412] 
A MEMOIR ON CUBIC SURFACES. 
385 
49. Putting the equation of the surface in the form 
W(l, 1, 1, l+), m + ~, n + -\X, Y, Z) 2 +°^ S XYZ = 0, 
l m n P 
where for shortness 
a = mn — l, 
/3 — nl — m, 
7 = Im — n, 
8 = Imn — 1, 
p = Imn, 
then taking X = 0 as the equation of the plane [12], Y= 0 as that of the plane 
[34], Z= 0 as that of the plane [56], the equations of the 30 distinct planes are 
found to be 
X =0, 
[12] 
T = 0, 
[34] 
Z =0, 
[56] 
+ 
N 
+ 
II 
© 
[23] 
m~ l X + l Y + Z = 0, 
[24] 
m X + l _1 Y + Z — 0, 
[13] 
w -1 X + l _1 Y + Z = 0, 
[14] 
X + n Y+m Z = 0, 
[45] 
X + ?i _1 F+ m Z = 0, 
[46] 
X + n Y + m _1 Z = 0, 
[35] 
X + 7i _1 F + to -1 Z = 0, 
[36] 
n X + Y+l Z = 0, 
[16] 
n -iX+Y+l Z = 0, 
[15] 
n X + Y+l- 1 Z = 0, 
[26] 
n~ l X + Y +l~ x Z = 0, 
[25] 
TP=0, 
[12.34.56] 
X + ¡3y W = 0, 
[12.36.45] 
X-a8 W=0, 
[12.35.46] 
Y + ay W = 0, 
[16.25.34] 
Y-/38 W = 0, 
[15.26.34] 
Z + a/3 W = 0, 
[14.23.56] 
Z-y8 W = 0, 
[13.24.56] 
mnX + nl 
Y + Im Z + a/3y8 W = 0, 
[16.23.45] 
pX + n 
Y + m Z + /3y8 W= 0, 
[13.26.45] 
nX + p 
Y+l Z + ya.8 lf = 0, 
[16.24.35] 
mX -f-1 
Y + p Z + 0.(38 W = 0, 
[15.23.46] 
X + Im Y + In Z — ¡3y8 W = 0, 
[15.24.36] 
ImX + 
F + mn Z — ya8 W = 0, 
[13.25.46] 
nlX + mn 
y+ Z — a/38 If = 0, 
[14.26.35] 
IX -j- m 
Y+n Z-a/3y W=0, 
[14.25.36]