412] 
A MEMOIR ON CUBIC SURFACES. 
431 
Section XII = 12 — U e . 
Article Nos. 150 to 156. Equation W(X + Y + Zf + XYZ = 0. 
150. The diagram of the lines and planes is 
Lines. 
00 to 1-^ 
05 tO H- 1 
XII = 12 - U 6 . 
Oil w 
CO 
X 
X 
h-» 
CO 
II 
II 
to I 
-a 1 CO 
0 
. 
1x32 = 32 
. . . 
Uniplane. 
1 
• 
• ■ 
W3 2 
Q) 
. 
~ ■ 
Planes each touching 
a 
c3 
3 x 4=12 
along a ray, and con- 
5 
taining a mere line. 
3 
• 
1'2'3' 
lx 1= 1 
Plane through the three 
5 45 
mere lines. 
g 
(t> 
•g.3 
p to 
5' 
p 3' 
W 
(D 
151. 
The planes are 
The lines are 
X + Y+Z= 0, 
[0] 
X = 0, Y + Z = 0, 
(1) 
X =0, 
[1] 
Y — 0, Z + X = 0, 
(2) 
Y = 0, 
[2] 
Z = 0, X + F = 0, 
(3) 
Z =0, 
[3] 
X = 0, 17 = 0, 
(10 
17 = 0, 
[1'2'3'] 
7 = 0, 17 = 0, 
(20 
Z = 0, 17 = 0, 
(so. 
152. 
The three mere lines are each 
facultative : p = b' = 3 ; 
t' = 
153. Hessian surface. The equation is 
(X + Y + Z)- (X- + Y 2 + Z 2 — 2YZ — 2ZX — 2XY) = 0, 
viz. the surface consists of the uniplane X +Y+Z = 0 twice, and of a quadric 
having its vertex at U 6 , and touching each of the planes X = 0, Y=0, Z=0. 
cone