MONTMORT. 125 



Nicolas Bernoulli gives at the end of his letter an example 

 )f summation of series. He proposes to sum p terms of the 

 jeries 1, 3, 6, 10, 15, 21, ... He considers the series 



1 + 3:c + 6aj' + lOa;^ + lox" + 21^' + ... 



which he decomposes into a set of series, thus : 



1 + 2a; + 3x' + ^x^ + 5;c* + ... 



+ a; + 2a;' + 3^' + 4ic*+ ... 



+ ir' + 2a;' + 3^'+... 



+ 33^+2^'+... 



+ 0;*+... 

 + ... 

 The series in each horizontal row is easily summed to p terms ; 



he expression obtained takes the form - when x = l, and Nicolas 



Bernoulli evaluates the indeterminate form, as he says, ...en me 

 servant de la regie de mon Oncle, que feu Monsieur le Marquis 

 de I'Hojoital a insere dans son Analyse des infiniment petits, ... 



The investigation is very inaccurately printed. 



21 -i. The next letter is from Montmort to Nicolas Bernoulli ; 

 it occupies pages 337 — 317. Besides remarks on the game of Her 

 and on Waldegrave's Problem, it contains some attempts at the 

 problems which Nicolas Bernoulli had proposed out of his uncle's 

 letter on the game of Tennis. But Montmort found the problems 

 difficult to understand, and asked several questions as to their 

 meaning. 



215. Montmort gives on his page 312 the following equation 

 as the result of one of the problems, 



4m'-8m'+llM + 6 = 3'"-'\ 



and he says that this is satisfied approximately by m = of-J^ ; but 

 there is some mistake, for the equation has no root between 

 5 and 6. The correct equation should apparently be 



which has a root between 51 and 5 •2. 



