136 
A MEMOIR OX QUARTIC SURFACES. 
[445 
5. Suppose that the orders of the surfaces P = 0, Q=0,... are a + 1, 6+1,... (so 
that the orders of the differential coefficients of P, Q... are a, 6,...), then we have 
for the orders of the several loci, 
J {P, Q) = 0, point-system, order a 3 + a 2 6 + ab 2 + 6 3 ; 
J(P, Q, R) — 0, curve, „ a 2 + 6 2 + c 2 + be + ca + ab; 
J(P, Q, R, S) = 0, surface, „ a+b + c + d; 
J (P, Q, R, S, T) = 0, curve, „ ab + ac ... + de; 
J (P, Q, R, S, T, U) = 0, point-system, „ abc + abd ... + def; 
see, as to this, Salmon’s Solid Geometry, Ed. 2, (1865), Appendix IV., “ On the Order 
of Systems of Equations” [not reproduced in the later editions]. In particular, if 
a = 6 = c ... = 1, then the orders are 4, 6, 4, 10, 20. 
As to the Surface obtained by equating to zero a Symmetrical Determinant. 
6. It is also shown (Salmon, Ed. 2, p. 495) that the surface obtained by equating 
to zero any symmetrical determinant has a determinate number of nodes ; viz., if the 
orders of the terms in the diagonal be a, b, c, &c., then the number of nodes is 
= \ (Za . Zab — Zabc), or, as this may also be written, (Za 2 b + 2Zabc). In particular, 
the formula applies to the case of the surface 
A, 
H, 
G, 
L 
= 0, 
H, 
B, 
F, 
M 
0, 
F, 
G, 
N 
L, 
M, 
N, 
D 
(a, 6, c, d) being here the orders of A, B, C, D respectively, and the orders of P, G, &c., 
being | (6 + c), i ( a + c )> ^ c - t'he terms are all of them linear functions of the 
coordinates, or a = b = c = d — l, then the number of nodes is =10. 
7. That the surface has nodes is, in fact, clear from the consideration that any 
point for which the minors of the determinant all vanish will be a node; and that 
(for the symmetrical determinant), by making the minors all of them vanish, we 
establish only a threefold relation between the coordinates. The expression for the 
number of the nodes is, I think, obtained most readily as follows: 
The nodes will be points of intersection of the curve and surface 
A, 
H, 
G, 
L 
= 0, 
B, 
F, 
M 
H, 
P, 
F, 
M 
F, 
G, 
N 
G, 
F, 
G, 
N 
M, 
B T , 
D 
these, however, contain in common the points 
H, 
B, 
F, 
M 
= 0 
G, 
F, 
G, 
R