412] 
A MEMOIR ON CUBIC SURFACES. 
393 
63. And the coordinates of the fifteen distinct lines are 
(a) 
(6) 
(o) 
(/) 
(9) 
(h) 
whence equations may be written 
0 
0 
0 
0 
-1 
1 
(1) X = 0, Y + Z = 0 
0 
0 
0 
1 
0 
-1 
(2) Y = 0,Z+X = 0 
0 
0 
0 
-1 
1 
0 
(3) Z = 0, X+Y = 0 
0 
0 
0 
0 
— n 
m 
(4) X=0, viY+nZ=Q 
0 
0 
0 
n 
0 
-l 
(5) Y=0,nZ + lX=0 
0 
0 
0 
— m 
l 
0 
(6) Z=0, IX+mY=0 
1 
0 
0 
0 
0 
0 
II 
II 
o 
0 
1 
0 
0 
0 
0 
(25) F = 0, W = 0 
0 
0 
1 
0 
0 
0 
(36) Z=0, W=0 
l 
n 
11 
rrv 
— nlv 
0 
(15) IX+ 11Y+ nZ = 0, W+nvZ= 0 
l 
m 
VI 
— VI 2 /A 
0 
IvifJL 
(16) IX + mY+mZ = 0, W+mp.Y= 0 
l 
m 
l 
0 
l 2 X 
— Imp. 
(26) lX+mY+ IZ = 0, W+l\X=0 
n 
m 
n 
mnv 
- 1VV 
0 
(24) nX+mY+nZ = 0, W+nv Z=0 
rn 
VI 
11 
— vin\x 
0 
Vl 2 p. 
(34) vxX + viY + nZ= 0, W+ vipY= 0 
l 
l 
n 
0 
1ll\ 
(35) IX+ lY+nZ= 0, W + l\X=Q 
64. The rays are not, the mere lines are, facultative; hence b' = p = 9 : t' = 6. 
65. The equation of the Hessian surface is 
— W (X + Y + Z) (IX + m Y + nZ) (p,vX + v\ Y + \p,Z) 
— k (l-X 4 + m-Y* + n-Z i — 2mnY 2 Z 2 — 2nlZ 2 X 2 — 2lniX 2 Y-) 
+ kX YZ {(l 2 + 3Ini 4- 3In + mn) X + (m 2 + 3inn + 3ml + nl) F+ (n 2 + 3nl + 3inn + bn) Z) = 0. 
The Hessian and cubic surfaces intersect in an indecomposable curve, which is the 
spinode curve; that is, spinode curve is a complete intersection 3x4; a = 12. 
The equations may be written in the simplified form 
W(X+Y+Z) (IX + mY + nZ) + kXYZ = 0, 
№ + m 2 F 4 + riW - 2mnY-Z- - 2nlZ-X* - 2lmX*Y* 
— 4A r \ Z \l (m + n) X + in (n +1) Y + u (l + in) Z] = 0. 
We may also obtain the equation 
k- (X+Y + Z) (IX + viY+ nZ) {IX 2 + m F 2 + nZ 2 - (in + n) YZ - (n + l) ZX - (l + in) X Y] 
+ X 2 Y-Z 2 + fx 2 Z 2 X 2 + v 2 X 2 Y 2 - 2XYZ(fivX + v\Y+ \pZ) = 0, 
C. VI. 
50