MR. SYLVESTER ON FORMULiE CONNECTED WITH STURM’S THEOREM. 485 
with my formulae for the Sturmian functions, and prove ab initio in the most simple 
manner, that the successive ascending- coaxal determinants (always of course supposed 
to be taken about the axis of symmetry) of the matrix to the form above written, or 
to the more general form (which I shall quote as G, viz.) 
2(f-A.)n?>l.(/il)*h + ?>2(/h)W2+.-+9m.(/0-Mm}' (G.) 
(where <p„ are absolutely arbitrary integral forms of function with real 
coefficients), will form a rhizoristic series in regard to fx {i, e. a series, the difference 
between the number of the continuations of sign between the successive terms of 
which corresponding to two different values of ^ will determine the number of real 
roots of f lying between such two assumed values), provided only that q be an odd 
positive or negative integer. Nothing can be easier than the demonstration, fcr 
whenever o is greater than any one of the real roots as (/q) 
1st. Any pair of imaginary roots will give rise to two terms of the form 
and {l—m^Zr\y.{v-Wy/ZI\y-, 
or more simply, 
and (L — — — 2vw^ — 1 ) ? 
where v and w are real linear functions of u^, u^, ...w„. The sum of which couple 
will be 
2 { L . (m^ - 1;^) - 2 } = I; . { (Lm - My) ^ - (LH M V } ; 
so that each such couple combined will for every value of x give rise to one positive 
and one negative square. 
2ndly. Any real root of the series b„ h^, when § is taken greater than such root, 
will give rise to a positive square of a real linear function of u^, u^, 
3rdly. Any real root of the same series, when § is beneath it in value {q being odd), 
will give rise to the negative of the square of a real linear function of the same. Hence 
the number of real roots between taken equal to one value {a), and § taken equal to 
any other value {h), will be denoted by the loss of an equal number of positive squares in 
the reduced form of the expression (G.) when § is taken (a) and when ^ is taken {h ) ; i. e. 
by virtue of art. (45.) will be denoted by the difference of the number of permanencies 
of sign in the successive minor determinants of the matrix corresponding to the 
quadratic form (G.)* (which we have taken as our generating function) resulting 
* The inertia of the quadratic form G is the measure of the number of real roots of/ir comprised between oo 
and p, and may be estimated in any manner that may be found most convenient. If p be made infinity, and 
be taken equal to and the inertia of the corresponding value of G be estimated by means of the for- 
mulae in ordinary use by geometers for determining the nature of a surface of the second degree, the criteria of 
the number of real roots inyic will be, or may be made to be, symmetrical in respect to the two ends of the 
expression This system of criteria, however, is not so good as that given by the Bezoutiant to the two 
differential coefficients oi f{x, 1) taken with regard to x and 1 respectively, which will also possess the like 
character of symmetrical indifference, and be one less in number than the former. 
