380 ON dr. wiener’s model of a cubic surface WITH 27 REAL LINES ; [521 
or say 
that is, 
viz. that is, 
1 — © 1 
F{0-b) + G*J® = ^-~--^ 
= -{0-b) + 
(0 — b) (abc F + a + c — 0)+G abc V© = 0, 
(e-b)(P-0) + (m-b)~(P-m) =0 
nM 
or, rationalising and throwing out the factor 0 — b, this is 
(e-W~py- (m -V) (m - =0, 
which is a cubic equation satisfied by 0 = m and 0 = m l \ so that throwing out the 
factors 0 — to, 0 — m 1 we have for 0 a linear equation. 
Putting for shortness 
A = (m — a) 2 — (a — b) (a — c), 
B = (to — b) 2 — (b — c) (b — a), 
C — (to — c ) 2 — (c — a) (c — 6), 
the value of 0 may be expressed in the forms 
a B 2 , \ a 7 ^ 2 / i\ a 4 (to — a) (to — b) (to -c)(b- c) (a - c) 
0-ct = ^ 2 (c-a), 0-b = Qi(c-b), 0-c= — -- v g2 - — 
We have moreover 
-r, 2 (a —c) (m — b)(m — c) „ (to — c) A 
P-c = G , P -vo = - ±, 
equations which express P in terms of to only; also 
a n —2 (a — c) (to — b)(rn — c) B 
0~ p= c* ■ 
and then 
whence 
V© = — Vil/ 
d-b P-0 
m — bP— to’ 
AT) 
V© = 2 ViF (b - c) (c - a) ~, 
so that 0, V© are now determined.