The collected mathematical papers of Arthur Cayley.

412]

A MEMOIR ON CUBIC SURFACES.

449

Reciprocal Surface.
197. The equation is at once found to be
27 (z2 + 4xw)2- 64wy = 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, z2 + 4xw = 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'= 1, ' = 2.
I give in an Annex a further investigation in reference to this case of the cubic
surface.
Section XXI = 12 - 3B3.
Article Nos. 198 to 201. Equation WXZ + Y =0.
198. The diagram of the lines and planes is

II II II
CD
If. C
XXI = 12 - 3B3. cl o
x
II
Planes are -
Y=0 0 - lx 27 =27 Common biplane containing
'0 ' ~ 1x7=7 the three axes.
X=0 1
Z=0 2 3x 6 =18 ~ Remaining biplanes, one for
each binode.
W=0 3 -
4 45
0
r >
0

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 X YZW= 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, -' = 0; whence also /'= 0.
C. VI. 57