141 
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
31. Denoting, as above, by M 0 , M u ..., M n the values which correspond to v 0 , v lt ..., v n 
respectively, and writing S y = ^+ A + ...+ A, & c „ we have «I £lÿ, &c„ all of 
them expressible as determinate functions of u; and we have moreover the theorem 
that each of these is a rational and integral function of u: we have thus the series 
of equations 
s lr A > .... st=H, 
M 
M 
where A, are rational and integral functions of u. These give linearly the 
different values of in fact, we have 
C% ~ v x )... (v 0 -v n )^ = H- GSv 1 + FSi\v, - ... ± Av,v 2 ...v n , 
where Sv 1} Sv } v,, &c. denote the combinations formed with the roots v u v 2 ,... ,v n (these can 
be expressed in terms of the single root v 0 ) ; and we have also (v 0 — (v 0 — v n ) = F'(v n ) : 
the resulting equation is consequently F'v 0 = R (u, v 0 ), R a determinate rational and 
integral function of (u, v 0 ) ; but as the same formula exists for each root of the modular 
equation, we may herein write M, v in place of M 0> r 0 ; and the formula thus is 
F' v -^=R(u, v), 
viz. we thus obtain the required value of | as a rational traction, the denominator 
being the determinate function F'v, and the numerator being, as is easy to see, a 
determinate function of the order n as regards v. 
32. The method is applicable when M is only known by its expression in terms 
of q ; but if we know for M an expression in terms of v, u, then the method trans 
forms this into a standard form as above. By way of illustration I will consider the 
case n — 3, where the modular equation is 
v 4 + 2 vhi 3 — 2 vu — it 4 = 0, 
and where a known expression of M is Here writing $_ u S 0 (= 4), S, &c. 
to denote the sum of the powers — 1, 0, 1, &c. of the roots of the equation, we have 
S—=S 0 +2u s S- u =0 , as appears from the values presently given,