510 MR. SYLVESTER 0>i THE THEORY OF INTERCALATIONS 
it seems not unlikely that the interval between the greatest and least of the roots of 
tl,e above equations will be a minimum when the intervals between any pair is the 
same for each pair, i. e. when 
1 
F-l 
1 ,1 
i«-2+~ f^3+;r 
IX,, 
fll «3 
If we assume these equations, and write the equation foi determining „ 
will be r V 5 . —n 
^ 3^3 • • • 5 
If w = 2 this equation becomes 
If n=3, rejecting the factor it becomes 
Ujaafl'af — (^1 + ^ 3 ) = 0* 
If n=4 it becomes 
If w=:5, rejecting the factor |, it becomes 
and so in general the equation in f being always of a degree measured by the integer 
nearest to and not exceeding and it is easy to be seen that for all values of «, the 
second coefficient divided by the first will be an inferior limit to f (of course actu- 
ally coinciding with it for the cases of n=2 and w=3). Hence we have the follou mg 
valuable practical rule for finding a superior and inferior limit to the cumulant 
[ai(fr— -Cl), aJyX—c^, c,,)]? 
where « 2 , ••.«« have the same sign, viz. if C be the greatest, and K be the least of 
the quantities c„C 2 ,...c„, C + A will be a superior, and K-A an inferior limit, A 
being taken equal to the positive value of 
]_ 
« 2-^3 « 3-«4 
and it may be noticed that C and K are the quantities which would themselves be the 
superior and inferior limits to the given cumulant if the series of terms a.,, 
instead of presenting only a sequence of continuations or permanencies, presente 
only a sequence of changes or variations of sign. 
Section V. 
On the Theory of Intercalations as applicable to two functions of the same degree, and on 
the formal properties of the Bezoutiant with reference to the method of Invariants. 
Art. (56.). If/^ and (px be any two given functions of x of the same degree ?n, u e 
may form a system of m Bezoutics to / and (p (as shown in the first section), the 
coefficients of the powers of ... in which will compose a square matrix 
of m lines of m terms each, which will be symmetrical in respect to the diagoiia 
