412] 
A MEMOIR ON CUBIC SURFACES. 
433 
156. For the cuspidal curve we have 
12 , w — 4/3, 
w — 4/3, 
4y 
4 7 , —123 
= 0, 
or say 
48 7 — (w — 4/3) 2 = 0, 
363 + y(w— 4/3) = 0, 
whence the cuspidal curve is a complete intersection 2x3; c =6. 
Section XIII = 12 - B ;t — 2G 2 . 
Article Nos. 157 to 164. Equation WXZ + Y*(Y + X + Z) = 0. 
157. The diagram of the lines and planes is 
Lines. 
o 
h- 1 
tO tO I-* O 05 Ol 
Aiii= 12 - 
■xl I-* 
X 
I-* 
II 
tO| 
l-l 
2x2= 4 
2x3= 6 
X 
II 
2x6 = 12 
1 
2 
056 
5 
$ 6 
a 
J3 
S 
84 
12 
0 
2 x 6 = 12 
• 
• 
Biplanes. 
1 + 12 = 12 
. 
* * 
Plane through the three 
axes. 
2 x 6 = 12 
• 
• 
• 
Planes each through an 
axis joining the binode 
with a cnicnode. 
lx 4= 4 
• • 
• 
Plane through the axis 
joining the two cnic- 
nodes. 
lx 3= 3 
• 
* * 
Planes through the bi 
planar rays. 
lx 2= 2 
8 45 
• 
• * 
Plane touching along the 
axis which joins the 
two cnicnodes. 
Transversal. 
Cnicnodal rays, one 
through each cnicnode. 
Biplanar rays, one in each 
biplane, and being each 
a transversal. 
Axis joining the two cnic- 
nodes. 
Axes, each joining the bi 
node with a cnicnode. 
C. VI. 
55