420 PROFESSOR A. CAYLEY ON THE 
Before going further it is proper to remark that, writing as above a=l, then if 
S=fiy, we have 
1 — fixfi-yx~ — Sx 8 =(l —fix)(l-\-yx°), 
1 -j- fix -f yx l -}- ox 3 = ( 1 -|- fix) ( 1 . -f- yxfi , 
and the equation of transformation becomes 
1 —y 1—x /l— /3lv\ z 
1+2/ l-\-x \ 1 + fix) 
viz., this belongs to the cubic transformation. The value of (3 in the cubic trans- 
or*> 
formation was taken to be fi= but for the present purpose it is necessary to pay 
u s 
attention to an omitted double sign, and write fi = + — ; this being so, 8 =fiy, and 
q$ 
giving to y the value d+d, 8 will have its foregoing value = — . And from the theory 
qjp> q fO 
of the cubic equation, according as fi—~ or = the modular equation must be 
w 4 — v 4 +2m’(l — u z v~) = 0, or u 4 — v 4 — 2wy(l ~vfiv z ) = 0. 
We thus see d priori , and it is easy to verify that the equations of the septic 
transformation are satisfied by the values 
qjro q/J 
a—\ , fi= — , y= u 4 ‘,S=—,a 1 nd u 4 —v 4 '-\-2uv(l—u 2 v 2 ) = 0; 
V V 
3 7 
a= 1 , fi=- 7 ~- ) y= — u 4 \$—— , and u 4 — v 4 — 2uv(L — u 2 v 2 ) = 0 ; 
and it hence follows that in obtaining the modular equation for the septic trans- 
formation, we shall meet with the factors u 4 — — nhd). Writing for shortness 
uv—0, these factors are u 4 — 'y 4 = k20(l — 9~) } the factor for the proper modular equation 
is v s -\-v s — ©, where 
0 = 80— 280M- 560 3 — 7O0 4 -f 560 5 — 280 G + 80 7 
[viz., the equation (1 — w 8 )(l — v 8 ) — (l — m>) 8 =0 is w 8 -j- / y 8 — • © = 0], and the modular 
equation as obtained by the elimination from the two quadric equations in fact 
presents itself in the form 
(u 4 - #+20-20 3 )V— v 4 - 20-f20 3 ) 2 (w 8 +i/ 8 — @) = 0. 
Proceeding to the investigation, we substitute the values 
«=i,Hi-iU=i'44 3=7 
in the residual two equations, which thus become 
