692] 
TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
335 
viz. the equation (1 — u 8 ) (1 — v 8 ) — (1 — uv) 8 = 0 is u 8 + v 8 — © = 0; and the modular 
equation, as obtained by the elimination from the two quadric equations, presents 
itself in the form 
(m 4 — v 4 4- 20 — 20 3 ) 2 (u 4 — v l — 20 + 20 8 ) 2 (u 8 + v 8 — ©) = 0. 
Proceeding to the investigation, we substitute the values 
in the residual two equations, which thus become 
the first of which is given p. 432 of the “Memoir,” [Coll. Math. Papers, vol. ix., p. 150]. 
Calling them 
we have 
2j^- 2 : : 1 = be' — b'c : ca' — c a : ab' - a'b, 
and the result of the elimination therefore is 
(ca' — c'a) 2 — 4 (be' — b'c) (ab' — a'b) = 0. 
Write as before uv — 0. In forming the expressions ca' — c'a, &c., to avoid fractions 
we must in the first instance introduce the factor v 2 : thus 
v 2 (ca' — c'a) = v [v (1 — u 8 ) — 4 (1 — 0) (v + u 7 )) {— 0 2 (1 + 0) (1 — 0) 3 } 
- {m 14 + 6u R 0 (1 - 6 2 ) - vW) {1 - v 8 }, 
= -&>(! +0)(1- ey [v 2 (- 3 + 46>) + u 8 (- 4(9 + 30 2 )} 
- {í¿ 14 + 6m 6 {6 - 0 s ) - v 2 0 2 } (1 - v 8 ); 
but instead of 0 2 v 2 writing wV, the expression on the right-hand side becomes divisible 
by u 2 ; and we find 
- 2 (ca' - c'a) = - (1 + 0) (1 - 0f {ü 4 (- 3 + 40) + u* (- 4<9 3 + 3<9 4 )} 
IXj 
- {m 12 -l- 6m 4 (0 - 0 s ) - V 4 } (1 - V 8 ),