g|2 MR. SYLVESTER ON THE THEORY OF INTERCALATIONS 
1 or 3 in the second, and 2 in the third. The most symmetrical (hnt least expeditions) 
method of finding the inertia of any quadratic form is that which corresponds to t e 
method of orthogonai transformations, and is, in fact, the usual method employed m 
geometrical treatises on lines and surfaces of the second degree. If we apply this 
method to the Bezontiant B considered as a homogeneous quadratic fanction of the 
(m) arbitrarily named variables u^, u^, u^, ...u^ in ordei to measuie its inertia, 
is to say, the number of effective interpositions between the two systems of roots, we 
must construct the determinant 
■(P.B 
du\ 
du^ . du^ 
d'^B 
du-^ . du:^ ’ 
d^B 
du^ . du^ ’ 
d^B _ 
du^ . duQ ’ 
d^B 
du^ . du„ 
d^B 
du ,^ . du„ 
d^B 
d^B, 
dm 
dum . dua ^ duyti • du^ 
dm 
dul. 
^ dujYi . dti^ 
All the roots of D(X) = 0, as is well known, are real ; the inertia of B, being measured by 
the number of positive roots of D(-X), will be equal to the number of continuations 
of sign in D{>.) expressed as a function of of the mth degi ee. 
If in /x and <px we reverse the order of the coefficients, and /x and <p.v so trans- 
formed become /. (x) and (x), it is obvious that the roots of/, and f. being the 
reciprocals of the roots of/ and p respectively, the number of effective intercalations^ 
to f, and fi, must be the same as for/ and f. Accordingly we find that the form o. 
the Bezoutiant to/ and ? is the same as that of the Bezontiant to / and the so e 
difference (one only of names) being that B(»„ u„ ... m,_„ u,.) for tlie one ecoines 
B(« u ...M„n,) for the other. The equation D(X), which determines the i««fw 
of b’,’ remains precisely the same as it ought to do for either of the two systems/ and 
<b or /, and (pj. . . i 
Art. ( 58 .). The theory in the preceding articles of this section may be ma 
embrace the case involved in Sturm’s theorem ; for if 
J' x~w,a^ ,x^~'^-\- {n— 1 .iT" .. . 
f^x-mfx-fx 
= a. . j:”- * + 2 0-2 . + . . . + w . 
the Bezoutian secondaries, or which is the same thing, the simplified Stiirmian resi- 
dues to fx and /a?, will evidently be the same as those to f ,x u.ndfx. According y, i 
we form the signaletic series 
and 
fXyfx, B„ Ba-.-Bm-iJ 
