PROFESSOR CAYLEY ON CUBIC SURFACES. 
301 
Section XI=12-B„. 
Equation WXZ+Y 2 Z+X 3 -Z 3 =0. Article Nos. 144 to 149. 
144. The diagram of the lines and planes is 
where the equations of the lines and planes are shown in the margins of the diagram. 
145. The edge is a facultative line counting 3 times ; this will appear from the dis- 
cussion of the reciprocal surface. Therefore g'=b'= 3; t’= 1. 
146. Hessian surface. This is 
Z(WXZ+Y 2 Z-3X 3 -3Z 3 )=0, 
breaking up into Z=0, the oscular biplane, and into a cubic surface (itself a surface 
XI=12 — B 6 ). The complete intersection with the cubic surface is made up of the line 
X=0, Z=0 (the edge) six times, and of a residual sextic ( = 3 conics), which is the 
spinode curve ; c'= 6. 
The equations of the sextic are in fact XZ-j-Y 2 =0, X 3 +Z 3 =0, so that this consists 
of three conics, each in a plane passing through the edge. 
The edge touches each of the three conics at the point X=0, Z = 0, Y=0; but it 
must be reckoned as a single tangent of the spinode curve, and then counting it three 
times, /3' = 3. 
