIC TRANSFORMATION [XIII. Application to Integrals of the third kind. Art. No. 436. 436. The transformation is applicable to elliptic integrals of the third kind, but the results are not of any particular interest. Writing down the equations 2 d _ Va2 cos2 + b2 sin2 Vof2 cos2^ + bx2 sin2 fa ’ a2 cos2 <£ + 62 sin2 <\> (a2 cos2 (j> + b2 sin2 ) (cos2 + sin2 ) -f 4n1a12 sin2 0 cos2 _ 1 1 + rq sin2 fa ’ the expression on the left-hand of this last equation is A BX a cos2 + bX sin2 aX cos2

* where a2 cos4 <\> + b2 sin4 <£ + (a2 -f 62 + 4n1a12) sin2 0 cos2 <£ = (a cos2 $ -f 6 A sin2 <£) cos2 <£> + sin2 ^ ; that is A + ^ (ci2 + 62 + 4/iia!2) 2 = gi (ai2 + Ci2 + 2ft1a12) ; = ^ K®i2 "b cf2 + 2?i1a12)“ — 6i4] = (1 + %) Ox2 + ^oq2); whenceXIII.] OF THE ELLIPTIC INTEGRALS. 337 that is and then X = A {a,2 + Ci + 2n1a12 + 2^ J(1 +n1j(ci- + n1a12)}, Ul 1 1 X = jy2 WJ + ci2 + 2n*ai2 ~ 2aiv/(l + nj (¿a2 + n^ai2)}; x ^ _ (aZ — 6) A" D _ a — bX A ~ ~XT-_T\~ ’ Z^ 1 ’ and we have f(aZ--6)Z 1____________(a-6Z)Z _1____ ( X'2 — 1 acos2 <£>+bX sin2 X2 — 1 aZ cos2 -f & sin2 2 d(j> V a2 cos2 + 62 sin2 <£ _ _______________________________ (1 4- n1 sin2 <£j) V cos2 ^ 4- 6a2 sin2 ! whence, integrating, the function f _________________#i________________ J (1 + Wj sin2 !) Vc?!2 cos2 0! 4* &i2 sin2 <£2 is expressed as the sum of the two elliptic integrals of the third kind having a common modulus but different parameters. Numerical instance for complete Functions Ely Fly and for an incomplete F. Art. No. 437. 437. As a numerical instance take (as in Legendres example, t. L p. 91), -----__ ^2 a=l, b = |v 2 + V3 = cos 75° (whence k = sin 75°), tan <ƒ> = ; v3 we have a b c k k’ (0) 1000,0000 0-258,8190 0-965,9258 0-258,8190 47° 3' 31" (1) 0*629,4095 0-508,7426 •370,5905 j 0-588,7908 0*808,2856 62° 36' 3" (2) 0-569,0761 0-565,8688 j •060,3334 0106,0200 0-994,3636 119° 55' 48" (3) 0-567,4724 0-567,4701 | 001,6037 0-002,8260 : 0-999,9959 240° 0' 0" (4) 0-567,4713 0-567,4713 ' •000,0011 0-000,0020 0-999,9999 480° 0' 0" c. 22338 QUADRIC TRANSFORMATION. [XIII. first as to the complete functions we have = =2-768,064. Z Cl>4 = axcx = *233,2532 -h 2a2c2 **068,6686 + 4 a3c3 *003,6402 + 8a4c4 *000,0051 = *305,5671 agreeing with Legendres values Fx = 2'768,0631, Ex = 1*076,4051, and thence ^ ^1 — ~^ = *305,5671. Also, we have F (Jc, ) = ^ —. <£4, or since ^ 4 = 30° = ¿7r, z d4 this is ) = ^F1 = 0*9226877: it is in fact easily verified that the assumed value of is such as to give exactly F (k} ) =-^Fx. The notion of the arithmetico-geometrical mean was esta- blished by Gauss in the memoir “ Determinatio Attractionis &c.” Comm. Gott. Rec. t. IV. (1818), but his later researches in relation to the subject were not published until after his death, Werhe, t. IV. pp. 361—403; a table is given p. 403, of the values of the arithmetico-geometrical mean M (1, sin 6) and of its logarithm, 0 = 0° to 90° at intervals of 30'.XIV.] 339 CHAPTER XIV. THE GENERAL DIFFERENTIAL EQUATION dx dy \/Z“ V?’ Integration of the differential equation. Art. Nos. 438 to 440. 438. In the present Chapter, writing X = a + bx + cx2 + daf + ex4, Y = a + by + cy2 + dyz + ef, I consider the differential equation =0 VX VF ' 439. A direct process for finding the algebraical integral as follows was given by Lagrange. Assume ^ = and therefore ^ = — VF; dt dt then 2 ^ = b + 2 cx + 3 daP + 4 ea?, 2 % = b + 2 cy + 3dy* 4- 4e«/3 ; and if p = x + y, q = x — y, then elf = W + cL* = b + Cp + ^ + 2') + ^ ^ + ^ ^ = X - F= 65 + cpi + (3^ + g2) + iejJ? (p2 + g2); whence hd - sin 0)2 + k2 sin20 sin20; and to introduce /u, instead of Q we must write cospAp = 1 -^(1 + k*) sin2/*-\G sin2/*, that is \C sin2/* = 1 - \ (1 + A;2) sin2/* - cos /*A/*. The equation thus is 1 - ^ (1 + A:2) (sin20 + sin20) + k2 sin20 sin20 - cos 0 cos 0A0A0 this is of course a form of the addition equation, and could be verified as such by substituting for cos /*, sin /*, A/* their values in terms of 0, 0: but the form is not a convenient one.341 XIV.] THE general differential equation. -1 viz. this is iC*(x-yf, - C(x- y f [a + \b(x + y) + \c (aP + y-) + \dxy (x + y) + eaPy% + a2.1 + ac. x2 + y2 -{-ad.xy (oc + y) + ae . 2x2y2 + bt.\(x+-yf + bc .\(x + y)(oP + ƒ) + bd.^xy(x + yf + be. aPy* (x + y) + cKi(x> + yX = 0, = 0, = 0, = -(x-yf(x + y), = -{x- yf (x + yf, =+}(*- yf, -aPy-xf· =+\(x-y¥(x + y), -xf-apy = -\xy{x-yf, - xyi - xpy xy(x- yf (x + y), -x-y — a? — y2 — x? — y3 — x4 — y4 -xy - x*y* = + \ (x - y)2 (x + yf, + cd. \xy (x + y) (&'2 + y2) — a?y3 — x?y2 = + \ (x — y)2 xy (x + y)> -f ce. x2y2 (x2 + y'2) — oc^y2 — x2y4 = 0, + d2. \x2y2 (x + yf — o^y3 = +1 (% — yf oc?y2, + de. afy3 (x 4- y) — ahf — = 0, + e2. afy4 — xty* = 0, =0; viz. the whole equation divides by (x — yf. Omitting this factor it is $C*(x-yf — C[a + \b (x + y) + \ c (x~ -+- y") + \dxy (x + y) + eapy-}, — etc? {x + y) — ae(x + yf + {b* + (« + y) -\bdxy — bexy(x + y) + \cP (x + yf + %cd xy (x + y) + \d>aPy* — 0,342 THE GENERAL DIFFERENTIAL EQUATION. [XIV. or what is the same thing, it is ( - Ca +±b* ) + (x + y) ( — £06 — ad + ^bc ) 4-(#a4-2/2) ( \02—£0c — ae 4-£c2) + xy (— \ 02 — 2ae — \bd 4- £c2 ) 4- xy(x +y) ( — £ Orf — be + £cd) + a?y2 ( — Ce 4-^rf2) =0. This may be written (a 4- 2h# 4- g#2) 4- 2y (h 4- 2b# + f#2 ) 4- 2/2 (g + 2f# 4- c#2) = 0, where the several coefficients have the values a = 62 — 4 aO, b = — 2ac — £6rf 4- £c2 —|02, c = rf2 — 4c 0, f = erf — 26c — Orf, g = — 4ac 4- c2 — 20c 4- O2, h = 6c — 2arf — 06. The result shows that the complete integral of the differen- tial equation is an equation u = 0, where u is a symmetric quadriquadric function of (#, y); that is, a symmetric function, quadric in regard to each variable separately. Further development of the theory. Art. Nos. 441 to 450. 441. This may be verified almost instantaneously: starting from u = (a 4- 2h# 4- g#2), 4- 2y (h 4- 2b# 4- f#2), 4- y2 (g 4- 2f# 4- c#2) = 0, we may write u = A 4- 2By 4- Cy2 = A' 4- 2J3'# 4- O'#2 = 0, A, Bt O being given quadric functions of #, and A\ B', O' the same quadric functions of y.XIV.] THE GENERAL DIFFERENTIAL EQUATION. 343 Then differentiating But du dx dx + du dy dy = 0. ■^ = 2{Cy + B) = 2 V#2 — AC, since u = 0 gives {Cy 4- B)2 = B? — AC, ^ = 2 {Cx + B') = 2 Vif-' - ¿'C', „ (G'x + BJ^B^-A'C, and the differential equation thus is dx dy ^ V52 - .4C + V#2 - A'C' ~ This will coincide with dx dy VT + vT = 0, if only the quadric functions A, B, G are determined so that B2 — AC = OX (which of course implies B'2 — A'C' = 0T). We have in all six disposable quantities a, b, c, f, g, h, that is five ratios; and the equation in question (h 4- 2b# 4- f#2)2 — (a 4- 2h# + g#2) (g 4- 2f# 4- c#2) = OX, establishes four relations between the five ratios, and thus leaves one indeterminate ratio serving as a constant of integra- tion : we in fact satisfy the equations by means of the before- mentioned values of a, b, c, f, g, h, which contain the arbitrary constant G; viz. we then have (h 4- 2b# 4- f#2)2 — (a 4- 2h# 4- g#2) (g 4- 2f# 4- c#2) = fh!-Jlgor f^cgw \ a el = 4 [ad2 4- b2e — bed 4- {— 4ae 4- bd 4- (C — c)2) G] X. As a partial verification, observe that the equation Jj2_g^cr f2___(*or ----- =------6 or eh2 — af2 = g (ea — ac), = (eb2 — ad2) g, a e is satisfied identically.344 . THE GENERAL DIFFERENTIAL EQUATION. [XIV. 442. u = 62 Regard u as a function of G ; we have + 2 G — 2a 4- C2. (x — y)2 -f (26c — 4ad) (x + y) + (c2 — 4ae) (æ2 + y2) -f· (— Sac — 26d -I- 2c2) xy + (2cd — 46c) xy (x + y) + d2 x2y2 -b(x + y) — c {a? + ƒ ) -dxy(x + y) — 2exLy2 say this is then we have a — X. -f- 2 fi, C + v G2 ; y? — \v = 4a2... + 4 eVy4, = 4XF : viz. calculating /¿2 — \z/, it will be found to have this value. 443. Now starting with the equation u=0, and treating it as before, except that we now regard C as a variable; that is, forming, and then reducing, the equation we obtain du j du 7 du, dxdx+drydy + ^dG = o, dC J®Ydx + J(&Xdy + JXYdG = 0, or, what is the same thing, dx dy dC •JX ' VT V® ’ where © = ad2 + b2e — bed + G {—4ae + bd + (C—c)2], a cubic function of G. 444. Write G = § c — 2o>; then © = ad2 + 62c — 6cd + {— 4ae + bd + (f c + 2g))2} (|c — 2g>) = — 8G>3 + (8ae — 26d *f |c2) a> + (- face + ad2 + b2e - f 6ccZ + ^c3).XIV.] THE GENERAL DIFFERENTIAL EQUATION. 345 But the invariants of a + bx + cx2 + daf + ear4 are / = _i_(12ae-3&d + c2), J = (72 ace — 27 ad2 — 27 b2e — 2c8 + 9bcd); whence CD = - 8 (a)3-1ft) + 21), V© = 2i V2 V&)3 — /(» + 21, = 2i V2 Vfi, suppose, dC = — 2da>; or the differential equation is dx dy da> _ ^ VZ+vT~iV2VS- ’ viz. writing for (7 its value |c — 2ft), the corresponding integral equation is a = X + 2ft (f c — 2ct>) + ^ (§c — 2ft>)2, = X + f Cfx + f c2z/ + 2co (— 2fi — f ci>) + ft)2.4d, = 0; or substituting for X, /x, z> their values and reducing, this is b2 — f ac + 2û) + (§ be — 4ac£) (& + 3/) + (ic2 - 4ae) (x2 + y2) + (— 8 ae — 2 bd + -^c2) xy -f (f cd — 4fbe) xy (,x + y) + (d2 — fee) x?y2 + 4a +û)2.4(# — 2/)2=0 + 25 (x + 3^) + f c (x2 4- y2) + f cxy + 2dxy (x + y) + 4 ex2y2 where the left-hand side is quadric in each of the variables a, y, <»: as there is no arbitrary constant, this is only a parti- cular integral. 445. We may by a linear substitution performed on the to bring the third radical JTi to a like form with the other two radicals.346 THE GENERAL DIFFERENTIAL EQUATION. [XIV. Write for convenience a + bx + cx? -f- daf + ea? = e(x — a) (x — /3) {x — 7)(# — 8); then the substitution may be taken to be 20.=i6|-/8y-73-aj8+28(a+^+7)-382}+e(g~g)^~g8^7~8)> which, as will appear, makes the radical Vn to depend on *JZ, where Z=a + bz + cz2 + dzz + ezl·. Some preliminary formulae are required. 446. Reverting to the formulae which contain G (= §c — 2o>), assume C1=e(/3 + 7)(a + 8), C'2=e(7 + a)(/9+S), G3 = e (a + (7 + S); then we have (i0-(7,) (0- C2) (C -C3) = C {((7- c)2 - 4ae + bd\ + ad? + b2e - bed, viz. G1} C2, C3 are the roots of the equation ® = 0. Hence writing for (7 its value = f c — 2o>, we have (f c — 2o) — <70 (§c - 2o> - (72) (f c - 2o> - <73) = — 8 (a)3 — la) + 2 J) = - 8(a) - 0)0(0) - o)2)(o> - o>3) suppose. We have "1 = 3 c — ^ ^, = £(2c —3i?0 = \e {2^7 + 2aS - ay8 - £8 - 017 - 7^}; or putting -4 = (y8 — 7) (a — 8) = a/3 + yB — £8 — «7, B = (y-a) (f3-8) = /3y+ a8-y8-/3a, C = (a - 0) (7- S) = 7a + /3S -aS - 7ft we have o>2 (R — (7); or, forming the analogous equations *>1 = ie C® “ ^2= ¿0 (^7 -4), o)3 = ^ (A-B).XIV.] THE GENERAL DIFFERENTIAL EQUATION. 347 447. Now writing as above 2et>=£e{-/87-7a-a/9+28(a+/S+y)-382} + ——&). then if 2 = a, ¡3, 7 we have g> = ®1, o>a, a>3 respectively: thus writing z —a we find 6 to = e {— /Sy — 7a — a/3 + 28a + 28/3 + 287 — 382 + 3/37 - 38/3-387 +38s) = e{2a8 + 2/87-(a + 8)(/9 + 7)j, = 6a>i, and so for the others. Hence 2 (w — wj) = - e (/S - 8) (7 - 8) > 2 (® - ®2) = - e (7 - 8) (a - 8) > 2 O - a>3) = - e (a - 8) (/3 - 8) Z-^% ’ and therefore 8 {to — o»i) (w — — &>3), = 8 (ta3 — Ia> + 2J), g = - {(a - 8)(/3—8)(7 or say 2iV2^ii= (a-8) (/9-S) (y"2) But from the expression for « 28« = - (a - 8) 08 - $) (7" a) * (' " 5)8 whence or the equation do) dz i V2 Vü--Vf: dx dy do) _ (348 THE GENERAL DIFFERENTIAL EQUATION. [XIV. is by the substitution 2®=4«{-07-7a-«0+28(a+/8+7) - 382} + transformed into dx dy dz VI VF vr ’ and if in the equation between x, y, o> we write for co the above value, we have the corresponding integral equation between x} y, z. (This will be presently given in the particular case a = 0, No. 449.) 448. But we may in a different way make the transforma- tion from to , U = a 4- bu + cu2 4- du3 + eu4. Take as before /, */ for the invariants, H for the Hessian, and for the cubi-covariant, {{Sac — 3b2) u° 4- (24ad — 4be) u 4- (48ae 4- 6bd — 4c2) u2 4- (246c — 4cd) u3 4- (8cc — 3d2) m4}, 4> = { (— 8a2d 4- 4a6c — 63 ) u° + (— 32a2c — 4a6d 4- 8ac2 — 262c) u 4~ (— 40a6c 4* 20acd 4~ &b2d ) v2 4- ( 20ad2 — 2062c )u4 4- ( 4i0ade — 206cc 4- 56d2 ) u5 4- ( 32ae2 4- 46rfc- 8c2e + 2cd2) it5 4- ( 86c2 — 4ccZc 4- d3 ) w6}; then identically JTJ3 — IU2H 4- 4ZT3 = — 2, i 2 H whence assuming 2 /7T i V2<1> s/U (o3 — /a) 4- 2J = —jjg , or say v H ^ ·XIV.] THE GENERAL DIFFERENTIAL EQUATION. 349 From the expression of ® we find 2 ( UH' — U’H) du da> = 17» and hence dto _ 2 ( UH' - U'H) da VS- " dit . where the multiplier of is a constant. We in fact have nU UH' - U'H= iV (8a2d - 4abc + 63) + &c. = -20>; that is dû, dw vB-2), leading to a relation between x, y, z, suppose in order to simplify that a = 0 ; that is, assume X = 6# + cx2-l· dx3+ ex4, = — a)(# —ft) {x—y), the value of 8 being thus zero. Then the integral of dx dy dw __ VJ + vT _«V2 Vii“350 THE GENERAL DIFFERENTIAL EQUATION. [XIV. becomes 62 + 26c (x + y) + & (a? + y2) + (2c2 — 2bd) æy + (2cd — 46c) xy (x + y) + d2x2y2 + 2 {— 6 (a? 4- y) — c (æ2 + y2) - dxy (x + y) — 2c#2y2} (§c — 2a>) + (tf-y)2.(|c-2o>)2 = 0, say this is \ + 2fi (|c — 2oj) + v (|c — 2©)2 = 0, and writing herein 2® = ie (— fty — ya. — aft) + t Z b z ’ that is §c — 2g> = c -f -, the equation becomes \z2 + 2¡i (cz2 + 6-z) + i/ (az 4* 6)2 = 0; or substituting for A, /¿, v their values 62 (#2 4- y2 4- z2 — 2yz — 2^# — 2#y) — %cxyz — 2bdxyz (pc + y + z) — 4tbexyz (;yz + #y) 4- (<^2 — 4cc) a?y2z2 = 0; viz. this is a particular integral of dx dy dz_ vz vf + ’ where Z = 6.r + car + cfe3 + ear4, &c. 450. It would be easy to verify this by writing the integral equation successively in the forms u = A + 2Bx + Ca?=A' + 2 By + G'tf = A" + 2_B"s + GV;XIV.] THE GENERAL DIFFERENTIAL EQUATION. 351 we then have B2 — AC, B'2 — A!G\ B"2 — A”C" proportional to YZ, ZX, XY respectively. Write b, c, d> e = 1,0, — I,2 J; then X becomes x — /¿e3 + 2 J#4, which putting therein - instead of x is —4 (¿r3 — lx + 2J); writing x x similarly -, - for y, z, and putting finally y z X = x? — lx + 2 J, Y = y* — ly + 2 J, Z= 2? — Iz + 2 J, we have /2 — 8 J (x +y + z) -+- 2/ (yz + zx + #y) + y2z2 + 22#2 -f ¿^y2 — 2xyz (x -f y + 5) = 0, as a particular integral of dx dy dz ■----1--— 4- — = 0 * vx Vf VI ’ this can of course be directly verified in the same manner.352 [xv. CHAPTER XY. ON THE DETERMINATION OF CERTAIN CURVES, THE ARC OF WHICH IS REPRESENTED BY AN ELLIPTIC INTEGRAL OF THE FIRST KIND. Outline of the Solution. Art. Nos. 451 to 453. 451. In Chapter ill. it was seen that the lemniscate was a curve such that its arc represented an elliptic integral of the first kind : but the problem of finding such a curve is obviously an indeterminate one ; we have to find x> y functions of such that da? + dy1 — dz2 f- .1 - k2z2 ; for this being so then, writing 2 = sin , the arc of the curve, measured from the point for which z= 0, will be s = F(Jc, ). Similarly if a, a are conjugate imaginaries, and x, y are functions of 5, such that 7 0 , . dz2 dx2 + dy2 = z2 — et?. ^ - or then the expression for the arc of the curve is then the relation is 1 £n—m = 0; this is an equation of the order m in f, giving for f, m values which (n being within certain limits) are all or some of them real and less than unity, and the corresponding values of a, a are then conjugate imaginary values, in accordance with the original supposition. Thus if m = 1, the equation is —(f — 1) = Ç dÇ O, bilclu IS (n + 1) f — ?i = 0 or f ^ ^ - ; which, n being positive, is positive and less than 1; if m= 2, it is £*(£— 1)2 = 0, viz. this is (n + 2)(w + 1) f* - 2 (n +1) nÇ + n (n - 1) = 0, (n + SK^ + V^. c. 23354 ANALYTICAL REPRESENTATION OF [XV. If n is positive and less than 1, one value of f; if n be greater than 1, each value of f; is positive and less than 1. It is to be observed that if n is integral, and less than m, the equation as above obtained contains the factor ^m~n) and throwing this out sinks to the degree n; the equation may in fact be written indifferently in the forms 1 *n—m dr At r(C-l)w = 0; 1 (1 - Ç)m~n £*(?- i)m = o, the degree being m or n whichever is least. The values of f are in this case all of them positive and less than 1. General Theorem of Integration. Art. Nos. 454 to 462. 454. The foregoing result depends on a general theorem of integration which is as follows: taking 0 any positive integer, the integral i(u +p)m+n~o (u -f- qy du J it™*1 (& + p q- q)n+1 has an algebraical value provided a single relation subsists between p, q, m, n: viz. writing ([m]p2+ [n] q2)9 to denote 0 [m]9p29 + — [ra]*-1 [ri]1 p2Q~2q2 4- ... 4- [?i]0 q29, where as usual [m]e represents the factorial m (m — 1) ... (m — 0 + 1), the required relation is ([m] p2 4- [n] q2)9 = 0. 455. If in this theorem, m being a positive integer, we take 0 = m, and writing u — z — a, take p = a + a, q = a — a, we have the integral f(z — a)m (z + a)n dz J (z — a)m+1(z+ajn+1 having an algebraical value, provided there is satisfied between a, a, m, n the relation iW (a + a)2 + [n] (a - a)2}w = 0.XV.] THE ARCS OF CERTAIN CURVES. 355 Or taking as before f ^ , we have f — 1 = ^ —, ° 4aa 4aa and the equation may be written {Mr+W(f-i)}"-0, which is the before-mentioned equation in f · thus, m = 2, the equation is [2p p + 2 [2]1 [»p £(? -1) + [nf (?- I)2 = 0, that is, 2^ + 4h£ (£- 1) + (^2 -»)(?-1)2 = 0; or (n2- 7i)(f2 - 2£ + 1) + 4(f2 - 0 + 2^ = 0, which is (w + 2) (n +1) f2 — 2 (w -f1) + w (n — 1) = 0, as above. 456. To prove the general theorem, write for shortness U=(u + p)m+n-9+1 (u+p + q)~n. The integral then is s, U ( u + q)e du lum+1(u + p) (u+p + q)’ which we assume to be = TJu~m (A + Bu + GuK..+ AV'1), say it is = TJQ. This will be the case if UQ' + U'Q = _____U(u + qf___ um+i (u + p)(u+p + q)’ or what is the same thing, U'n.g- + jZ-y + W um+1(u+p)(u+p + q)’ viz. substituting for V its vahie, this is [(m + n — 0 + l)(u + p + q)~ n{n + J>)] Q + (u+p)(u+p + q)q (« + qf Mm+1 ’ 23—2356 ANALYTICAL REPRESENTATION OF [xv. 1 f\ where Q' denotes ^. The question therefore is to express that this differential equation has an integral Q = u~m(A + Bu + Cu2... -f-Kufl_1). Substituting this value and equating coefficients, we have between the 6 coefficients A, B, C ... K, a system of 0 + 1 equations implying one relation between the quantities m,n,p, q: and this condition being satisfied, the coefficients A, B... K will be completely determined, or we have for Q an equation of the form in question. 457. For instance, if d = 1, the equation is {mu + mp + m 4- nq) Q+{u2 + u (2p 4- q) + p2 + pq] Q' = > u to be satisfied by Q = Au~m : this gives {mp + (m+ n)q +mu } Au~m + {p2 + pq + (2p + q) u + w2}. — mAu~m~x = qu~m~1 + ic™, that is irm-\ | — m (p2 + pq) A — q} u~m {[mp + (m + n)q} A — m (2p + q) A — 1} irm+1 { + mA — mA } = 0, viz. the equations are m(p2+pq) A +q = 0, (mp — nq) A + 1 = 0, whence eliminating A we have m (p2 +pq) — q (mp — nq) = 0, that is mp2 + nq2 = 0, as the required relation in the case in hand. 458. Similarly if 0= 2, the differential equation is [m - Ip + m + n^- lq + m- lte] Q +[p2+M+u(2P+q)+u2]Q'=^>Sr >XV.] THE ARCS OF CERTAIN CURVES. 357 satisfied by Q = iw-® + Bu~m+1. This gives vr™-1 u~m vr™*1 ' (m—lp+m+n—lq)A (m — 1 p+m+ti—1 <7)5 yy tf m—1.4 (ra—1)5 —m(pt+pq)A —(m — l)(p2+pq)B jy i) —m(2p+q)A —(771—l)(2p+£)5 — TuA —(ra—1)5 -2* that is -2 q -1 1 4- [m — lp — nq] B -f 1A = 0, 2q + (m — 1) (p2 +pg)} = q2 {(m2 — m)p2 — 2mnpq + (ft2 — ft) #2}, = q2 ([m] — [?i] q)2. And similarly the third determinant is composed of terms in 1, 3q, 3q2, q3, which are the four terms in the first reduced expression of the determinant: and so in other cases. These first reduced expressions give without difficulty the final forms ([m] p2 + [ft] q2)\ ([m] p2 + [ft] q2)2y &c. 462. Writing 0 = n, and z— a, a — a, a + a for u, p, q re- spectively, we have the originally mentioned theorem in regard to the integral f(z — a)m (z + a)n dz J (z — a)w+1 (z + a)n+1 ’ and thence, as already mentioned, the expressions of x, y as functions of a parameter z such that the arc of the curve is given by the formula r-f-— J *Jz2 — dz a*. & -2 — «2 viz. as an integral in the nature of an elliptic integral of the first kind.360 [XVI. CHAPTER XVI. ON TWO INTEGRALS REDUCIBLE TO ELLIPTIC INTEGRALS. of x, is not in general reducible to elliptic integrals; but Jacobi has shown (Crelle, t. VIII. (1832) p. 416) that if P has the par- ticular form are reducible to elliptic integrals: and that by means of the theory an elliptic integral of the first kind 2 ' where k is a complex imaginary quantity, say k = sin (a 4- ¡3i), can be reduced to the form G 4* Hi, where G and H are real integrals of the above-mentioned kind. Investigation of the Formulae. Art. Nos. 464 to 467. 464. Considering the integral where P is a quintic function P = x (1 — x) (1 + kx) (1 4- Xx) (1 — tcXx), i dx J V# V1 — x .1 + tcx .1 - viz. X used to denote dx for shortness, — #.1+/m?.1+\#.1— k\x ’ 1 — x.1+kx.1+\x.1 — kXx ;XVI.] write ON TWO INTEGRALS &C. tcr/iniMtNI Ul- MAIHtMAiltHS CORNELL UNIVERSITY 361 6=V* + V\ (+), C=JK-Jx (*), b'=l -x/kX (+), c' = l + J*x (-), denom. = */! + «. 1 + X; and therefore b- + b'~ = 1, c2 + c'2 = 1. Assume Jx = ---------(¿>' + 0 sin,/, Vl — b'2sin2 + Vl — c2 sin2 (b' + c') sin e t =-----~—r for shortness, B + G we have (1 + kx) (1 + Xx) = (B + C)* +(tc + X)(B + Of (6' + c')2 sin2 + k\ (b' + c'Y sin4 (-r), (1 - X) (1 - tcXx) = (B + cy - (1 + kX) (B + cy- (6' + cj sin2 4> + kX (b' + try sin4 O), denom. = (B + (7)4; which after all reductions become (1 -I- kx) (1 + Xx) = (1 — x) (1 — kKx) = (B + C) 1 .4 (B+cy, (B + G) 4 (B+C)* cos? = /^ '(B+cy _ 4 cos2 (.B+cy■ 465. We have in fact c'-b' A (d - b' y c' + b’~V/iX’ KX~\c'+b') ’ b2 ■+· c2 _ k 4- \ 6M-T2 ” r+ /c\; and thence /c + X _ f /c' - 6 V[ b2+c2 2 (62 + c2) " | + Vc + 67 j 6'2 + c'2 ’ (6' + c')2 ’ and (1 + *)(1 + X)=(y_7y2.362 ON TWO INTEGRALS REDUCIBLE [XVI. Hence observing that tc\ (£>' + c )4 sin4

'2 — c'2)2 sin4 = (5s — C2)2, we have (5 + C)4 + (/c + A) + G)2 (i> + c )2 sin2 <£> + /cA (6 + c )4 sin4 <^> = (B + cy + 2 (b2 + c2) (5 + cy sin2 0 + (52 - O2)2, = (B + cy {(.B + C)2 + 2 (62 + c2) sin2 0 + (5 - C)2} = 2(5+ <7)2 {52 + C2 + (b2 + c2) sin2 } = 4(5 + C)2. Also (5 + (7)4 — (1 + /cA) (5 + (7)2 (6' + cj sin2 <£ + tc\ (b' + c')4 sin4 = (B + C)4-2 (b'2 + c 2) (5 + Gy sin2 + (52 - O2)2 = (5 + O)2 {(5 + C)2 - 2 Q>2 + c'2) sin2 + (5 - C)2} = 2(5 + C)2 \B2 + <7* -(b’2 + c2) sin2 = 4(5 + C)2 cos2 , and we have thence the formulae in question. 466. Moreover from the equation we have V# = (b' + c') sin B + C 5 dx 2 x (b' + c ) cos (f> " (5 + 5 + C + am**(|+^})C + (C"2+°2 sin2 *>£} d _ (6' + c) cos <£d<£ ~ ~ (B+CfBC ’ and combining herewith the foregoing equations V1 - a?. 1 — tc\x — 2 cos ~bVc> V1 + kx . 1 + A# = 2 5 + C’XVI.] whence also TO ELLIPTIC INTEGRALS. 363 VX = Vl— x.1 + kx.\+\x.\ — kXoc = zi-—: (B + Lf we have therefore dx *JxX 467. Moreover X — and thence (bf + c')2 sin2 cp (B+Cf ' Vxdx . /7, ,v„ . „ , 1 -j= a (b + c) sin BCjB+C) d Jg _ Q ” 2 4· c )3 sin2^> Or since 52 - O = - (62 - c2) sin2 (f>} = (6'2 - c'2) sin2 <£>, Vxdx ! (6' + c')2 B — C ' = Vw- -d VX 2 (6'-c) ’ 5(7 this is c' —6' V5 <7, viz. we have the two equations ^=1 («'+*) (¿4)#, where X = V1 — #. l + tf#.l-fX2.1 — k\x, B = Vl — b2 sin2 , (7= \/l — o2 sin2 <£, ,/- (b' + c')sw va!- jS + a · and as above364 ON TWO INTEGRALS REDUCIBLE [XVI. We thus see that the two integrals in question , J vxX fVxdx J depend upon the two elliptic integrals of the first kind, f d [ d(f> J Vl — b2 sin2 ’ iVl -c2 si ' Vl — 62 sin2 5 which is the theorem in question. sin2 Further Developments. Art. Nos. 468 to 472. 468. We may express as a function of x\ viz. the last equation gives B i e^(&' + c/)sin, the whole equation divides by (&'+ c')2 sin2 <£, and throwing out this factor it becomes (iV - cf sin2 4>-2(&- + Ci)-+ ^ + c')*sin,j£ = 0, X X that is (bf — c')2 sin2 cj> — 2x {2 — (i>2 -f- c2) sin2 <£} + (&' + c')2 sin2 <¡> = 0, viz. sin2 <£ {(6' + c')2 + 2 (62 + c2) # + (&' — c')2 #2} = 4>x; that is sin2 a. =_______________^_________________ v (b' + c'Y + 2 (£>2 + c2) a; + (6' — c')2 ¿r2 4# 1 = (&'+c')2, , 2 (¿>2 + c2) , /6' - c'\« , 1 + -WT7r‘ + {F+?)‘- __ (1 + k) (1 +X) X 1 + kx + Xx ’XVI.] TO ELLIPTIC INTEGRALS. hence we may write sin2<£ = 1 + * . l+\.# (-). cos2 (f> = 1 — x . 1 — k\x (-). B2 = (1 — V/tX#)2 (-0, O2 = (1 + V /e\#)2 (-0. where denom. = 1 + /t# . 1 + \x. 469. It may be remarked that writing 1 + 0x = {(B + cy + 0(b' + c'y sin2 } - (B + <7)2 = {2 — (b2 + c2) sin2 (£ + 6 (6' + cj sin2 + 2BG} + (B + C)\ and endeavouring to make the numerator a square, it will be the square of Vl + 6 sin + ^1-6 sin Vi- c sin Vl — c sin + Vl — b sin Vl H- csin ; viz. in the first case we must have 2 — (62 + c2) sin2 + 0 (6' + c')2 sin2 (f> = 2 + 26c sin2 <£, that is - (62+ e2) + 0 (6' + c')2 = 26c, or 6 = , = k : and in the second case 2 — (62 + c2) sin2 <£ + 0 (6' + c')2 sin2 = 2 — 26c sin2 that is -(b*- + &) + 6 (6' + c')2 = - 26c, or 6 = , = X.366 ON TWO INTEGRALS REDUCIBLE [XVI. Hence the two equations are 1+ K#=={Vl+&sin0Vl-i-csin<£-f V 1—¿>sin^>Vl —csin^>}2~-(.B+C')2> 1+\#=={Vl+&sin0Vl—csin<£4Vl—&sinVl+csin<£}2-i-(2?-f C*)2* leading to the before-mentioned equation l + /ar.l-f-\# = 4-r-(i?+ Gf, but there are no analogous values of 1 — x, 1 — kx to lead to 1 — x * \ — tc\x = 4 cos2 -T- (B + Of. 470. Write now b = sin (a + /8), c — sin (a — /9), and therefore B = Vl — sin2 (a + ft) sin2 , C? = Vl — sin2 (a — #) sin2 , X = (1 — #) (1 + # tan2 a) (1 + # tan2 /3) (1 — x tan2 a tan2 /3), and therefore b' = cos (a + £), c' = cos (a — 0); we hence obtain /- 2 sin a cos £ V#G = ~-----------Q 2 cos a cos p = tan a, \/x = 2 cos a cos y8 sin BTC _ 2 sin /3 cos a ~~ 2 cos a cos /? .To — tan p, vx Writing this last equation in the form XVI.] we have TO ELLIPTIC INTEGRALS. 367 r.dS dx J ; 2 cos a cos P = —,= (1 + sc tan a tan p), nxX 2 cos a cos ¡3 (1 “ x tan a tan /3). If in these equations we write ¡3i for X continues a real function, viz. we have X = (1 — x) (1 + x tan2 a) (1 + x tan2 /3i) (1 — x tan2 a tan2 /3i); and the formulae are /— 2 cos a cos Bi sin 6 . . . Nx =--------jg- g--------- , giving rise to 2 cos a cos /3i^ = -^L· (l+x tan a tan /3i), 2 cos a cos Bi ^ (1 — # tan a tan /3i), where observe that B + (7 = Jl— sin2 (a + /&*) sin2 +J1 — sin2 (a — fii) sin2 is real; viz. these formulae give the values of dcf> f d >’ JVl — sin2 (a — f___________ in terms of the integrals (a - Bi) sin2 f dx . fV x dx We may change the form by writing tan Bi = i sin y, whence we thus have cos Bi =-----, sin Bi = i tan y: cos 7 k = tan2 a, X = — sin2 7368 ON TWO INTEGRALS REDUCIBLE [XVI. ,— _ 2 cos a sin cosy'B+C’ X = (1 — x) (1 + x tan'2 a)(l — x sin2 7) (1 + x tan2 a sin2 7), 2 cos a. dé dx . -----. = -7= (1 4- ix tan a sm 7), cos 7 B VicX 1h 2 cos a d __ da; cos 7 ' (7 “ Viz (1 — ix tan a sin 7), and observing the equation sin2 . = (!+«)(!+\)a; = 1 ____ xcos27 . ^ (1 + /or) (1 + \x) (cos2 a + x sin2 a) 1 — x sin2 7 ’ we see that to real values of there correspond values of x which are positive and less than 1, and that as x passes from 0 to 1, sin2 passes from 0 to 1, or from 0 to 90°, X being thus always real and positive. Writing sin = y, the relation between , x gives a re- lation between x, y: viz. this is x = Q>' + c')y Vl -by + Vl - c2y2 3 or what is the same thing = (1 + k) (1 + X) x y (1 H- kx) (1 + \x) ’ viz. this is a quartic curve; and introducing z for homogeneity, or writing the equation in the form y2 (z + kx) (z + \x) — (1 + k) (1 + \) xz3 = 0, we see that x = 0, z = 0 is a fleflecnode, the tangents being z + rex = 0, z + Xx = 0 ; y = 0, z = 0 is a cusp, the tangent being y = 0 ; x = 0, y = 0 is an ordinary point, the tangent being x = 0 ; hence the curve, as having a node and cusp, is bicursal.XVI.] TO ELLIPTIC INTEGRALS. 369 471. The transformation of a given imaginary modulus into the form sin (a -f fii) presents of course no difficulty: assuming that we have ]c = e+ fi, then we have to find a, such that e+fi — sin (a + fii), or writing sin a = £, sin = ¿17, to find 17 from the equations e =fVl + 972, ƒ = 77 \/l - f2: these give e2 = f2 + f y, /2 = V2 - fy, whence e2 + ƒ* = f2 + -17s, and thence easily f* = i{ 1 +e2+/a — VV), -7!! = i{-l + e2+/2 +VV}, where V = 1 -f e4 + /4 — 2e2 + 2/2 + 2e2/2. If as above sin /3i = i tan 7, then tan 7 = 97, or the equations give f, = sin a, and 97, = tan 7. 472. The integrals J^p> Jare a^so reducible to elliptic integrals when the quintic function P has the form P = x (1 — x) (1 + kx) (1 + A#) (1 + fc + A + *A x), as shown by Prof. M. Roberts in his “ Tract on the Addition of Elliptic and Hyper-elliptic Integrals,” Dublin, 1871, p. 63; and in the Note, p. 82, to the same work, a simple demon- stration is given of the theorem (due to Prof. Gordan) that the like* integrals, wherein P denotes a sextic function the skew invariant of which vanishes, are reducible to elliptic integrals. c. 24370 [xvi. ADDITION. FURTHER THEORY OF THE LINEAR AND QUADRIC TRANSFORMATIONS. The Linear Transformation. Art. Nos. 473 to 477. 473. We consider the transformation of the differential expression dx *s/x — a.x— fi.x—ry.x —h’ where the new variable y is given by an equation of the form xy + Bx + Gy + D = 0. The coefficients B, C9 D might be expressed in terms of any three pairs of corresponding values of the variables x, y, say the values at /3, 7 of x, and the corresponding values d, ft, 7' of y : but it is better to consider in a symmetrical manner four pairs of corresponding values, viz. the values a, ¡3, 7, S of x and the corresponding values a', $y 7, 8' of y. We have thus four equations from which B, C, D may be eliminated, and we obtain the relation ad, a, a', 1 fip, A P, 1 77'. % 7.1 Stf, 8, 8', 1 = 0, which in fact expresses that the two sets of values (a, /3, 7, 8) and (a, /3', 7', 8') correspond homographically to each other. 474. Writing for convenience a, b, c, f, g, h = l3-y, 7-a, a-ft a-8, £-8, 7-8, a', V, c', f, g', A' = ft-7', 7' - «> «' S', ft - 8', 7' - S',ADD.] LINEAR AND QUADRIC TRANSFORMATIONS. 371 so that identically af+ bg + ch = 0, af' -f b'g' + c'h! = 0; then, as is well known, the relation in question may be ex- pressed in the several forms af : bg : ch — a'f' : b'g' : ch': or, what is the same thing, there exists a quantity N such that «t=^i=ct= af bg ch 475. The relation between (x, y) may now be expressed in the several forms, y-a'_ pX-a y-p_nx-l3 y-y _ y y — S' x — h* y — 8' ^ x — b’ y— S' x — S’ and writing for (x, y) their corresponding values, the values of P, Q, R are found to be p_ bji_ eg . 0 _ c/ _ a7i. n bh!~ eg’ H~ cf~ aK' ag'~bf' and we thence obtain f-PN2=f'-QR, g2QN2=g'-RP, h?RN2 = h'*PQ, \/PQR = # f'g'h' 476. Differentiating any one of the equations i^ for instance the first of them, we find y\ fdy fPdx and then forming the equation *Jy — a.y — j3'.y — f__ VPQP \/x — a. x — ft. (y — S') *ly — S' (x — S) *lx — 8 3 or if we please Jy-a .y-0'.y-y .y-8' _\/PQR\/x-a.X- — (y ~ S')- (oo - S)- 24^2372 FURTHER THEORY OF THE [ADD. and attending to the relation f2PN2=f'2QR, we obtain Ndy __ dx sly — - y — fi'. y — y - y — & *Jx — ol.x — fi.x — y.x — 8 which is the required formula: (a, fi, 7, 8) and any three, say (a', fi', 7'), of the other set of quantities are arbitrary, and the values of 8', N in terms of these are given as above. 477. It is proper to remark that in this and similar formulae the sign of the multiplier N may be assumed at pleasure: only, this being so, the radicals s/X and VF of the formulae are not in general both positive ; we have between the radicals a relation of the form F VX = ± G \!Y (F, G rational functions) wherein the sign + has a determinate signification ; in fact the last-mentioned relation combined with the differ- ential equation gives ± NGdy = Fdx, which equation sub- stituting therein for ^ its value, obtained by differentiation as a rational function of (x, y), is a rational equation equivalent, when the sign is taken properly, to the given rational equation between the variables (x, y). The sign ± of the equation Fs/X = ±G s/Y might have been assumed at pleasure, and the sign of N would then have been determinate ; but this is less convenient. Transformation of a form into itself Art. No. 478. 478. The homographie relation is satisfied by writing therein a'> 0', 7> S' = (a, fi, 7, S), (fi, a, 8, 7), (7, 8, a, fi), or (8, 7, fi, a) : these values in fact give a, v, c', 9, h’, a, b, c, /> 9> h, ƒ. ~9> ■ -C, a, -b, - -K “ƒ> -b, h, ■ -a, ~9> c, -a, g>·- -A, b, -c, respectively, so that in each case a'f : b'g' : ch' = af : bg : ch.ADD.] LINEAR AND QUADRIC TRANSFORMATIONS. 373 We have thus four solutions of the equation dy _ dx *Jy — a.y — fi.y — y.y — 8 *Jx — a.x — ¡3.x — y .x — 8’ viz. these are = y-h y-ff _ y-y y-y _ y-P x — a x — 8 ’ /3 — 8 x — a y — a æ — 8 ’ 7 — 3 x — a a —/3 x — 8’ y — S _ £ — 8 .y — 8 x — a y — a y—a.a — fix —8' the first of them being the self-evident solution y = #. In particular there are four solutions of dy _ dx that is \A2-1-a;2-p dy dx v/l - y2. ! - ¿y Vi - ¿e2. 1 - k2x2 ’ viz. these are y = x, y = — x, V — and y = — , respectively. te5 Application to the standard form. Art. Nos. 479 to 481. 479. Considering now the equation Ndy____________dx______ y/y2_l.y2_I ¡f _ 1 . *■ - I or, writing A' = —-, say A Mdy dx Vl - y2.1 - xy Vl - #2.1 - k2x2 ’374 FURTHER THEORY OF THE [ADD. if in the general form we assume a, A 7. S=l, “I. p -p then we have in any one of the twenty-four orders «',/3',7>S'=l, -1, p -p and since, for any one of these orders, A will be determined by a quadric equation, it would at first sight appear that there might be in all twenty-four pairs of solutions, belonging to forty-eight different values of A, M. But the solutions corre- sponding to two orders in which 1, — 1 are interchanged, are A. A equivalent; and moreover y = $(%) being a solution belonging to determinate values of A, M, then we have, belonging to the same values of A, Mt the four solutions y = (x)y y = (— x), y = and y = : we have thus only three pairs of solutions, or say six solutions, belonging each to a different set of values of A, M; and which correspond to the three orders S' = 1’-1’ v i, p-1. 1 A ’ 1 ’A’ 1 1-1-1 ’ A’ A ’ 480. Forming for each of these the equation which de- termines A, say in the form , we have successively the three equations /1 + AY2_ (1 + kV (1 + A)2 _/l + A?y 4A /l+*\2 [l-x) ” Vl - k) ’ 4A "Vi-*/ ’ (1 + ^)2 U-*/ ’ giving for A the values k’ 1 /1-Viy /I + V&y _ /I - i \/k y /I + i’Jk'v* k) ’ vi — Vfc/ \i + i V&/ ' Vi — i V& i + VfcADD.] LINEAR AND QUADRIC TRANSFORMATIONS. 375 The corresponding values of N are derived from the equa- tion N2 = , viz. we thus obtain for , that is for M, the aj fc values 1 4“ A % (1 -|- A) 2% Va 1+ìc ’ 1 4-A; 5 viz. substituting for A its values, these are ^ 1 2 i 2 i 2 i 2 i ’ (14-V^’ (1-Vi)2’ (1 + tVii)1’ (1-iVi)2’ 481. The six transformations Jfcfa/ _ dx Vi - 2/2. l - Ay ~ Vi^Ti-^2 ’ then are y= A = M = a k 1, 1 1 1 x ’ k’ k’ 1 4- V& 1 — a? Vi n - Viy 2i 1 — Vi 1 + a; Vi ’ Vi + ViJ ’ (1 + Vi)2 ’ 1-Vi 1+icVi /l + Viy 2 i 1 4- Vi 1 — # Vi ’ U-ViJ ’ (1- Vi)2’ 1 4- i Vi 1 — ix Vi /l-i Viy 2i 1 — i Vi 1 4- ¿a? Vi ’ VI + i Vi/ ’ (1+iVi)2’ 1 — i Vi 1 4- ioc Vi /l + i Viy 2» 1 4- i Vi 1 — ix *Jk9 \1 — i Vi/ (1-iVi)2’ where it is to be remarked that the last four transformations are included under the form 1 4- a 1 — olx ^ f1-aY M- 2t y 1 — a 1 4- ax9 U + J’ '(l+«)2’ where a is a fourth root of i2. These are in fact Abel’s results referred to No. 420.376 FURTHER THEORY OF THE [ADD. The Quadric Transformation for the standard form. Art. Nos. 482 to 486. 482. Reckoning the number of linear transformations as six, that of the quadric transformations is reckoned as eighteen; viz. these are Abel’s eighteen transformations referred to No. 422. Taking as before the differential relation to be Mdy _ dx Vl - y2.1 - \y Vl —x?. 1 have, Four transformations y= X = M = „ - ^ ——V ** V (i ·+■ k) x 2 Vi 1 1 4* kx2 ’ 1 + i ’ 1 +i ’ (1 — k) x 2i Vi 1 1 — kx? 9 1 — i ’ 1-i’ 2 fkx 1 + i 1 1 4- kx2 ’ 2 Vi ’ 2 Vi’ 2 i fkx 1 — i 1 1 — kx? 9 2i V/fc 9 2 i Vi ’ 483. Six transformations -A. M= x— 1 —- ^ 1 +kaf 1 — i i — 1 4- kx2 9 1+k ’ 1 + k ’ — 1 + kx? 1+k i 1 + ka? ’ 1 - i ’ l^k ’ 1 — (1 + i') x2 1-i' 1 l-(l-i')a?’ 1+k" 1 + k" 1 - (1 -k’)x> 1 + i' 1 1 - (1 + k')x‘i 1-k" 1-k'9 — (k' + ik) + ikx2 i' — ik k 4- ik\ (Jc — ik) + ik a? 9 k' + ik9 (Jc — ik) 4- ik a? — (V + ik) + ika?9 k' 4- ik k'-ik’ — k 4- ik']ADD.] LINEAR AND QUADRIC TRANSFORMATIONS. 377 484. And lastly, Eight transformations VI + & + V2V& 1 + Icoc? + os V2 \/k Vl + & 2 i Vl + A? — V2^A; l-f&ic2 — a; V2 \/A;Vl H-Ar* . _ /Vi + & — V2 vXy2 ____ \Vl + A? 4-V2 VAy * (Vi 4- A; 4- V2 */ky Do. with — Z/k for ifk, a jj » )) i t/k — i\/k Do. with for \/k and —k, Vl— k for k, Vl+A*, V2 -- J'm )) )) V2 J> jj » J> >> )> -1 +»' «T » » » » » 5) J) V2 j> » >> » » » 485. The last formulae, writing for shortness, /8 an eighth root of 16A;2, and a = Vl 4- ^/34, are included under the form _ a + j8 1 + a@oc + £/34#2 _ /a - £\2 M_ 2i V ' (a+/3)2’ and the verification may be effected as follows: we have (-). .!Vi+*w (+). ~--^0 ( 1 + a/3* + i/S4^2) a — p (-). (1 +*)(!+ i/34*) (-),378 FURTHER THEORY OF THE [ADD. 1 + \y = 1 — a&e + Ipa? + (1 + a/3x + |/3V) (-), M, l-\y = l-afix + (1 + a/3x + \^x-) (+), -¿^gd-^a-l/S4*) (-). where denom. = 1 — a fix + J/34#2. Hence Vl - f. 1 - xy = ^ + ^ (1 - iffV) Vl - x\l - tea? (-), where &2 is written instead of its value -^fi8: and moreover W, in which two formulae the denominator is equal to the square of its above-mentioned value; we hence find the required formula, Mdy _ dx Vl - f .l-Vy2 ~ Vl -xKl- k2x2 ’ 2 i where M has its proper value =--^ . r (« + £)2 486. It is, as regards all the formulae, convenient to remark that the value of M may be verified by taking x small; thus, if when x is small the equation for y becomes y = fix, then ob- viously M = 5 if the equation becomes y — + 1 + fix2, then we have M = —=^-; and so in other cases. V+2/3ADD.] LINEAR AND QUADRIC TRANSFORMATIONS. 379 Combined Transformations: Irrational Transformations. Art. Nos. 487 to 491. 487. By combining two linear transformations, we obtain a transformation which is linear, and as such is a transformation belonging to the system; viz. it is either one of the six trans- formations, or it is at once reducible to one of these. Similarly, by combining a quadric transformation with a linear one, we obtain a transformation which is quadric, and as such is equiva- lent to one of the system. For instance, changing the letters, if with the quadric transformation _ (1 + k) x Z~ 1 + km? ’ 1 + kdz dx 2 \/& 6 6 fl-zKl-XW \fl — z2. 1 - k2x2 ’ 1+jfe’ we combine the linear transformation 2 i giving (1 + *Jkf y= - dy 1 + \Tk 1 —x*Jk 1 — *Jk 1 + x\/k ’ dx Vl -yKl- y-f Vl-it+l-/>■'-.+ we have z a quadric function such that (1 + s/hf (\ - Vfcy 7 = tiwli ’ 2 i (1 + k) dz dy VI-2M-W Vl-2/2.l-7y’ and this must be one of the series of quadric transformations. We in fact find . - ^ _ \J fy vk =-----, and thence X = 1+V7 1 — 7 (1 + V&)2 _ — ' 1+7’ 2i(1 + k) 1+7’ 1 + V7 1 - y V7 sVA; = -—yfl,orx = --7= _ 1 +yvy 1 — Vyl-byV7380 FURTHER THEORY OF THE [ADD. and thence _(1 +k)x _l+7l —7 y2 Z’ ~ l + kaf ’ ~ T^y 1+yf ’ or, what is the same thing, 1 ^ 1+7f Xz — 1 + yy2 ’ giving 1 + 7 dz dy Vl - - XV Vl-y2.1 - yY , where \ = 1 ~y 1+7’ which ^with ^ in place of — is one of the series of quadric transformations. 488. If we combine two quadric transformations we obtain in general an irrational transformation: viz. neither of the two variables is a rational function of the other of them, but the two are connected by an equation: for instance, if the two transformations are _ 2 *Jlcx Z 1 + kx2 ’ giving — _ dz _ dx X=L+i'; and Vl-*2.1-X2z2 Vl-icM-Av’ 2 V* giving y- 1-* dy -1 + kx2 l+AA»2 ’ dx 1 + k Vl — y2.1 — rfy* Vl -a?.!-tea?’ 7 1_^’ 11 . 1 72 — 1 then we have here ƒ + 22 = 1; — + — = 1, giving — = ■ that is X2 = — _ 7s. or \ = --t if 7', = Vl - t2, is the comple-ADD.] LINEAR AND QUADRIC TRANSFORMATIONS. 381 1 — jk \ 1 mentary modulus to 7 ; also —~ = -r- = —., and the relation is — ^,dz 7 1 + dy_______. V1 — y2.1 — 72^/2 or, what is the same thing, changing the letters, the trans- formation arrived at is a? + y2 = 1, giving Vl - y2 . 1 - xy dx Vl- ¿c2.1 — k2x2 ’ X ih T” which is at once verified, since from the assumed relation a? 4- y* = 1 we have dx _ — dy V1 — #2 Vi — y2 * - xy. 489. Observe that the equations which define the two new variables y} z in terms of x are in general of the form A C y~ B ’ z~ D’ where A, B, C, D are quadric functions of x. Writing these equations in the form y : 2 : 1 =AD : BC : BD, then regarding (y, z) as the co-ordinates of a point of a plane curve, these expressions of y, z in terms of the arbitrary para- meter x show that the curve in question is a unicursal curve, and, being of the order four, it is a trinodal quartic; viz. the equation (y, z) = 0, obtained as above by combining any two quadric transformations, being a solution of the equation Mdz dy382 FURTHER THEORY OF THE [ADD. we have the theorem that this equation (y, z) = 0 represents a curve which is in general a trinodal quartic. It has been seen how in one case the curve is a circle. 490. It appears to be a conclusion of Abels, that if for any given values of (A, M) the equation Mdy _ dx Vl ~ Vl - xl. 1^1fa? admits of an irrational solution, then there is always an integer number n such that the equation Mdy _ ndx Vl — y2.1 — \2y2 Vl — x2. 1 — k2x2 admits of a solution y = rational function of x. So that, in fact, the general problem of transformation reduces itself to the problem of rational transformation. For instance, as just seen, the equation ______________i, has the irrational solution y = Vl — x2; the equation ~ k'dy 2 dx 4 A _„2 , Iff ~~ v y · + A'» has a solution y = rational function of x. To verify this, ob- serve that the first equation is satisfied by y — cn u, x = sn u (which are such that y = Jl — a?2) : hence the second equation is satisfied by the values y = cn 2w, # = sn u; we have cn 2w a rational function of sn u, ante No. 104, and writing therein x for sn u we obtain 1 — 2x2 + k2x* y= 1 - l:W as a rational solution of the second equation: the solution can of course be at once verified.ADD.] LINEAR AND QUADRIC TRANSFORMATIONS. 383 491. It appears from the formulae given No. 98, inter- changing therein (z, x) and also k, k\ that the equation — idz _ dx Vl — z2.l — k'2z2 V1 — x1.1 — k2x2 1 has the irrational solution z = V1 — kV — idz 2 dx ; hence the equation Vl — z2.1 — k’2z2 Vl — #2.1 — jfc2#2 has a solution ^ = rational function of x; viz. the first equation being satisfied by x = sn u, z = —, the second equation is satisfied by x = sn u, z = ; or dn 2u being a rational func- tion of sn u, see No. 104, replacing sn u by x, we find 1 -to4 1 — 2 k2x2 + tor1 as a rational solution of the second equation.INDEX The figures refer to the pages: where the reference extends to the whole or the bulk of a Chapter, the Chapter is also referred to. A heading “Function,” with divisions (1), (2)...(13) has been introduced. Abel, linear and quadric transforma- tions 322, 325, 375, 376: irrational transformation, 382. Addition, see Function, (2)...(9). Arc of curve ; representing integrals E (k, ). See also (9) infra, addition, 104. outline of further theory, 108. reduction of parameter to form sn (a+j8t), 114. addition of.parameters, and reduc- tion to standard form, 117. interchange of amplitude and parameter, 133. (6) gd w, sg w, eg u. addition and other properties, 56. (7) sn u, cn u, dn m, 63 (Chap. iv.). addition and subtraction formulae, 63. periods 4K, 4iK\ 66. imaginary transformation, 68. functions of tt + (0, 1, 2, 3) K + (0, 1, 2, 3) iK'y 69. duplication, 71. dimidiation, 72. triplication, 77. multiplication, 78. factorial formulae, 92. new form of same, 97. anticipation of doubly infinite product forms, 101. connexion with 0m, Huy 155, 156. quadric transformations, 183. w-thic transformation, 251 (Ch. x.). (8) Euy Zu. connexion with-E (k, 0), 107. addition, 107. connexion with 0m, 113. further theory, 147. (9) II (m, m), 142 (Chap. vi.). connexion with II (to, k, SONS LONDON: YORK STREET, COVENT GARDEN NEW YORK: 66, FIFTH AVENUE; AND BOMBAY CAMBRIDGE : DEIGHTON, BELL & CO December, 1894CONTENTS. 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