[654 
x values ; 
654] 
ON CERTAIN OCTIC SURFACES. 
91 
ij Pi> Ci- 
it is the 
ve 
/3 2 )} A> 2 
or, denoting for a moment the coefficient of A 2 /* 2 by K, and writing also <yv- = X — a\~ — /3/a 2 , 
v = P — X — /i, this is 
= xa> 2 
-4a (/3 + 7) 2 A/a 2 
-2 (/3 + y) 2 /a 2 (X-aA 2 -/3/a 2 ) 
+ 4/3 (7 + a)- A 2 /a (P — A — /a) 
+ 2a (7 + a) 2 A 4 , 
= — 2 (/3 + y) 2 /a 2 X + 4/3 (7 + a) 2 A 2 /aP 
+ 2a (7 + a) 2 A 4 
— 4/3 (7 + a) 2 A'/a 
+ {— 4/3 (7 + a) 2 + K + 2a (/3 + 7) 2 } A 2 /* 2 
— 4a (/3 4- 7) 2 A/a 3 
+ 2/3 (/3 + y) 2 /a 4 , 
and here the coefficient of A 2 /a 2 is found to be 
= 2 {a/3 (a + /3) + 7 (a — /3) 2 - 37 2 (a 4 /0)}. 
Hence, the terms without X or P are = 2 V, where 
V = a (7 + a) 2 A 4 
- 2/3 (7 + a) 2 A> 
+ {a/3 (a + /3) + 7 (a - /3) 2 - 37 2 (a + /9)} A> 2 
— 2a (/3 + 7) 2 fyi 3 
+ Æ (Æ + y)V*> 
and this is identically 
where observing that 
(a 4- 7) A 2 ^ 
= + 27A /Jb 
+ (/3 + 7) /¿ 2 
h x 
a (7 + a) A 2 
— 2 (/37 + 7a + a/3) A/a 
+ /3 (/3 + 7) /¿ 2 , 
aA 2 + /3/a 2 + 7 (P — A — /a) 2 = X, 
we have the first factor 
(a + 7) A 2 + (/3 + 7) /a 2 + 27A/a = X — 7P 2 4- 27P (A + /a), 
and consequently 
A 2 /a 2 (A — P) = — 2 (/3 4- y) 2 /x 2 X + 4/3 (7 + a) 2 A 2 /aP 
+ 2 {X — 7P 2 + 2yP (A + /a)} {a (7 + a) A 2 — 2 (/3y + 7a + a/3) A/a + /3 (/3 4- 7) /a 2 }; 
viz. in virtue of P = 0, X = 0, we have A = P. And thus 
A = B = C = - A 1 = - P, = - C,: 
so that the only values of il are, say, A and — A. 
12—2