PKOFESSOE 'CAYLEY ON CUBIC SURFACES. 
299 
139. The facultative lines are the edge counting twice, and the mere line ; 
Z=V= 3; t'= 1. 
140. Hessian surface. The equation is 
X(X + Z)(ZW + 3X 2 — XZ) + Y 2 (X — Z) 2 = 0. 
The complete intersection with the surface consists of the line (X=0, Y=0), the axis, 
4 times; the line (X=0, Z=0), the edge, 2 times; and a sextic curve, which is the 
spinode curve ; c'=6. 
Writing the equations of the surface and the Hessian in the form 
X(ZW+Y 2 )-X 3 +Z(Y 2 -X 2 )=0, 
X(X+Z)(ZW+Y 2 )-|-(Z-3X){-X 3 + Z(Y 2 -X 2 )}=0, 
we see that the equations of the spinode curve may be written 
ZW+Y 2 =0, 
— X 3 + Z(Y 2 — X 2 ) = 0, 
viz. the curve is a complete intersection, 2x3. 
Y 
There is at B 4 a triple point yy— 
( z \ 
I 2 X 
( z \\ 
Vw, 
/ 5 w — 
(w ) ’ 
tangents coinciding with the nodal rays W=0, Y 2 — X 2 =0. 
The edge and the mere line are each of them single tangents of the spinode curve. 
But the edge counting twice in the nodal curve, its contact with the spinode curve will 
also count twice, that is, we have j3'=2.1 -f-1, =3. 
Reciprocal Surface. 
141. The equation is obtained by means of the binary cubic 
4w 2 X(X-}-Z) 2 + 4wZ(X + Z)(xX+zZ) -f ?/ 2 XZ 2 ; 
or calling this (=&JX, Z) 3 , the coefficients are 
(12w 2 , 8w 2 -f 4m\ 4w 2 +4m£-f-4 ivz-\-f, 12 wz), 
and thence the equation is found to be 
16 w 4, [p 2 —{x—z) 2 ~\ 
+ 1 6w 3 [ 2x - 5z)f — 2(x — 2 z){x - z) 2 ] 
+ 8w 2 [y -j- (x*—xz -j- 6s! 2 )^ 2 — 2x\x'—zY\ 
+ ±w[(2x+3z)y 4 -2x%x+z)y*l 
+ f(f-x 2 )=0, 
where the section by the plane w = 0 (reciprocal of binode) is # 2 ) = 0, viz. this 
is the line w = 0, y=0 (reciprocal of the edge) four times, and the lines w=Q, f—x' 2 = 0 
(reciprocals of the biplanar rays). 
The section by the plane z=0 is found to be (f—x 2 )(f-j-4xw + 4w 2 ) 2 =0, viz. this 
2 s 2 
