REPORT ON CERTAIN BRANCHES OF ANALYSIS. 347 



Another method of approximation to the roots of equations 

 by means of recurring series was proposed by Daniel Ber- 

 nouUi *, and very extensively illustrated and applied by Euler f . 

 If we write down m ai-bitrary numbers to form the first m terms 

 of the series, and if we assume, for the scale of relation, the 

 coefficients of an equation of m dimensions, and form by means 

 of it and the assumed terms the other terms of the series which 

 may be indefinitely continued, and if we also form a series of 

 quotients by dividing each succeeding term (after the arbitrary 

 terms) by that which precedes it, then the terms of the se- 

 ries of quotients which thence arise, will converge continually 

 towards the value of the greatest root of the equation ; and if 

 we form the equation whose roots are the reciprocals of those 

 of the original equation, and proceed in a similar manner, we 

 shall obtain a series of quotients which will converge to the 

 greatest root of this equation, whose reciprocal will be the least 

 root of the original equation, considered without reference to 

 its algebraical sign. 



Lagrange, in the 6th Note to his Resolution des Equations 

 Numeriques, has analysed the principles of this method, and 

 has shown that its success will depend upon the greatest real 

 root, without reference to algebraical sign, being greater than 

 the modulus of any of the imaginary roots. If this condition be 

 not satisfied, the quotients will not approximate to the value of 

 any root of the equation, a consequence which Euler had also 

 pointed out. 



The recurring series which is formed by dividing the first 

 derivative function /' {x) by / {x), which is equal to 



ginary. Some of these have been noticed in the note to p. 267, in connexion 

 with our observations upon Mr. Graves's researches upon the theory of loga- 

 rithms ; another is noticed by M. Clausen of Altona, in the second volume of 



Crelle's Journal, p. 287; it is stated as follows :— Since e^'"' '^"^ = 1, 



when ra is a whole number, we get g^ + ^ " '^ '^^ = e and therefore 



consequently e""*" '^' = 1, whenever w is a whole number, — a conclusion 

 which M. Clausen characterizes as absurd. Its explanation involves no other 



2»i sr \/ —\ . , 



difficulty than that which is included in the equation e =1, and 



must be sought for in the circumstances which accompany the transition from 

 a function to its equivalent series, when a strict arithmetical equality does not 

 exist between them. It must be confessed, however, that these difficulties 

 arc of a very serious nature, and are in every way deserving of a more care- 

 ful examination and analysis than they have hitherto received. 



* Comment. Acad. Petrop., vol. iii. 



t IntroducHo in Analysim Infinitorum, vol. i. cap. xvii. 



