426 
A MEMOIR ON CUBIC SURFACES. 
[412 
136. The like investigation applies to the general surface, and we have thus 
6' = 16 ; the points in question are still the points (x = 0, y = 0, z = 0), (x = 0, y = 0, w = 0); 
viz. these are the points of intersection of the surface by the line (x = 0, y = 0), which 
points are also the common points of intersection of the four conics which compose 
the cuspidal curve, that is, they are quadruple points on the cuspidal curve; it does 
not appear that the points are on this account, viz. qua quadruple points of the 
cuspidal curve, off-points of the surface, nor does this even show that the points should 
be reckoned each eight times. As already remarked, the singularity requires a more 
complete investigation. 
Section X = 12 — B 4 — C 2 . 
Article Nos. 137 to 143. Equation WXZ + (X + Z)(Y 2 - X-) = 0. 
137. The diagram of the lines and planes is 
Lines. 
to K* to h-i 00 o 
X=12- B 4 -C 2 . 
1x1= 1 
7 27 
2x2= 4 
2x4= 8 
1 X 6= 6 
h-i 
X 
00 
II 
00 
0 
3 
11' 
œ 22' 
<0 
a 
oj 
s 
3' 
1'2' 
1x12 = 12 
* 
m 
Biplane touching along axis, 
and containing edge. 
1x12 = 12 
• • 
• 
• 
Other biplane. 
2 x 8 = 16 
• 
• 
Planes each through the 
axis and containing a bi 
planar ray and a cnicno 
dal ray. 
lx 3= 3 
• 
* * 
Plane touching along the 
edge and containing the 
mere line. 
lx 2= 2 
6 45 
* 
• * 
Biradial plane through the 
two cnicnodal rays. 
Mere line, being a trans 
versal. 
Cnicnodal rays. 
Biplanar rays in the non- 
axial biplane. 
Edge of binode, being a 
transversal. 
Axis, through the two 
nodes.