182 PROFESSOR H. J M. HILL ON THE LOCUS OF SINGULAR POINTS 
equation only. Similarly (J 6 ) and (18) are equivalent to one independent equation 
only. 
H ence the following ratios hold 
[£ : K y\ ■■ [£ a : [£ «] : K P] 
= bh f\ ■ [y> y\ '■ [y, Q ■ [ y > a ] : iy> P\ 
— [£> £] ; [L y] : [£, £] ■ [£, «]: [£, P] ...... (47). 
Applying (47) to (16) and (28), it follows that these two are equivalent to one 
independent equation only. 
Similarly (16) and (29) are equivalent to one independent equation only. 
Hence the five equations (16), (17, (18), (28), (29), are equivalent to one indepen¬ 
dent equation only, and, therefore, the following ratios hold 
K a 
K 
Q 
[£ 
«] 
= [>?. a 
[y> y] 
Iv, 
a 
Ab 
a] 
[>?> A] 
= k, a 
[£. y\ 
[£> 
Q 
[£> 
a] 
K. /3] 
= o, a 
[«> 17 ] 
[*» 
£3 
[a, 
a] 
[a, /3] 
= [p a 
[A y] 
[A 
a 
[A 
a] 
:[AA • 
Art. 1 .-—Examination of the Form of the Discriminant, and Calculation of its 
Differential Coefficients of the First and Second Orders. 
Let a } , b l ; a 2 , b 2 ; ... be the common roots of (4) and (5), and let it be supposed, 
in the first instance, that at points in the loci considered these sets of common roots 
are all distinct. 
Then if 
A = A f(x, y, 2 , a„ b 1 )f(x, y, z, a 2 , b. 2 ) . . . (49), 
where A is a factor introduced to make the discriminant of the proper order and 
weight, the result of eliminating a and b between (3), (4), and (5) is 
A = 0. 
Writing for brevity 
A — Rf(x, y, X, &]) = B/.(50), 
3a _ 0R f -p (W , ff , Vf &A . 
ox ox J 1 \] ),x I )c, ?jx L)b l dx) 
To determine dajdx, dbjdx there are the equations 
jy 
Dcq 
- 0 , 
1 )/ 
= o 
(51). 
