192 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
[£ £ a] [ft ft - [£ £ ft [a, ft 
+ 2 | {[#, a, a,] [fi, 13] - [f, «, /3] [a, /3]) + 2 | ([#, a, /3] [fi, /3] - [f, fi, /3] [a, /3]} 
+ (|t) {[a, a, a] [/3, /3] — [a, a, fi] [a, /3]} 
+ 2 (|) (!){[«, «, /3] [yS, 13] - [a, 13,13] [a, /3]] 
+ (|)"{[“.Affl[A«-[A/3,/3][a,^] j = 0.(79), 
and this equation with either of the equations derived from (52) by changing therein 
x into £ y into y, z into £ and therefore a x into a, b x into ft determines in general two 
values for 3<x/0£ and two corresponding values for 0/3/3£ 
The second of equations (52) gives 
[/3,f] + [«,ffl| + [A/3]| = 0.(.80). 
Eliminating 0/3/0£ between (79) and (80), it foliows that 
0«\2 
a* 
+ 2 
[a, a, a] [ft ft 3 — 3 [a, a, ft [ft ft 3 [a, ft+ 3 [a, ft ft [ft ft [a, ft 2 —[ft ft ft [a, ft 3 
[£ a] [ft ft 3 — 2 [£ a, ft [ft ft 2 [a, ft + [£ ft ft [ft ft [a, ft 2 
- [a, a, ft [ft ft 2 [ft £] + 2 [a, ft ft [ft ft [a, ft [ft ft - [ft ft ft [a, ft 2 [ft ft 
+ 
[£ £ a] [ft ft 3 — [£ £ ft [ft ft 2 [ a > P] 
-2 [£ a, ft [£ ft [ft ft 2 + 2 [£ ft ft [£ ft [ft ft [a, ft 
+ [£ ft 3 [«, ft ft [ft ft - [£ ft 2 [(8, ft ft [*, ft 
= 0 
(81). 
This is in general a quadratic for 8a/8£ 
The two corresponding values of 3ft3£ are given by (80). 
The case of exception, when the quadratic for 3a/3£ reduces to an equation of the 
first degree, viz., when 
[a, a, a] [ft ft 3 — 3 [a, a, ft [ft ft 2 [a, ft+3[a, ft ft [ft ft [a, ft 2 — [ft ft ft [a, ft 3 =0, 
will now be considered. 
(B.) The meaning of the condition may be determined by means of Art. 1, Pre- 
iminary Theorem D. 
