jnctions. [578 
125 
578] a MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
sin a= 0, 8 = 0; the 
21. As regards the non-existence of the factor il-1, I further verify this by 
writing in the equations 0 = 1; they thus become 
k s oi- = 8\ 
k (2ay + 2 a/3 + /3 2 ) = 7 2 + 2y8 + 2 /38, 
y 2 + 2/3y + 2a8 + 2/38 = k (2ay + 2/3y + 2aS + /3 2 ), 
8 2 + 2y8 = k 3 (a 3 + 2 a/3), 
which it is to be shown cannot be satisfied in general, but only for certain values of k. 
Reducing the last equation, this is y8 = tea/3, which, combined with the first, gives 
ay = /38\ and if for convenience we assume a = l, and write also 6 = + \/k (that is, k = 6~), 
then the values of a, [3, y, 8 are a = 1, /3 = yd~ 3 , 7 = 7, 8 = 6 3 ; which values, substituted 
in the second and the third equations, give two equations in y, 6\ and from these, 
eliminating y, we obtain an equation for the determination of 6, that is, of k. In fact, 
the second equation gives 
& 2 (2y + 2yd 3 + y-6~^) = 7 2 + 2yd 3 + 2y ; 
= ka, 8 = kß ; in fact, 
or, dividing by 7 and reducing, 
that is, 
7 (1- 0 4 ) = 2d 3 (6 2 -1) (d- -0 + 1), 
7 (1 + 0 3 ) = — 20 3 (0 3 -0 + 1), 
or, as this may also be written, 
(7 + 0 3 ) (1 + d-) = - d 3 (d — l) 2 , 
that is, 
Moreover the third equation gives 
dentically; and these 
the factor {(il, l) 4 }. 
have P = a (1 + kx 2 ), 
alue 
y- + 27 2 0“ 3 + 2d 3 +2y = 0 2 (2 7 + 2y"d~ 3 + 20 3 + y 3 d 6 ), 
7 2 ( 0 4 — 20 3 + 20 — 1) — 2 (7 + d 3 ) d 4 (d 2 - 1) = 0 ; 
whence also 
-2d 7 
Also 
? 2= >~+r 
40 6 (0 2 — 0 + l) 2 = 7 2 (d 2 + !) 2 , 
) or 8 = 0 reduce the 
shows that if either 
we have then the 
mation. 
wherefore 
2(0»- 0 + 1) 2 = -0(0 2 + l) or 2 (d 2 - d +1) 2 + d(d' 2 +1) -0, 
d (d 2 + l) 2 + 2 (0 2 — 0 + l) 2 = 0, 
2d 4 — 3d 3 + 6d 2 -3d+ 2 = 0, 
(2d 2 — d +1) (d 2 — d + 2) = 0. 
or 
that is, 
or finally 
1 = 0 or ß = 0, 7 = 0, 
LCtOr n — 1.