320 
PEOEESSOE CAYLEY OJST CUBIC SUEEACES. 
up of X = 0, Y=0 (the ray) 10 times, and of a residual conic, which is the spinode 
curve; <r'= 2. 
The equations of the spinode conic are Y=0, XW-f-Z 2 =0, viz. the plane of the conic 
passes through the ray. Since there is no facultative line, (3'=0. 
Reciprocal Surface. 
197. The equation is at once found to be 
2 7 (z 2 -f- ixw ) 2 — 6 kufy = 0 . 
The section by the plane w = 0 (reciprocal of the Unode) is w= 0, z — 0 (reciprocal of 
ray 4 times. 
There is no nodal curve; 0. But there is a cuspidal conic, y= 0, 2 2 +4ot=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' = l, %'=2. 
I give in an Annex a further investigation in reference to this case of the cubic 
surface. 
Section XXI=12-3B 3 . 
Equation WXZ+Y 3 =0. Article Nos. 198 to 201. 
198. The diagram of the lines and planes is 
Planes are 
Y=0 0 
X=0 1 
Z =0 2 
W=0 3 
II tr< 
® 5' 
= 12-3B 3 . 
Oil Co 
X 
CO 
II 
1x27=27 
Common biplane containing 
the three axes. 
3x6 =18 
4 45 
. . . 
Remaining biplanes, one for 
each binode. 
Axes each joining 
two binodes. 
where the equations of the lines and planes are shown in the margins. 
199. There is no facultative line; §'=b'= 0, #=0. 
200. The Hessian surface is XYZW=0, the common biplane and the other biplanes 
each once. The complete intersection with the surface consists of the axes each 4 
times ; there is no spinode curve, a'=0 ; whence also /3'=0. 
