548 LAPLACE. 



Ce mode d'election serait sans doute le meilleur, si des considerations 

 6ti'angeres au merite n'influaient point souvent sur le clioix des elec- 

 teurs, meme les plus honnetes, et ne les determinaient point a placer 

 aux derniers rangs, les candidats les plus redoutables a celui qu'ils pre- 

 ferent; ce qui donne un grand avantage aux candidats d'un merite 

 mediocre. Aussi I'experience I'a-t-elle fait abandonner aux etablissemens 

 qui I'avaient adopte. 



It would be interesting to know where this mode of managing 

 elections bad been employed. The subject had been considered by 

 Borda and Condorcet ; see Arts. 690, 719, 806. 



991. Thus we close our account of the second Chapter of 

 Laplace's work which we began in Art. 970 ; the student will see 

 that comparatively a small portion of this Chapter is originally 

 due to Laplace himself. 



992. Laplace's Chapter III. is entitled Des his de la proha- 

 hiliU, qui resultent de la midtiplication indefinie des evenemens : it 

 occupies pages 275 — 303. 



993. The first problem is that which constitutes James Ber- 

 noulli's theorem. We will reproduce Laplace's investigation. 



The probability of the hap]3ening of an event at each trial 

 is p; required the probability that in a given number of trials 

 the number of times in which the event happens will lie between 

 certain assigned limits. 



Let q = 1 — p and yu, = m + r^ ; then the probability that the 

 event will happen m times and fail w times in fi trials is equal to 

 a certain term in the expansion of (/; + q^, namely 



\m\n -^ 



Now it is known from Algebra that if m and n vary subject 

 to the condition that m + n is constant, the greatest value of 



the above term is when — is as nearly as possible equal to 



- , so that m and n are as nearly as possible equal to yup and y^q 

 respectively. We say as nearly as possible, because jbLp is not 



