334 
ADDITION TO THE MEMOIR ON THE 
[692 
1 1 /1 u i\ 
M=l+2 /3, that is, = 1, then the last equation gives 2<y = u 3 v 3 f -jj- — — J ; and 
a, /3, y, 8 having these values, we have the residual two equations 
m 6 (2a7 + 2a/3 +/3 2 ) = v 2 (y 2 + 278+ 2/38), 
7 2 + 2/3<y + 2a 8 + /38 = v 2 u 2 (2ay + 2/3y + 2a 8 + /3 2 ), 
viz. each of these is a quadric equation in JL; hence eliminating we have the 
modular equation; and also (linearly) the value of and thence the values of 
a, ¡3, 7, 8 in terms of u, v. 
Before going further it is proper to remark that, writing as above a = 1, then 
if 8 = /3y, we have 
1 — (3x + yx 2 — 8x 3 = (1 — ¡3x) (1 + yx 2 ), 
1 + /3x + yx 2 + 8a? = (1 + /3x) (1 4- yx 2 ), 
and the equation of transformation becomes 
1— y _1 — x /1 — (3x\ 2 
1 + y 1 + x \1 + ¡3x) 
viz. this belongs to the cubic transformation. The value of /3 in the cubic transforma- 
u s 
tion was taken to be /3 = ~, but for the present purpose it is necessary to pay 
u s 
attention to an omitted double sign, and write ¡3 = ± —; this being so, 8 = ¡3y, and 
• • ... 
giving to 7 the value + u 4 , 8 will have its foregoing value = —. And from the 
theory of the cubic equation, according as /3 — ^ or = — —, the modular equation 
must be 
u 4 — v 4, + 2uv (1 — u 2 v 2 ) = 0, or u 4, — ?j 4 — 2uv (1 — u 2 v 2 ) = 0. 
We thus see a priori, and it is easy to verify that the equations of the septic 
transformation are satisfied by the values 
a — 1, ¡3 = — , y— u 4 , 8 = —, and w 4 — v* + 2uv (1 — u 2 v 2 ) = 0; 
V V 
« = 1, /3 = — —, 7 = — u 4 , 8 = —, and u 4 — v i — 2uv (1 — u 2 v 2 ) = 0; 
V V 
and it hence follows that in obtaining the modular equation for the septic transform 
ation, we shall meet with the factors u 4, — v 4 + 2uv (1 — u 2 v 2 ). Writing for shortness 
uv = 9, these factors are u 4 — v 4 ± 26 (1 — 6 2 ); the factor for the proper modular equation 
is u 8 + V s — ©, where