250 EULEPu 



The memoir consists of explanations of psn't of that memoir 

 by Lagrange to which we have aUuded in Art. 371 ; nothing new 

 is given. The explanations seem to have been written for the 

 benefit of some beginner in Algebra, and would be quite un- 

 necessary for any student unless he were very indolent or very 

 dull. 



4i8. The next contribution of Euler to our subject relates to 

 a lottery ; the problem is one that has successively attracted the 

 attention of De Moivre, Mallet, Laplace, Euler and Trembley. 

 We shall find it convenient before we give an account of Euler's 

 solution to advert to what had been previously published by 

 De Moivre and Laplace. 



In De Moivre's Doctrine of Chances, Problem xxxix. of the 

 third edition is thus enunciated: To find the Expectation of J., 

 when with a Die of any given number of Faces, he undertakes 

 to fling any numxber of them in any given number of Casts. The 

 problem, as we have already stated, first appeared in the De Men- 

 sura Sortis. See Arts. 251 and 291. 



Let 71 be the number of faces on the die ; x the number of 

 throws, and suppose that m specified faces are to come up. Then 

 the number of favourable cases is 



,f _ ,„ u -Vf-\- ^^'^lH (,, _ 2)-^ - . . . 

 ^ ^ 1.2^ ^ 



where the series consists of m + 1 terms. The whole number of 

 possible cases is if, and the required chance is obtained by di- 

 viding the number of favourable cases by the whole number of 

 possible cases. 



44^9. The following is De Moivre's method of investigation. 

 First, suppose we ask in how many ways the ace can come up. 

 The whole number of cases is 7f ; the whole number of cases 

 if the ace were expunged would be {n — iy ; thus the whole number 

 of cases in which the ace can come up is ;?/*"— (n — ly. 



Next, suppose we ask in how many ways the ace and deux 

 can come up. If the deux were expunged, the number of ways 

 in which tlie ace could come up would l)e [n — ly — {n — 2y, by 



