different Tables of Mortality. 347 



XX 



D. Living compared with s = 1 — — . 



cc 



Deparc. Carlisle. Eq. Offi, 



Agear. c=87 c=90 c=93 



.954 



.896 



.815 



.711 



.584 



.433 



.260 



.064 



It is obvious that the formulas approach, in both these com- 

 parisons, much nearer to the tables than in A and B. The column 

 of Deparcieux is best represented by the divisor 87, at least from 

 25 to 80, and the same is true of the Carlisle table, except just 

 about 60 ; while the supposed experience of the Equitable Office, 

 after 40, agrees best with the divisor 90, or even 93. 



13. The expectation of life, or the value of a life annuity without 

 interest, is next to be determined for each hypothesis. The fluxion 

 of the expectation is evidently equal to the fluxion of the age, mul- 

 tiplied by the chance of surviving to that age, which is expressed 



by the quotient of the survivors, — , supposing k to be the initial 



k 



number of the living at the given age, and the fluxion of the ex- 

 pectation is-LdcF, that is, {\-±\'h or A - —\ —, ac- 

 Ic \^ c y fc \ cc y fC 



cording to the hypothesis to be employed, and the fluents are 



and — — respectively, takino- the values from 



k 2ck k 3cck 



s =Ar or a? = q, the given age, to j? = c, the extreme period of life 



assumed in the hypothesis. 



X XX 



14. Now, in the arithmetical hypothesis for — — , we have 



^* k 2ck 



-i- — -ii_ and — — , the difference beinff_( c — — — o 



k 2ck k 2ck "^ k\ 2 ^ 



+ ^^=££z5^i±Ii=(ili>!. But;t=l-X = £Zi,and 

 2c/ 2ck 2ck c c 



the expectation e becomes = 2llJL , as is well known. 



