DANIEL BERNOULLI. 235 



( 



1 - - = e '* 

 n 



Hence when n and x are very large, we find that the value of u^ 

 ofiven in Art. 419 reduces to 



«e-"U + ,4-f-T+7^l'-U...^ 



6 



13 \nj ' 16 V**> 



Daniel Bernoulli sums the series in the brackets by the aid of 

 the Integral Calculus. We know however by the aid of the 

 theorem relating to the value of the sums of the powers of 

 a, A 7, that this series is equal to 



Hence the analogy of the value of u^, when x and n are in- 

 definitely large, with the value of u in Art. 420 is sufficiently 

 obvious. 



Daniel Bernoulli gives some numerical applications of his 

 general results. 



Daniel Bernoulli's memoir has been criticised by Malfatti, in 

 the Meniorie ... della Societa Italiana, Vol. I. 1782. 



422. The next memoir by Daniel Bernoulli is entitled, 2Ien- 

 sura Sortis ad fortuitam successionem rerum naturaliter contin- 

 gentium applicata. This memoir is in the same volume of the 

 I^ovi Comm Petrop. as the preceding; it occupies pages 26 — 45. 



423. The memoir begins by noticing the near equality in the 

 numbers of boys and girls who are born ; and proposes to consider 

 whether this is due to chance. In the present memoir only thus 

 much is discussed : assuming that the births of a boy and of a girl 

 are equally likely, find the probability that out of a given 

 number of births, the boys shall not deviate from the half by 

 more or less than a given number. The memoir gives some calcu- 

 lations and some numerical examples. 



Daniel Bernoulli seems very strangely to be unaware that 

 all which he effects had been done better by Stirling and Do 

 Moivre long before ; see De Moivre's Doctrine of Chances^ 

 pages 243—254. 



