OF THE REAL ROOTS OF TWO ALGEBRAICAL EQUATIONS. 
489 
and a third and final reduction brings it to the form 
h'/jh'/j’. 
and accordingly we shall find for such an arrangement of the h and n system 
I(&)-I(a) = + 2. 
3jfx 
Art. (50.). If we suppose (px— by a well-known theorem of algebra, any two 
consecutive roots of fx will contain between them an odd number of roots of px, 
and the number of real roots of f’x greater than the greatest root of fx, and the 
number of real roots of f’x less than the least root of fx will each be even. Hence 
the effective intercalation scale between any two limits (a) and {b) will be formed by 
merely reducing the n groups to single units, and the number of A’s in the scale so 
formed will be the total number of A’s between the limits (a) and (h). Moreover, 
since such scale commences always with a root of fx, or with an even number of roofs 
of/ 'x followed by a root of fx, if the number of A’s and ;?’s cut off be even, and with 
a root oifx or an even number of roots of fx followed by a root of fx, if the number 
so cut off be odd, it follows that for this case 1(a) — I(Z>), (a) being the superior limit, 
will be always positive, and will measure the total number of real roots of fix) lying 
between (a) and {h) ; this, then, is Sturm’s theorem, treated as a corollary to the 
Theory of Intercalations. 
Art. (51.). If we write down the last syzygetic equation between fx of m and p(x) 
of n dimensions, viz. 
r„_ , . (t)/(^) — C- . (J:^) Slo = 0, 
it has been shown that the succession of signs in the series formed with fx, px and 
their successive Bezoutian secondaries will contain the same number of continuations 
and variations as the series formed with f(x), RRcl their successive Bezoutian 
secondaries. This indicates that the effective scale of interpositions for fx and 
px will contain an equal number of roots of fx with the effective scale fovfx and 
; the two scales however will not necessarily be identical, because the roots of 
px will not necessarily be in the same order relative to the A’s in the one scale as those 
of O’ relative to the /fs in the other scale. This equality is perfectly well ex- 
plained a posteriori by the form of which by the formula in Section II. will be 
represented by 
^{x-h^){x-h^f.{x-}i^^_;). 
^f/2) ' 
•■Km Km- 1 
Now, whenever x is indefinitely near to any one of the roots of fx, as this sum 
reduces to the simple expression 
pK^ ph^^...pK^_={pK.pK...phj-^, 
T Qm 
and consequently in the immediate neighbourhood of every real root of fx, p(x) and 
tr„^i.x will have always the same or always a contrary sign, according as phg^.phg^...ph^^ 
is positive or negative, which will depend upon the relative disposition of the real 
roots in f and p ; in either case the effective scale of interpositions for/r with px and 
3 s 2 
