“OX EQUAL ROOTS OF EQUATIOXS.’ 
407 
778] 
Similarly, when v — 2, the condition for n — 2 equal roots is found to be 
dfn Ctr+m —1 ^ 3 ®7rt+2 ®r+)/X—2 ^m+3 d r ^.fYi—3 
r —l.r —2.r —3 r — l.r — 3.r — 4 r — 2.i—S.r—5 
3. r — 4. r — 5 
= 0 
or, what is the same thing, 
r - 4 . r - 5 . a m cu+m 
3.r 2.?' O. i 
+ 3. r 1 . r 4. cLr jf.y/1—2 
. 1. 2 . (Ln+3 u J .-f m _ 3 = 0, 
where ?i = 6 at least, and to, r have the values 
TO = 
o, 
1, 
n — 6 
r = 
6, 
6, 
6 
7 
n — 1 
n 
Observe that the sum of the coefficients is = 0, viz. 
(r-4) (r — 5) — 3 (r — 2) (r — 5) 4- 3 (r — 1) (r — 4) — (r — 1) (?' — 2) = 0, 
this should obviously be the case, since the conditions for n — 2 equal roots must 
be satisfied when the roots are all of them equal; and the property serves as a 
verification. 
It is to be remarked that the equation -ifr (r, v +1, m) = 0 does not in all cases 
give all the conditions for the existence of n — v equal roots in an equation of the 
order n; thus when n = 3 and v = l, we cannot by means of it obtain the condition that 
a cubic equation may have 2 equal roots. The problem really considered is that of 
the determination of those quadric functions of the coefficients which vanish in the 
case of n — v equal roots; and in the case in question (^ = 3, v = 1) there is no 
quadric function which vanishes, but the condition depends on a cubic function. 
The question of the quadric functions which vanish in the case of n — v equal 
roots, and to a small extent that of the cubic functions which thus vanish, is considered 
in Dr Salmon’s “Note on the conditions that an equation may have equal roots,” 
Camb. and Dublin Math. Jour., t. v. (1850), pp. 159—165, and in particular the 
equation there obtained p. 161 is the equation ^(0, v + 1, r?) = 0.