406 
ADDITION TO MR HUDSON’S PAPER 
[778 
that is, 
ClmClr+m ®)n+i®r+«i—l = 0, 
satisfied when the equation has all its roots equal. 
The values of m are 0, 1, 2, — 2, and those of r are 2v + 2, 2v+3, n — m; 
in particular, if m — 0, the values of r are 2, 3, ..., n, and the corresponding conditions 
are 
a 0 a 2 — Uj 2 =0, 
a 0 a 3 — a x a 2 = 0, 
Ct {l (t n ttittn—l — 0, 
and so for the different values of m up to the final value n — 2, for which r = 2, 
and the condition is 
tt'n—2 ®'/J Q? n—1 = 0 , 
we have thus, it is clear, the whole series of conditions included in 
= 0, 
which are obviously satisfied in the case in question of the roots being all equal. 
Again, when v = 1, the condition for n — 1 equal roots is 
-, 1 
V . I . t o &r+m 
11 do ) 
ftx, 
u 2 , .. 
., Ct ,i— 2 , 
1 | 
|| ai, 
a 2 , 
«3, •• 
• j l > 
i 
r. r — 1.1— 2 
(r — 2) . 2 . ■= ^ s 
v ; r — l.r — 2 . r — 3 
1 
r — 2 . r — 3. r — 4 
®m+l m—l f 6, 
+ (r - 4). 1. 
2 Ur+rn—2 
that is, 
d'un Clr+m l , ®m+2 ^-r+m—2 q . 
r — 1. r — 2 r — 1.7— 3 ?— 2 . r — 3 ’ 
or, what is the same thing, 
(?’ — 3) Ct m Cl r + m — 2 (r 2) Ur/i-fittr+m—l H" 0’ 1) ^m+2®r+w—2 6, 
where n = 4 at least, and m, r have the values 
m == 
0, 1, 2, .. 
., ?i — 4 
r = 
4, 4, 
4 
5, 5 
. . 
j n — 1 
n 
thus, when n = 4, the only values are m = 0, r = 4, and the condition is 
a 0 a 4 — 4axa 3 + 3a 2 a = 0.