PROFESSOR CAYLEY ON CUBIC SURFACES. 
295 
and the lines are 
X=0, Y =0, (0) 
/iX— Y=0, Z =0, (1) 
/ 2 X- Y=0, Z =0, (2) 
/ 3 X- Y=0, Z =0, (3) 
/,X— Y=0, W=0, (4) 
/ 2 X- Y=0, W=0, (5) 
/ 3 X-Y=0, W=0, (6) 
128. There is no facultative line; fJ=b'=0, #=0 ; and hence also /3'=0. 
129. Hessian surface. The equation is 
X{ZW(cX + <TY)-3X(«c-5 2 , awZ— fo, bd-c 2 JX, Y) 2 } = 0, 
so that the Hessian breaks up into the plane X=0 (axial or common biplane) and into 
a cubic surface. 
The complete intersection of the Hessian with the cubic surface is made up of the 
line X=0, Y=0 (the axis) four times; and of a system of four conics, which is the 
spinode curve ; d = 8 . 
In fact combining the equations 
WXZ + («, b, c, d£X, Y) 3 =0 
and 
ZW(<?X+dY)-3X(«c-5 2 , ad-bc, bd-c 2 JK,Yf=Q, 
these intersect in the axis once, and in a curve of the eighth order which breaks up 
into four conics ; for we can from the two equations deduce 
(a, b, c, dJX , Y) 3 (cX+ «!Y) -f- 3X 2 (ac—b 2 , ad-bc , bd—c%Y, Y) 2 =0, 
that is, 
(4ac— 3b 2 , ad , bd, cd , c£ 2 3(X, Y) 4 =0, 
a system of four planes each intersecting the cubic XZW + («, b, c, Y) 3 =0 in the 
axis and a conic ; whence, as above, spinode curve is four conics. 
It is easy to see that the tangent planes along any conic on the surface pass through 
a point, and form therefore a quadric cone ; hence in particular the spinode torse is 
made up of the quadric cones which touch the surface along the four conics respectively. 
Reciprocal Surface. 
130. The equation is obtained by means of the binary cubic 
X(£X+_yY) 2 +4zw(«, b , c, djfl, Y) 3 , 
viz. calling this (*£X, Y) 3 the coefficients are 
(3a, ,2 +12azw, 2xy -\-12bzw, y 2 + 12czw, 12 dzw). 
