126 
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
126 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578 
578] 
We have thus (20- + l) 2 = 6 2 , that is, 40 4 +30 2 + l=O or 4& 2 + 3k + 1 = 0, or else 
(0- + 2) 2 = 6 3 , that is, 0 4 + 30 2 + 4=0 or k 2 + 3k + 4 = 0; viz. the equation in k is 
The 
great im 
(4k 2 + 3k + 1) (k 2 + 3k + 4) = 0, 
expressio 
rational j 
these being in fact the values of k given by the modular equation on putting therein 
0 = 1. 
24. 
The equation of the order 32 thus contains the factor {(fi, l) 4 } at least twice, and 
it does not contain either the factor 0 — 1, or the factor {(O, l) 6 } belonging to the 
quintic transformation; it may be conjectured that the factor {(O, l) 4 } presents itself 
six times, and that the form is 
1(0. 1)?(0, l) 8 = 0 ; 
say this 
so that 
but I am not able to verify this, and I do not pursue the discussion further. 
having a 
volving s 
22. The foregoing considerations show the grounds of the difficulty of the purely 
algebraical solution of the problem; the required results, for instance the modular 
equation, are obtained not in the simple form, but accompanied with special factors of 
high order. The transcendental theory affords the means of obtaining the results in 
the proper form without special factors; and I proceed to develop the theory, repro 
ducing the known results as to the modular and multiplier equations, and extending 
it to the determination of the transformation-coefficients a, /3, .... 
g 
form q H l 
for the s 
n, 2n, 3r 
Hen 
integral 
The Modular Equation. Art. Nos. 23 to 28. 
g having 
known 1 
jtA'' 
23. Writing, as usual, q = e K , we have u, a given function of q, viz. 
known 1 
thus for 
--'/‘-i-'A- 
1 l+q.l+q 3 .l + q 5 .. 
= V2q* (1 — q + 2q 2 — 3q 3 + 4>q* — 6q 5 + 9(f — 12q 7 + ...) 
= *J2q*f(q) suppose ; 
where A 
coefficier 
25. 
and this being so, the several values of v and of the other quantities in question are 
all given in terms of q. 
Sohnke’s 
properth 
the unk 
The case chiefly considered is that of n an odd prime; and unless the contrary 
is stated it is assumed that this is so. We have then n +1 transformations corre 
sponding to the same number n +1 of values of v; these may be distinguished as 
v 0 , v 1} v 2 ,...,v n ] viz. writing a to denote an imaginary ?ith root of unity, we have 
simplifie 
&c., whi 
minatior 
(althoug 
U 3 ~\ n 111 111 
v 0 = (—) 8 V2q H f(q 11 ), v x = V2 (aq n ff(aq n ), v 2 = V2 (a 2 q n )Y(a. 2 q n ), &c., 
high nu 
of the 
i i 
v n = V2 q 8n f(q n ). 
reprodm 
n 3 -1 
(Observe (—) 8 = + for n = 8p ± 1, — for n — 8p± 3.)