STURM'S THEOREM. There is a branch of the theory of equations, containing the celebrated theorems of Descartes, Fourier, and Sturm, which it is advisable to place in an article by itself, and the present heading has been chosen because Sturm's theorem is at once the most conclusive and the latest of the three. It has long been a problem of much interest and notoriety to find, in a given equation, how many roots, if any, are contained between two given limits ; how many roots are positive, how many negative, how many imaginary.
The first step the solution of the preceding problem was made by Descartes, though it is asserted by Cossali and Libri, that Carden came very near to the same step. Cossali, after collecting a table of Cardan's cases, and putting them in a form which Carden did not use (an equation with 0 on the second side), then says that an analyst who. should look at this table would be able to rise to Des carttais theorem. This is true enough, but it does not prove that Carden either could or did make the invention, but the contrary. All the world knows that mathematical discoveries are recognised often enough by analysts of a later day, in rudiments from which the fabri cators of them could evolve nothing.
The theorem of Descartes, expressed in his own words, is as follows (` Geometria ' lib. ii): " Ex quibus ctiam cognoscitur, quot vertu et quot falsie g i radices in unaquaque "Equatione haberi possint. Nimirum, tot in ea veras haberi posse, quot vaxiationes rcperiuntur eignorum + et — ; et tot Lime quot vicibus ibidem deprehenduntur duo signa vel duo signa rime se invicem sequuntur." That is, that an equation may have as many positive roots as there are changes of sign in passing from term to term, and as many negative roots as there are continuations of sign; but not more of either kind. It has been doubted whether Descartes knew the true meaning of his own theorem as to the case of imaginary roots ; this doubt is as early as the time of Descartes himself, who replies in a letter which we cannot find by means of Rabuel's reference to it. This is however of little consequence, as the following sentence (also from the Geometry) shows in what manner Descartes understood his own words : " Crater= radices tam vertu quam falsie non seniper aunt reales, fled aliquando tantum imaginative • hoc est, scraper quidcm in qualibet "Equations tot radices quot dui, imaginari licet ; verum nulla inter dum est quantitas quai illis, quas respondet." It would seem then that Descartes not only remembered the limitation of the theorem arising from the possible existence of imaginary roots, but proposed to divide those last roots themselves into two classes corresponding to the true and false (or positive and negative) of the real roots. The next step was made by De Gua (1741), who showed
that the roots of an algebraical equation 4. x= 0 are never all real, unless the roots of the derived equations 4/ x=0, Is" x=0, &c. be also all real ; x, 41" x, &c. being the derived functions, or differential coefficients, of 4. x. He also showed how to determine the conditions of the reality of all the roots. (Lagrange, 'Res. des Equ. Numer.', note viiii.; Peacock, Report,' &c., p. 327.) Descartes's theorem would be perfect if the roots of equations were always real. For example, take If the roots be real, they are both positive ; write x+ 6 for x [Invotorms AND EVOLU TION], and we have = 0, of which the roots are less by 6 than those of the former equation. But in the second equation, one root is negative and one positive; consequently the roots of the first equation are one greater and one less than 6. In the same manner a more com plicated case might be treated.
The theorem of Descartes, and the notion derived from it, that the order of signs of coefficients regulates the signs of the roots ; with the step made by De Gua, and the notion derived from it, namely, that the derived functions must be consulted upon the question whether the roots of an equation be real or not ; and the common theory of equal roots, namely, that when is x=0 has m equal roots, m-1 of its derived functions (neither more nor fewer) vanish at the same time, or with the same root,—were the hints on which FOURIER [Broo. Div.] was able to make an advance upon his predecessors. The coefficients of the equation are themselves nothing but the divided derived functions on the supposition that x=0. Thus, if sex = 3z' — + Ilx + 4, we have Let cpx, thx, &c. be the function in question, and its divided derived functions. If we make x great enough and negative (say infinite and negative), the signs of these functions are all alternate, that is, the series yields nothing but changes of sign in passing from term to term. But if we make x great enough and positive (say infinite and positive), the series yields nothing but permanences of sign. Thus, in the preceding expression we have x= x=0 0.r (Pre txr nothing two no but changes. changes. changes.