206 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
From the above 
[y (x — a) + h{y — b)J + (s 2 - c 3 ) — 0, 
therefore 
y(x — a) + S (y - b) = v / (c 3 — z~) . 
Hence by (111) and (112) 
x-a = ± \/{-(V-0' ,! } 
y-b = ± 
(112) 
Substituting in (112), 
±y -\/{“ = 4 cW ) l,2 }± 
therefore 
{- a (« + S ±2r ® 
therefore 
therefore 
7° 
|8/ 
+ ^l±2 r S 
a/ 
4yS 
«/3 
(c~ - 2 2 ) 
1 2 
4 (y 3 /3 4~ S 3 a) 2 -j- 16a/3y 3 S 3 4: 1 6 (y 3 /3 4" S 3 a) a 1 " /3 1 2 y 3/2 S 3 2 = a 2 /3 2 (c 3 — 2 2 ). 
Therefore 
| a 2 /3 2 (z 2 — C 2 ) d~ 4 (y 3 /3 4" 8'V/)' 4~ 16«/3y 3 S 3 ] 2 = 2Soapy 'S 3 (y 3 /3 4" 8 3 a) 2 . 
This reduces to 
[a 2 /3 2 (z 2 - c 2 ) 4- 4 («S 3 - (Sy 3 ) 2 ] 2 4- 64a 3 8 3 y 3 8 3 (z 2 - c 2 ) = 0. 
This accounts for the remaining- factor in the discriminant, 
ordinary envelope. 
It corresponds to an 
