490 
MR. SYLVESTER ON A NEW METHOD OF FINDING 
iovfx with must contain the same number of h's , ; but the difference will be, 
that if is positive an h will occupy the first place in each scale, or the 
second place in each scale ; but if negative, then in one scale an {h) 'will occupy the 
first place, and in the other scale the second place. 
Art. ( 52 .). The same process of common measure or residues which serves to furnish 
a rhizoristic series for/(:r) or a synrhizoristic series for fx and <px, will serve also to 
furnish superior and inferior limits to the real roots of any proposed equation. Thus 
suppose fx to be any rational integral function of {x) of the degree (^0 and (p{x) any 
other function of x, which I shall begin with supposing to be of the degree («— 1 ), 
and let the successive quotients resulting from the process of finding the greatest 
common measure of fx, px continued until the last remainder is not a constant but zero, 
be supposed to be (as they may generally be taken, but subject to cases of exception, 
which will hereafter be alluded to) n linear functions q, then we shall have 
fx qi+ g,i+"'qn-l+ Qn 
^x=K.N 
and therefore 
/x=K.D, 
where N is the numerator and D the denominator of the fraction 
1 1 
9 l+ ?2+ 
and K is a constant (the value of which is immaterial to be considered, but in fact equals 
+ &C., 
— T2t2t2 ^ 
JUi l-i3 i-<5 
Lo, Lj, La, L3, &c. being the leading coefficients of the last, the last but one, the last 
but two, &c. of the Bezoutian secondaries tofx and (px). Accordingly, 
if w=l, let J) = qi=[/Ji ; 
and in general let 
where 
D pi. p2 ‘(^3 • • • 
Now suppose X to be so taken that 
qi does not 1 
does not lie between + 1 and — 1 " 
V2 . 
qs • 
q, . 
. -j-2 and —2 
. +2 and —2 
, +2 and — 2 )> , 
2 and —2 
1 and — 1 
