322 
PEOFESSOE CAYLEY ON CUBIC STTEFACES. 
Section XXII=S(1, 1). Equation X 2 W+Y 2 Z = 0. Article No. 203. 
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 Y 2 =0; the complete intersection with the cubic surface is 
made up of X— 0, Y=0 (the nodal line) 8 times, and of the lines X=0, Z=0, and 
Y=0, W=0, each twice. 
The reciprocal surface is x 2 z—y 2 w= 0; viz. this is a like scroll, XXII = S(.L, 1) ; 
c'=0, It— 1. 
Section XXIII=S(1, 1). Equation X(XW+YZ)+Y 3 =0. Article No. 204. 
204. This is also a scroll; there is a nodal line X=0, Y=0, and a single directrix 
line united therewith. 
The Hessian surface is X 4 =0 ; the complete intersection with the cubic surface is 
X=0, Y=0 (the nodal line) 12 times. 
The reciprocal surface is w(xw-\-yz) — z 3 = 0 ; viz. this is a like scroll, XXIII = S(1, 1) ; 
c'= 0, V= 1. 
Annex containing Additional Researches in regard to the case XX =12 — U 8 ; eguation 
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 XX, 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 
AY, cZ their values in terms of X, Y, we have 
(-gX+fY)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 -cg), 2ab+cf, 3 c 2 XX, Y) 3 =0, 
viz. the condition of contact is obtained by equating to zero the discriminant of the 
cubic function. We have thus 
27 c\a 2 -cg) 2 
+ 4 b\a 2 —cg) 
+ 4c 2 (2«5 + cff 
- 5 4 (2a5+c/) 2 
— 18b 2 c 2 (a 2 —cg)(2ab+cf)=0, 
