REPORT ON CERTAIN BRANCHES OF ANALYSIS. 327 



mining the absolute number either of one class or of the other, 

 in the absence of any means of ascertaining the number of ima- 

 ginary roots. If the roots of the equation were all of them real, 

 and could be shown to be so by any independent test, it would 

 be easy to determine the limits between which the roots were 

 severally placed ; for the number of changes of sign which are 

 lost upon the substitution ofx + e for x would show the number 

 of roots which are included between and e ; and if, therefore, 

 we should assume a succession of values of e, whether positive 

 or negative, such as to destroy one change of signs and rio more, 

 upon the substitution of any two of these successive values, we 

 should be enabled to obtain the limits of every root of the 

 equation. 



It was chiefly with a view to this consequence of Des Cartes's 

 theorem that De Gua investigated and assigned the conditions 

 of the reality of all the roots of an equation. If we suppose 

 X = to be the equation, and X', X", X''', X'% X\ &c., to 

 denote the successive differential coefiicients of X, then, if all 

 the roots of X = be real, the roots of the several derivative 

 equations X' = 0, X" = 0, X"' = 0, &c., must be real like- 

 wise ; and if the roots of any one of these equations X^*"' = 

 be substituted in X^''"'' and X'''+'', it will give results affected 

 with different signs. If we form, therefore, a succession of 

 equations in y by eliminating successively x from the equations 

 t/ = XW . XC-^' and X'»-'' = 0, 



1/ - X(»->^ . X(» 3) and XC- ) = 0, 



y = Xi X"' and X" = 0, y = X X'' and X' = 0, 



the coefficients of all these equations must be positive, forming 



from + to + and from — to — as it has real and negative roots." It is very 

 doubtful, notwithstanding the assertions of some authors, whether Des Cartes 

 himself was aware of the necessary limitation of the application of this theorem, 

 which is required by the possible or ascertained existence of imaginary roots. 



The demonstration which was given by De Gua of this theorem in the Me- 

 moires de I'Academie des Sciences for 1741, founded upon the properties of the 

 limiting equation or equations, has been completed by Lagrange with his 

 usual fullness and elegance, in Note viii. to his Resolution des Equations Nu- 

 meriques. The most simple and elementary, however, of all the demonstra- 

 tions which have been given of it, and the one, likewise, which arises most 

 naturally and immediately from the theory of the composition of equations, is 

 that which was given by Segner in the Berlin Memoirs for 1756. The few im- 

 perfections which attach to this demonstration, as far as the classification of 

 the forms which algebraical products may assume is concerned, have been 

 completely removed in a demonstration which Gauss has published in the 

 third volume of Crelle's Journal. 



This theorem is included as a corollary to Fourier's more general theorem 

 for the separation of the roots, as we shall have occasion to notice hereafter. 



