440 
A MEMOIR ON CUBIC SURFACES. 
[412 
The complete intersection with the surface is made up of X — 0, Y= 0 (torsal 
ray) six times; X = 0, Z = 0 (single ray) twice; and of a residual quartic, which is 
the spinode curve; a = 4. 
The equations of the spinode curve are XZ — Y 2 — 0, XW + 2Z 2 = 0; the first 
surface is a cone having its vertex on the second surface; and the curve is thus a 
nodal quadriquadric. 
The mere line is a single tangent of the spinode curve; /3'= 1. 
Reciprocal Surface. 
175. The equation is obtained by means of the binary cubic 
(— 3y 2 , 2yz, 4aw, QywfX, Yf, 
viz. throwing out the factor y, the equation is 
w 2 (— 64a 3 ) + w (— 16x 3 z 2 + 72xy 2 z 4- 27y*) + 1 Qy 2 z 3 = 0. 
The section by the plane w = 0 (reciprocal of U 7 ) is iv = 0, z = 0 (reciprocal of 
torsal ray) three times, and w — 0, y = 0 (reciprocal of single ray) twice. 
Nodal curve. This is the line a = 0, y— 0, reciprocal of the mere line: b' = l. 
Cuspidal curve. The equation of the surface may be written 
(64a;, — 1 62, — 3iu\z 2 + 3aw, 9 y 2 + 4zx) 2 = 0, 
where 
4. 64a; (— 3w) — 2o6z 2 = — 256 {z- + 3xw). 
This exhibits the cuspidal curve z- + Saw = 0, 9y 2 + 4^a = 0, where the surfaces are 
each of them cones; the vertex of the second cone is on the first cone, and the two 
cones have at this point a common tangent plane; the curve is thus a cuspidal 
quadriquadric. 
176. {The equation 
(64a, — 16^, — Swjz 2 + 3xw, 9y 2 + 42a) 2 = 0 
resembles that of a quintic torse, viz. the equation of a quintic torse is 
( a, — 4 2, 8 iv\z 2 — 2 wx, y 1 — 2 zx) 2 = 0, 
which equation, writing 9y for y, — 2a for x, and fw for w, becomes 
(— 2a, — 4>z, 0>w\z 2 + 3aw, 9y 2 + 42a) 2 = 0, 
or, what is the same thing, 
( a, 22, — 3w\z 2 + Saw, 9y 2 + 42a) 2 = 0 ; 
and developing, this is 
a 3 , w 2 
+ a 2 . — 2 z 2 w 
4- a . — 18 y 2 zw + 2 4 
— 27i/ 4 w 4- 2y 2 z 3 = 0, 
Avhich, however, differs from the equation of the reciprocal surface, not only in the 
numerical coefficients, but by the presence of a term xz 4 .}