80 
ON CERTAIN OCTIC SURFACES. 
[654 
If in this equation we write 
aY = a, df 2 =/, 
6Y=6, b'g' 2 =g, 
c' 2 h' = c, 6b! 2 = h ; 
and therefore 
bc'-b'c 
ca' — c'a 
bd'-b'd= J-:, 
bd' — b'd = 
and consequently 
ab' — a'b 
(&/)* + (bgf + (ch)* = 0 ; 
then the equation becomes 
d 2 y 4 z! + b 2 z 4 ofi + c 2 x 4 y 4 
+ f 2 x 4 w 4 + g 2 y 4 w 4 + h 2 zhv 4 
4- 2bcx 4 y 2 z‘ 2 — 2cfx 4 y 2 w 2 + 2 bfa6z 2 w 2 
+ 2cay 4 z 2 x 2 — 2agy 4 z 2 w 2 + 2cgy i xhv 2 
+ 2abz 4 x 2 y 2 — 2bhz 4 xhu 2 1- 2ahz 4 y 2 w 2 
— 2ghw i y 2 z 2 — 2hfyj i z 2 x 2 — 2fgw 4 xhf 
+ 2 {(bg) H — (ch)*} {(chy — (a/)»} {(a/)- 1 — (ch)*} x 2 y 2 z 2 w 2 = 0. 
This same equation, without the relation 
{off + (bgf +(chf = 0, 
and with an arbitrary coefficient for x 2 y 2 z 2 w 2 ; or say, the equation 
a?y 4 z 4 + b 2 z 4 x 4 + dxty 4 
+ f 2 o6w 4 + g 2 y*w* + h 2 z*w* 
+ 2bcx i y 2 z 2 — 2cfx 4 y 2 w 2 + 2bfx 4 z 2 w 2 
+ 2 cay i z 2 x 2 — 2agy 4 z 2 w 2 + 2cgy 4 xhu 2 
+ 2abz 4 x 2 y 2 — 2bhz i x 2 w 2 + 2ahz 4 y 2 vu 2 
— 2ghw 4 y 2 z 2 — 2hfw 4 z 2 x 2 — 2fgw i x 2 y 2 
+ 2kx 2 y 2 z 2 w 2 = 0, 
where a, b, c, f g, h, k are arbitrary coefficients, is the general equation of an octic 
surface having the four nodal curves 
x = 0, . hz 2 w 2 — giu 2 y 2 + ay 2 z 2 — 0, 
y — 0, — hz 2 w 2 . + fw 2 x 2 + bz 2 x 2 = 0, 
z = 0, gy 2 w 2 — fw 2 x 2 . + cx?y 2 — 0, 
w = 0, — ay 2 z 2 — bz 2 x 2 — copy 2 . = 0.