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then, taking the derived-functions of both the members of (19), we have

f\x) = raa?-1-(n-l)j»1ar»-»+ ... ±pn-i,

= (x a2) (x—a3)... (x— an) + (x—«i) (x—a3)... (x—an) + (x—ai) {x—di) ... (x—an) + ...

... + (ar-ai) (x-a2) ... (x-an_{). (22)

Let x=ai; then observing that all the parts, except the first, of the second member of (22) disappear, and that by virtue of the arrangement of the roots, as indicated by (21), every factor of the first part is positive, it follows that, if x = ai, f'(x) is positive; similarly, if x = a2, f'(x) is negative; there is therefore some value of x between ax and a2 which makes f'(x) vanish; a root therefore of f'(x) lies between ai and a2. Similarly we have the following results;

if x = Oi, f'(x) is positive,
x = a-i, f'(x) is negative,
x = as, f'(x) is positive,

x = a„, f'(x) is positive or negative, according as n is odd or even. Hence the roots of f'(x) are real, and lie between the roots of f(x).

Let the roots of f'(x) be oi, a2, ... a„_i, arranged in order of descending magnitude, then they stand to the roots of f(x) in the relation indicated in the following table;

«i a> a3 . . . «„_!, a„, <*i a-i a3 . . . o„_ij whence it appears, that the greatest root off(x) is greater than the greatest root of f'(x), and the least root of f(x) is less than the least root of f'(x). It is on account of this particular relation of the roots of f'(x) to the roots of f(x), that f\x) is sometimes called the limiting equation of f(x).

Hence also it follows, that if all the roots of an equation are real, all the roots of each of its successive derived-functions will be real also.

These results admit of the following geometrical interpretation: Let the curve represented in fig. 23 be that whose equation is y =f(x). As f(x) has n real roots, f(x), and therefore y, = 0 at n points corresponding to them; that is, if Oai = au Oa2 = a2, ... the curve cuts the axis of x at Au A2, ... that is, in n points. As f(x) only = + x , when x = + so , the ordinate

is finite for all values of x between ax and a2, a2 aQd a3,;

and by the last Chapter, as f\x) = 0, and changes sign when x = oi, = 02, = ..., f(x) is a maximum or minimum corresponding to these roots of f'(x), and therefore we have maxima or minima ordinates at points intermediate to At and A2, A2 and A3, ... A„_i and A„; that is, Ob, = oi, Ob2 = a2, — Also as n is odd or even, so will the curve towards the left, when x = — oo , be below or above the axis of x.

177.] Hence it appears that, if f(x) has m roots equal to each other, /'Or) has (m 1) roots equal to each of the equal roots of/(#); for if ax = a2 = ... am, then ai a2 = ... = Om-i, as the o's are intermediate to the a's; which proposition is also thus manifest.

Let m roots of f(x) be equal to one another and to a, and let Qx symbolize the product of the other n m roots, then

f(x) = (x-a)mcix;
.-. /'(at) = (x-a)'—1 jmq.,-1- (x-a);

and f'(x) has m 1 roots equal to a, that is, to each of the m equal roots of f(x).

Hence if f(x) has equal roots, they may be determined by the method of finding the greatest common measure of f(x) and f'(x), and f(x) may be depressed by as many dimensions as it has equal roots.

The latter proof of this proposition manifestly reaches the case of equal imaginary roots which the former may not resolve.

178.] Such information does Infinitesimal Calculus afford as to the continuity of algebraical expressions, as to the possibility of the resolution of f(x) into simple and quadratic factors corresponding to the roots, and as to the relation between the roots of f(x) and of its derived-function. I proceed now to a theorem of great importance) one indeed which theoretically completes this part of the subject; inasmuch as we are hereby enabled to determine the number of real roots, and that of the imaginary roots of an equation. It was discovered a few years ago by M. Sturm, and the memoir containing it was presented by him to the French Institute, and published in the "Memoires presentes par divers savants a l'Academie des Sciences/' Tom. VI, 1835. It is now generally known by the name of "Sturm's Tlieorem," and is one of the greatest modern discoveries in Algebraical Analysis. The problem is as follows:

Given an equation /(x) 0, which has real coefficients; it is required to determine the number of real roots which it contains, and the limits of them.

First let f(x) be cleared of equal factors by means of the last Article, so that no two roots of /{x) are equal to each other: then we have the following theorem:

Let /(x) be a function of x of real coefficients, of which f'(x) is the derived-function; let /(r) be operated upon by f'{x) in the way of finding the greatest common measure, but with the peculiarity of the sign of a remainder always being changed before it is made a divisor; and let this process be continued until it terminates by giving a remainder independent of x, and which does not vanish, because f(x) has no equal factors; let the successive remainders thus modified be symbolized by

/lW, fi(x), ...f„-i(x);
so that we have the following system of equations;

J{x) =f'(.r)ql-Mx), ^
fix) =fi(x) q2-/2(x),

. Mx) = M*) q*-Mx), y (23)

fn-s(x) = f„-i(x) (/„_! — f„-i(x); J

fn-\(x) being the last factor and consequently independent of x.

Let a and /3 be two numbers of which a is the less, regard being had to its sign; substitute a for x in the scries

fix), fix), /iW.-Z.-iW. (24)

and write down in the same order the signs of the results; and count the number of sequences of two terms having contrary signs in this series of results; and suppose A to be that number. Again, substitute /3 in the same series of functions, and count as before the number of sequences of two terms with contrary signs, and suppose it to be equal to B,

Then there are A B roots of fix) lying between o and ft; that is, the number of the excess of variations of signs in successive terms, when o is substituted for x, over that when /3 is substituted for x, is the number of real roots of f(x) greater than a and less than ji.

This is Sturm's Theorem; but before I proceed to the proof of it, I must make two observations on the series of functions in (24). (1) No two consecutive functions can vanish for the same value of x; for if two consecutive terms vanish, when x = c, say ft(c) and /_i(c), then by reason of (23), /i_2(c) = fis(c) = ... = 0, and ultimately f'(c) and /(c) vanish; which last values can coexist only when f(x) has two or more equal roots, and this circumstance is contrary to our first assumption: and therefore we conclude that no two successive functions of the series (24) can simultaneously vanish. (2) No function of the series can be identically zero; for suppose/;(x) to be always zero; then from (23)

fi-t{m) = g</<-i(«),

and thus the value of x which makes fi-t{x) to vanish will also make/_i(,r) to vanish; that is, two consecutive functions will simultaneously vanish; and this has just been shewn to be impossible.

Now I will proceed to the proof of the theorem; let us suppose a to be a root of f(x); and in the series (21) for a? let a —A be substituted; then by equation (16), Art. 114, the series of functions becomes

/(a) -/'(«) 'l+f"(a + 0h)-^,
f'(a) -f"(a + 0h)h,
fx (a) -A'(a + 0h)h,
Ma) -M(a + 0h)h,

in each of which we may suppose h to be such an infinitesimal, that the terms involving it must be neglected when added to a finite quantity; then, since a is a root of f(x), f{a) 0, and therefore the signs of the series of terms are the same as those of

-/'(«), /'(«), Ma), (25)

Again, let a + h be substituted for x in the series (24); then the signs of the series of results are the same as those of

/'(«), /'(«), Ma), Ma),; (26)

whereas then of (25) the first two terms are affected with oppo

PR1CE, VOL. l. v p

site signs, and the first two terms of (26) have the same signs; therefore by making x increase from a quantity a little below a real root to a quantity a little above it, a variation of sign in the series of functions (24) is lost; that is, what was a sequence of opposite signs has become a sequence of the same signs.

Also a similar loss of variation of signs takes place whenever x passes through a root; and therefore, if we make x to grow by infinitesimal increments from a to /3, every time that its value becomes that of a real root of the equation the series of signs of f(x), f'(x), fi(x), ... loses a variation. And a loss of change of sign occurs only under that circumstance. For let us suppose it to occur otherwise; it is evident that it can be only when a sign of one or more of the other functions is changed, and that is, when one or more of them become zero. We have above shewn that two consecutive functions do not simultaneously vanish; let us now suppose that/<(x) = 0, when x—c, and that it changes its sign at that critical value; then since from (23)

ft-i(c) = qi+\ fi(c)-fi+l(c),

■ •• fi-x(c) = (27)

neither of which can vanish; and therefore as x passes through c their signs are the same whether x is infinitesimally less or infinitesimally greater than c. But /<_i(c) and/+j(c) are, as is clear from (27), of opposite signs; and therefore, whatever are their signs respectively, and whatever change of sign/ (x) undergoes as x passes through c, the only possible arrangements and

changes are (1) from + -\ to H , (2) from 4- to

+ H , (3) from 1- + to 1-, (4) from y to

— + -f ; and in each of these cases no change of sign is lost: in all there is one and only one sequence of two terms affected with contrary signs. We conclude therefore that a loss of change of sign can take place only in the first two functions of the series, and only when x passes through a root of f(x).

Hence we finally conclude that there are as many real roots between a and /3 as there are more variations of sign for a than for 0.

In calculating the successive values of the series of functions (24) we may observe that any function may be multiplied or divided by a positive but not by a negative quantity; because

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