LAPLACE. 473 



indeed superseded by Laplace's later researches; but what we 

 have given from Art. 877 inclusive might have appeared in 

 Chapter VIL of the Theorie...des Prob. 



881. Laplace's next memoir on our subject is in the Memoires 

 ...par divei^s Savans... 177 S', the date of publication is 1776. The 

 memoir is entitled Recherclies sur ^integration des Equations dif- 

 ferentielles aux differences Jinies, et sur leur usage dans la theorie 

 des hasards, &c. 



The portion on the theory of chances occupies pages 113 — 163. 

 Laplace begins with some general observations. He refers to the 

 subject wdiich he had already discussed, which we have noticed 

 in Art. 877. He says that the advantage arising from the w^ant 

 of symmetry is on the side of the player Avho bets that head 

 will not arrive in two throws : this follow^s from Art. 879 ; for to 

 bet that head will not arrive in two throws is to bet that both 

 throws will give tail. 



882. The first problem he solves is that of odd and even; see 

 Art. 865. 



The next problem is an example of Compound Interest, and 

 has nothing connected with probability. 



The next problem is as follows. A solid has p equal faces, 

 which are numbered 1, 2,...^:?: required the probability that in 

 the course of n throws the faces will occur in the order ], 2,...^. 



This j)roblem is nearly the same as that about a run of events 

 Avhich w^e have reproduced from De Moivre in Art. 325 : instead 

 of the equation there given we have 



^'n+i = ^n + (1 - ^n^i-p) «"; whcre a = - . 



883. The next problem is thus enunciated : 



Je suppose un nombre n de joueiirs (1), (2), (3), ... (?/), jouant de 

 cette maniere ; (1) joue avec (2), et s'il gagne il gagne la partie ; s'il ne 

 perd ni gagne, il continue de jouer avec (2), jusqu'a ce que I'un des 

 deux gagne. Que si (1) perd, (2) joue avec (3) ; s'il le gagne, il gagne la 

 partie ; s'il ne perd ni gagne, il continue de jouer avec (3) ; mais s'il 

 perd, (3) joue avec (4), et ainsi de suite jusqu'a ce que I'un des joueurs 

 ait vaincu celui qui le suit; c'est-a-dire que (1) soit vainqueur de (2), 



