MONTMORT. 113 



It is to be observed that the case in which r = l does not 

 occur, from the nature of the game ; for the player, if not arrested 

 at the very beginning, will certainly be able to remove tivo pairs. 

 We may how^ever if we please consider the summation to extend 

 from r = lio r = n-l, since/. = when r = 1. 



We have then 



u„=.T\n[l + 2J^. 



The summation for w„_, extends to one term less ; thus we 

 shall find that 



But «„., +/„_, = 



71-1 



therefore 



2/2 I 2;i - 2 



n — l 



\1n 2\2?i-2 I 2/1 ,j_i 



Hence /„ = i= - z/„ =-==-- ; and /„ -^ 



[w " 1 71 -2 ' ^" • |_^ 2/4 -l' 

 , This is Montmort's result. 



19^. We noAv arrive at what Montmort calls the fifth part 

 of his work, which occupies pages 288 — 41-i. It consists of the 

 coiTcspondence between Montmort and Nicolas Bernoulli, together 

 with one letter from John Bernoulli to Montmort and a reply 

 from Montmort. The whole of this part is new in the second 

 edition. 



John Bernoulli, the friend of Leibnitz and the master of Euler, 

 was the third brother in the family of brothers of whom James 

 Bernoulli was the eldest. John was born in 1667, and died in 

 IT-iS. The second brother of the family was named Nicolas ; his 

 son of the same name, the friend and corres^oondent of Montmort, 

 was born in 16S7, and died in 1759. 



195. Some of the letters relate to Montmort's first edition, 

 and it is necessary to have access to this edition to study the 

 letters with advantage ; because although Montmort gives re- 

 ferences to the corresponding passages in the second edition, yet 



8 



