412] 
A MEMOIR ON CUBIC SURFACES. 
451 
Article No. 203. Section XXII = 5(1, 1). Equation X 2 W+Y 2 Z = 0. 
203. As this is a scroll there is here no question of the 27 lines and 45 planes; 
there is a nodal line X = 0, Y= 0, (5 = 1) and a single directrix line, Z = 0, W = 0. 
The Hessian surface is X 2 F 2 = 0; the complete intersection with the cubic surface 
is made up of X — 0, F=0 (the nodal line) eight times, and of the lines X = 0, Z = 0, 
and F= 0, W = 0, each twice. 
The reciprocal surface is x 2 z — y 2 w = 0viz. this is a like scroll, XXII = 5(1, 1); 
c' = 0, V = 1. 
Article No. 204. Section XXIII = 5(1, 1). Equation X (XW + YZ) + Y 3 = 0. 
204. This is also a scroll; there is a nodal line X = 0, F = 0, and a single directrix 
line united therewith. 
The Hessian surface is A 4 = 0; the complete intersection with the cubic surface is 
X = 0, F = 0 (the nodal line) twelve times. 
The reciprocal surface is w (xw + yz) — z 3 = 0 ; viz. this is a like scroll, XXIII = 5(1, 1) ; 
c' = 0, V = 1. 
Annex containing Additional Researches in regard to the case XX = 12— U s ; equation 
WX 2 + XZ 2 + Y 3 = 0. 
Let the surface be touched by the line (a, b, c, f, g, h), that is, the line the 
equations whereof are 
( 0, h, — g, a \X, Y, Z, W) = 0. 
- h, 0, /, b 
9, ~f, 0, c 
— a, — b, — c, 0 
Writing the equation in the form cW. cX 2 + X (cZ) 2 + c 2 Y 3 = 0, and substituting for 
cW, cZ their values in terms of X, F, we have 
(_gX +/F) cX 2 + X(aX + bY) 2 + c 2 Y 3 = 0, 
that is 
( a 2 — eg , 2ab + cf b 2 , c 2 \X, Y) 3 = 0, 
or say 
(3 (a 2 — eg), 2ab + cf, b 2 , 3c 2 F) 3 = 0, 
viz. the condition of contact is obtained by equating to zero the discriminant of the 
cubic function. We have thus 
27 c 4 (a 2 — eg) 2 
+ 4b 6 (a 2 — eg) 
+ 4c 2 (2a5 + cf) 3 
— 5 4 (2a5 + c/) 2 
— 185 2 c 2 (a 2 — eg) (2 ab + cf) = 0, 
57—2