OF ELLIPTIC FUNCTIONS. 
431 
We then have 
Moreover 
1 ( 1 — v^vF'v 
n (1— u s )uF'u 
liE'u— -u s (l—v*)-\-uv{\—uv ) 7 
— I _'ls (l — uv) e -f- uv{ 1 — UV y 
(1— uv) 7 . . 
= \ r- u(v — u 7 )- 
1— ir v ' 
and similarly 
, (1— uv) 7 . 
Y'v—> l _ v& v(u—v 7 ), 
whence 
1 — 7w ( V — u 7 ) 
M 2 ~ v u—v 7 ’ 
Writing this under the form 
1 — *Juv 
(v— u 7 )(u— v 7 ) 49w 2 (l- 
1 
IS 
(u— v 7 )* ’ — 
I find, as will appear, that the root must be taken with the sign — , and that we thus 
have 1 = _ 7.<(l-^)(l-« . + »V) whence ako «( l-y,)(l- w , + ,V) - 
M « — v 7 v —u 7 
44. Recurring now to the fundamental equations for the septic transformation, the 
coefficients are a, 3, y, o, and we have 
:=1 
2 ^=m- 1 
2y=»v(i-^) ; 
so that the coefficients are all given in terms of v, M. The unused equations are 
M 6 (2ay+2«3+3 2 ) =v 2 (y 2 +2yl+2&), 
u~\y 2 + 23y + 2«H 2fi) =v 2 (2 ay + 23y + 2aS+3 2 ), 
which, substituting therein for a, 3, y, c> the foregoing values, give two equations ; from 
these, eliminating M, we should obtain the modular equation, and then M in terms 
of U , V. 
Substituting in the first instance for cc, h their values, the equations are 
« 5 ( 2 p+ 2 r +| 3*)=*, ! {/+2 £@+y)j 
f+2Hy + (2+2H)~=uv\2y+2H 7 +2~+^. 
The first of these is 
4 ( 1 — «)( 2 £+ 2 y )+ 4^— 4 ^ /= 0 , 
viz. this is 
4(1— »)(m~1 + T— v) + (h— ?)=° i 
3 l 2 
