Hence, as a change of sign is lost in passing from x — — 2 to x = —1, a root lies between —2 and —1; and as changes are again lost in passing from x = 0 to x = 1, and from x = 2 to a? = 3, two other roots lie severally between 0 and 1, and between 2 and 3. Thus the equation has three real roots, the positions of which have been determined. 180.] Hence follows a method of determining the whole number of real and imaginary roots of an expression of the form f(x) = 0. Whatever is the number and value of the real roots, they must be between — oo and -f x; let us therefore form as above the series of functions /(*), /'(*). fl(*), /iW, .../,-!(#), the number of which is generally n +1, f(x) being of n dimensions; and let + oo or a very large quantity be substituted for x, so that the signs of the functions are the same as the signs of the first terms; let To be the number of variations of signs of consecutive terms of this series. And now let — oo be substituted for x, then the signs of the functions of even dimensions will be the same as before, but the signs of those of odd dimensions will be the contrary. There will therefore in the latter substitution be as many variations of succeeding terms as there were permanences in the former, that is, there will be n — m variations. And as all the real roots are comprised within these limits, their number, by Sturm's Theorem, is n — m — m or n — 2 To, and therefore the number of imaginary roots is 2 To. There exist therefore as many pairs of imaginary roots as there are variations in the signs of the first terms of the functions /(*), f'(x), .../»-!(*). Ex. 1. f(x) = x3-3px+2q, f(x) = 3x*-3p, Mx) = p3-q3. The series of signs of the first terms is the same as that of 1, 1, p, p3-q*. If therefore p is negative, there is one variation, and therefore only one real root; and if p is positive, there is one real root when p3 is less than q2, and when p3 is greater than q- all the roots are real. Before I pass on to another subject, I would observe, as it was originally remarked by Sturm himself, that the preceding process would be equally applicable if f'(x) were replaced by any other function of x of n — 1 dimensions, of which no two or more roots are equal, and of which no root is equal to a root of f(x). But as such a function could only generally be found, if we knew the roots of /(x), and as these are not known, so the only available function with which we may first operate on f(x) in the way of finding the greatest common measure is f'(x), because its relation to /(x) fulfills all the conditions which the process requires. Hence it is not quite left to our choice to take for its first divisor any function of n — 1 dimensions that we please. The functions in (24) are called the Sturmian functions. The general forms of them were given without proof by Mr. Sylvester in the Philosophical Magazine, Dec. 1839: and were subsequently demonstrated hy Sturm himself in Liouville, Tome VII, p. 356. Other inquiries connected with them will be found in Liouville, Tome XII, p. 54, by M. Borchardt; Tome XIII, p. 269, by Mr. A. Cayley; in a paper by Mr. A. Cayley lately (Feb. 17,1857) read to the Royal Society. The insertion of the results is evidently beyond the scope of the present work, and the student desirous of further information must have recourse to the original memoirs. 181.] Fourier's Theorem. The following process was arranged by Fourier to separate the real and impossible roots of an equation; but as it only indicates a number which the sought number of real roots does not exceed, the discovery of M. Sturm renders it almost useless; however, as it advantageously exhibits the relations between the successive derived-functions in an algebraical point of view, it is right to insert it in a treatise on Infinitesimal Calculus. Let f(x) be a function of x of the form (1), Art. 170, and let it be cleared of equal factors; and let us suppose all its coefficients to be real j let the several derived-functions of it be formed, whereby we have a series, /<*), /'(*), /», •../"<*); (28) and let o and /3 be two numbers of which a is the less; then there cannot be more real roots of f(x) between a and /3 than the excess of the number of alterations of sign in the above series of functions, when a is substituted for x, over the number resulting from the substitution of /3. For suppose a to be root of the equation f(x) = 0; then, since /(a) = 0, by the substitution of a — h for a? in the above functions, the series (28) becomes, when h is infinitesimal, -/'(a) A, f\a), .../"(a); and when a 4 h is substituted for x, f'(a)h, /'(a), .../-(a); and thus, supposing none of the derived-functions to vanish, the passage of the substituted quantity, from a value a little below a real root to one a little above it, causes a variation of sign to be exchanged for a permanence; and pari ratione as such a variation will be lost whenever the substituted quantity passes through a root, it follows that, as many real roots as there are lying between a and /3, so many losses of variations of signs at least will there be in the series of functions, when we pass gradually from a to ft. At least, I say; for the series of signs may also be affected by the vanishing of any of the subsequently derived-functions; for suppose b to be such as, when substituted for x, to cause fr(x) to vanish, and h to be an infinitesimal; then, if b — h is substituted for x, we have for fr~1(x), fr(x), fr+l(x), fr-li!>), -fr+Hb)h, (29) and when b+ h is substituted for x, /-H*), fr+1(b)h, f^(b). (30) If fr~l{b) and /r+1(6), viz. the first and last terms of (29) and (30), are of contrary signs, then we shall have a variation and a continuance both in (29) and (30), so that no change will be lost. But if fr~Hb), /r+1(*) are of the same sign, then in (29) we shall have two variations, and in (30) two continuations, so that two changes of sign will be lost. Similarly it may be shewn, that if many successive derivedfunctions vanish for a particular value of x, an even number of variations of sign may disappear. There may therefore be losses of variation of sign in the series of functions given in (28) at other values of x than roots of f(x), but variations must nevertheless be exchanged for continuations at the roots; therefore the Theorem gives only a number which is not less than the number of real roots. The advantage of Sturm's Theorem is, that it gives the exact number of real and of imaginary roots. 182.] The rule commonly known by the name of Des Cartes' Rule of Signs is a particular case of Fourier's Theorem; viz. in the general equation f(x) = 0, the number of positive roots cannot exceed the number of variations of signs of the successive terms, and the negative roots the number of continuations of signs. Firstly, let x = 0, in the series of functions (28); then the signs off(x),f'(x), ...fn(x) are the same as those of the several and successive terms off(x) taken from right to left; and when x = 00, the signs of the functions are all positive. Hence there can be no more real positive roots than there are changes of sign in the successive terms of f(x). Secondly, let x = — ■ 00 , then the series of functions form only variations of sign, of which there are of course n, and therefore the number exceeds the number of variations, when x = 0, by the number of permanences in the terms of the equation. Hence the number of negative roots cannot exceed the number of continuations of signs. 183.] Taylor's Series also furnishes a method, which was invented by Newton, for finding a number greater than the greatest root of an equation. Let f(x) — 0 be an equation of n dimensions of the form (1), Art. 170, and for x let us substitute y + h; so that f(y + h) = f(h) H f'(h) \ +f'(h)^ + ... +f(h) i^l^. Suppose such a value to be given to h as to make f(h), f\h), ...f"(h) all positive, then by Fourier's Theorem no root of the equation can lie between h and -+ oc; therefore h is greater than the greatest positive root. 184.] Taylor's Series is also useful for approximating to a root of an equation. Suppose two values a and j3, of which a is the less, and the |