PROFESSOR CAYLEY ON CUBIC SURFACES. 
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cone having its vertex on the second surface, we see moreover that the spinode curve is 
a nodal quadriquadric. Instead of the last equation we may write more simply 
, 4Y(X+Z)+16XZ+3W(3Y+4X+4Z)=0. 
The equations of the transversal are W=0, X-f- Y-j-Z = 0, and substituting in the 
equations of the spinode curve we obtain from each equation (X— Z) 2 =0, that is, the 
transversal is a single tangent of the spinode curve; /3'=1. 
Reciprocal Surface. 
162. The equation of the cubic is derived from that belonging to VI=12 — B 3 — C 2 
by writing therein a=b = 0, c=^, d= 1. Making this change in the formulse for the 
reciprocal surface of the case just referred to, we have 
L =y 2 +4(x+z)w, 
M.=2x(y-{-2w), 
N=-4^ 2 , 
P =1 6x 2 (y+w—x—z ) ; 
and substituting in the equation 
L 2 P + 8zM 3 - 9zLMN - 2 7z 2 wN 2 = 0, 
the equation divides by x 2 ; ' or throwing this out, the equation is 
(y 2 + 4 xw + 4 zw)\y -\-w—x—z) 
— %xz{y-\-2w) 3 
+ 9 xz{y 2 + 4xw + 4 zw)(y + 2 w) 
— 27#Vw=0 ; 
reducing, this is 
w 3 . l§(x—z) 2 
4-w 2 f y\x+z) 1 
| +2y(x 2 —4xz+z 2 ) 1 
[+(x-\-zX2x-z)(-x+2z) J 
-\-w 
+ 8 y\x+z) 
— 2y\4x 2 2 3 xz + 4^ 2 ) 
+ 3 6xtjz(x+z) 
— 27x 2 z 2 
+ f(y-x)(y-z)=o. 
The section by the plane w = 0 (reciprocal of B 3 ) is w=0, y=0 (the edge) 3 times; 
and w — 0, y—x= 0; w=0, y—z— 0 (reciprocals of the CB-axes). 
163. Nodal curve. This is the line y—x—z; wherefore V — \. To put the line in 
2 t 2 
