450 MR. SYLVESTER ON THE RESIDUES AND SYZYGETIC MULTIPLIERS TO fx 
Now make x—h, then/=0, and ^ becomes 
^24 ^23 
^1 ^2 ^3 ^ 4 _ 
.K K j 
i. e. 2 
[A, 1 
n 
A 4 A 5 
% % ^ 4 _ 
Ai /22 
KK 
Ai being kept constant in the above sum, but h^, h^, /l, h being partitionable in all the 
six possible ways into two groups, as into Jh, h„ ih, h in the term above expressed. 
This sum is evidently identical with 
^1 ^2 ^3 ^4 
hi h^ h^ 
_ ^1 ^2 ^3 ^4 _ 
, i. e. 
[Kh-^ 
_ ^1 ^2 ^3 ^ 4 _ 
/l h 
x2 
' h . 2 A3 
\_Kh. 
Again, ® becomes 
Hence t=- becomes 
<P 
^1 ^2 ^3 ^4. 
A2 A3 
_ ^1 ^2 ^3 ^ 4 _ 
But when a?=A] 
G 
(-) 
• becomes 
i. e. 
A2 A3 
A4 A5 
A, ■ 
_ A2 A3 
'A, 
A„ Ao 
A 2 A3 
_ ^2 ^3 ^4 
A2 A3 
A 2 A3 
lAi A4 A, 
~hh 
. _??1 ^2 >?3 ^ 4 j , 
Ag A3 
LA4A5 
Jh 
Thus when x=h^, t=.G. In like manner, when x=h,, or A3, or h^, or A5, t always =G ; 
but t and G are both functions of x of the same degree, and of only two dimensions 
in Hence t is identical with G. So in general it may be proved, that whenever 
x—K or A2 or A3... or A„, t and G, which are each of only (w— 1 — i) dimensions in .r, 
