444 MR. SYLVESTER ON THE RESIDUES AND SYZYGETIC MULTIPLIERS TO (pJ', fjc 
weightlower than the lowest-weighted simplified residue of the degree i—a. Hence 
a is not greater than i—n or {m — i){i—n) is not greater than i—n, i. e. m — i cannot 
be greater than 1, i.e. i when intermediate between m and n cannot be less than 
otherwise will vanish identically. Moreover, when i=m— 1, u — i—n, and 
i—ai—n, and accordingly is not merely, as we might know, a priori an alge- 
braical, but more simply a numerical multiple of for all values of v. The same is 
of course true also, m being greater than n, for every form since this is always 
a conjunctive of / and <p, of which the former is of a degree higher than the ^ in 
question, so that the multiplier of/ in this conjunctive must be zero*. 
Art. (25.). To enter into a further or more detailed examination of the values 
assumed by\, for the most general values of m, n, i, would be to transcend the limits 
I have proposed to myself in drawing up the present memoir. What we have esta- 
blished is, that to every form of appertaining to a value of i between 0 and n, 
there is a sort of conjugate form for which i lies between m+n and m-, that for 
i=m—] or i=n, \i-„ becomes a numerical multiplier of <p ; and that when i lies in 
the intermediate region between n and m — 1, vanishes for all values oft'. I 
pause only for a moment to put together for the purpose of comparison the forms 
corresponding to i and to m-\-n — i. By art. (16.), making i=v-{-v, 
hj (x - A, J ...(x~ hj X{x- %) {x ~ ...{x- 
X 
X 
+ l + 2 
^^Qv + 1 + 
X 
% % •••%! 
The conjugate form for which and m—v n~v take the places of t' and v 
{m~v){n—v) will be got by taking 
hq^ 
% % _ 
X 
^Iv+l ^?»+2'"^^?»i 
• • • ^IV 
_ ^<lv+ l^qv+2‘ 
X 
% % •••% 
which it will be perceived are identical, term for term, in the fractional constant 
factor, and differ only in the linear functions of x, which in and in are complemen- 
tary to one another. Our proper business is only with those forms for which i<n. 
Art. (26.). It will presently be seen to be necessary to ascertain the numerical rela- 
tions between ^o,i and o when i<n, and this naturally brings under our notice the 
* It thus appears that if the indices m and n do not differ by at least 3 units, ^ will have an actual quanti- 
tative existence for all values of i between 0 and ki + k; or in other words, the failure in the quantitative 
existence of the forms only begins to show itself when this difference is 3 ; thus if 3, exists, an 
exists, but ^^+ 1 = 0 . 
