198 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Hence the roots of (90) and (91) give rise to the portion 
(W(9£ + ^)f, 
and the roots of (92) and (93) to 
Hence the discriminant is 
(Z,z»wp( 8H? + + ,») + i'iAIA 
(zww («ip+ : '£ (/ 3 * s - «j/ 8 ) + “y’y 
Z 1 Z 2 Z 3 Z t \ r r 
*/9 
£ ,; [{9a/3£ + 4 n (fix 6 — a?/ 3 )} 2 -f 64?i 2 «/Trk/ 3 ]. 
This agrees with the result previously stated. 
Returning now to the part of the discriminant arising from the two systems of roots 
of (90) and (91), it will be shown that the factor £ 3 arises entirely from one of the 
systems only. 
Consider, in fact, 
(£Zi + xX x - 2/Yi) (HZ, + xX 1 - yYJ, 
which is the part of the discriminant due to the system of roots X 1? Y 1} Z v 
It is equal to 
where 
Therefore 
(£Zi + &,) (3CZ, + £X,) = Zf 2 {3? + 4ft (Xj/Zj) + f (X^Zj) 2 } 
« {XJZ{f — Znxi; (Xj/Zj) — 2??x£ = 0. 
<£Z, + - j/Yj) (S{Z, + xX, - !/¥,) = S{» + ■A'S? + A ( 4 ft + AA 3 
Now, 
X 1 __ nxg_ y/ (nh s g 2 + 2 nxu%) 
rj —-I... 
yyf ny| r «£ __ 2 » 3 £ 3 
« « | ~ 2 ?ivc 2 £ 1 2 
If the root corresponding to the positive sign be taken, then 
(£Zi + ajX x — yYj) (3£Zi + xX Y — yY x ) 
is not divisible by £, 
