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

52

THEORY OF EQUATIONS

and from the first equation that x - c is a factor of f(x).
Hence f(x) and f'(x) would have a collmon factor and (~ 21)
f(x) would have equal roots, which is contrary to hypothesis.
(2) When any auxiliary function vanishes the two adjacent
functions have opposite signs. Suppose, for example, that
f3(x) is zero for x= c. By (1), f2(x) and f4(x) cannot be zero
when f3(x) is zero. The third equation, above, then reduces
to f,(x) = -f4(), showing that f,(x) and f4(x) have contrary
signs.
(3) When x, in passing from the value a to the value b,
passes through a value which makes an auxiliary function
vanish, Sturm's functions neither gain nor lose variations in
sign. For, suppose that, for x = c, f,.() = 0, then f,. 1(c) and
f,.+(c) have opposite signs. As f.(x) passes through zero, it
changes its sign from + to -, or from - to +. Thus the
three functions f.;-(x), f.(x), )f.+l() will have one variation in
sign just before x=c and also just after x= c. In other
words, no matter which sign is placed between two unlike
signs, we have only one variation. Hence no variation is
either gained or lost among Sturn's functions.
(4) When x, in passing from the value a to the value b,
assumes a value which is a root of the equation f(x) - 0, then
Sturm's functions lose one variation in sign. By Taylor's
Theorem, ~ 18,
f(C - 1) -f() =- f'(c) + 2f (c)-
f(c + h) f(c) = +f + hf ()....
For very small values of h the sign of the right member of
each expansion will be the same as the sign of its first term.
If f(x) vanishes for x = c, so that f(c) = 0, and f f'(c) is positive, f(c - h) is negative and f(c +- h) is positive. That is, the
signs of f(x) and f'(x) will be - + just before x = c, and -}- +