IklALLET. JOHN BERNOULLI. 325 



597. We have next to notice a memoir by Mallet, entitled 

 Recherches sur les avantages de trots Joueurs qui font entreux une 

 Poule au trictrac ou a un autre Jeu quelconque. 



This memoir is published in the Acta Helvetica... Basilece, 

 Vol. V. 1762 ; the memoir occupies pages 230 — 248. The problem 

 is that of De Moivre and Waldegrave ; see Art. 211. Mallet's 

 solution resembles that given byDe Moivre in his pages 132 — 138. 



Mallet however makes some additions. In the problem as treated 

 by De Moivre the fine exacted from each defeated player is con- 

 stant; Mallet considers the cases in which the fines increase in 

 arithmetical progression, or in geometrical progression. A student 

 of De Moivre will see that the extensions given by Mallet can be 

 treated without any difficulty by De Moivre's process, as the series 

 which are obtained may be summed by well-known methods. 



598. The same volume which contains Euler's memoir which 

 we have noticed in Art. 438, contains also two memoirs by Beguelin 

 on the same problem. Before we notice them it will be convenient 

 to consider a memoir by John Bernoulli, which in fact precedes 

 Beguelin's in date of composition but not in date of publication. 

 This John Bernoulli was grandson of the John whom we named 

 in Art. 194. John Bernoulli's memoir is entitled Sur les suites ou 

 sequences dans la loterie de Genes. It was published in the volume 

 for 1769 of the Histoii^e de VAcad Berlin; the date of pub- 

 lication is 1771 : the memoir occupies pages 234 — 253. The fol- 

 lowing note is given at the beginning : 



Ce Memoire a ete In en 1765, apres le Memoire de Mr. Euler sur 

 cette matiere insere dans les Memoires de I'Academie pour cette annee. 

 Comme les Memoires de Mr. Beguelin imprimes a la suite de celui de 

 Mr. Euler se rapportent au mien en plusieurs endroits, et que la Loterie 

 qui I'a occasione est phis en vogue que jamais, je ne le supprimerai pas 

 plus longtems. Si ma methode ne mene pas aussi loin que celle de 

 Mrs. Euler et Beguelin, elle a du moins, je crois, I'avantage d'etre plus 

 facile a saisir. 



599. In the first paragraph of the memoir speaking of the 

 question respecting sequences, John Bernoulli says : 



Je m'en occupai done de terns en tems jusqu'a ce que j'appris de 

 Mr. Euler qu'il traitoit le meme sujet; e'en fiit assez pour me faire 



