d'alembert. 2G9 



The mean duration of life for persons of the age a years is 



I -v/r (x) dx 



a 



The probable duration is a quantity h such that 



This is D'Alembert's mode. We might however use another 

 curve or function. Let cf) (x) be such that <^ (x) dx represents the 

 number who die in an element of time dx. Then the mean dura- 

 tion of life for j)ersons aged a years is 



I (a? — a) (f) (x) dx 



a 



I </) (x) dx 



J a 



The probable duration is a quantity h such that 



I (f) (x) dx = I cj) (x) dx, 



that is I (f> (x) dx = - j (f) (x) dx. 



Thus the mean duration is represented by the abscissa of the 

 centre of gi'avity of a certain area ; and the probable duration is 

 represented by the abscissa corresponding to the ordinate which 

 bisects that area. 



This is the modern method of illustrating the point ; see 

 Art. 101 of the Theory of Probability in the Encyclopcedia Metro- 

 politana. 



486. We may easily shew that the two methods of the pre- 

 ceding Article agree. 



For we have <f)(x) —— h -^fr' (x), where Jc is some constant. 

 Therefore 



I {x — a)<f> (x) dx j (x — ^) ^' (^) <^'^ 



. ' a J_a . 



I <^ (x) dx I -v/r' (x) dx 



J a -'a 



