508 MR. SYLVESTER 
ON A DEVELOPMENT OF THE METHOD OF ASSIGNING 
we have 
h(px- r{x—a)] -~~qr{x-\-'B-\~a){x~a)-\ 
1 = — — ci^ — aB+i?}' 
= —qrix^-^^x—a^ — «B) 
Hence is of the form 
w(^^+Bx~a^~aB + K+«B + C-r)) = m(j:^+Bj: + C-r). 
If we call the three roots of/r, a, b, c respectively, we have 
1 1 . 
^~~r{a^ + 'Ba + C—r)~~r{{a—b){a — c)+r) ’ 
and since q and r are both to be positive, we see that (a) must be taken the greatest or 
least of the three roots if they are all real, so that a^+Ba+C may be positive, whic it 
will of course necessarily be if b and c are imaginary ; we must also have a^+Ba+C-r 
positive, so that the form of ^x is m{{x^-a^)+Bix~a)~t), t being necessarily posi- 
tive, but otherwise arbitrary, a form containing two arbitrary constants, one of which 
is subject to satisfy a certain condition of inequality ; whereas when /r is of such a 
form as to admit, and.^5(^) is supposed to be so assumed as to came it to come to 
pass that the quotients to ^ form a single sequence, then the three coefficients in fa 
remain exempt from all conditions of equality but are subject to two conditions of 
inequality. And so in general when the degree offx is x and the number of sequences 
it is to be inferred that the n coefficients of (px will be subject to satisfy n-i-\ 
conditions of inequality and i conditions of equality. 
Art. (|.). The theory of the determination of the minimum interval between either 
limit determinable by this method and the nearest root, or between the two limits 
so determinable when px is so assumed that ^ gives rise to a defined even number of 
sequences (which will include the theory of the case where all the roote of /rare 
imaginary), must be deferred to an opportunity more favourable for leisurely con- 
templation. As regards the application of the theory to the very interesting case ot 
all the roots being imaginary, the principal point remaining to be cleared up is the 
determination of the least value that can be assigned to the greatest, and the greaUsr 
value that can be assigned to the least root of the algebraical product X,.X,.X,...X,„, 
where X„ X„ ...X,. are all of them real linear functions of x, subject to the condition 
that the cumulant [X., X„ X....XJ shall (to a numerical factor prks) be equal to a 
given function of the degree 2n in x incapable of changing its sign, which condition 
implies, as a necessary consequence, that the coefficients of x in each of the terras 
X„ X2,...X2„ must be affected with the same algebraical sign. 
ArLlo.). It should be observed that in the application of the above method, the 
division of the series of quotients into distinct sequences governed by the signs of the 
coefficients of ir is introduced for the purpose of drawing the limits closer to the roots, 
but is not necessary for the mere object of assigning limits. 
