338 Rule for Jindiw^ the Value of an Annuiiyon three hives. 



proximation when a /3y is not a small fraction of/*; and this 

 may account for De Moivre not having noticed a rule which 

 would, it might be imagined, present itself for trial at least, to 

 any one who was considering the subject for the first time. 



A much more satisfactory account of this rule can be given 

 from the expression for the law of human mortality which was 

 given to the Royal Society, by Mr. Benjamin Gompertz, in 

 1825. That law is as follows : The number living at x years 

 from a given age (upwards of 10) may be represented by 



00 



Kg^ where A^ is the number living at the given age. 

 The constants do not retain their value during the whole of 

 life ; Mr. Gompertz, for instance, finds one set of values re- 

 presenting the Carlisle tables very nearly from 10 to 60, and 

 another from 60 to 100. The closeness with which this theory 

 represents the Northampton, Sweden, Carlisle, and Depar- 

 cieux's tables may be seen in the paper cited ; and it adds 

 not a little to the speculative value of the formula, that its 

 author has deduced it from so simple a principle as that the 

 po^'doer of the human constitution to oppose decay loses equal pro- 

 portions in equal times. 



If Mr. Gompertz's theory were accurately true, with a uni- 

 form value of the constants, throughout the whole remainder 

 of life, Thomas Simpson's rule would no longer bean approxi- 



00 



mation, but an exact method. Let A^ be the number li- 

 ving at the age of A, and let B be j/ years older than A, and C 

 z years older than B. Consequently the numbers now alive 

 in the table at the ages of B and C will be 



A g; and A ^ or Kh^ and A ^ 



where h — g'^ ^ = 6 



/c = g 



Hence the chances of the parties living t years are 



^(/-i) ^^''(y'-i) .^''(/-i) 



g i h , k 



and, V being the present value of £l to be received in one 

 year, the value of the annuity is 



{ghk) .V + (ghk) .v^ + (I.) 



Determine w and I from hk = g^"" = I, then it will be 

 evident that if (1.) be called ^ {g h k), we have <p{gh/c 

 ^ <P{g 0> ^^^ annuity on the joint lile of A and another, whose 



