594 LAPLACE. 



40 years ; these values are given in Art. 902. Laplace finds that 

 the probability is approximately 



1 - -0030761 



where /a is a very large number, its logarithm being greater than 

 72. Thus Laplace concludes that the probability is at least equal 

 to that of the best attested facts in history. 



With respect to a formula which occurs in Laplace's solution 

 see Art. 767. With respect to an anomaly observed at Vitteaux 

 see Arts. 768, 769. 



1026. Laplace discusses in his pages 381 — 384 the problem 

 which we have noticed in Art. 902. 



He offers a suggestion to account for the observed fact that the 

 ratio of the number of births of boys to girls is larger at London 

 than at Paris. 



1027. Laplace then considers the probability of the results 

 founded on tables of mortality : he supposes that if we had observa- 

 tions of the extent of life of an infinite number of infants the tables 

 would be perfect, and he estimates the probability that the tables 

 formed from a finite number of infants will deviate to an assigned 

 extent from the theoretically perfect tables. We shall hereafter in 

 Art. 1036 discuss a problem like that which Laplace here considers. 



1028. A result which Laplace indicates on his page 390 sug- 

 gests a general theorem in Definite Integrals, which we will here 

 demonstrate. 



Let u^ = 



let e~^^ be integrated with respect to each of the n — 1 variables 

 z^, z^, ... Zn_^, between the limits — oo and oo : then the result 

 will be 



n-l 



Cf|i*2 • • • ^'n_i^7i 



e-Y^^„', 



where -^ = — ^^ + ~^ + —H-fj}^ + . . . + -i-^ — 2—^ 



