An introduction to the modern theory of equations, by Florian Cajori.

54

THEORY OF EQUATIONS

We find that gl(x) differs from g'(x) only by the presence in
gl(x) of the positive coefficient m. If x= r, then gi(x) and
g'(x) have the samne sign; for, g()() -n m( - r1)(r — ). and
g(r')= (r-ri1)(- ' 1 2) *.. They have like signs also for
X -= ', or 01'2...
We may therefore find the situation of the roots of g(x) =0
by taking g(x) and gy(x) as the first two of Sturm's functions
and forming from these two the rest of them. This is permissible, since by applying the reasoning of the First Case it
may be shown that this new set of functions possesses the two
fundamental properties that as x passes from a to b no variations of signs are gained or lost when an auxiliary function
vanishes, and that one and only one variation is lost when
g(x) vanishes.
The numlber of variations in sign will always be the same for
the seriesf(x),f' () (x)
f (X), f'(X), I2(X),... Af%,
as for g(x), g(X), g2(x)..g,(.
For, corresponding terms of the two series of functions differ
always only by the factor (x - r)'-1, so that, for any value of
x, the signs of the terms in the first series are all the same as
those of the second series, or the signs are all unlike.
Hence, by examining the variations in signs of the first series
we can find out how many real roots of the equation g(x) = 0
lie between a and b, and this number of roots is the same as the
number of real and distinct roots of the equation f(x)= 0
between those same limits. This proves the second case when
r is a multiple root. If f(x) = 0 has, besides r, the multiple
root rr,, then a slight and obvious modification of our proof is
necessary.
49. In the application of Sturm's theorem, the following
point must be borne in mind. In finding the functions f(x),
fs(x), * * it is allowable to introduce or suppress any monomial