REPORT ON CERTAIN BRANCHES OF ANALYSIS. 337 



equation, which are included between two Hmits which are con- 

 secutive whole numbers. The formation, however, of these 

 transformed equations, and the determination of the next infe- 

 rior integral limit of their roots, even when no further separation 

 of the roots is required, is excessively laborious, and Lagrange 

 has pointed out methods by which the operations required for 

 both these objects may be greatly simplified. Legendre also, 

 in the 14th section of the first part of his Theory of Numbers, 

 has given a considerable practical extension to these methods 

 of Lagrange. If we combine their processes for finding the 

 neai'est inferior limit of the root with the theorems of Budan * 

 for the formation of the transformed equations, we shall proba- 

 bly have arrived at the greatest simplification which the practi- 

 cal solution of numerical equations, by means of continued frac- 

 tions, is capable of receiving. 



Lagrange has pointed out the principal defects of the me- 

 thod of approximation to the roots of numerical equations which 

 was given by Newton f. It is only under particular conditions 

 that it is competent to attain the object proposed, and in no 

 case does it immediately furnish a measure of the accuracy of 

 the approximation. But notwithstanding these objections to 

 this method, in the form under which it has been commonly 

 applied, it is unquestionably that which most naturally arises 

 out of the analytical conditions of the problem, and which is also 

 capable of the most immediate and most simple application in 

 almost every department of analysis. Lagrange had demon- 

 strated that this method could only be applied with safety to 

 find the greatest and least roots of an equation, and in those 

 cases only in which the moduli of the imaginary roots, if any ex- 

 isted, were included in value between such roots. But Fourier 

 has shown, by considering the superior and inferior limits of 

 every real root, and by a proper examination of certain condi- 

 tions which those limits may be made to satisfy, and by insti- 

 tuting the approximation simultaneously with respect to both 

 those limits, that all sources of ambiguity may be removed and 

 the accuracy of the approximation determined J. We shall now 

 proceed to give a short notice of these researches. 



* Nouvelle Methode pour la Resolution des Equations Numiriques. It con- 

 tains the exposition of exceedingly simple and rapid rules for the formation 

 of the transformed equation whose unknown quantity is a; — e, where e is 

 any integral or decimal number. In other respects, however, this publication, 

 though announced with great pomp and circumstance, is a very superficial 

 production, and is only remarkable for having received the charitable notice 

 and approbation of Lagrange. 



t Resolution des Equations Numeriques, Note v. 



X Analyse des Equations determinees, livr. ii., Calcul des Racines. 



1833. z 



