420 
A MEMOIR ON CUBIC SURFACES. 
[412 
120. The three transversals are each facultative; p' = b'= 3; t' = 0. 
121. Hessian surface. The equation is 
4>aXZW(S Y+X + Z+W)+Y 2 (X 2 ±Z*+ If 2 - 2 XZ-2XW-2ZW) = 0. 
The complete intersection with the cubic surface is made up of the lines (Y= 0, 
X = 0), (F = 0, Z= 0), (F=0, W = 0) (the axes) each twice, and of a sextic curve which 
is the spinode curve ; a = 6. 
The spinode curve is a complete intersection 2x3; the equations may in fact be 
written 
Y 3 + Y*(X + Z+ W) + 4>aXZW=0, 
3F 2 + 4F (X + Z+ F) + 4(XZ+XTF + ZIF) = 0; 
the nodes D, C, A are nodes (double points) of the curve, the tangents at each node 
being the nodal rays. 
Each of the transversals is a single tangent of the spinode curve; in fact for 
the transversal Y + Z + X -- 0, W = 0, these equations of course satisfy the equation of 
the cubic surface; and substituting in the equation of the Hessian, we have 
F 2 (X — Z) 2 = 0. But Y + Z + X = 0, IF = 0, F=0isa point on the axis W = 0, F = 0, 
not belonging to the spinode curve ; we have only the point of contact Y + X + Z = 0, 
IF = 0, X — Z = 0. Hence /S' = 3. 
Reciprocal Surface. 
122. The equation is found by means of the binary cubic, 
aT(T-yUy + (T-xU)(T-zU)(T-wU), 
viz. writing for shortness 
/3 = x + z + w, 
7 = xz + %w + zu), 
8 = xzw; 
this is a binary cubic (*$F, U) 3 , the coefficients whereof are 
3(a+l), —2ay — /3, ay 2 + y, — SB, 
and the equation is hence found to be 
4>a 3 y 3 (y 3 - /3y 2 + 7y-8) 
+ a 2 {(12 7 - /3 2 ) y* - (8#y + 368) y 3 + (30/38 + 8 7 2 ) y 2 - 36 7 8y + 278 2 } 
+ 2a {(6 7 2 - /3 2 7 - 9/38) y- + (12/3 2 8 - 2/3f - I878) y+2f + 278 2 - 9/3 7 8} 
- (/3y + 18/878 - 4/3 3 8 - 4 7 3 - 27 8 2 ) = 0;