MONTMORT. 121 



1 n —J) {n —p) (n — 2p) 



i " 1 . 2 (7i - 1) "^ 1 . 2 . 3 (?i - 1) (n - 2) 



(n —p) (n — 2p) (n — 8/;) 

 ""1.2. 3. 4(7-^-1) (w-2) (/i^Ts) +••• 



205. Nicolas Bernoulli then passes on to another game in 

 which he objects to Montmort's conclusion. Montmort had found 

 a certain advantage for the first player, on the assumption that the 

 game was to conclude at a certain stage ; Nicolas Bernoulli thought 

 that at this stage the game ought not to terminate, but that the 

 players should change their positions. He says that the advantage 

 for the first player should be only half what Montmort stated. 

 The point is of little interest, as it does not belong to the theory of 

 chances but to the conventions of the players ; Montmort, however, 

 did not admit the justice of the remarks of Nicolas Bernoulli ; see 

 Montmort's pages 309, 317, 327. 



206. Nicolas Bernoulli then considers the problem on the 

 Duration of Play which had been suggested for him by Mont- 

 mort. Nicolas Bernoulli here gives the formulaa to which we have 

 already alluded in Art. ISO; but the meaning of the formuloB was 

 very obscure, as Montmort stated in his reply. Nicolas Bernoulli 

 gives the result which expresses the chances of each player when 

 the number of games is unlimited ; he says this may be deduced 

 from the general formulae, and that he had also obtained it pre- 

 viously by another method. See Art. 107. 



207. Nicolas Bernoulli then makes some remarks on the 

 summation of series. He exemplifies the method which is now 

 common in elementary works on Algebra. Sujipose we require 

 the sum of the squares of the first n triangular numbers, that is, the 



sum of n terms of the series of which the r^^ term is \---^ — —^ 



Assume that the sum is equal to 



an^ + hn^ + cn^ + dn^ + en + /*; 



and then determine a, h, c, d, e, f by changing n into n-\-\ in 

 the assumed identity, subtracting, and equating coefficients. This 

 method is ascribed by Nicolas Bernoulli to his uncle John, 



