170 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
It is given that all the values of x. y, z which make <f>(x, y, z) = 0 , also make 
0 ” + 1 'yjr 
dx r + 1 dy s dz n ~ T ~ s 
Now let £ + S£, y + &y, £ + §£ be a point near to £, y, £ on <f> ( x, y, z) = 0 . 
Therefore <f> (£, y, £) = 0 . 
(f + y 4- l + K) — °.' ■ 
(!)• 
And since £ y, £ make 
0 ,l + 1 i|r 
d £ r+1 dr) s d£“- 
= 0 
0” yfr 
dx r dy s 02 ”" 
0 , 
( 2 ). 
for this is a differential coefficient of the n th order, £ + S£ 77 + By, £ + S£ must also 
do the same. 
Hence 
(§0 
0 ” + 1 
dgr + 1 dyS 
a;-, + ( s v) 
0" +1 -<|r 
0£ r 0?? s + 1 d£‘ 
UAA + ( S 0 
0»+i ^ 
d£ r dr/ s d£’‘- 
-s +1 
= 0 . 
Also from ( 1 ) 
(«) | + (&») | + (»»| = 0- 
Since this is the only relation between 8 £, 817 , §£, it follows that 
/ c u + 1 \fr \ / d >l+1 ifr \ / 0” + 1 y/r \ 
\0p +1 0j^0f“- r_ 7 _ \0£ r 0/? s+1 d£ n '~ r ~ s ) _ \8|y 077* 8g”~ r ~ * +1 / 
“ST = "IT -= ' m 
Hence, by means of ( 2 ), 
d n + l yjr 0” + 1 i/r 
d£ r dr) s+1 d£ n ~ r ~ s ~ ’ 0| r 0>f 0£'' i ~ r-,s + 1 ~ 
In this way it is possible to pass from any one partial differential coefficient of 
order (n + l) by successive steps to any other of order (n + 1 ); at each step always 
diminishing by one the number of differentiations with regard to one variable, and 
increasing by one the number of differentiations with regard to another variable. 
Hence all the differential coefficients of the (n -f- l) th order vanish when x = £, 
y=zy,z=£. 
