The collected mathematical papers of Arthur Cayley.

412] A MEMOIR ON CUBIC SURFACES. 439
171. Cuspidal curve. The equation of the surface may be written
(x, - y, 3wl12xw - y2, 9zw + 8xy) = 0,
where 4x. 3w - y2 = 12w - y. This exhibits the cuspidal curve 12xw - y2= 0, 9zw + 8xy = 0,
breaking up into the line w =O, y= (reciprocal of edge) and a skew cubic; the line
is really part of the cuspidal curve, or c'= 4.
The equations of the cuspidal cubic may be written in the more complete form
12,, z =0.
y, w, -8x
Section XV = 12 - U.
Article Nos. 172 to 176. Equation WX2 + XZ2 + YZ =O.
172. The diagram of the lines and planes is

II 11
0o
-1-1
o O
x x
x x
I 0
iiII
iI 0

II
o
11
II
o
x
cgr
11
II
03

yl
CD
02>

XV=12- U7.

ClI

Planes are
X=O

21

1 x 40=40

Uniplane.

Z=0 23

Ix 5= 5
2 45

Plane touching along
the single ray.

CD
CD
p

H.
5 -OP
CD
Y
w
*~t

0
g3
Y:1

where the
173.
174.

equations of the lines and planes are shown in the margins.
The mere line is facultative: p'=b'= 1; t' = 0.
The Hessian surface is

X2 (XZ- Y2) = 0,
viz. this is the uniplane X =0 twice, and a quadric cone having its vertex at U7.