d'alembert. 289 



This is D'Alembert's principle, only he uses it thus: he shews 

 that all the terms beginning with 



are less than 



2{l-a){l-a-'h){l-a'-h'-e){l-a-h-c-d)s, 

 where s denotes the geometrical progression 



r being = l^a — b-c — d. * 



532. Thus on his arbitrary hypotheses D'Alembert obtains a 

 finite result instead of an infinite result. Moreover he performs 

 what appears a work of supererogation ; for he shews that the suc- 

 cessive terms of the infinite series which he obtains form a con- 

 tinually diminishing series beginning from the second, if we suppose 

 that a, h,Cjd, ... are connected by a certain law which he gives, 

 namely, 



where p is a small fraction, and m — 1 is the number of the quan- 

 tities a, h, c, d, e, ,.. Again he shews that the same result holds if 

 we merely assume that a,h,c,d,e... form a continually diminish- 

 ing series. We say that this appears to be a work of supereroga- 

 tion for D'Alembert, because we consider that the infinite result 

 Avas the only supposed difficulty in the Petersburg Problem, and 

 that it was sufficient to remove this without shewing that the 

 series substituted for the ordinary series consisted of terms con- 

 tinually decreasing. But D'Alembert apparently thought differ- 

 ently ; for after demonstrating this continual decrease he says, 



En voila assez pour faire voir que les termes de Tenjeu vont en 

 diininuant des le troisieme coup, jusqu'au dernier. Nous avons prouve 

 d'ailleurs que I'enjeu total, somme de ces termes, est fini, en supposant 

 meme le nombre de coups infini. Ainsi le resultat de la solution que 

 nous donnons ici du probleme de Petersbourg, n'est pas sujet a la diffi- 

 culte insoluble des solutions ordinaires. 



583. We have one more contrilnition of D'Alembert's to our 

 subject to notice; it contains errors which seem extraordinary, 



19 



