50 
[647 
647. 
ON THE QUARTIC SURFACES REPRESENTED BY THE EQUATION, 
SYMMETRICAL DETERMINANT = 0. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), 
pp. 46—52.] 
Consider the equation 
V = 
a, h, g, l =0, 
h, h , f, m 
g, f, c, n 
l, m, n, d 
where for the moment (a, b,...) denote linear functions of the coordinates (x, y, z, w). 
This is a quartic surface having 10 nodes; viz. if we write {A, B, ...) for the first 
minors of the determinant, then the cubic surfaces A = 0, B= 0, ... have in common 10 
points which are nodes of the quartic surface. 
Suppose that (a, b, c, f, g, h) are linear functions of the form (x, y); then, observing 
that every term of V contains as a factor 
a, h, g , 
h, b, f 
g> f> c 
or one of its first minors, it is clear that the line x — 0, y = 0 is a double line on 
the surface. But the number of nodes is now less than 10; in fact, writing (x = 0, 
y = 0), we make each of the first minors of V to vanish; that is, the cubic surfaces, 
which by their intersection determine the nodes, have in common the line (x = 0, y — 0), 
and there is a diminution in the number of their common intersections. I do not 
pursue the enquiry, but pass to a different question.