276 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
87. And the lines are 
a 
b 
C 
f 
9 
h 
equations may be written 
0 
0 
0 
0 
1 
0 
( 3 ) 
X 
II 
_p 
CS3 
II 
o 
1 
0 
1 
0 
0 
0 
( 4 ) 
X+Z=0, w=o 
0 
0 
0 
0 
y/b 
1 
(1) 
X=0, Y — Zy/b—O 
0 
0 
0 
0 
-y/b 
1 
(2) 
X=0, Y+Zy/b=ti 
° 
0 
0 
1 
\/ a 
0 
(!') 
Z=0, — X\/«-}-Y=0 
I 0 
0 
o' 
1 
— %/ a 
0 
(2') 
Z = 0, Xy«+Y=0 
1 
Vb 
1 
V ab 
1 
V a 
2 
2( 
s/ a— y/b) 
2 
(IT) 
but for the other lines the co- 
ordinate expressions are the 
1 
Y b 
1 
V ab 
I 
V a 
2 
2(- 
-\/ a— y/b ) 
2 
(120 
more convenient. 
I 
V b 
1 
V ab 
1 
~ V a 
2 
2( 
y/ a+y/ b) 
— 2 
(210 
1 
Vb 
1 
V ab 
1 
V a 
2 
2(- 
— \/ a-\-\/ b ) 
2 
(220 
88. The four mere lines and the transversal are each facultative; the edge is also 
facultative, counting twice', 7, t'= 3. 
That the edge is as stated a facultative line counting twice, I discovered, and accept, 
a posteriori, from the circumstance that on the reciprocal surface the reciprocal of the 
edge is (as will be shown) a tacnodal line, that is, a double line with coincident tangent 
planes, counting twice as a nodal line. Eeverting to the cubic surface, I notice that the 
section by an arbitrary plane through the edge consists of the edge and of a conic 
touching the edge at the biplanar point ; by what precedes it appears that the arbitrary 
plane is to be considered, and that twice, as a node-couple plane of the surface : I do 
not attempt to further explain this. 
89. Hessian surface. The equation is 
(X+ Z)XZW+(X-Z) 2 Y 2 + (X+Z)(3a, -a, -l, 3 bJX, Z) 3 =0. 
Combining with the equation 
XZW+ (X + Z)(Y 2 - «X 2 - b7J) = 0, 
and observing that from the two equations we deduce 
- XZY 2 + (X+ Z)(«X 3 + bZ 3 ) =0, 
it appears that the complete intersection of the Hessian and the surface is made up of 
the line X=0, Z=0 (the edge) twice (that is, the two surfaces touch along the edge), 
and of a curve of the tenth order, which is the spinode curve ; c'=10. 
