VOL. XLVIII.] PHILOSOPHICAL TRANSACTIONS. 449 



before. Thus too it may be proved, that the probability of his receiving the 3d, 

 4th, 5th, &c. payment, will be 4-|-, f|-, -§-^, &c. and therefore, if the annual 

 payments were each ll. and if the interest of money was not to be considered, 

 we might conceive these several probabilities, as the present worths of the several 

 payments, and the sum of them would be the value of an annuity of the first kind. 



But since the interest of money necessarily enters the process, and since the 

 payments become due at the end of the 1st, 2d, 3d, &c. year, the first of these 

 payments must be discounted for 1 year, the 2d for 2 years, the 3d for 3 years,. 

 &c. and the sum of their present worths will be the value of an annuity of the 

 first kind, to continue during the life of a person who may possibly live 36 

 years ; and this sum may be found by an easy and well-known process, from the 

 common tables of compound interest and annuities, which need not be inserted 

 here. 



The annuity secured by land must necessarily be of greater value than the 

 above; because, though the annuitant dies before the payment becomes due, 

 yet his heirs are to receive a part of it ; the atniuitant therefore, in this case, has 

 not only the probability 4-^ of receiving the first payment, but he has also an ex- 

 pectation on part of the probability -^y which in the first case was wholly against 

 him. Now it may be esteemed an equal chance, supposing him to die in the 

 first year, whether that decease happens before the expiration of half that year, 

 or after it ; and if it happens before, he is to receive less than half the annual 

 payment ; but if after it, more. 



The annuitant may therefore be supposed to have an equal chance, if he fails 

 of receiving the whole first payment, yet of receiving half of it; and consequently 

 half of the probability, ^, which was before totally against him, will in this case 

 be favourable to him ; and his expectation of receiving either the whole, or at 

 least half of the first payment, will be -y- + ^. So since the probability of 

 his dying in the 2d year, is also -3-'^; we may in the same manner prove, that 

 one half of it will in this case become favourable to him ; and consequently that 

 ■§-|- -j- ^ will be the probability of his receiving the whole, or at least half, of 

 the 2d payment. It appears therefore that for every year which he has the pos- 

 sibility of living, he will in this case have the probability y^, or half of ^, in his 

 fevour, more than he had in the former case; and therefore, if the present 

 worths of the constant sum ^vl- (supposed to be due at the end of 1, 2, 3, &c. 

 years) be found, and added to the value of the annuity, according to the former 

 case, the sum will be the value of an annuity, secured by land, to continue 

 during the life of a person who may possibly live 36 years. 



Now since the sums by which the former annuity is to be increased, consist 

 of the present worths of that fraction of a pound sterling, whose numerator is 

 unity, and denominator twice the number of years that the annuitant can pos- 



VOL. X. 3 M 



