234 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Art. 18.— To prove that under the conditions stated at the head of this Section , every 
Surface of the System touches the Locus of Ultimate Intersections along a Curve. 
Consider the surface (142), the values of a, b being now supposed to he fixed. 
Consider any point y, £ on the curve in which the surface (143) meets the locus 
of ultimate intersections; then, by (158), these coordinates also satisfy the surfaces 
(144), (155). 
Multiplying (143) by a, (144) by b, (155) by 1, and adding, it follows that these 
coordinates also satisfy (142). 
Hence any point on the curve of intersection of (143) with the locus of ultimate 
intersections lies on (142) and (144) also. 
Hence the surfaces represented by the three fundamental equations meet the locus 
of ultimate intersections in the same curve. 
It is necessary to prove that the surface of the system (142) will touch the locus of 
ultimate intersections along this curve. 
N o w, 
D 
Dx 
(ua z + 2W ab + vb z -f- 2Ya + 2U b -f w) 
JD 
D,r 
-{(ua + Wb + V) 2 -f b z (uv - W 2 ) + 2 b (U u - VW) + (uw - V 2 )} 
- {(ua + W b + V f + IT (uv - W 2 ) + 2 b (Uw - YW) + (uw - V 2 )} 
+ - 12 (ua + Wb + V) ~ (ua + W b + V) 
T)x 
+ U L p ; (uv - W 2 ) + 2 b // (Uw - YW) + ~ (uw - Y 2 ) 
D,r 
Hence, at a point on the locus of ultimate intersections, this is equal to 
1 
u 
W 2 ) + 2b £ (U u - YW) + (uw 
Hence the tangent plane to the surface at the point x, y, z is 
(X - x) | U ^ (uv - W 2 ) + 2b ~ (Uw - VYV) + ~ (uw ~ Y 2 ) j 
+ (Y - v) {^ 2 1 (uv - W 2 ) + 2b | (Uw - YW) 4- | (uw - V 2 )} 
4- (Z - z) j& 2 f y (uv - W 2 ) 4- 2 b~ (Uw - YW) 4 | (uw - \~) j 
= 0. 
