226 
PROFESSOR M. J. M. HILL ON THE LOCHS OF SINGULAR POINTS 
Art. 17.— The relations ivliich hold good at points on the Locus of Ultimate Intersections. 
(A.) The analytical condition (76) which holds, becomes with the notation of this 
section 
uv — W 2 =0.(153). 
Hence the values of a, h given in (145) are either infinite or indeterminate. 
Excluding the cases where they are infinite, it is necessary to have 
WD - vY = 0 ) 
WV - uJJ = 0 J 
Again, by substituting from (143) and (144) in (142), it follows that 
Y a + U6 -f- w — 0.(155). 
Solving (144) and (155) for a, b, it follows that 
a = 
b = 
vw — U 2 
WU - Yv 
uv - Ww 
WU - Yv 
(156). 
J 
Hence by (154) these values will be infinite unless 
Hence by (153), (154), (157) 
vw — U 2 = 0 
UV - W w = 0 
u : W : Y 
= W : V : U 
= V : U : w 
(157). 
(158). 
Now if P = 0, Q = 0 represent any two of the five equations (153), (154), (157), 
then these are satisfied at every point of the locus of ultimate intersections. 
Let £ g, C and £ + g + hrj, £ + S£ be neighbouring points on the locus of 
ultimate intersections. 
Then 
0Q 
0 £ 
0P 
0 ? 
0Q 
0C 
S£ = 0, 
sc = o 
