412] A MEMOIR ON CUBIC SURFACES, 
455 
and substituting these values in the equation X = 0, it becomes 
27A 2 . 
H 2 
16/3 2 S 
+ 2g*. 
n 
2/3 
= 0, 
viz. multiplying by 16/3 2 S, and omitting the factor O, this is 
or finally 
21h 2 il + 16,0 8g i = 0, 
U/38f - 27 ( 7 2 + 4aS) fit + 27/3 2 /d = 0, 
a pencil of four lines, each passing through the point g = 0, h = 0, and therefore inter 
secting the conic 
(f + 4aS) f + ¡3 2 lt — 2/3ygh — 4 /3Shf= 0 
at that point and at one other point ; and we have thus four points of intersection, 
which are the required four points. 
Recapitulating, the conic a 2 — 4c<7 = 0 meets the sextic (/ 2 — bh) X = 0 in the two 
points 
\hf 2 -ah 2 + yhf= 0, 
{2/38f — (f + 4a8) g + fiyh = 0 
each three times, in the point g = 0, h = 0 twice, and in the four points 
j'16/3fy 4 - 27 (y 2 + 4<x8) g 2 h 2 + 27/3 2 /d = 0, 
|( 7 - + 4 a8)f + ¡3 2 h 2 — 2/3ygh — 4 /38hf = 0 
each once. Or reverting to the proper significations of (a, b, c, f, g, h), instead of 
points we have 2 lines each three times, a line twice, and 4 lines each once; the 
line /7 = 0, h = 0, that is, <7 = 0, h = 0, a = 0, being, it will be observed, the line 
V = £ 
/3 7 
w 
8 
drawn 
from 
(a, /3, 7 , 8) to the point y = 0, z = 0, w = 0, which is the 
reciprocal of the uniplane X = 0: the twelve lines are the a'c' lines of intersection of 
the circumscribed cone a with the cuspidal cone c', viz. a'c = [uV] + 3a + % ; [a'c 7 ] = 4 
referring to the last-mentioned four lines; a' = 2 to the two lines; and = 2 to the 
line g = 0, h — 0, a = 0, which it thus appears must in the present case be reckoned 
twice.