TRANSFORMATION OF ELLIPTIC FUNCTIONS 
421 
L | — «V(1 — uvf(\ + uv) j- +fj | u ' v '( 1 — >l? ) +'f ( 1 + uV ) ( u *~ O j- 
+ {^+6“ 7 (l -uV)-uvj =0, 
the first of which is given p. 432 of the £ Memoir.’ Calling them 
we have 
(a, b, c]& 1 ) =0, (a', b', c 1 ) =0, 
I 2 
, 77 : — : 1 =bc' — b e : ca" -—ha : ab — a b, 
M 3 M 5 
and the result of the elimination therefore is 
(ca / — ha) 3 — 4 (be' — b'c) (ab' — a'b) = 0 . 
Write as before uv— 0. In forming the expressions ca' — ha, &c., to avoid fractions 
we must in the first instance introduce the factor v~, thus 
v 2 (ca' — c'a) = v { v ( 1 — u 8 ) — 4( 1 — 6) (v -j- u 7 ) } { ~ 0 2 ( 1 -f- 6) ( 1 — 0 ) 8 } 
-{u u -\-6u 8 0(l-d 2 )-v 2 0 2 }{l -v 8 }, 
= -0 2 (l + 0)(l-0) 8 {v 2 (~3 + i6)+u 6 (-4:d+3 6 2 )} 
- {u u +6u G (0-6 8 )-v 2 0 2 }{ 1-v 8 ) ; 
but instead of 0 2 v 2 writing u 2 v^, the expression on the right hand side becomes 
divisible by u 2 ; and we find 
^(ca' - ca) = - ( 1 + 0) ( 1 - 0) : 8 { v\ - 3 + 4 0) + u\ - 4 6> :j + 3 0 ±) } 
- {u 12 +6u 4 (0-0 8 ) -v*}{l -v 8 ), 
and thence 
2 
- - 3 (ca' -ca) = # + u\ 6 0 - 1 0 0 8 + 1 1 0 A - 6 0 h — 8 -f 1 0 0 7 - 4 0 8 ) 
+#( — 4 + 1O0-80 3 - 60*4- 110 1 - 100 s + 60 ^ + v 12 , 
and similarly we have 
(be' — b'c) = u 12 (5 — 5 0 + 4 6 2 — 5 0 :3 + 2 0±) + u± (9 0— 160 2 -\-0 i -]-lOd iJ f0 :, — lG 0'' + 9 0 7 ) 
+ v\2 — 50+4:0-~-50i-\-50 i ), 
2 
~ 2 (ab' — a'b) = n%6+ 0>-6 v ) +#(2 — 50+4 0 2 + 3 0 3 - 1 0 0 l + 3 0 5 + 40 (l - 5 0 7 + 
Lb ' 
-f-r 1 -! — 1 + 0+0 3 ) ; 
