REPORT ON CERTAIN BRANCHES OF ANALYSIS. 343 



separation of the roots which we have previously described 

 may have left in the first instance uncertain. We refer to the 

 end of the second book of Fourier's Analyse des Equations 

 determin^es, for a very complete examination of the theory of 

 such approximations*. 



It has been a question agitated on more than one occasion, 

 whether the tests of the reality of the roots of equations of finite 

 dimensions which De Gua established, or rather the principles 

 of the much more general theorem of Fourier, were applicable 

 likewise to transcendental equations. In a discussion of the 

 transcendental equation 



y ^^ '^ ""'■gi g2 02 ">" 02 33 22 ~ OCC, 



which presents itself in the expression of the law of propagation 

 of heat in a solid cylinder f of infinite length, Fourier ventured 

 to apply the principles in question to show that all its roots were 

 real ; but M. Poisson J has disputed the propriety of such an 

 application, both in this case and in others : thus, if we suppose 



X = e*— be"", 



we shall find 



* The rule for the determination of the nature of two roots included in a 

 given interval, which is given in page 333, is merely the expression of a con- 

 sequence of the application of the method of linear approximation to the di- 

 stinction of those roots ; and whatever difficulties in certain extreme cases 

 may attend the successful application of that rule, M'ill necessarily present 

 themselves likewise in the application of the linear approximation under the 

 same circumstances. This character, however, is not confined to the Newto- 

 nian or linear method of approximation. If the interval of the roots be deter- 

 mined, by the application of Fourier's theorem of the succession of signs of 

 the original function X and its derivatives, so that no more than two roots 

 may be said to exist in that interval, whose nature is unknown, whether real 

 or imaginary, then the application of the method of continued fractions, as 

 well as of other equivalent modes of approximation, will be competent to de- 

 termine the values of those roots when real, and their nature, when imaginary. 

 Such, at least, is the assertion of Fourier, who refers to the third book of his 

 work on equations for its demonstration. It is unfortunate, however, that 

 only two books of this work, which is full of such remarkable researches upon 

 the theory of equations, were fully prepared for publication at the time of his 

 death. Our knowledge of the contents of the other five books, which were 

 left unfinished, is derived from an Expose Synoptique prefixed to those which 

 are published, and which contains a general review and analysis of their prin- 

 cipal contents. It is to be hoped, however, that the materials which he has 

 left behind him will be found to be sufficient at least for their partial, if not 

 for their complete restoration. 



t Theorie de la Chaleur, p. 372. 



X Journal de I' Ecole Poly technique, cahier xix. p. 381; M^moires de I'ln- 

 stitut, torn. ix. p. 92. 



