298 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
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 
8 times. As already remarked, the singularity requires a more complete investigation. 
Section X=12-B 4 -C 2 . 
Equation WXZ + (X+Z)(Y 2 — X 2 )=0. Article Nos. 137 to 143. 
137. The diagram of the lines and planes is 
Lines. 
u> 
2— B 4 — C 2 . 
^ X 
iT 
ssi - 
to 
X 
if 
U> 
X 
<X 
X 
o 
II 
c; 
X 
CO 
II 
o> 
1x12 = 12 
Biplane touching along axis, 
and containing edge. 
1x12=12 
: : 
Other biplane. 
2x 8 = 16 
• 
* 
• 
Planes each through the 
axis and containing a bi- 
planar ray and a cnicno- 
dal ray. 
IX 3= 3 
• 
. . 
Plane touching along the 
edge and containing the 
mere line. 
lX 2= 2 
6 45 
• 
. 
Biradial plane through the 
two cnienodal rays. 
o - 
5' 
CTQ 
o 
"1 
Biplanar rays in the iion- 
j axial biplane. 
Edge of binode, being a 
transversal. 
Axis, through the two 
nodes. 
138. The planes are 
and the lines are 
X =0, 
[0] 
Z =0, 
[3] 
X— Y=0, 
[11 # ] 
X +Y=0, 
[22'] 
w=o, 
[3'] 
x+z=o, 
[l'2'l 
X=0, Y =0, 
(0) 
X=0, Z =0, 
(3) 
X— Y=0, Z =0, 
(1) 
X+Y=0, Z =0, 
(2) 
X— Y=Q, W=0, 
(!') 
X+Y=0, W=Q, 
(2') 
x+z=o, w=o, 
(12) 
