478 MR. SYLVESTER ON THE QUOTIENTS RESULTING FROM THE EXP.1N5ION 
[Received Nov. 3, 1853.] 
Art. (n.) The expressions obtained for the quotients to ^ may be generalized and 
extended to the quotients to where px and fx are two functions of x of any de- 
grees m and n, whose roots are respectively, and h, If we suppose 
where Q(^) is of w-m dimensions, and q^{x), q^{x)...q^+,{x), each of one dimension 
in X, it may be proved that on writing 
1 1 _1 N;(a^) 
QR— “ qi{x)~Di{xy 
we shall have 
= (A.) 
= (B-) 
where C4:C'=0, 
C^i+iR) being the (^+ l)th simplified quotient. When QR) is a linear function of .r, 
in finding q,x from the formula B, we must take Doa:=l. The proof of this theorem 
being generally true, may easily be shown to depend upon its being true in the special 
case* when and n {m being supposed less than ?^), and h„ h,, ... h„ 
become 4.../, h, and k, k,...k^ become 4 h and the truth of 
the theorem for this special case (if for instance we wish to prove the formula (B)] 
depends upon the expression 
hi h2...hi,-i 
hi h^ ... h^i 1 
k\ k^ . . . km 
hi' . • ’hji 
hi ^2 • • • hh 
/q ^2 .••hn 
k\ ^"2 . . . kill 
hii + i h^i^2'’'hn 
being identical with the expression 
rj hi ^2 . . . 
hi h^ . 
Ij ki k2...km 
hi, hji j . 
. • h,ji 
X ( hi, — A, ) ( /i;- — /i.,) . . . ( A i' — 1 ) I 
X 
1 
K 
ki k^. 
hi, 
hi Ih 
...hi,^ 
hi'+x 
..Jin 
* By virtue of tlie Lemma, that when fx and fx are two algebraical functions &c.) ; {x’^'^^ + ax»+(‘' &;c.) 
no function of the coefficients vanishing identically when i roots oifx coincide with i roots of (px respecth ely c^ 
be formed, in which there are fewer of the coefficients of / and <p respectively than appear in the leading coeffi- 
cient of the (n — f-|-l)th residue of 
