486 
MR. SYLVESTER ON THE THEORY OF THE INTERCALATIONS 
from the substitution respectively of a and h in place of p, which gives a theorem 
equivalent to that of M. Sturm, transformed by my formulae, when we choose to 
adopt the particular suppositions 
This method of constructing a rhizoristic series to/r by a direct process is deserving 
of particular attention, because it does not involve the use of the notion of continuous 
variation, upon which all preceding proofs of Sturm’s theorem proceed. It completes 
the cycle of the Sturmian ideas. Happily this cycle was commenced from the other 
end, for it would have been difficult to have suspected that the root-expressions for the 
terms in the rhizoristic series could be identified with the residues, had the former 
been the first to be discovered, and much of the theory of algebraical common mea- 
sure laid open by means of this identification would probably have remained unknown. 
Art. ( 48 .). I proceed now to consider a theorem concerning the relative positions 
of the real roots of two independent algebraical functions as indicated by the suc- 
cession of signs presented by their Bezoutian secondaries; this more general theory 
of intercalations or relative interpositions will be seen to include within it as a corollary 
the justly celebrated theorem of M. Sturm. 
Let the real roots of/x taken in descending order of magnitudes be h, h...hp, and 
the real roots of <px taken in the like order so that 
fx= {x~Jh) hp) H 
(px={x—ni){x~n^^’’-{^—nq)^. 
H and K being functions of x incapable of changing their signs. Now, as m 
M. Sturm’s method, let us inquire what takes place in respect to the sign of 
which I shall call the Indicatrix, as a: descends the scale of real magnitude trom 
_|_ oo to - oo . If between -foo and i real roots of ^x are contained,^ it is obvious 
that as X travels from to the superior brink of /q, the Indicatrix will change its 
sign from -j- to - and from - to -j- altogether i times, so that at the moment when 
X is about to pass through /q, it will be positive if i is zero or even, and negative if 
i is odd ; but the moment after x has passed through the value the indicatiix vi 
be negative on the first supposition, and positive on the other supposition. Hence 
immediately after the passage of x through hi the indicatrix will have been oiiCw 
oftener negative than positive on the one supposition, and as often negative as posi 
tive on the other. Again, in like manner as traverses the interval between /q and 
the inferior brink of if no ;? or an even number of ??’s occupy this interval, the sign 
which the Indicatrix had at the beginning of this interval will have been reveised 
once oftener than restored; but if there be an odd number of h's so interposed, the 
number of reversals and restorations will have been identical ; and so for each 
successive interval, reckoned from a value for x immediately subsequent to one lea 
root offx, down to a value immediately subsequent to the next less real root of the 
