212 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Next, when x = Or y = y, z = 0 a 2 = a + Sa', b 2 = /3 + S/3', 
y(a?, y, z, a 2 , +) becomes 
/(£ y, L v-, (3) 
+ [f] (§0 + M (&?) + [£] + [“] ( Sa ') + [Z 3 ] (W 
+ 
1 
2 
[£ £] ( 8 £) 2 + b ’ ( §7 ?) 2 + [£> £] ( 8 0 2 
+ 2 fo, {] (&,) (so + 2 [0 0] (so (80+ 2 [0, (so (so 
+ 2 ( So O {[£ «] ($£) + [y, «] (%y) + [£> «] (§01 
+ 2 (S/3) {[0 /3] (SO + [17, £] (SO + [0 £] (SOI 
+ [a, a] (Sa) 3 + 2 [a, /3] (Sa) (3/3') + [/3, /3] (S/3 ') 2 
+ terms of the third and higher orders in SO By, SO Sa', S/3' 
(terms of the second order in SO, Sr;, SO 
+ (Sa') (terms of the first order in SO, Sr;, SO 
+ (S/3') (terms of the first order in SO, Sr;, SO 
+ 2 ^ 0 d * ^ a ’ ^ + t a » W 2 
+ terms of the third and higher orders in SO, Sr;, SO Sa', S/S'. 
Now, the terms of the second order in SO, Sr;, S£ are of order eh 
The terms containing Sa' or S/3', multiplied by terms of the first order in SO, Sr;, SO 
are of order e 8/ h 
The terms - 7 - 
2 [«, « 
[a, a] (Sa') + [a, /3] (S/S') is of order e, by the same argument as the one which was 
applied to show that [a, a] (Sa) -fi [a, /3] (S/3) was of order e. 
The most important terms of the third and higher orders in SO, Sr;, SO Sa', S/S' are 
of order e 3/3 . 
Hence f (x, y, z, a 2 , b 2 ) is of order e 3/3 , when x = 0, y — y, z — 0 the coordinates of 
a point on the conic node locus. 
F urther 
K, K, &]] 
! [ ft i> ^ 1 ] [^n ^ 1 ] 
becomes 
1 
{[a, a] (Sa') + [a, f3] S/3' 
are of order e 3 , since 
[a, a] [a, /3] 
[a, /SJ [(3, /S] | 
20 
Dr, 
+ (K) 
10 
cr 
+ (Sa) 
D 
D ct 
[a, a] [a, /3] 
[a, /S] [/3, /3] 
+ terms of the second and higher orders in SO, Sr;, SO Sa, S/S. 
