434 MR.SYLVESTEa ON THE RESIDOES AND SYZYGETIC MULTIPLIERS TO fx 
where P 
?1. 9! ■ 
is used to denote 
+ + • • • ihq^^—rin) 
R(^ h^...h,) denoting any rational symmetrical form of function whatever of the 
quantities preceded by the symbol R, and q. q,...q, q,.....q^ being any permutation 
of the OT indices 1, 2 , ... 
Suppose /=0 and ?)=0, then a? is equal to one of the series of roots 
hi 
and also to one of the series of roots 
^ 2 * * 
Suppose then that x—K—ri^, 
and consider any term of 
If in any such term (a) is found in the series qi q^-.-qi, then 
(^-A,.)GT-Aj...(.r-y=o. 
But if not, then (a) must be found in the complementary series ■■■> ^r„- 
and consequently will contain a factor K-n. and P,„„..,=0; in every case 
therefore i \ r h \__n 
and therefore 3, as expressed in equation (18.) is a syzygetic function of/and ?; 
accordingly we have found a function of the ith degree in x, and of course expre^- 
ible by calculating the symmetric functions as a function only of x and of the coe. 
cients off and <p, which will satisfy the equation 
T, .f — ^,.<P + ^) = 0. 
[It will be remembered that by virtue of art. (2) we know a priori that all the 
values of satisfying this equation are identical, save as to an allotrious factor, 
which is a function only of the coefficients in / and <p.] It is clear that we may 
interchange the h and % m and n, and thus another representation of a value ot 
satisfying the equation (17-) will be 
i^q+l h\) {llq+2 ^ 2 ) ••• (^ 5+1 
i’lq + 2 ~ hi'){fiq + 2 ” * (^^9 + 2 ^*n‘) 
a, = 2R(>Jg,»7^,...>jOX {fl^ + s—hi){f]q + 3~h2)...{flq + 3 — h,n) 
>{X-riq^{x-Tlqf..{x %). 
Art. (15.). If we employ in general the condensed notation 
~l, m,n, ... p 
X, p, ... p _ 
