778] 
405 
778. 
[ADDITION TO MR HUDSON’S PAPER “ ON EQUAL ROOTS OF 
EQUATIONS.”] 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvm. (1882), 
pp. 226—229.] 
It seems desirable to present in a more developed form some of the results of 
the foregoing paper. 
Thus, if the equation (a 0 , a 1} ..., cy$#, l) ,l = 0 of the order n has n — v equal 
roots, where v is not > \n — 1, then we have yjr (r, v +1, m) = 0, where m has any one 
of the values 0, 1, ..., n— 2v — 2, and r any one of the values 
2v + 2, 2v + 3, ..., n — m. 
The signification is 
r 
1 
' [ r ]«+2 
a m 
a r +m 
-0-2) 
V + 1 
1 
a m +i 
ttr+m—x 
1 
• [r _ l]»+2 
1 . 
+ 
v + 1. v + 2 
1 
ef-m+2 
^»•+»1—2 
1.2 
' [r _ 2p+ 2 
+ (~) s O - 2s ) 
[v + l] s 
1 
Mr+m—s 
№ 
’ [r — Sp +2 
+ (—) v+1 (r — 2v — 2) . 
1 
. Q-r+m—v- 
' [r — V — 1]" +2 
Thus, when v = 0, the condition is 
r . r — 1 
«1» ttr+m 
J. 
— (r — 2) _ i r _ £ a m+l a r+m-i 
[ = 0,