REPORT ON CERTAIN BRANCHES OF ANALYSIS. 323 



of the progress and existing state of the diiFerent branches of 

 the mathematical sciences. In the following report we shall 

 commence by a general account of the state in which the pro- 

 blem was left by him, and of the practical difficulties which 

 attend the use of his methods, and we shall then proceed to 

 notice the important labours of Fourier and other authors, with 

 a view to bring its solution within the reach of arithmetical 

 processes which are at once general and easy of application. 



The resolution of numerical equations involves two principal 

 objects of research : the first of them concerns the separation 

 of the roots into real and imaginary, positive and negative, and 

 the determination of the limits between which the real roots 

 are severally placed ; the second regards the actual numerical 

 approximation to their values, when their limits and nature have 

 been previously ascertained. Many different methods have been 

 proposed for both these objects, which dilFer greatly from each 

 other, both in their theoretical perfection and in their practical 

 applicability. We shall begin with a notice of the first class of me- 

 thods, which have been proposed for the separation of the roots. 



If the coefficients of an equation be whole numbers or rational 

 fractions, their real roots will be either whole numbers or ra- 

 tional fractions, or otherwise irrational quantities, which will be 

 generally conjugate* to each other and which will generally pre- 

 sent themselves, therefore, in pairs. The method of divisors 

 which Newton proposed, and which Maclaurin perfected, will 

 enable us to determine roots of the first class, and they are also 

 determined immediately and completely by nearly all methods 

 of approximation. It will be to roots of the second class, there- 

 fore, that our methods of approximation will require to be ap- 

 plied, though such methods will never enable us to assign them 

 under their finite irrational form, nor would our knowledge of 

 their existence under such a form in any way aid us, unless in a 

 very small number of cases, in the determination of their ap- 

 proximate numerical values. 



The equal roots of equations, if any exist, may be detected 

 by general methods ; and the factors corresponding to them 

 may be completely determined, and the dimensions of the equar 



* An irrational real root may be conjugate to the modulus of a pair of im- 

 possible roots ; and there may exist, therefore, as many irrational real roots 

 which have no corresponding conjugate real roots as there are pairs of im- 

 possible roots in the equation. It is not true, therefore, generally, as is some- 

 times asserted, that such irrational roots enter equations by pairs. It would 

 not be very diflScult to investigate the different circumstances under which 

 roots present themselves, and the different conditions under which they can 

 be conjugate to each other ; but the inquiry is not very important, in as much 

 as the knowledge of their form would not materially influence the application 

 of methods for approximating to their values. 



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