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

LOCATION OF,THE ROOTS OF AN EQUATION

51

f2(x), 3(x), *.., f,c(x), and call them auxiliary functions. The
functions f(x), f'(x), f (), f3(x),..., f, (x) are called Sturmn's
functions.
48. Sturm's Theorem. If f(x) 0 has cno equal roots, let
any two real quantities a ancd be substituted for x in Sturm's
finctions, then the difference between the anumber of variations
of sign in the series when a is substituted for x and the number
when b is sutbstituted for x expresses the number of real roots of
f(x) = 0 between a ad b.
When f(x)= 0 has multiple roots, the difference between the
number of variations of sign wzhe acndl b are substituted for x
in the series, f(x), f'(x), f2 (x, *., f (Z), w(here fr(x) is the highest
common ftctor of f(x) and f(x), is equal to the number of real
roots between a and b, each mulltiple root counting only once.
First Case. No Equal Roots. In ~ 21 the operation of finding the highest common factor between f(x) and f'(x) was used
for finding multiple roots of the equation f(x) = 0. If there is
no highest common factor involving x, there are no multiple
roots, and we are able to find all of the n + 1 Sturm's functions.
The last function, f,(), is numerical and not zero.
From the mode of formation of Sturm's functions we obtain
the following equations, in which q%, q2, *., q,,_ are the successive quotients in the process:
f(x)= q,f'(x) -f2(x),
f'(x) = q2, (x) — f3(x),
f2(x)= q3f3(X)-f4() t I
fn-2(X)= qn(-lf-1().fx(=)
(1) Two consecutive auxiliary functions cannot vanish for
the same value of x. For, if f2(x) and f,(x) vanish together
when x = c, each would contain the factor x - c. From the
second equation it would follow that x- c is a factor of f' (x),