412] 
A MEMOIR ON CUBIC SURFACES. 
449 
Reciprocal Surface. 
197. The equation is at once found to be 
27 (z- + 4xuif — 64w 3 y = 0. 
The section by the plane w = 0 (reciprocal of the Unode) is w = 0, z = 0 (reciprocal 
of ray) four times. 
There is no nodal curve; b' = 0. But there is a cuspidal conic, y = 0, z 3 + 4<xw = 0. 
The point y = 0, z = 0, w— 0 (reciprocal of the uniplane X = 0) is a point which 
must be considered as uniting the singularities B'—l, % = 2. 
I give in an Annex a further investigation in reference to this case of the cubic 
surface. 
Section XXI = 12 — SB 3 . 
Article Nos. 198 to 201. Equation WXZ + Y 3 = 0. 
198. The diagram of the lines and planes is 
XXI = 12-35, 
Oi| 
Planes are 
r=o 
0 
1x27 = 27 
• • • 
Common biplane containing 
the three axes. 
x=o 
1 
• • • 
z=o 
2 
3x6 =18 
. . . 
Remaining biplanes, one for 
each binode. 
W=0 
3 
4 45 
• 
O' 
B or 
where the equations of the lines and planes are shown in the margins. 
199. There is no facultative line; p — b' = 0, t' = 0. 
200. The Hessian surface is XYZW = 0, the common biplane and the other 
biplanes each once. The complete intersection with the surface consists of the axes 
each four times; there is no spinode curve, a' = 0; whence also ¡3'= 0. 
C. VI. 
57