LAPLACE. 607 



And there is, by Art. 1002, tlie probability —r- I e'^"^ dt that 

 Se will lie between 



t (5i?i + ?i^i) - 2t/c and t {t^, + ft^^) + 2T/t. 

 There has been no limitation as to the sign of f^ or l^i- 



■ This result will be found to agree with that given by Laplace 

 on his page 423 ; he had previously, on his page 420, treated the 

 particular case in which the function /^ {£) is supposed the same at 

 every trial, so that the suffix i becomes unnecessary, and the result 

 simplifies in the manner which we have explained towards the 

 end of Art. 1002. 



1038. An important consequence follows so naturally from the 

 investigation in the preceding Article, that in order to explain it we 

 will interrupt our analysis of Laplace. Suppose that fi = 1 and 

 fi = 0, for all values of i : thus 



and Sfi becomes equal to the number of times in which an event 



happens ovit of s trials, the chance of the happening of the event 



2 /''■ 

 being pi at i}^ trial. Thus we have the probability -r- e'^ dt 



that the number of times will lie between 



%Pi — T^/'2Spiqi and S^i + r V22^i^i. 



This is an extension of James Bernoulli's theorem to the case 

 in which the chance of the event is not constant at every trial ; if 

 we suppose that pi is independent of i we have a result practically 

 coincident with that in Art. 993. This extension is given by 

 Poisson, who attaches great importance to it ; see his Recherches 

 sur la Prob. ..., page 246. 



1039. If instead of two values at the i*^ trial as in Art. 1037, 

 we suppose a larger number, the investigation will be similar to 



