PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
317 
Section XVIII=12-B 4 -2C a . 
Equation WXZ+Y 2 (X+Z)=0. Article Nos. 186 to 189. 
186. The diagram of the lines and planes are 
Kj Kj 
II II 
O O g 
N M § 
■ II II " 
Planes are 
Y =0 
0 
X =0 
01 
Z =0 
02 
x+z=o 
34 
W=0 
04 
-B t -2C,. 
1 X 
IT 
5SI - 
X 
05 
II 
05 
X 
X 
CO 
05 
1x16=16 
• 
• • 
Plane through the three axes. 
2x12=24 
• 
• 
Biplanes. 
IX 3= 3 
Plane touching along edge and con- 
taining the mere line. 
IX 2= 2 
5 45 
• 
* * 
Plane touching along axis through 
the two cnienodes and containing 
the mere line. 
Mere line, being 
also a trans- 
versal. 
Edge of the bi- 
node. 
Axis through the 
two cnienodes 
Axes, each through 
the binode and a 
cnicnode. 
where the equations of the lines and planes are shown in the margins. 
187. The mere line is facultative ; the edge is also facultative counting twice (this 
will appear from the discussion of the reciprocal surface): b'=g'= 3, tf= 1. 
188. The Hessian surface is 
(X + Z) WXZ + (X - Z) 2 Y 2 =0. 
The complete intersection with the cubic surface is Y=0, Z = 0 and Y=0, X=0 
(the CB-axes) each 4 times; Y=0, W=0 (BB-axis) twice; and X=0, Z = 0 (the edge) 
twice. There is no spinode curve, </— 0; wherefore also (3' = 0. 
Reciprocal Surface. 
189. The equation is obtained from the binary quadric 4w(X+Z)(X# + Z 2 )-f ^rXZ, or 
say 
(8m, 4 iv(x+z)-\-f, 8 wz£K, Z) 2 . 
