442 MR. SYLVESTER ON THE 
RESIDUES AND SYZYGETIC MULTIPLIERS TO (px, fx 
Hence the weight of the leading coefficient in the lowest-weighted conjunctive of 
y’and <p of the degree / in x is (m — i){n — i), m being the degree of/* and n of p. 
From this we infer that any conjunctive of / and <p of the degree /, of which the 
leading coefficient is of the weight (all the coefficients being of course 
understood to be integral functions of the roots of / and (p), must, to a numerical 
factor pres, be equivalent to any other of the same weight ; and furthermore, any 
supposed functionof 0 ^ of the /th degree which possesses the property characteristic of a 
conjunctive of vanishing, when/ and <p vanish simultaneously, but of which tbe weight 
of the leading coefficient would be less than {m—t)(n—i), must be a mere nugatoiy 
form and have all its terms identically zero*. 
Art. (21.). We have previously shown, art. (16.), that S-, as defined by equation 
(21.), is an integral function of the roots/ and <p, and vanishes when/ and p vanish. 
Moreover, its weight in the roots has been proved to be (m— /)(w— /), and consequenth , 
if by way of distinguishing the several forms of we name that one where ; in the 
equation above cited is supposed to be divided into two parts, and v, we have 
for all values of v and ^ such that v-{-v is not greater than n, to a constant nume- 
rical factor prh identical with the («;+0th simplified residue to (/, <p), so that the 
form of depends only upon the value of v-\-v. 
Art. (22.). It must be well borne in mind that this permanency of the value of 
for different values of v has only been established for the case where i can be 
the degree of a residue to /and <p, that is to say, when i is less than the lesser of the 
two indices m and n. When i does not satisfy this condition of inequality, the 
theorem ceases to be true. It is clear that when m-n and v-^v=7n=n, which 
always remains a conjunctive of/ and <p, can only be a numerical linear function of/ 
and (p ; and I have ascertained when m~n on giving to v and v the respective values 
successively (0, n), (1, w—l), (2, (w— 2)), ... (w, 0) 
that 
Thus, by way of a simple example, let 
ax b — (.r — hi^i^x ho) 
(p = x^+ax-\-(5 = {x — 7^i){x — ri.,) 
X,= {x-h,){x-h,') 
lljl^ 
X 
_ _ 
~ hdi^~ 
=z{x—h,){x—h.^=f 
* And more generally it admits of being demonstrated by precisely the same course of reasoning, that tlie 
number of arbitrary parameters in a conjunctive of the degree i, and of the weight (m-i)(n-i) + s in the roots 
cannot (abstraction being supposed to be made of an arbitrary numerical multiplier) exceed the number s. 
