412] 
A MEMOIR ON CUBIC SURFACES. 
437 
Section XIV = 12-55-0,. 
Article Nos. 165 to 171. Equation WXZ + Y 2 Z + YX 2 = 0. 
165. The diagram of the lines and planes is 
Planes are 
Z=0 
x=o 
12 
Y=0 
023 
- c 2 . 
lx 2= 2 
4 27 
lx 5= 5 
1x10 = 10 
1x10 = 10 
1x15 = 15 
• 
* 
Torsal biplane. 
1x20 = 20 
• 
• 
Ordinary biplane. 
1x10=10 
3 45 
• 
Plane through axis and 
the two rays. 
o 
5. 
¡3 
o 
fil 
E- 
p 
Biplanar ray in 
torsal biplane. 
Edge. 
> 
a. 
where the equations of the planes and lines are shown in the margins. 
166. The edge is a facultative line, as will appear from the discussion of the 
reciprocal surface : p = b' — 1; £' = 0. 
167. Hessian surface. The equation is 
WXZ 2 + Y 2 Z 2 - 3X 2 YZ + X i = 0. 
The complete intersection with the surface is made up of the line X = 0, F = 0 
(the axis) five times, the line X = 0, Z = 0 (the edge) four times, and a skew cubic, 
the equations of which may be written 
II Z, F, W | = 0. 
|| 4Z, X, - 5F Ij 
In fact from the equations U=0, H = 0 we deduce H — ZU = X 2 (Z 2 + 4YZ) = 0; and 
if in U = 0 we write X 2 = 4YZ, it becomes Z (XW + oY 2 ) = 0; and then in 5U=0, 
writing 5 F 2 = — Z W, we have 
5 WXZ + Z (- XW) + 5X 2 Y= X (5ZF + 4ZW) = 0.