168 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS, 
where Sv 0 , &c. are the coefficients of the equation 
v 6 4- 4 v 5 u 5 + ov 4 u 2 — 5 vhi 4 — 4 vu — w 6 = 0, 
Sv 0 , V 0 V lt V ü V{üo_, V 0 V{Ü 2 V 3 , VoV^VsVt 
— 4m 5 , + ou 2 , 0 , — 5m 4 , 4u ; 
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VIZ. 
are 
or the equation is 
F'v. = 20u (1 — u 8 ) 
or, say 
where 
— 10 m 4 (— 4 m 5 — v) 
— 10m 2 ( — 5u 2 v — 4 mV — v 3 ) 
— 10 ( 4 u + 5iih) — 5v 3 u 2 — 4 v*ii 5 — v 5 ), 
\F'v = 5 [v 5 4- 4v 4 u 5 + 6v 3 u 2 + 4vhi 7 + vu 4 — 2u (1 — m 8 )}, 
^F'v =3 v 5 + 10v 4 u 5 4- lOvhi 2 — 5vu 4 — 2u. 
Hence also, reducing by the modular equation, 
\vF'v ~ = 5m [v 4 u + 4v 3 u 4 + 6vhi 3 + 2v (1 4- m 8 ) + u 5 ), 
the one of which forms is as convenient as the other. 
71. Making the change u, v, ^ into v, —u, —5M, we have 
— \F'u. bM = 5 {— u 5 + 4vhi 4 — 6vhi 3 4- 4v 7 u 2 — v 4 u — 2v (1 — v 8 )}; 
and comparing with the equation 
we obtain 
5 Jf._ 
(l-w')wi'V 
v (1 — v 8 ) _ — 2v (1 — v 8 ) — v 4 u + 4v 7 u 2 — 6v 2 u 3 + 4v 5 u 4 — u 5 
u (1 — u 8 ) — 2m (1 — m 8 ) 4- u*v 4- 4mV + 6m V 4- 4mV 4- v 5 ' 
Writing for a moment M = u 4 4- 6?/V + v 4 , N=u 2 + v 2 , this is 
v (1 — v 8 ) — 2v (1 — v 8 ) — uM + 4 v 5 u 2 N 
u (1 — m 8 ) — 2m (1 — m 8 ) 4- vM + 4v 2 u 5 N ’ 
that is, 
— 4uv (1 — m 8 ) (1 — v 8 ) — {u 2 (1 — m 8 ) — v 2 (1 — i> 8 )] M + 4v 3 u 3 [u 2 (1 — v 8 ) 4- v 2 (1 — m 8 )} N = 0. 
But we have 
u 2 (1 — u 8 ) — v 2 (1 — v 8 ) = (m 2 — v 2 ) (1 — m 8 — u 6 v 2 — mV — mV — v 8 }, 
u 2 (1 — v 8 ) + v 2 (1 — m 8 ) = (m 2 4- v 2 ) {1 — u 2 v 2 (u 4 — mV 4- fl 4 )}.