AS APPLICABLE TO TIVO FUNCTIONS OF THE SAME DEGREE. 
513 
where B„ are the Bezoutian secondaries to f^x and fx, the number vari- 
ations sign between consecutive terms in this series, when x is made +oo, will 
measure the number of pairs of imaginary roots in fx\ and fx and fx forming always 
a continuation, and the coetBcient of f{x) being supposed positive, we see that the 
terms of the rhizoristic series will be 1, (B,), (B 2 )...(B„_i) consisting of positive unity, 
and the successive ascending coaxal determinants of the Bezoutian matrix to f and 
f^x. Hence then the form of the Bezoutiant to fx and f^x will serve to determine the 
number of pairs of imaginary, and consequently also the number of real roots to fx. 
It should be remarked that the form of the Bezoutiant to f'x and fx, considered as 
a quadratic function of Mj, and of the coefficients in fix), will remain unal- 
tered when for/a? we write /, a-, for this will change the signs throughout of /a- and 
f^x, and consequently the coefficients in the Bezoutiant, which contain in every term 
one coefficient from fx, and one from f^x, wall remain unaltered in sign. 
Art. (59.). It appears then from the preceding article, that for every function of a; 
of the degree m, there exists a homogeneous quadratic function of (m~l) variables, 
the inertia of which augmented by unity will represent the number of real roots in 
the given function. Now this inertia itself may be measured iy the number of posi- 
tive roots of a certain equation in X formed from the quadratic function (in fact the 
well-known equation for the secular inequalities of the planets), all whose roots will 
be real. Hence then we are led to the following remarkable statement. ‘‘ An alge- 
braical equation of any degree being given, an equation whose degree is one unit lower 
may be formed, all the roots of which shall be real, and of which the number of positive 
roots shall be one less than the total number of real roots of the given equation'' 
Let us suppose /a’ written in its most general form, the first and last as well as all 
the intermediate coefficients being anything whatever : by reversing the order of the 
coefficientsy'a: will become /ixand fx will become fx-, the Bezoutiant to fx and fx 
(which we may term the Bezoutoid to fx) will remain unaltered except in sign, and 
the equation of the (m— l)th degree in X formed from the Bezoutoid remain un- 
changed, consequently the equation in X enables us to substitute, for the purpose of 
calculating the total number of real roots in fx) in lieu of Sturm’s auxiliary func- 
tions tofx), another set of functions which remain unaltered when the order of the 
coefficients is completely reversed, i. e. in effect, when we consider the number of real 
roots of f(^ in lieu of those of fx). And of course more generally the equation of 
the mth degree in X formed from the Bezoutiant to any two functions fx and (px of 
the mth degree each in x, supplies a set of functions for determining the total number 
of effective intercalations between the roots of fx) and p{x), which do not alter when 
we consider in lieu of these, the roots 
This substitution of func- 
tions symmetrically formed in respect to the two ends of an equation for the purpose 
of assigning the total number of real roots in lieu of the unsymmetrical ones furnished 
3x2 
