PROFESSOR CAYLEY ON CUBIC SURFACES. 
319 
192. The Hessian surface is 
Z(WXZ + Y 2 Z - 3X 3 )=0, 
viz. this is the oscular biplane Z = 0 and a cubic surface. 
The complete intersection with the cubic surface is made up ofX=0, Z = 0 (the edge) 
6 times, and X=0, Y=0 (the axis) 6 times. There is no spinode curve, a' = 0 ; 
whence also (3' = 0. 
Reciprocal Surface. 
193. The equation is at once found to be 
6 4 zw 3 +(y 2 + ixw f = 0. 
The section by the plane w = 0 (reciprocal of B 6 ) is w = 0,^=0 (reciprocal of edge) 
4 times. The section by the planer =0 (reciprocal of C 2 ) is z= 0, y~ + 4#w = 0 (reciprocal 
of nodal cone) twice. 
Nodal curve. The equation gives 
w=-^f± z -^y 3 + &c., 
‘tJ, 
showing that the line w= 0, y= 0 (reciprocal of edge) is an oscnodal line counting as 3 
lines; b'= 3. 
There is no cuspidal curve; c'= 0. 
Section XX=12-U 8 . 
Equation X 2 W+XZ 2 +Y 3 =0. Article Nos. 194 to 197. 
194. The diagram of the lines and planes is 
Kj 
II S' 
XX = 12— XL. 
X=0 0 
1x45=45 
* * * 
Uniplane. 
T 45 
£ 
►A 
*5 
where the equations of the line and plane are shown in the margins. 
195. There is no facultative line; b'=g'= 0, ^'=0. 
196. The Hessian surface is X 3 Y=0, viz. this is the uniplane X=0, 3 times, and the 
plane Y = 0 through the ray. The complete intersection with the cubic surface is made 
