144 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
But the conclusion fails if 
w 
I>£ 
= 0 . 
(12). 
Hence the work itself suggests the examination of this exceptional case, i.e., where 
a locus of singular points or lines exists. 
Example 1.— Envelope Locus. 
Let the surfaces be 
[<£ {x, y, z) — of + y ( x, y, z) = 0. 
(A.) The Discriminant. 
The discriminant is found by eliminating a between the above equation, and 
— 2 [</> ( x , y, z ) — a] = 0. 
Hence the discriminant is y(a?, V’ z )- 
Hence the locus of ultimate intersections is 
X ( x > V> z ) = o. 
(B.) The envelope locus is y [x, y, z) = 0. 
For let 7], £ be any point on y (x, y, z) = 0. 
Let a = y, £), and consider the single surface 
[(f) (x, y,z) — cf) (£ 7j, £)] 3 + x ( x > V> z ) = °- 
Put x = £ +X, y = 7} + Y, z = C + z. 
Then the lowest terms in X, Y, Z are 
X 
* _P Y ^ -f Z 
dr) 
Hence the surface considered touches y (x, y, z) — 0 at y, £, 
Hence y (x, y, z) = 0 is the envelope. 
It touches the surface at every point of the curve 
y ( x , y, z) = 0, 
<f)(x, y, z) = (f> (£ 7), £). 
Hence this curve is the characteristic. 
