412] 
A MEMOIR ON CUBIC SURFACES. 
429 
Section XI =12 — B 6 . 
Article Nos. 144 to 149. Equation WXZ + Y 2 Z + X 3 — Z 3 = 0. 
144. The diagram of the lines and planes is 
II 
►*! 
II 
II 
© 
© 
© 
N 
1 
+ 
II 
Ni 
N 
o 
II 
1! 
o 
o 
tO 
o 
XI = 12 
Planes are 
-R 6 . 
ool to 
X 
© 
II 
to. h-* 
-al to 
1x15 = 15 
X=0 0 
1x15 = 15 
* * * 
Oscular biplane. 
X=0 3 
1x30 = 30 
2 45 
• • 
• 
Ordinary biplane. 
Rays in the ordi 
nary biplane. 
Edge. 
where the equations of the lines and planes are shown in the margins of the diagram. 
145. The edge is a facultative line counting three times; this will appear from 
the discussion of the reciprocal surface. Therefore p =b' = 3; t'= 1. 
146. Hessian surface. This is 
Z(WXZ+ Y-Z — 3 A 3 — HZ 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+Y 2 = 0, X 3 + Z 3 = 0, so that this con 
sists 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, F=0; but 
it must be reckoned as a single tangent of the spinode curve, and then counting it 
three times, /3' = 3.