82 ON CERTAIN OCTIC SURFACES. [654 
If for a moment we write af= a 3 , bg = {3 3 , ch — y 3 , and, therefore, a+ /3 + 7 = 0; then 
for the twofold root, we have A : g : v = a : /3 : 7, and consequently 
k = a} (7 — /3) + /3 2 (a - 7) + 7 2 (¿0 — a) 
= (a-/3)(/3- 7) (7 - a), 
that is, 
* = {(«/)* - %) 4 } K%) A “ №"} {(ch) 3 - (a/)*}, 
which agrees with the result in regard to the octic torse. 
If in the octic equation we write (x, y, z, w) in place of (x\ y 2 , z 2 , w 2 ), then we 
have the quartic equation 
a 2 y 2 z 2 + b 2 z 2 x 2 + c 2 x 2 y 2 
+ f 2 x 2 w 2 + g 2 y 2 w 2 + h 2 z 2 w 2 
+ 2 bcx 2 yz — 2cfx 2 yw + 2 bfx 2 zw 
+ 2cay 2 zx — 2agy 2 zw + 2 cgy 2 xw 
+ 2 abz 2 xy — 2bhz 2 xw + 2 ahz 2 yiu 
— 2 gliw 2 yz — 2hfiu 2 zx — 2fgw 2 xy 
+ 2 kxyzw = 0, 
which is the equation of a quartic surface touched by the planes x = 0, y= 0, s=0, 
w = 0, in the four conics 
x —0, . hzw — gwy + ayz = 0, 
y = 0, — hzw . + fiux + bzx = 0, 
z =0, gyw — fwx . + cxy = 0, 
w = 0, — ayz — bzx — cxy . = 0, 
respectively. 
II. The octic surface 
U = b 2 c 2 f 2 af + c 2 a 2 g 2 y 8 + a 2 b 2 h 2 z s + f 2 g 2 h 2 w s 
— 2d 2 eg (bg — ch) y 8 z 2 — 2b 2 ah (ch — af) z 6 x 2 — 2c 2 bf (af — bg) x (i y 2 
+ 2a 2 bh ( „ ) y 2 z 6 + 2b 2 cf ( „ ) 2V 5 + 2c 2 ag ( „ ) ¿r 2 y fl 
— 2/ 2 6c ( „ ) x 6 w 2 — 2g 2 ca ( „ ) yhu 2 — 2h 2 ab ( „ ) 2 6 w 2 
+ 2f 2 gh( „ )x 2 w 6 +2g 2 hf( „ ) y 2 w 6 + 2h 2 fg ( „ ) 
+ / 2 (6 2 ^ 2 + c 2 /i 2 — 4>bgch) w i x i + y 2 (c 2 ^ 2 + ay 2 — 4<chaf) w 4 y 4 + h 2 (a 2 f 2 + 6 2 // 2 — 4abfg) w i z 4 
+ a 2 ( 
) y*z* + b 2 ( 
) z*x 4 + c 2 ( 
)x 4 y 4 
— 2gh (begh — a 2 f 2 — Zafbg — 
2 afeh) iv 4 y 2 z 2 
- %bh ( 
) z 4 x 2 w 2 
+ 2cg ( 
) y 4 x 2 w 2