436 MR. SYLVESTER ON THE RESIDUES AND SYZYGETIC MULTIPLIERS TO (px, fx 
=0 for all values of v and v which make v-\-v=i, and for all symme- 
trical forms of the function denoted by the symbol R( ; )• 
Art. (16.). 1 shall now proceed to show how to assign the arbitrary function whose 
form is denoted by this symbol in such a manner as to make become identical 
with a simplified residue to f and (p. To this end I take for 
the value 
'KK-K~^ 
\K 
^92 
K • 
h 
h .../?„ 
X 
h. 
.X 
1 ^tJ + 2 _ 
L 
1 lv + 2 
5„J 
we shall then have 
(20.) 
hqhq^.. .hq 
K 
...h„ 
y + l ®i/ + 2 •‘m 
Tj. ^ VI ...\ 
&+1 iv+2 in 
K K. ' 
■X 1 
X 
% 
\+1^9„ + 2- 
s 
yir 
lv+\ |i/ + 2 
{{x-hq){x-~hq)...{^~hq)}{{x-r,^){x-r,Q...x- 
I shall first show this sum of fractions is in substance an integral function of the 
quantities For greater conciseness write in geneial x h E, 
j:— we have then, since h—n=zY{—R, 
"H|^H|,.. 
•HiH 
H| Hg ...H| 
‘3V-1-2 n 
&-2< 
_E„E,^.. 
l'^ _ 
X 
E, E„ ...E, 
_ ^v + l ^v + 2 ^mj 
..E, 
i 
X 
rHs H| ...H„ 
^v + l ^v + 2 
1 
[Hi. H?, ...H|J 
(22.) 
On reducing the fractions contained within the sign of summation to a common 
N . j f I w— I 
denominator, will take the form where D will be the product oi the in . g 
differences of E„ E^, ...E,„ subtracted each from each, and A the corresponding 
product of the differences inter se of Hi, Hj, ...H„. 
Hence, unless the sura in question is an integral function of the E’s and H s, it will 
become infinite when any two of the E series, or any two of the H series of quantities 
are made equal. Suppose now Ei=E 2 ; the terms in (22.) which contain Ej— E, m 
the denominator will evidently group themselves into pairs of the respective forms, 
(E,.E^^...EJx(H|jH|,...H|^) X 
"E, E,...E, 1 
X 
■E.3 E„ ...E, 1 
9, + j 9„, 
Hg Hg ...Hg 
-E.E.. ...E,/ 
_e,e,.^,...e,^_ 
X 
H|, H|„ ...] 
Hg Hg ... 
L ^*’+1 ^J' + a 
