602 LAPLACE. 



The constant is unity, if we suppose the lower limit of the 

 integral to be 0, for q and v are each unity when a; = ; thus 



gy = 1— I irv dx. 



The differential equation obtained in Art. 405 becomes when 

 expressed in our present notation 



1 dz 1 d^ _ ir _ irv 



z dx ^ dx q - r . , ' 



1 — \irv dx 



therefore, by integration, 



z _ constant 



1—1 irv dx 

 As before the constant is unity ; thus 



z = 



1 — I irx) dx 



This result agrees with that on Laplace's page 414. 



Laplace intimates that this would be an advantageous formula 

 if i and r were constants ; but as these quantities may vary, he 

 prefers another formula which he had previously investigated, and 

 which we have given from D'Alembert in Art. 483. He says that 

 by using the data furnished by observation, it appears that the 

 extinction of the small-pox would increase by three years the 

 mean duration of life, provided this duration be not affected by 

 a diminution of food owing to the increase of population. 



1036. Laplace discusses in his pages 415 — 418 the problem 

 of the mean duration of marriages which had been originally 

 started by Daniel Bernoulli ; see Arts. 412, 790. 



Laplace's investigation is very obscure : we will examine various 

 ways in which the problem may be treated. 



Suppose yu, men aged A years to marry yu, women of the same 

 age, /I, being a large number : determine the probability that at 

 the end of T years there Avill remain an assigned number of un- 



