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TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
337 
the third and fourth equations should then be verified identically. Writing down the 
coefficients of the different powers of 6, we find 
2p = 0 + 12 0- 20 + 22-12- 16 + 20-8 {6°, ..,6 s ) 
- 4\p = 0 - 20 + 20 - 36 + 60 - 44 + 36 - 28 + 8 
6 = 0- 8 + 20 - 56 + 82 - 56 + 20 - 8 0 
<H> = 0 + 8- 28 + 56- 70 + 56- 28 + 8 0 
that is, 
.'. B = 0 0- 8 0 + 12 0- 8 0 0 
B = - 86 2 + 120 4 - 86“; 
and in precisely the same way the fifth equation gives 
jD = - 86 2 + 120 4 -80 6 . 
We find similarly G from the second equation: writing down first the coefficients of 
p\ 2q6 i , — 4A<70 4 , and —4pp, the sum of these gives the coefficients of c; and then 
writing underneath these the coefficients of and of — 6 8 , the final sum gives the 
coefficients of G : the coefficients of each line belong to (6°, 6 1 ,.., 6 16 ). 
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- 16 
- 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 
000 0+ 16 0- 48 0+ 70 0- 48 0+ 16 0 0 0 0, 
that is, 
G = 160 4 - 48 d 6 + 700 s - 48<9 10 + 160 12 ; 
and in precisely the same way this value of G would be found from the fourth 
equation. There remains to be verified only the fourth equation (D + B) 6 s — ©(7 = d, 
that is, 
26 s (— 86 2 + 120 4 — 86 6 ) — ®G = (2 — 4\t) 0 12 + (2pq — 4¡xa — 4vp) 6 4 , 
and this can be effected without difficulty. 
The factor of the modular equation thus is 
u 16 + v 16 + (- 86> 2 + 12<9 4 - 86 e ) (u 8 + v s ) + 16<9 4 - 480 6 + 700 s - 48d 10 + 160 12 , 
C. X. 
43