412] 
A MEMOIR ON CUBIC SURFACES. 
427 
138. 
The planes are 
X =0, 
[0] 
Z =0, 
[3] 
X -7=0, 
[IF] 
X +Y = 0, 
[22'] 
W = 0, 
[31 
X +Z =0, 
[121 
and 
the 
lines 
are 
X = 
0, 
F 
= 0, 
(0) 
x = 
:0, 
£ 
= o, 
(3) 
X- 
F= 
o, 
£ 
= 0, 
(1) 
x + 
Y- 
= 0, 
£ 
= 0, 
(2) 
X- 
Y = 
0, 
w 
= 0, 
(!') 
x + 
F = 
0, 
w 
= 0, 
(2') 
x + 
0, 
w 
= o, 
(12). 
139. The facultative lines are the edge counting twice, and the mere line ; 
p = b' = 3 ; t' = 1. 
140. Hessian surface. The equation is 
X (X + Z) (ZW + 3X 2 - XZ) + F 2 (X - zy = 0. 
The complete intersection with the surface consists of the line (X = 0, Y = 0), the 
axis, four times; the line (X = 0, Z = 0), the edge, twice; 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 + F 2 ) - X 3 + Z(F 2 - X 2 ) = 0, 
X (X + Z) (Z W + F 2 ) + (Z- 3X) {— X 3 + Z (F 2 — X 2 )} = 0, 
we see that the equations of the spinode curve may be written 
ZW+ F 2 = 0, 
- X s + Z (F 2 — X 2 ) = 0, 
viz. the curve is a complete intersection, 2x3. 
There is at B i a triple point ^ = -(^j , W = _ (w) ’ and at a double P oint > 
the tangents coinciding with the nodal rays W = 0, F 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 $' = 2.1 + 1, =3. 
Reciprocal Surface. 
141. The equation is obtained by means of the binary cubic 
4w 2 X (X + Zf + 4mZ (X + Z) (xX + zZ) + fXZ>; 
or calling this (*$X, Zf, the coefficients are 
(12w 2 , 8w 2 + 4wx, 4iv* + 4vox + 4wz + y 1 , 12wz), 
54—2