[647 
QUATION, 
iv. (1877), 
3 (x, IJ, z, w). 
for the first 
n common 10 
hen, observing 
ouble line on 
vriting (x = 0, 
cubic surfaces, 
(SB-0, y = 0), 
as. I do not 
647] ON QUARTIC SURFACES REPRESENTED BY A PARTICULAR EQUATION. 51 
I, in fact, take the terms (a, ...) of the determinant to be homogeneous functions 
of (x, y, z, w) of the degrees 
0, 
1, 
1, 
0, 
1, 
2, 
2, 
1, 
1, 
2, 
2, 
1, 
respectively, viz. a, d, l are constants, g, h, m, n linear functions, and b, c, f quadric 
functions of the coordinates; V = 0 still represents a quartic surface; and it appears 
by a general formula that the number of nodes is = 8. But we can easily show this 
directly; and further, that the 8 nodes are the intersections of three quadric surfaces; 
or say that the quartic surface is octadic. For denoting as before the first minors by 
A, ..., then B, C, F are each of them a quadric function of the coordinates, viz. we 
have 
B = d (ac — g-) — cl- — an- + 2gin, 
C = d (ab — h-) — am 2 - bP + 2him, 
F=d (gh — af) + l 2 f + mna - nlh — ling, 
and we have identically 
BG — F 2 = (ad — l 2 ) V, 
so that throwing out the constant factor ad — P, the equation of the surface is 
BG — F 2 = 0, 
and it has 8 nodes, the intersections of the three quadric surfaces B= 0, (7=0, F= 0. 
By equating to zero any other minor of the determinant V, we have a surface passing 
through the 8 nodes; we have for instance the quartic surface 
h, 
a, 
h, 
9> 
9 
b, f 
f> G 
= 0. 
Suppose now (and in all that follows) that, the degrees being as already mentioned, 
we further assume that b, c, f are quadric functions of the form (x, yf, g, h linear 
functions of the form (x, y); then since each term of V contains either 
a, h, g 
h, b, f 
g> f> c 
r one of its first minors, it is clear that the line (x = 0, y = 0) is a double line on 
le surface. But in the present case there is not any diminution in the number of 
le nodes; in fact, writing x=0, y = 0, and therefore b, c, f, g, h each =0 (but not 
— 0^ the minors B, G, F none of them vanish, that is, the line x 0, y 0 is not 
line on any one of the quadric surfaces, and the quadric surfaces intersect as before 
i an octad of points. 
7—2