52 ON THE QUARTIC SURFACES REPRESENTED BY [647 
The equation V = 0 thus represents a quartic surface having a double line, and 
also 8 nodes forming an octad. 
We may without loss of generality write d = 0; in fact, the determinant is unaltered 
if we add to the fourth column 6 times the first column, and then to the fourth 
line 6 times the first line; the determinant is thus of the original form, but in place 
of d it has d + 261 + 0 2 a, which by properly determining 6 can be made = 0. And 
then changing the original l, m, n, the equation is 
Or, writing for shortness, 
a, 
h, 
9> 
l 
h, 
b, 
/> 
m 
9, 
f> 
o, 
n 
1 l, 
m, 
n, 
0 
K = 
a, 
h, 
9 
h, 
b, 
f 
9> 
f> 
c 
(a, b 
c, 
( > g> 
h), 
= 0. 
V = (a, b, c, f, g, h$7, m, n) 2 = 0, 
where the degree of K is 4, and the degrees of a, b, c, f, g, h are 4, 2, 2, 2, 3, 3 
respectively, those of l, m, n being 0, 1, 1 respectively. 
The nodes are, as before, the intersections of the quadric surfaces B = 0, 0=0, 
F = 0, viz. (d being now = 0) the values are 
— B = cl 2 — 2gin + an 2 , 
— O = bl 2 — 2 him + am 2 , 
F = fl 2 — glm — kin + amn. 
But, according to a previous remark, the nodes lie also on the quartic surface K— 0; 
viz. this is a set of four planes intersecting in the line x = 0, y — 0. 
Now, in general, any plane through the line x = 0, y = 0 meets the surface in this 
line twice and in a conic; if the plane is y = Ox, we have 
a, b, c, f, g, h= a', b'a?, do?, fa?, g'x, h'x, 
where a, b', d, f, g', h! are functions of 0 of the degrees (0, 2, 2, 2, 1, 1) respectively; 
and thence also 
a, b, c, f, g, h = a 'a?, h'x 2 , da?, f'x 2 , g'a?, h'x 3 , 
where a', b', d, f', g', h' are functions of 6 of the degrees 4, 2, 2, 2, 3, 3 respectively; 
the equation of the surface thus becomes (a', b', c, f', g, W^lx, m, n) 2 = 0; viz. this 
is a quadric equation which, combined with the equation y — Ox = 0, determines the