430 
PROFESSOR CAYLEY ON THE TRANSFORMATION 
42. The quintic transformation, n= 5. 
Here there are the three coefficients a, 
coefficients g> , a ; we have 
a=l, 
0, y, or 0, y are the last but one and last 
Comparing the two values of 8, we have - , and then 
r & M v(l—v 3 )u 
-i ctn u{v 4 — U 4 ) U b 
a=l, 20 = -A -g-A, y=_, 
1 0 ( 1 — v d u) v 
so that only the modular equation remains to be determined. 
The unused equation is 
2ay-f-2a0+0 2 =-^ (2ay + 20y + 0 2 ), 
which, putting therein a=l, may be written 
(2y+0 2 )(M 2 — y 2 )=2 0(yw 2 — O ; 
attending to the value of 0, this divides byM 2 — V; in fact the equation maybe written 
2y+0 2 = 
w(v 2 + m 2 ) 
®(I— « 3 m) 
(yy 2 — U 2 ) ; 
and then completing the substitution, and integralizing, this becomes 
{ 8yzt 3 (l — y 3 w) 2 + (y 4 — u 4 ) 2 } = 4 uv(u 2 -f- y 2 )(l — u 3 v)( 1 — uv 3 ), 
\iq. this is 
4(1— v 3 u)uv{2ii 2 (l— v 3 u)— (w 2 -|-y 2 )(l — vu 3 )} -j- (y 4 — m 4 ) 2 — 0 ; 
and the term in { } being = — (v 2 —u 2 )( 1 -\-vu 3 ), the whole again divides by v 2 — u 2 , and 
the equation thus becomes 
(v 2 +w 2 )(y 4 — u 4 ) — 4 lUv(1— v 3 u)(1 -\-vu 3 )= 0, 
which is the modular equation. 
43. The septic transformation, n— 7. 
I do not propose to complete the solution directly from the fundamental equations 
for a, 0, y, &, but resort to the known modular equation, and to an expression of M 
which I obtain by means thereof. 
The modular equation is 
(1 - 0(1 -v s ) - (1 ■ — uv) 3 - 0, 
which may also be written 
(y — u 7 )(u — v 7 ) + 7 uv( 1 — uvf{ 1 —m-\- mV) 2 , 
as can be at once verified ; but it also follows from Cauchy’s identity, 
{oc+y) 7 -x 7 -y 7 =lxy(x-\-y)(x 3 -\-xy+tff. 
