404 
A MEMOIR ON CUBIC SURFACES. 
[412 
87. And the lines are 
a 
b 
c 
f 
9 
h 
equations may be written 
0 
0 
0 
0 
1 
0 
(3) 
X = 0, 
Z=0 
1 
0 
1 
0 
0 
0 
(4) 
x + z 
= 0, W=0 
0 
0 
0 
0 
41 
1 
(1) 
x=o, 
Y-Z4b = 0 
0 
0 
0 
0 
-41 
1 
(2) 
x=o, 
Y+Z4b = 0 
0 
0 
0 
1 
4 a 
0 
(!') 
Z = 0, 
-X4h+Y= 0 
0 
0 
0 
1 
— 4 a 
0 
(2') 
z= 0, 
X4a+ Y= 0 
1 
- 4b 
1 
4 ab 
1 
4 a 
2 
2 ( 4a - 4b) 
- 2 
(11') 
but for the other lines the 
coordinate expressions are 
i 
1 
1 
2 (— 4a — 4b) 
the more convenient. 
~ 4b 
Vab 
4 a 
2 
- 2 
(12') 
i 
■fb 
1 
Vab 
1 
4 a 
2 
2 ( 4a + 4 b) 
— 2 
(21') 
1 
41 
1 
4 ab 
1 
4 a 
2 
2 (— 4a + 4b) 
- 2 
(22') 
88. The four mere lines and the transversal are each facultative; the edge is 
also facidtative, counting tiuice; p' = h' = 7, t'= 3. 
That the edge is as stated a facultative line counting twice, I discovered, and 
accept, d 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. Reverting 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-Zy-Y n - + (X + Z)(Sa, -a, -b, 3b\X, Zf = 0. 
Combining with the equation 
XZW+ (X + Z) (Y* - aX 2 - bZ 2 ) = 0, 
and observing that from the two equations we deduce 
- XZY 2 + (X + Z) (aX 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; d = 10.