PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
303 
The curve may be put in evidence by writing the equation of the surface in the form 
(3y 2 -i-5s 2 +l2,ra;, 24s, 16^3?/ 2 — 4s 2 +12^w, s 3 -{-27w 3 ) 2 =0, 
where 
1 6 (3^ 2 + 5s 2 + 1 2xw) — 1 44s 2 = 1 6 (3y 2 — 4s 2 + 12ww ) . 
Section XXX=12-U 6 . 
Equation W(X+Y+Z) 2 +XYZ=0. Article Nos. 150 to 156. 
150. The diagram of the lines and planes is 
w 
Lines. 
H- CO «— 
XII = 
= 12-U 6 . 
CO 
X 
X 
<x> 
|i 
-I 
* co 
to 
0 
lx 32 =32 
: : : 
Uniplane. 
1 
□5 2 
& 
3 
3 X 4=12 
; 
Planes each touching 
along a ray, and con- 
taining a mere line. 
1'2'3' 
lx 1= 1 
5 45 
Plane through the 
three mere lines 
Rays in i 
plane. 
■ S' 
151. The planes are 
X+Y+Z=0, 
X=0, 
Y=0, 
Z =0, 
AY z=0, 
The lines are 
[0] 
X=0, Y +Z =0, 
[1] 
Y =0, Z +X=0, 
[2] 
Z=0, X -J-Y =0, 
[3] 
x=o, w=o, 
[ '2'3'] 
o 
II 
£ 
o' 
Z =0, W=0, 
152. The three mere lines are each facultative: g'=b'= 3; t'= 1. 
153. Hessian surface. The equation is 
(X-f Y+Z) 2 (X 2 + Y 2 + Z 2 — 2YZ — 2ZX— 2XY) = 0, 
( 1 ) 
( 2 ) 
(3) 
(!') 
( 2 ') 
3') 
