of M. Fourier in his Analyse des Equations. 39 



Institute : I am not aware that any succeeding part of this 

 work has been published since that date. 



The first part commences with a synoptic exposition of the 

 whole work, and from this Expose, which is rather diffuse, but 

 clear in its language, and which contains the announcement 

 of several theorems, the demonstrations of which are reserved 

 for the succeeding parts, there can be no doubt that the 

 author has committed some grave errors in the application of 

 recurring series to the solution of numerical equations. 

 M. Navier's attention may, through this medium, be directed 

 to expunge or correct those parts of the unpublished manu- 

 script referred to from p. 7 1 to 75 of the Expose. 



(1.) Let a recurring series A, B, C, D, E, F... be formed 

 by the known method of Bernoulli and Euler, the successive 



R C D 

 quotients of which, viz. —r- > -^- 3 -pr 9 & c - converge to the 



A 15 v^ 



first (when real, greatest) root of a proposed equation ; from 

 this let a second series be formed, of which the terms are 

 AD— BC, BE-CD, C F-DE, &c; the successive quo- 

 tients of this series, says M. Fourier, converges towards the 

 sum of the two first roots. This is incorrect, for when these 

 quotients are convergent, they give the product instead of 

 the sum of the two first roots. 



To prove this, let a, /3, y, &c. be the roots of the proposed 

 equation, a, /3 being the two first, that is, greatest, abstracting 

 from sign, when real, and when conjugate imaginary quantities, 

 such that a /3 >y 8 , &c. 



Let u, represent the general term of the first recurring 

 series, and v x of the second, formed as above indicated, and 

 let C, C, C", &c, c, c, &c. represent quantities invariable 

 with x, 



then v s = 2/^+3 — u x+1 u s+2 



and u s = Caf+C S* + CT/ + , &c. 



Hence v x = c(a/3)* + c' (ay)*+c"(/3y)*+, &c. 



that is, making for abridgment 



P,= c + C ' [f) +«-.(£)* + , &c; 



and as x increases it is clear that P^ converges to c, and there- 



P b 



fore the limit of -§^- is unity, and consequently that of -— t? 



is a |3 and not « + £. 



Example. Given x 2 — 2 x— 3 = 0; roots —1 and 3. 



