MONTMORT. 117 



deal the player on Lis right hand takes it ; and so on in order. 

 B is on the left of A, C is on the left of B, and so on. Let the 

 advantages of the players when A deals .be a, h, c, d, ... respec- 

 tively; these advantages are supposed to depend entirely on 

 the situation of the players, the game being a game of pure 

 chance. 



Let the chances of A, B, C, D, ... bo denoted by z, y, x, u, ... ; 

 and let s = 7n + 7l 



Then Nicolas Bernoulli gives the following values : 



z = a + —^ + ~, + -, +..., 



, 7nh + nc m^h 4- 2mnc-\-n^d m%-^Snfnc + 2mn^d+ n^e 

 2/ = i + -^- + p + p +..., 



. mc + 7id 7n'^G-\-27n7id-{-7i^e 7ifc-h2m^7id+Sm7i^e + 72^f 



^ = + ^— + 7 + — ? - + ■■■' 



_ , md + 7ie m^d-\-2miie+ri^f 7n^d-\-Sm^ne + Sm7}^f+7i'g 

 and so on. 





Each of these series is to continue for I terms. If there are 

 not so many as I players, the letters in the set a, h, c, d, e,f,(/,... 

 will recur. For example, if there are only four players, then 

 e = a, f=h, g = c,.... 



It is easy to see the meaning of the separate terms. Take, for 

 example, the value of z. A deals ; the advantage directly arising 

 from this is a. Then there are m chances out of 5 that A will hav 

 the second deal, and 7i chances out of s that the deal will pass o. 

 to the next player, and thus put A in the position originally hek 



by B. Hence we have the term . Again, for the third 



deal ; there are (rti + 7iy, that is, s^ possible cases ; out of these 



there are 711^ cases in which A will have the third deal, 2mn cases 



in which the player on the right of A will have it, and n"^ cases in 



which the player next on the right w^ill have it. Hence we 



, ,1 , iii^a -\- Imnh -\- 7i^c . , 



nave the term z . And so on. 



