90 ON CERTAIN OCTIC SURFACES. [654 
The value of fl is unsymmetrical in its form, and there are apparently six values; 
viz. writing 
A =(0 + 7 )» ja(-3-^)-|i 7 } + (,3- 7 )(S + /3 7 + n 
B = ( 7 + a)*{/3(-3-^)-^a} +(y-*)(S + y «+n 
G = <« + /3)4 7 f- 3 - + (« -/3)(S+ «/3 + 7 a )> 
A = 03 + 7) a { « (- 3 - - (/9- 7 ) (S + /3 7 + «»), 
A = (7 + «) 5 {/3 (-3-^)-^ 7 l-( 7 -a)(S + 7 a+/3 i ), 
C, = (a + fiy 17 (- 3 - ^ a | - (a - /3) (S + a/3 + t>), 
where for shortness S= 2(/3y + ya+ aft), the six values would be A, B, G, A ly B 1} G^. 
But we have really 
A=B = C = -A 1 = -B 1 = -C 1 ; 
so that il has really only two values, equal and of opposite signs, or, what is the 
same thing, fl 2 has a unique value. In fact, writing for shortness 
X + yu, + v = P, aX 2 + ft/A + <yv 2 = X, 
we find at once the identity 
X 2 (A + A,) = (ft + 7 ) 2 (- 2X - 4XaP), 
so that A = — A 1} in value of P = 0, X = 0. And similarly B = — B ly G = —G l . 
But the demonstration of the equation A =B is more complicated. We have 
.1 -B = -3« (£ + 7 ) ! -4a03 + 7 )3£-2 7 (/3 + 7 ) 2 £ + os - 7 )(S + £ 7 + tf) 
+ 3/3 ( 7 + af + 4(3 ( 7 + a) 2 - + 2a (7 + of \ — (y — a) (S + 7 a + /3-), 
fM /¿" 
that is, 
^V 2 (A — B) = {— 3a (ft + y) 2 + 3/3 (7 + a) 2 + (ft — 7) (S + fty + a 2 ) — (7 — a) (S + 7a + ,3 2 )} X 2 /u/ 2 
— 4a (ft + y) 2 \/jl 3 
— 27 (/3 + y) 2 v 2 ft 2 
+ 4/3 (7 + a) 2 v\ 2 /u, 
+ 2 a (iy + a) 2 X 4 ,