56 On the Transformation of Ellipic Functions. [Jan. 8, 



III. "A Memoir on the Transformation of Elliptic Functions." 

 By Professor CAYLEY, F.R.S. Received November 14, 1873. 



(Abstract.) 



The theory of Transformation in Elliptic Functions was established by 

 Jacobi in the ' Fundamenta Nova ' (1829) ; and he has there developed, 

 transcendentally, with an approach to completeness, the general case, n 

 an odd number, but algebraically only the cases n=3 and n=5 ; viz. in 

 the general case the formulae are expressed in terms of the elliptic func- 

 tions of the nth part of the complete integrals, but in the cases n=3 and 

 w=5 they are expressed rationally in terms of u and v (the fourth roots 

 of the original and the transformed moduli respectively), these quantities 

 being connected by an equation of the order 4 or 6, the modular equation. 

 The extension of this algebraical theory to any value whatever of n is a 

 problem of great interest and difficulty. The general case should admit of 

 being treated in a purely algebraical manner ; but the difficulties are so 

 great that it was found necessary to discuss it by means of the formulae of 

 the transcendental theory, in particular by means of the expressions in- 



TrT^'X 

 volving Jacobi's q (the exponential of -^- I, or, say, by means of the 



^-transcendants. Several important contributions to the theory have 

 since been made : Sohnke, " Equationes Modulares pro transformatione 

 functionum Ellipticarum," Crelle, t. xvi. (1836), pp. 97-130 (where the 

 modular equations are found for the cases n = 3, 5, 7, 11, 13, 17, & 19); 

 Joubert, " Sur divers equations analogues aux equations modulaires dans la 

 theorie des fonctions elliptiqu'es," Comptes Eendus, t. xlvii. (1858), 

 pp. 337-345 (relating among other things to the multiplier equation for 

 the determination of Jacobi's M) ; and Konigsberger, " Algebraische Un- 

 tersuchungen aus der Theorie der elliptischen Functionen," Crelle, t. Ixxii. 

 (1870), pp. 176-275; together with other papers by Joubert and by 

 Hermite in later volumes of the ' Comptes Rendus,' which need not be 

 more particularly referred to. In the present Memoir I carry on the 

 theory, algebraically as far as I am able ; and I have, it appears to me, 

 put the purely algebraical question in a clearer light than has hitherto 

 been done ; but I still find it necessary to resort to the transcendental 

 theory. I remark that the case n=7 (next succeeding those of the ' Fun- 

 damenta Nova'), on account of the peculiarly simple form of the modular 

 equation (1 u 8 )(l v 8 )=(l uv} s , presents but little difficulty; and I 

 give the complete formulae for this case, obtaining them as well alge- 

 braically as transcendentally; I also to a considerable extent discuss 

 algebraically the case of the next succeeding prime value n= 11. For 

 the sake of completeness I reproduce Sohnke's modular equations, exhi- 

 biting them for greater clearness in a square form, and adding to them 

 those for the non-prime cases n=9 and ^ = 15; also a valuable table 

 given by him for the powers of / (q) ; and I give other tabular results 

 which are of assistance in the theory. 



