446 MR. SYLVESTER ON THE RESIDUES AND SYZYGETIC MULTIPLIERS TO (px, fi 
So in like manner, again, if ?;=2 v=2, the coefficient of will be similarly 
reducible to the form 
(C.) 
^1 ^2 
A, h^ 
X 
hs hi As he / 
Aj A2 
% ^4 
Jiq h-j 
X 
^3 ^4 
in 
2 , 2 * 
So, again, when i;=3 p=1, the coefficient of will be 
(D.) 
i 1 
S' 
1 1 
X 
^2 ^3 ^4 
h h h hy_ 
h A 5 h hy 
hi h^ A3 _ 
X 
^2 ^3 ^4 
in ^3, 1 ; 
and finally, the coefficient of will be 
(E.) 
^1 ^2 ^3 ^4 
^5 ^6 ^'7 
U 
h, he hy 
hi h^ h^ h^ 
n 
in ^0,45 
out of all which sums it is to be remembered that % and, are supposed excluded 
from appearing. All these several coefficients being numbers in disguise, we may 
determine them by giving any values at pleasure to the terms in the h and , system. 
Let now ,.=A. then in (B.) it will readily he seen that all the 
terms included within the sign of summation vanish identically, except the following. 
1 1 
bd 
1 1 
X 
hi A2 A3 A5 Ag hy _ 
hi h^ A3 As he hy 1 
U J 
^4 
^2 i^ 3 _ 
ni J?2 ^4 
A3 
X 
^3 
Ai A2 A4 As Ag Ay_ 
Aj A2 hi As Ag hy 
_ A3 _ 
X 
^3 
.'t?! ^2 ^ 4 _ 
^1 ^3 ^4 
A2 _ 
r 
^ 1 hi A3 A4 As Ag hy 
Ai A3 A4 As Ag Ay 
X 
^2 
A2 
_?Ji ^3 ^4 
1 1 
1 _l 
X 
nx 
A2A3 A4A5 AgAy_ 
A 2 A 3 A 4 Aj Ag Ay 
X 
% ^ 4 _ 
In each of these expressions the first factor of the numerator is identical in value 
