[649 
PAPER, 
7. (1877), 
onstrated by 
investigated 
ration is in 
and let v} 
3 of the t-th 
Iving higher 
like manner 
is the same 
the formula 
is a rational 
3d in terms 
ave any one 
! factor i — s 
vanishes for 
Inch case it 
649] ADDITION TO MR GLAISHER’s NOTE ON SYLVESTER^ PAPER. 59 
where the summation extends to the several values s=0, 1, 2,.., n; or, what is the 
same thing, it is 
u l = £ 
{i .i-l .i-2 ...i-n 
l—S 
or, changing the sign of i, it is 
(i. i 4* 1. i + 2 ... i + n 
u~ i = £ 
i +s 
1.2 ... s. 1.2 ... (n — s) 
til *1 
1.2...S.1.2 ...n-s J ’ 
where, as before, s has the values 0, 1, 2,.., n successively. Or, what is the same 
thing, we have 
C—i. n = 2 
\i. i + 1. i + 2 ... i + n 
i+ s 
(-)* 1 n I 
1.2...*.X.2...»-s S '”J' 
where the term corresponding to s = 0, as containing the factor C 0i n vanishes except 
in the case ?i = 0 (for which it is =1); and omitting this evanescent term, this is in 
fact the formula (11). 
8—2