445] 
A MEMOIR ON QUARTIC SURFACES. 
137 
and not only so, but they touch at the points in question; so that, multiplying 
together the orders of the curve and surface, and subtracting twice the order of the 
point-system, we obtain the expression for the number of nodes. In the particular case 
where the functions are all linear, we have a sextic curve and cubic surface inter 
secting in 18 points; but the curve and surface touch in 4 points, and the number 
ol nodes is (18 — 2.4) = 10. And in the same way the formula may be established for 
the general case. 
8. The subsidiary theorem of the contact of the curve and surface requires, how 
ever, to be proved. Seeking for the equation of the tangent plane of the surface at 
any one of the points in question, we have first 
8B, 
BF, 
8M 
+ 
B, 
F, 
M 
+ 
B, 
F, 
M 
F, 
G, 
N 
8F, 
8C, 
8N 
F, 
c, 
N 
M, 
N, 
D 
M, 
N, 
D 
8M, 
8N, 
8D 
where, in virtue of the equations 
H, 
B, 
F, 
M 
= 0, 
G, 
F, 
G, 
N 
the last term vanishes. Expanding the other two terms, the equation becomes 
D(C8B + B8C - 2F8F) - (N*8B - 2MN8F + M8C) + 8M (FN - CM) + 8N (BN - MF) = 0; 
but, in virtue of the same equations, the coefficients of 8M and 8JS r each of them 
vanish, and we have also 
AT2 
N*8B + M-8C - 2MN8F = ^ (C8B + B8C - 2F8F); 
so that the equation becomes finally C8B + B8C - 2F8F = 0. Investigating by a like 
process the equation of the tangent of the curve 
A, 
H, 
G, 
L 
= 0, 
H, 
B, 
F, 
M 
G, 
F, 
G, 
N 
we find between the differentials 8A, 8B, &c., a twofold linear relation, expressible by 
means of the foregoing equation CSB + BSC - 2FSF= 0, and one other equation; that 
is, at each of the points in question the tangent of the curve lies in the tangent 
plane of the surface, or, what is the same thing, the curve and surface touch at these 
points. 
18