268 d'alembert. 



The result is not of practical use because the value of the 



C Ciu 

 integral 1— is not known. D'Alembert gives several formulie 



which involve this or similar unfinished integrations. 



484. D'Alembert draws attention on his page 74 to the two 



distinct methods by which we may propose to estimate the espe- 



7'ance de vivre for a person of given age. The mean duration of 



life is the average duration in the ordinary sense of the word 



average ; the j^^^^ohahle duration is such a duration that it is an 



even chance whether the individual exceeds it or falls short of it. 



Thus, according to Halley's tables, for an infant the 7nea)i life is 



26 years, that is to say if we take a large number N of infants 



the sum of the years of their lives will be 2QN; the probable 



N 

 life is 8 years, that is to say ^ of the infants die under 8 years 



N 

 old and - die over 8 years old. 



Li 



The terms mea/i life and probable life which we here use have 

 not always been appropriated in the sense we here explain ; on the 

 contrary, what we call the mean life has sometimes been called 

 the probable life. D'Alembert does not propose to distinguish the 

 two notions by such names as we have used. His idea is rather 

 that each of them might fairly be called the duration of life to be 

 expected, and that it is an objection against the Theory of Proba- 

 bility that it should apparently give two different results for the 

 same problem. 



485. We will illustrate the point as D'Alembert does, by means 

 of what he calls the curve of mortaliti/. 



Let X denote the number of years measured from an epoch ; let 

 yjr (x) denote the number of persons alive at the end of x years 

 from birth, out of a large number born at the same time. Let 

 'yjr (x) be the ordinate of a curve ; then yjr (x) diminishes from 

 X = to X = c, say, where c is the greatest age that persons can 

 attain, namely about 100 years. 



This curve is called the curve of mortality by D'Alembert. 



