OF ELLIPTIC FUNCTIONS. 
449 
or say 
£F / y^j-=5{fl 5 + 4yV+ 6uV+4yV+ vif— 2u(l — if)), 
where 
iF'y = 3y 5 + 10uV+10yV —5vif—2u. 
Hence also, reducing by the modular equation, 
^ =5w{ v 4 u + 4 u V + 6 y V +2y(l + if) + if } , 
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. 5M=5{ — w 5 +4Hw 4 — 6vV+4y 7 «i 2 — y 4 ?&— 2y(l— y 8 )} ; 
and comparing with the equation 
we obtain 
5M 2 = - 
(1 — v 8 )vWv 
(l—u 8 )uF'u* 
v ( 1 — v s ) — 2v ( 1 — v s ) — v 4 u + 4i -7 w 2 — Gv*u 3 + 4d 5 m 4 — u b 
m(I — u 8 ) — 2m(1 — u s ) + + 4w 7 y 2 -f 6ifv 3 + 4u b v* + v 5 
Writing for a moment M = «i 4 + 6 ifv 2 + v\ N = if + y 2 , this is 
v(l — v s ) — 2v{\ —v 8 ) — mM + 4w 5 m 2 N 
u (1 — u 8 ) — 2«(1 — u 8 ) + + 4t?*M 5 N ’ 
that is 
— iuv(l — u 8 )( 1 — v 8 ) — { if( 1 — if) — y 2 (l — v 8 ) } M -J- 4yV { ?6 2 (1 — v s ) 4- y 2 (l — if ) } N = 0. 
But we have 
v?(\ — if) — y 2 (l — v 8 ) = (if — y 2 ) { 1 — if — if V 2 — ifv 4 — ifv 6 —v 8 }, 
u 2 (l—v 8 )+v%l — u 8 ) = (if+v‘ 2 ){l — vfv\ u 4 — if y 2 -f • v 4 )}. 
Hence, replacing M, N by their values, this is 
- 4uv(l — if)( 1 — v 8 ) 
- (lf — v 2 )(l — lf— ifv 2 — U 4 v 4 — if 13 6 — V 8 )(lf + Qlfv 2 + V 4 ) 
+ 4 ifv 3 (if -f- y 2 ) 2 { 1 — ifv 2 (if — wV-j-y 4 ) } = 0 ; 
viz. writing if—v 2 = A, uv= B, this is 
_ 4B { 1 - A 4 - 4 A 2 B 2 — 2B 4 + B 8 } 
- A { 1 — A 4 — 5 A 2 B 2 — 3B 4 } (A 2 + 8B 2 ) 
+ 4B 3 (A 2 + 4B 2 ) { 1 - A 2 B 2 - B 4 } = 0, 
that is 
- 4B { (1 - A 4 - 4 A 2 B 2 — 2B 4 + B 8 ) - B 2 (A 2 + 4B 2 )(1 - A 2 B 2 - B 4 ) } 
- A(l-A 4 — 5A 2 B 2 — 3B 4 )(A 2 +8B 2 )=0; 
Hz. 
- 4B(1 - A 4 - 5 A 2 B 2 - 3B 4 )(1 - B 4 ) 
- A(1-A 4 -5A 2 B 2 -3B 4 )(A 2 +8B 2 )=0; 
