SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF AN EQUATION. 507 
under the form of an improper continued fraction, may be made to take the form 
Qj : Qg ; Qg; where Q^; Q4 ; ...Qai ai’e linear functions of x, and i is 
any number assumed at will, not less than [/j, and of course not greater than m ; and 
where <^ 1 ; co^; will have in common a root X, which may be made at will the 
greatest or the least root of the investigation, however, according to 
the present light which I possess on the subject, appears complicated and tedious, 
and therefore, in order that the press, which is waiting for the completion of these 
supplemental articles, may not be kept standing, must be adjourned to some future 
occasion. For the present I content myself with showing the truth of the law for the 
simple case where fx is a cubic function of x. 
0CC 
1st. If ^ gives rise to a single sequence of quotients Q, we know, from the theory 
of intercalations, that it is necessary that all the roots of fx shall be real, and in order 
that when this is the case the quotients may form a single sequence Q, it is only 
necessary so to assume <px, that its roots may be intermediate between those of fx. 
2nd. If the roots of fx are not all real, or if they are all real, but do not compose 
the roots of fx intercalated between them, and if for greater brevity of ratiocination 
we stipulate that (px shall have its leading coefficients of the same sign as that of the 
leading coefficient o{fx, the leading coefficients of the three quotients will either bear 
the respective signs +-| — , or the respective signs -| or the respective signs 
d ; in the first and last of these cases there would be two sequences, and there- 
fore, by what has been shown above, the method of limitation of the text could not 
give a limit coincident with a root. Let us then look to the remaining case, and 
inquire whether, and how, <px may be assumed so that fx shall become representable 
to a constant factor pres by the cumulant a), —q{x—^'), r(x— a)], where p, q, r 
are all positive, and a is a root of fx. 
Let this cumulant be called hfx. 
Nothing in point of generality will be lost if we suppose the leading coefficient of 
hfx to be —1. We then have 
hfx—\_p{x—a), —q{x—^); r{x—a)^ 
= — pqr{x — aY{x — h) — {p-\-r){x — a) 
hfx 
and writing ^^=x^-\-Bx-{-C and making x=a, we find from the above identity that 
and 
hence 
j,»-f r — i. e. /? = a^-l-Ba-l-C— r, 
pqr{x—^) =x-\-a-YB, 
/3 -|- <2 “1“ B = 0, i. e. ^ — — B — <2, 
and 
Hence if (px be so assumed that the quotients to ^ dive p{x—a ) ; —q{x—f ) ; r{x—a) 
jx 
