306 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
planes are 
The lines are 
X =0, 
[1] 
X =0, Y =0, 
(5) 
Z =0, 
PI 
2 =0, Y =0, 
(6) 
Y =0, 
[056] 
Y =0, W=0, 
(0) 
Y +X =0, 
[5] 
X =0, Y+Z =0, 
(1) 
Y +Z =0, 
[6] 
Z =0, Y+X=0, 
(2) 
1 
II 
[34] 
W= Y=-Z, 
(3) 
X+Y+Z=0, 
[12] 
M 
1 
II 
>X 
II 
£ 
(4) 
w=o, 
[0] 
W=0; X+Y+Z=0, 
(012) 
159. The transversal is facultative ; ^=V = 1, £'=0. 
160. The Hessian surface is 
WXZ(3Y-f-X+Z)+Y 2 (Z-X) 2 =0. 
The complete intersection with the surface is made up of the line Y=0, X— 0 (CB-axis) 
3 times; the line Y=0, Z=0 (CB-axis) 3 times; line Y=0, W=0 (CC-axis) twice, 
and of a residual quartic, which is the spinode curve ; a' =4. 
161. Representing the two equations by XJ=0, H=0, we have 
(3Y+X+Z)U-H=Y 2 (3Y 2 +4YX+Z+4XZ), =MY 2 suppose, 
and 
27(X+Z)U+9H= 9 WXZ(3Y + 4X + 4Z) + 36 Y 2 (X 2 + XZ + Z 2 ) + 2 7Y 3 (X+Z) ; 
but 
(— 9(X+Z)Y + 16XZ)M= 
64X 2 Z 2 +28YXZ(X+Z)-Y 2 (36X 2 +28XZ+36Z 2 )-27Y 3 (X+Z), 
whence 
27(X+ Z)U + 9H + ( — 9X + ZY + 1 6XZ)M 
=ZX{12Y 2 +28YX+Z+64XZ + 9W(3Y-f4X+4Z)}; 
or, as this may also be written, 
27Y 2 (X+Z)U +9Y 2 H 
+ ( - 9 YX+Z + 1 6XZ)( 3 Y + X + Z)U + (9 YX+Z - 1 6XZ)H, 
that is, 
{-9Y(X+Z) 2 + 48YXZ + 16XZ(X+Z)}U + {9Y 2 + 9YX+Z-16XZ}H 
=' Y 2 ZX{ 12Y 2 + 28YZ+X+ 64XZ + 9W(3Y+ 4X+ 4Z) } =0 ; 
and we thus obtain the equation of the residual quartic, or spinode curve, in the form 
3 Y 2 + 4 Y(X +Z)+ 4XZ = 0, 
12 Y 2 + 2 8 Y(X + Z) + 6 4XZ + 9 W( 3Y+ 4X + 4Z) = 0 . 
The spinode curve is thus a complete intersection, 2x2; and since the first surface is a 
