272 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
The locus is, therefore, generally a finite number of points. 
The values of x, y, z are independent of the parameters. 
(B.) If two of the three expressions xJj, <f>, y, say xfj, <f>, have a common factor 6, then 
the curve 6 = 0 , y = 0 is a part of the locus of ultimate intersections. 
(C.) If the equation of the system of surfaces be transformed to plane coordinates, 
then a point has an equation, and the locus of ultimate intersections would have 
an equation, which could be determined as a factor of the discriminant. 
III. The Fundamental Equations cannot be satisfied by any values of the coordinates 
which make the Discriminant vanish, the 'parameters being finite. 
Example 20. 
Let the surfaces be the spheres 
(z + c) (a° + b~) — (c + cl) (2 ax + 2 by — x 2 — y z — (z — c) (z — d)} = 0, 
where c, d are fixed constants; a , b are the parameters. 
They all touch the plane z = d, and the sphere x 3 + r + * 3 = c 3 . 
(A.) The Discriminant. 
It is 
z c 0 — (c + cl) x 
0 z -f c — (c + d) y 
— (c + d) x — (c + d)y (c + cl) {x z + y 2 + (z — c) (z — d)} 
= (c + d) (z — d) (x z + y* + z 2 — c 3 ) (z + c). 
(B.) The Plane z — cl = 0 is a part ofi the Envelope. 
(C.) The Sphere x 2 + y 1 -j- s 2 — e 3 = 0 is a part ofi the Envelope. 
(D.) The remaining factor z -j- c requires explanation. It is on account of this 
factor that this example is introduced. 
Ifz + c = 0, the left-hand side of the equation of the system of surfaces, which is 
of the'Second degree in a, b breaks up into two factors, one of the first degree in a, b. 
the other of degree zero. 
But the fundamental equations being equivalent to 
a (z + c) — x (c + d) — 0, 
b (z + c) — y (c + d) =0, 
— a (c + cl) x — b (c + cl) y + (c + cl) [cc 3 + y 1 + (z — c) (z — d)~\ = 0, 
cannot be simultaneously satisfied by finite values of a, b when z -\- c — Q. 
