. 
870] ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS (SEQUEL). 
555 
U 
70—2 
with a formula of transformation 
U 
y = 
T 2 
T = x v + aj«" -1 + a 2 x v ~ 2 + ... + a v , where v = ^(n — 1), 
U — x n + cf 1 ir n-2 + a 2 x n ~ 3 + ... + a„. 
The general theory for any value of n is developed to a considerable extent, and it 
would without doubt give very interesting results for the case n — 7 ; but the formulae 
are only completely worked out for the preceding two cases n = 3 and n= 5. For 
these cases the formulae are as follows: 
Cubic transformation : n = 3, 
x 3 + a, x 2 + a 2 x + a 3 
^ (it + tti) 3 
Corresponding to the modular equation, we have 
«i 4 - + g s ai - -hgi = 0, 
G 2 -9g 2 = 6 (20a x 2 - 3g 2 ), G 3 + 27g 3 = - 14 (20a a 2 - Sg s ) a u 
and then 
whence also 
_ _ 3 <?» + 27^3 
111 — J 
G 2 - 9g 2 ’ 
and by the general theory a 1} a 2 , ct 3 are given rationally in terms of a 1} g 2 , g 3 . 
Quintic transformation: n — 5, 
cc 5 + a y x 4 + a,,x 3 + a A x- + a^x + a 5 
We have 
where 
The first of these gives 
(x 2 + a y x + a 2 ) 2 
a y X — 2F=0, (12a 1 2 + ^ 2 )^ — 30«^= 0, 
X = cl?- 6a?a 2 + \g 2 a x — g 3 , 
Y = 5a 2 2 - a?a 2 + \g 2 a 2 - g s a y + T \g. 2 . 
( h = ~^( a i + hg* a i - g-.d; 
then eliminating a 2 , we have, corresponding to the modular equation, 
cl? - 5g 2 a? + 40# 3 a 1 3 - 5g?a? + Sg 2 g 3 a y - 5g? = 0. 
We then have 
G 2 - 25g 2 = - (lOoq 3 - 8g 2 a x + 5g 3 ), G 3 + 125g 3 = - 14 (lOaq 3 - 8g 2 a x + 5g 3 ); 
whence also 
Go + 125y 3 _ 
G 2 — 25g 2 ’ 
and by the general theory eq, ot 2 , a 3 , cr 4 , a 5 are given rationally in terms of a 1} g 2 , g 3 . 
These results are contained in the former of the papers above referred to; the 
latter contains some properties of these modular equations. 
®i 7