338
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS.
[692
viz. this is
(u s + y 8 ) 2 + (- 40 2 + 60 4 - 40«) 2 O 8 + v 8 ) 4- 160 4 - 480« + 680 8 - 480 10 + 160 12
= (u 8 + y 8 — 40 2 4- 60 4 — 40«) 2
= {(u 4 -v 4 ) 2 - 40 2 (l-0 2 ) 2 } 2 ,
that is,
{u 4 - if 4 - 20 (1 - 0 2 )} 2 {w 4 - v 4 + 20 (1 - 0 2 )} 2 ;
or the modular equation is
[u 4 - v 4 - 20 (1 - 0 2 )} 2 [u 4 - v 4 + 20 (1 - 0 2 )} 2 (u 8 + y 8 - ®) = 0;
viz. the first and second factors belong to the cubic transformation; and we have
for the proper modular equation in the septic transformation ii 8 4-y 8 — © = 0, or what
is the same thing (1 — u 8 ) (1 — v 8 ) — (1 — 0) 8 = 0, that is, (1 — u 8 ) (1 — v 8 ) — (1 — uv) 8 = 0,
the known result; or, as it may also be written,
(0 - u 8 ) (0 - v 8 ) + 70 2 (1 - 0) 2 (1 - 0 4- 0 2 ) 2 = 0.
The value of M is given by the foregoing relations
1 2
¥ 2 : M : 1 = ^ 12 + ^ + vv * : — ( u12 + pu 4 4- qv 4 4- v 1 -) : pu* 4- cry 4 4- tv 12 ;
but these can be, by virtue of the proper modular equation u 8 4- y 8 — ® = 0, reduced
into the form
¿5 : \ : 1 =7(0-0 : 14 (0 - 26'- + № - P) : -6+v>,
viz. the equality of these two sets of ratios depends upon the following identities,
(— 0 4- v 8 ) (u 12 +pu 4 + qv 4 4- y 12 ) + 14 (0 — 20 2 4- 20« — 0 4 ) (pu 4 + crv 4 4- tv 12 )
= {- du 4 4- (1 - 0) (- 4 - 0 4- 50 2 - 0 :i - 40 4 ) y 4 + y 12 } (u 8 -0 4- v s ),
— 7 (6 — u 8 ) (pu 4 + (tv 4 4- rv 12 ) — (0 — y 8 ) (\u 12 4- ¡m 4 + vv 4 )
= {(20 + 50 2 4- 30 s - 20 4 - 20 s ) u 4 4- (2 4- 20 - 30 2 - 50« - 20 4 ) y 4 } (u 8 -0 4- v 8 ),
— 2 (0 — 20 2 4- 20 3 — 0 4 ) (\u 12 4- pM 4 4- vv 4 ) 4- (u 8 — 0) (a 12 4- pu 4 4- qv 4 4- y 12 )
= [u 12 4- 0 (1 — 0) (3 4- 50 4- 30 2 ) u 4 — 0y 4 } (u 8 -0 4- v s ),
which can be verified without difficulty: from the last-mentioned system of values,
replacing 0 by its value uv, we then have
1 2
M 2 ' M ' 1 = ( v ~ u7 ) : 14wy (1 — uv) (1 — uv 4- u 2 v 2 ) : —v(u — v 7 ),
which agree with the values given p. 482 of the “ Memoir ”; and the analytical theory
is thus completed.
