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XI. A Memoir on the Transformation of Elliptic Functions. 
By Professor Cayley, F.B.S. 
Received November 14, 1873, — Read January 8, 1874. 
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=o and n— 5 ; viz. in the general case the formulae are expressed in terms of the 
elliptic functions of the nth. part of the complete integrals, but in the cases n= 3 and 
n— 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 equa- 
tion 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 : such 
theory should admit of being treated in a purely algebraical manner; but the diffi- 
culties 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 involving Jacobi’s 
q /the exponential of — j , or say by means of the ^-transcendents. 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, 1 1, 13, 17, & 19) ; 
Joubert, “ Sur divers equations analogues aux equations modulaires dans la theorie des 
fonctions elliptiques,” Comptes Rendus, t. xlvii. (1858), pp. 337-345 (relating among 
other things to the multiplier equation for the determination of Jacobi’s M) ; and 
Koxigsberger, “ Algebraische Untersuchungen aus der Theorie der elliptischen Func- 
tionen,” Crelle, t. lxxii. (1870), pp. 176-275; together with other papers by Joubert 
and by Hermite in later volumes of the 4 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 
‘ Fundamenta Nova ’), on account of the peculiarly simple form of the modular equation 
(1— w 8 )(l — y 8 ) = (l— uvf, presents but little difficulty; and I give the complete formulae 
for this case, obtaining them as well algebraically as transcendentally ; I also to a con- 
siderable extent discuss glgebraically the case of the next succeeding prime value n— 11. 
For the sake of completeness I reproduce Sohnke’s modular equations, exhibiting them 
MDCCCLXXIV. 3 G 
