TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
423 
that is 
B= — 80 3 +120 4 — 80 6 ; 
and in precisely the same way the fifth equation gives 
D=-80 3 +12^-80 6 . 
We find similarly C from the second equation : writing down first the coefficients of 
qr, 2qO h , — 4 XaO and — 4/xp, the sum of these gives the coefficients of c; and then 
writing underneath these the coefficients of B@ and of — 6 s , the final sum gives the 
coefficients of C : the coefficients of each line belong to (6°, 6 l , . . . 
0 0 36 0 — 120 + 132+ 28 — 316 + 361— 20 — 340 + 396 — 144—112+164 — 80 + 16 
— 8+ 20— 16— 12+ 22— 20 0+ 12 
— 40+140 — 212 + 140+ 80 — 188+168- 92- 64+176-164 + 80- 1 6 
-36 + 64— 40+ 60- 72+ 28 0+ 68-100+ 36 
0 0 0 + 64 — 208 + 352 — 272 — 160+463 — 160 — 272 + 352 — 208+ 64 0 0 0 
0 0 0-64 + 224 — 352 + 224+160 — 392+160 + 224 — 352 + 224- 64 0 0 0 
— 1 
0 0 0 0+ 16 0— 48 0+ 70 0— 48 0+ 16 0 0 0 0 
that is 
C=16l9 + -48<9 6 +7O0 8 -48f9 lo +16<9 13 , 
and in precisely the same way this value of C would be found from the fourth 
equation. There remains to be verified only the fourth equation ( I ) + B) 6 8 — ©C~ cl, 
tllcltj IS 
2 6 s ( - 8 6 2 + 1 2 0 l -S6 Cl )-®G = (2 — 4Xr) 6 l2 -\- (2 pq - 4/xa- - 4 vp) 6\ 
and this can be effected without difficulty. 
The factor of the modular equation thus is 
u™ + v w + ( ■ - 8 0 2 + 1 2 d 1 - 8 0°) (u s + v 8 ) + 1 6 6 l - 4 8 6 t] + 7 0 6 s - 4 8 6 W + 1 6 6 l \ 
viz., this is 
(u s + v s y 3 + ( ■ — 4 6 2 + 6 & l — 46>°)2 (u s + +) + 1 6 6 '• — 4 8 &' + 6 8 6 s - 4 8 6 K) + 1 6 ( 9 + 
= (++ v 8 - 4 6 2 + 6 6 [ - 4 6% 
= 4^(1 _+)2}2 
that is 
{ u 4 ' - # - 2 0 ( 1 - 6 2 ) } ! 3 { u 4 - + + 2 6 ( 1 - 6 2 ) } 2 ; 
3 i 
MDCCCLXXVIIl, 
