[654 
654] 
ON CERTAIN OCTIC SURFACES. 
87 
we find 
TT / A* . 2 , t>\ A 2 (A - 1 y . , 
n + gy*) = Ay\ 
Il (kef) — l) 2 = "'^-¿2^ (Aæ 2 — ay 2 + hw 2 ) ~ , 
a? 
and thence 
n (fx 2 ef 2 + gy-) (kef) - 1 ) 2 = - ^ A (A# 2 - ay 2 + Aw 2 ) 2 ^, 
and consequently 
© = h {k — l) 2 A (bx 2 — ay 2 + Aw 2 ) 2 . S \ . , — . 
y ' CW + ^Xty-i) 1 
Hence, writing 
4> 2 
4 5 
+ — i + 
a 
D 
(fiW + 9tf) № - 1) ! (H - 1)’ T k-f, - 1 ' r a* V(/) + ¿2/ VO) + ¿0 V(7) ty VO) ’ 
we may calculate separately the terms 
and 
The first of these is 
V i A ^ B 
)(kef> — l) 2 A kef) — lj ’ 
( 
G 
+ • 
D 
{X(f> V(/) + iy f(g) x<f) V(/) - iy f(g)j ' 
1 
(A — l) 2 (fx 2 + k?gy) 2 (bx 2 — ay 1 + Aw 2 ) 2 y> ’ 
if for shortness 
y, w] 6 = (fa? + A% 2 ) [4 {(2 - k) bx 2 - ay 2 + A (1 - A) w 2 } 2 
— 2 (A« 2 — ow/ 2 + Aw 2 ) {(6 — 6k + A 2 ) A« 2 — ay 2 + A (1 — A) 2 w 2 }] 
the second is 
if for shortness 
+ 4A 2 (A — 1) gy 2 (bx 2 — ay 2 + Aw 2 ) {(2 — A) A« 2 — ay 2 + A (1 — A) w 2 } 
2 
(A — l) 2 (yir 2 + k' 2 gy 2 ) 2 A 
(«» y, w )\ 
(x, y, w) 6 = {(fx 2 - k 2 gy 2 ) (cfx 2 - egy 2 —fgvfi) - ^ckfgxf 2 } 
x {fgbk 2 x 2 + [2cA (A — l) 2 + a/] gry 2 + fgh (A — l) 2 w 2 } 
+ 2 {A 2 A$r — cA (A — l) 2 } fgx % y 2 {c (A + 1) (/¿c 2 — A$q/ 2 ) — kfgw 2 } ; 
and hence 
which must be 
In partial 
0 = (fa? + Bgy 2 ) 2 & y> + 2 ( bx * ~ ay2 + ^ w2 ) 2 (®» V’ W ) 6 1> 
a rational and integral function of (x, y, w). 
verification of this, observe that, because U contains the terms 
2b 2 cf(ch — of) xPz 2 + 2Glxhfz 2 w 2 ,