The expressions f(x), f1(x), f2(x),---fn(x) are called Sturm’s Functions.
Let Q1, Q2,--- Qn-1 denote the successive quotients obtained; then the steps in the operation may be represented as follows :
f@) = A) —h@), AQ) =@ AL) —h), £@) = f(r) —A@),
ae) — Quah (2) — f, (2).
From these equalities we obtain the following:
(1) Two consecutive functions cannot vanish for the same value of x.
For if they could, all the succeeding functions would vanish, including fn(x), which is impossible, as it is independent of x.
(2) When any function except the first vanishes for a particular value of x, the two adjacent functions have opposite signs.
Thus in fi(@)=Qh (x) —fi(2) if f,(@)= 0, we have A@)=— A).
We may now state Sturm’s Theorem.
If in Sturm’s Functions we substitute for x any particular value a and note the number of variations of sign; then assign to x a greater value b, and again note the number of variations of sign; the number of variations lost is equal to the number of real roots of f(x) which lie between a and b.
(1) Let c be a value of x which makes some function except the first vanish; for example, fr(x), so that fr(c)= 0. Now when x =c, fr+1(x), and fr-1(x) have contrary signs, and thus just before x=c and also just after x =c, the three functions fr-1(x), fr(x), fr+1(x) have one permanence of sign and one variation of sign, hence no change occurs in the
