420 PHILOSOPHICAL TRANSACTIONS. [aNNO 1794. 



From this table it appears, that the approximations and exact values do not 

 differ much from each other till the last years of b's life, and that the principal 

 inaccuracy in adopting the approximation will arise after the extinction of the 

 life of B, when it becomes necessary to multiply the fraction expressing the pro- 

 bability of his dying after a into the remaining series of the solution. But this 

 perhaps will be better understood from the following problems, and from the 

 computations which are made to prove the correctness of the general rules. 



Prob 1. — To find the value of an annuity on the life of c after a, on the 

 particular condition that a's life when it fails shall fail before the life of b. 



As the approximation appears from the preceding table to be always suffi- 

 ciently correct, except in the last 2 or 3 years of b's life, it is evident, that if 

 the fractions which express the probability of b's dying after a in those years, be 

 either confined only to the value of the annuity during that short period, or be 

 not involved at all in the computation, no great inaccuracy will arise from having 

 recourse to the ordinary method of determining that probability, provided the 

 solution be founded on real observations of life, and not on Mr. De Moivre's 

 hypothesis. In the present problem, when c or a is the oldest of the 3 lives, 

 the above-mentioned fractions either never enter into the computation, or are 

 confined to the last years of a's life ; and in both cases they are combined with 

 another contingency, which necessarily renders them of less consequence. The 

 solution therefore, particularly in the former case, becomes very easy ; and even 

 in the latter, by the assistance of the table in my first paper, in vol. 78, it be- 

 comes equally simple and correct. But when b is the oldest of the 3 lives, the 

 above fractions are combined with a series which is often of considerable im- 

 portance, and consequently the common method of solution fails in this case. 

 Yet even here, being possessed of the table deduced from the foregoing lemma, 

 it is attended with little or no difficulty, and a general rule as short and accurate 

 is obtained as in the other cases. Mr. M. then gives an analytical solution of 

 the problem. 



Pkob. 2. — To find the value of an annuity during the life of c, after the 

 decease of a, provided a should survive b. After the analytical solution of this 

 problem, Mr. M. adds the following 



Carol. If the solution of either of these two problems be given, the solution 

 of the other problem may be immediately derived from it ; for the value of the 

 reversion in one is no more than the difference between the value of the rever- 

 sion in the other, and the value of an annuity on the life of c after a. In 

 other words, let the value found by either of these problems be called q, and 

 the required value of the reversion in the other problem, supposing the ages of 

 A, B, and c to be the same in both, will be = c — AC — a. This deduction 



