THEOREMS. [577 
the normal-variables and 
578] 
113 
■ - (au' + ßv') ; introducing 
') 2 
578. 
ly ; then <£ (du) is the 
-t 0 = 0, and thence u = 0, 
ponding value of f(u) is 
that the function </> (du) 
»} 2 ]> 
ds du, dv; the value in 
2 + dw 2 ), 
f wdw) 2 I ; 
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clxiv. (for the 
year 1874), pp. 397—456. 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= 3 and n = 5 ; viz. in the general case the formulse are expressed in terms of 
the elliptic functions of the ?ith 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 equation of the order 4 or 6, the modular equation. The extension of this alge 
braical 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 
difficulties are so great that it was found necessary to discuss it by means of the 
formulse of the transcendental theory, in particular by means of the expressions 
involving Jacobi’s q (the exponential of —), or say by means of the q-transcendents. 
Several important contributions to the theory have since been made :—Sohnke, “ Equa- 
tiones modulares pro transformatione functionum ellipticarum,” Grelle, t. xvi. (1836), 
pp, 97—130, (where the modular equations are found for the cases n = 3, 5, 7, 11, 13, 
17, and 19); Joubert, “Sur divers équations analogues aux equations modulaires dans 
la théorie 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 Königsberger, “ Algebraische Untersuchungen aus der Theorie der 
elliptischen Functionen,” Crelle, t. lxxii. (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 
c. IX. 15