The collected mathematical papers of Arthur Cayley.

578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 149

which may also be written
(v - z7) (u - v7) + 7uv (1 - uv)2 (1 - uv + U2V2)2 = 0,
as can be at once verified; but it also follows from Cauchy's identity
(x + y)7 - y- 7 = xy (x + y) (2 + sy + y2)2.
We then have
M2 = 1 (1 - v8) vF'
n (1 - u8)uF F
Moreover
F'u = - u8s (1 - v8) + v (1 - U)7
1 -, u - (1 - u)8 + ZV (1 - UV)7
(1 -uv)7
I - u u(v- u);
and similarly
F'v = (1 - mv)7,
F 1-vv8 V (8 - v7),
whence
1 -7z6 (v - )
IM2 v u-v7
Writing this under the form
1 - 7uv (v - u7) (u - v7) 49u2 (1 - uv)2 (1 -U v+ u22)2
M2 v2 (t - V7)2 ' (u - V7)2
I find, as will appear, that the root must be taken with the sign -, and that we
1 7u (1 - mv) (1 - UV + 2v2) v (1- ) (1 v + U2v2)
thus have 7= - -7, whence also M= v —
M it - v V1 - u7
44. Recurring now to the fundamental equations for the septic transformation, the
coefficients are a, /3, y, 8, and we have
1 /u7
a = 1 (- ),
2,= -1, 8 -
so that the coefficients are all given in terms of v, M. The unused equations are
u6 (2ca + 2a/3 + /32) = v2 (72 + 2y8 + 2/8),
U-2 (72 + 2/8y + 2a8 + 2/3) = v2 (2az + 2/3y + 2a8 + /2),
which, substituting therein for a, /, y, 8 the foregoing values, give two equations;
from these, eliminating M, we should obtain the modular equation, and then M in
terms of u, v.