40 On an Error o/M. Fourier in his Analyse des Equations. 



The first series is recurring; take 1, 2 as the first two terms 

 and 2, 3 as the constants of relation. 



First series 1, 2, 7, 20, 61, 182, &c. 



Second derived series 6, —18, 54-, —162, &c. 

 The successive quotients of the first series converge to the 

 greatest root 3, and those of the second series are exactly the 

 product —3 of the two roots, and not the sum which would 

 be +2. 



(2.) A, B, C, D, E, &c. being as above the primitive recur- 

 ring series, if another be formed such as A C— B 2 , B D — C, 

 &c., the successive quotients converge to the product of the 

 first two roots (p. 72.). This is correct, and may easily be 

 proved as in the preceding case. 



(3.) From the primitive series, Fourier adds, 3 other recur- 

 ring series may be deduced by rules we have announced [in the 

 MS. of sixth book]. The first will give by its successive con- 

 verging quotients the sum of the 3 first roots, the second will 

 determine the sum of their products two by two, and the third 

 their continued product, (ib.) 



The first two announcements here contained must be wrong ; 

 the third is most probably right ; for if the successive quotients 

 of a converging series gave a 4 - /3 4- y, its general term would 

 bev s = Cj (a + j3 + y)*+C 2 (a + jS +&)* + ,&<•. Such a series 

 cannot be derived in the above manner from u x = Ca*+ 0/3* 

 + C" y*, &c ; and if it could it would not give the three first 

 roots, but the three roots of greatest sum, which may be very 

 different; for the same reason a/3-fay + /3yis not to be thus 

 obtained from a recurring series deduced from the primitive, 

 but u y may be easily so found, and the value belong to the 

 three first roots, for their product must stand also first amongst 

 the products, though their sum may not amongst the sums. 



The knowledge of a + j8 as well as a/3 would, no doubt, 

 give at once the real and the imaginary parts of a conjugate 

 pair of impossible roots ; but Fourier's method does not ob- 

 tain the first, as has been shown, and therefore his observa- 

 tions (p. 74-.), " et, ce qui est remarquable, on connaitra pour 

 chaque racine imaginaire, la partie reelle, de cette racine, et 

 le coefficient de l'imaginaire. Voila Pusage le plus etendu 

 que Ton puisse faire de la methode de series recurrentes," &c. ; 

 and (p. 75.) ■* Les proprietes que nous venons d'enoncer sont 

 incomparablement plus generates que celles qui ont ete connues 

 des inventeurs, et des auteurs qui ont traite depuis la meme 

 question," must be received with considerable deductions. 



2, Bateraan's Buildings, Soho Square, Robert MuRPHY. 



London, May 20, 1837. 



