424 
PROFESSOR A. CAYLEY ON ELLIPTIC FUNCTIONS. 
or the modular equation is 
[w 4 —#— 2d(l-d 3 )} 3 {w 4 — v 4 +2d(l — 9 2 )}*(u 8 +v 8 — ®) = 0; 
viz., the first and second factors belong to the cubic transformation ; and we have for 
the proper modular equation in the septic transformation u 8 v 8 — © = 0, or what is 
the same thing (1— u 8 )(l — v 8 ) — (1 — d) s =0, that is (1 — w 8 )(l — v 8 ) — (l — uv) 8 —0, the 
known result; or as it may also be written (d — u 8 )(9 — v 8 )-\-70 2 ( 1 — d) 3 (l — d+d 3 ) 2 — 0. 
The value of M is given by the foregoing relations 
™ ^ : 1 — pu 4 -\-vv 4 : — (u 12 -\-pu 4 -\-qv 4 -\-v 12 ) : pu 4 '-\-crv 4 -\-TV 12 ; 
but these can be, by virtue of the proper modular equation, w 8 -f- , r 8 — 0 = 0, reduced 
into the form 
A : 1 = 7 ((?-«*) : lW-W+W-V) : -0+v\ 
viz., the equality of these two sets of ratios depends upon the following identities, 
( — 9-{-v 8 )(u lz -\-pu 4l ~\-qv 4! -\-v l2 )-\- 14(0 — 2d 3 d-2d 3 — 0 i )(pi^+o-y 4 +ry 12 ) 
= •[ — 0a { '-\- ( 1 — 6) ( — 4 — 0- f- 5 d 3 — 9 ; — 4d 4 ) v 4 -}- r 13 } (it 8 — © -f- v s ) , 
— 7(9— u 8 ) (pu 4 '-\- av 4 + tv 12 ) — ( 9 — v 8 ) (A 13 — b pu 4 - \- kw 4 ) 
= { (2 9 + 5 d 3 + 3d 3 — 2d* — 2 d 5 ) id+ (2 -f 2 9 - 3 d 3 — 5 d 3 — 2 d 4 )?; 4 } (w 8 — © + r 8 ) , 
— 2(d— 2d 3 + 2d 3 — d 4 ) (An 13 + pu 4j rvu 4 ) -j- (h 8 — d) (id 3 fi-pud-|- qy 4 + y 12 ) 
= |td 3 d-d(l — d)(3fi-5dd-3d 3 )id— 9v 4 } { it 8 — @-f-r 8 ), 
which can be verified without difficulty : from the last-mentioned system of values, 
replacing d by its value uv, we then have 
12 
: — : 1 = 7u(v—i0) : liuv(l—uv)(l — uv-\-u z v 2 ) : —v(u—v 7 ), 
which agree with the values given p. 482 of the £ Memoir,’ and the analytical theory 
is thus completed. 
