Various Tables of Annuities. 327 



an individual died in the year in which he is recorded to have 

 died, ei the year after, e„ the n"' year after, &c., and if the Table 

 of mortality be founded upon a population observed from birth 

 throughout life, upon the same hypothesis of probability a priori 

 as before, the foimula which I gave in my former Paper on 

 this subject, p. 152, shews that if d,^ be the number of deaths 

 recorded to have taken place at the m"' age, the probability at 

 the birth of a child, that he will die at the n"' age is 



^d„+p 



2rf„ being the total number of persons observed, and p the 

 number of cases possible, or ages at which deatlis are supposed 

 to take place. The values of e are to a certain extent arbitrary. 



If e be supposed to be constant and = -, and that the values 



of e are e_,„...e-i, e, e^...e,„, this amounts to taking the mean of 



2 2 



the deaths which are recorded to have taken place within -- 



years of the age n. Generally, however e be supposed to vary, 

 2e,„ = 1. This theory shews how a Table of mortality should be 

 corrected for the irregularities which present themselves, wljen 

 the observations are not numerous. 



The number p may also be considered ais arbitrary, and by 

 altering this which amounts to increasing the deaths at every 

 age by an arbitrary quantity, the Table may also be corrected, 

 but the former method is simpler. 



With the assistance, of Mr. Deacon I have calculated the 

 Tables of Annuities, at the end of this Paper, by the approxi- 

 mate method given above, and the data or table of observations 

 from which they are taken is prefixed to each. 



