412] 
A MEMOIR ON CUBIC SURFACES. 
435 
The spinode curve is thus a complete intersection, 2x2; and since the first surface 
is a 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 
4iY(X + Z) + 1QXZ+SW(SY+4<X + 4>Z) = 0. 
The equations of the transversal are W — 0, X + Y + 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 s — C 2 
by writing therein a = b = 0, c = d = 1. Making this change in the formulae for the 
reciprocal surface of the case just referred to, we have 
L = y 2 + 4 (x 4 z) w, 
M =2x(y + 2 w), 
N = — 4a? 2 , 
P = 16a? 2 (y + w — x — z); 
and substituting in the equation 
L-P + 8zM 3 - 9zLMN — Tlz-wN- = 0, 
the equation divides by # 2 ; or throwing this out, the equation is 
(y 2 + 4>xw + 4.sw) 2 (y + w—x — z) 
— 8 xz (y + 2 wf 
4- 9xz (y 2 4 4xw 4 4zw) (y 4 2w) 
— 27 x 2 z 2 w = 0; 
reducing, this is 
w 3 .16 (x — z) 2 
+ w 2 (y 2 (x + z) 
•; 4 2y (x 2 — 4xz 4 z 2 ) 
[ 4 {x 4 z) (2x — z)(—x 4 2z) / 
4 w ( y* ] 
j 4 8y 3 (x 4 z) 
■ — 2y 2 (4ic 2 4 2%xz-\-bz 2 ) 
4 3 Qxyz (x 4 z) 
^ — 27 x 2 z 2 J 
+ y 3 {y- x) (y-z) = 0. 
55—2