438 MR. SYLVESTER ON THE RESIDUES AND SYZYGETIC MULTIPLIERS TO (px, fx 
and thus is seen to depend only on the degree ; in of and not upon the mode of 
partitioning . into two parts and for the purpose of representing by means of 
formula (19.) ... i • i 
Art. (18.). I shall now demonstrate that every form m this scale (to a numerical 
factor prh) is identical with a simplified residue to /, <p of the same degree i in 
Any such simplified residue is like a syzygetic function, or to use a briefer form 
of speech, a conjunctive of/, <p; and if we agree to understand by the “weight of any 
function of the coefficients of / and ^ its joint dimensions in respect of the roots of 
/•and ^ combined, I shall prove,— 1st, that any simplified residue of/ and (p of a given 
degree in x is that conjunctive, whose weight in respect of the roots of/ and p is le«^s 
thL the weight of any other such conjunctive ; and 2nd, that as determined above 
(in equation 24.), is of the same weight as the simplified residue, and can therefore onh 
differ from it by some numerical factor. For the purpose of comparison of weights, 
it will of course be sufficient to confine our attention to the coefficients of the 
highest (or any other, the same power, for each) in of the forms whose weights are to 
be compared. _ , 
Suppose/ to be of m dimensions, and p to be of w dimensions in x ; and let m = e. 
Suppose 
A./+L.(p=Aa?'+B.r'~*4-&c. +K 
+ « + + +lg + e, 
the number of homogeneous equations to be satisfied by the q + l quantities .,X,, 
and the j+S+l quantities will be m + 5-i, and therefore ?+> “' 
g4-e+ 1 taken together must be not less than m+y— 1, i- e. 2y+e+*2 must e no 
Ls than q+m-i+l, i. e. q not less than and if this inequality e 
satisfied 4+e+2-(,+— + 9+i+^-»+2 -‘‘1 be the number ot 
arbitrary constants entering into the solution of equation ( 23 .). 
Uq be greater than (w-1), let 5 = (w~l)+^; 
and let (A) = (Xo).a)”-' + (X0j:"-"+... + (^«-i) 
and let (A), (L) be so taken as to satisfy the equation 
(A)/+(L).(p=A.r'+Bsr-^+...+K; 
and make H=(A) + (/+g.r+ 
X=(L)-(/+ga:+-- + /i‘»'"‘)/ 
/ being arbitrary constants; 
then H/+X.®=(A)/+(L)9=A.r'+B,r-'+... + K. 
Now the total number of arbitrary constants in the system (A) and (L) wUl be 
m+2, i. e. i+ 1 ; hence the total number of arbitrary constants in ^ an 
