412] A MEMOIR ON CUBIC SURFACES, 
453 
values which satisfy the relation 
( 0, h, - g, a $a, ¡3, 7, 8)=0, 
- K 0, / b 
9, ~f, 0, c 
— a, — b, - c, 0 
then the equation in (a, b, c, f } g, h) is that of the circumscribed cone, vertex 
(a, ft, 7, 3); the order being (as it should be) a'=6. 
The cuspidal conic is y = 0, 4fxw + z 2 = 0, and w r e at once obtain a 2 — 4tcg = 0 as 
the condition that the line (a, b, c, f, g, h) shall pass through the cuspidal cone. 
Hence attributing to (a, b, c, f, g, h) the foregoing values, we have 
a 2 — 4>cg = 0 
for the equation of the cone, vertex (a, /3, 7. 8), which passes through the cuspidal 
conic; this is of course a quadric cone, c = 2. I proceed to determine the intersections 
of the two cones. 
Representing by 0 = 0 the foregoing equation of the circumscribed cone, and putting 
for shortness 
X = 27h 2 (/ 2 - bh) - 2g 2 (2fg + ah), 
I find that we have identically 
0 - (/ 2 - bh) X + (g* - 4ah 3 - 8fgh 2 ) (a 2 - 4eg) - (‘№fg 2 h + I6agh?) (af+ bg + ch) = 0 : 
whence in virtue of the relation af+bg + ch= 0, we see that the equations © = 0, 
a 2 — 4>cg = 0, are equivalent to 
(/ 2 — bh) X = 0, ci 2 — <icg = 0, 
or the twelve lines of intersection break up into the two systems 
f 2 — bh = 0, a 2 — 4c$ = 0, 
and 
(X =) 27h 2 (f 2 - bh) - 2g 2 (2fg + ah) = 0, a 2 - 4>cg = 0. 
To determine the lines in question, observe that we have 
o, 
h, 
-9, 
-h, 
o, 
f> 
9> 
-/> 
0, 
-a, 
-b, 
— c, 
/3, % S)=0; 
b 
c 1 
0