482 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1788. 



From this table it appears that Mr. Simpson's approximation in the middle 

 stages of life is sufficiently accurate; but that it is exceedingly defective when the 

 life of a is very young. It should also be remembered, that these values have 

 been computed at a low rate of interest, and from the Northampton Table of 

 Observations, in which the decrements of life come nearer to M. de Moivre's 

 hypothesis than in any other table. But if the computations be made at a higher 

 rate of interest even from this table, the approximation does not always agree so 

 well, as will appear from the following specimens calculated at 5 per cent. 



bO 

 < 



20 

 20 

 40 

 60 

 60 



Value of 100/. 

 payable on the 

 death of a if B 

 survives him, by 

 the Northamp- 

 ton table at 5 

 per cent. 



True 

 value. 



25.09 

 15.49 

 23.57 

 19.83 

 18.73 



Simp- 

 son's 

 value. 



18.46 

 17.54 

 15.97 

 11.75 

 19.61 



Value of 100/. payable on the death of a if b sur- 

 vives him, according to the Sweden Table of 

 Observations, and at 4 per cent. 



< 



20 

 20 

 20 

 40 

 40 

 40 

 40 

 60 



In order further to compare Mr. Simpsoh's approximation with the true value, 

 I have inserted in the foregoing table a few computations deduced from the 

 Sweden Table of Observations, in which the decrements of life are unequal. 

 From these instances the approximation appears to be more defective in propor- 

 tion as the probabilities of life differ from the hypothesis. 



Pkob. 3. — The ages of a and e being given ; to determine the value of the 

 sum s, payable on the extinction of one life in particular, should that happen 

 after the extinction of the other life. 



iSo/m/Zow.— Supposing b to be the older of the 2 lives, and the sum s to be- 

 come payable on his decease ; it is evident that this payment at the end of the 

 first year must depend on the contingency of both lives being extinct before this 

 period and of b's dying last. Retaining the same symbols, and reasoning as in 

 the solution of the first problem, this value will be expressed by the fraction 



*'" 9 fe • ^^^ payment of the sum s at the end of the 2d year will depend 

 on either of 2 events hap[>ening. First, that a and b both die in the 2d year after 

 having survived the first, restrained, as above, to the contingency of b's having 

 died last ; 2dly, that b dies in the 2d year and a in the 1st year. The value 



