448 MR. SYLVESTER ON THE RESIDUES AND S\ZYGET1C MULTIPLIERS TO (px, fx 
the same degree. A glance at the form of the Bezoutian square will show that if we 
form the Bezoutian secondary of the degree {n—i) m x, the coefficient of its leading 
term will contain the term (0, iy ; (0, i) as usual denoting the product of the 
coefficient of s;" in / by the coefficient of in p, less the produffi of the coefficient 
of in 9 bv that of a?”"' in /; and as we suppose the first coefficients in / and p to 
be each 1, if we term the other coefficients last spoken of a, and respectivel^q this 
said coefficient of the leading term of the ith Bezoutian secondary will contain the 
term ( — and consequently (- and ( — y-ra:. ^ 
Now by the like reasoning as that employed in the preceding article, the coefficient 
of the leading term in o *’• 
2 {x—h^.+ j) {x — . . . (t - 
L^7i + l hqi+2'”^qm_ 
will contain the quantity and therefore will contain a term 
Le. (”)% which is equal to since (i—l)? is alwa\s even. 
Hence o=(”)‘'^X corresponding Bezoutian secondary. 
Art. ( 28 .). The above applies to the case where we have supposed m = n. 'When 
this equality does not exist we may proceed as follows. Prefix to ®(a), the first coeffi- 
cient of which is still supposed to be 1, a term where £ is positive and indefi- 
nitely small, and let px so augmented be called Then if A-/’2...A-„ are the roots 
oi<px, hX-h, together with the {m~n) values of (7)’"“”, will be the roots of ^{x). 
But it has already been proved that when (as here supposed) the first coefficient 
o^fx is 1 , the Bezoutian secondaries to/ and <p will be identical with those to/ and O 
respectively ; at least it has been proved that these latter, when £=0, but the form of 
O is preserved, become identical with the former, and consequently the same is true 
when £ is taken indefinitely small. Now if we call the (m-w) roots of O which do 
not belong to <p, aod make 
• ' 
fll % 
h ^ 
■'tqi 
o=2(a:-/q,^.)(x- A.,,^ J . .. (t- A J ^ 
^^ 7 i+l ^ki + 2' 
..A 
we have T^_i,o = 2P(Ay, Ag,...Aj.) 
Ag, 
..x: 
P(A,,Ag,...A,J = (^-Ag,,,)(^-A,,,,)...(T-A„„)pT 
'K 
Ki' 
n -i • 
_ 
K 
K 
...A, - 
Ao-^, 
_ 9i + l 
where 
