AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 175 
where a, b are independent arbitrary parameters, and f is a rational integral 
indecomposable function of x, y , 2 , a, b. 
The locus of ultimate intersections is obtained by eliminating a and b between 
(3) and 
D/ (x, y, z, a, h) _ 
Da 
D/ {x, y, z, a, b ) _ 
D6 ~ 
where D denotes partial differentiation when x, y, z, a, b are treated as independent 
variables. 
Let the result of the elimination be A = 0, then A is called the discriminant. 
If x, y , z are chosen so as to make any factor of the discriminant vanish, it indicates 
in general that it is possible to satisfy equations (3), (4), (5) by the same values 
of a, b. Hence x, y, z can be expressed as functions of a, b. 
In this case x — cp {a, b), y = ip (a, b), 2 = x ( a > &)• 
Eliminating a and b a surface locus is obtained. 
This is the general case. The exceptional cases are noticed in Section VI., Art. 30. 
(5), 
Art. 3.— The Loci of Singular Points of the System of Surfaces, 
The equation of the locus of singular points on the surfaces (3) can be obtained by 
eliminating a and b, between (3), and 
D/ (a. y, 2 , a, b) _ 
Dx ~ 
D f (pc, y, g, a, b) . 
% ~~ 
D/ (a;, y, z, a, b) _ 
I)z ~ 
The singular points are in general conic nodes. 
The locus of conic nodes is therefore a curve, whose equations are given by 
eliminating a, b between (3), (6), (7), (8). 
It follows, also, by eliminating x, y, z between the same equations, that there is a 
definite relation between a, b. 
If y, t, be the coordinates of the conic node on the surface 
( 8 ). 
( 6 ), 
f{x, y, z, a, 0) = 0 
( 9 )> 
