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 to show how such tables may be calculated with much greater facility.

41. By the method of No. 34, it will be found that, reckoning interest at 5 per cent., the present value of L. 1 to be received at the expiration of a year, provided that a life, now 89 years of age, survived till then, is $$\frac{142 \times 0{\cdot}95238.}{181}$$ But the age of that life will then be 90 years, and the proprietor of an annuity of L. 1 now depending upon it, will, in that event, receive his annual payment of L. 1 then due; therefore, if the value then of all the subsequent payments, that is, the value of an annuity on a life of 90 be 2·339 years’ purchase, the present value of what the title to this annuity may produce to the proprietor, at the end of the year, will be the same as that of L. 3·339, to be received then, if the life be still subsisting, or $$\frac{142 \times 0{\cdot}952381}{181} \times \text{L. } 3{\cdot}339 = \text{L. } 2{\cdot}495$$; which, therefore, will be the present value of an annuity of L. 1 on a life of 89 years of age. That is to say, an annuity on that life will now be worth 2·495 years’ purchase (7).

42. In the same manner it appears generally, that, if unity be added to the number of years’ purchase that an annuity on any life is worth, and the sum be multiplied by the present value of L. 1, to be received at the end of a year, provided that a life one year younger survive till then, the product will be the number of years’ purchase an annuity on that younger life is worth in present money.

43. But according to the table of mortality, an annuity on the eldest life in it is worth nothing; therefore, the present value of L. 1 to be received at the end of a year, provided that the eldest life but one in the table survive till then, is the total present value of an annuity of L. 1 on that life. Which, being obtained, the value of an annuity on a life one year younger than it may be found by the preceding number; and so on for every younger life successively.

Rate of Interest 5 per cent.

44. Proceeding as in No. 36, it will be found, that at 5 per cent. interest, and according to the Carlisle table of mortality, the present value of L. 1 to be received at the expiration of a year, provided that a person now 89 years of age, and another now 99, be then living, is $$\frac{142 \times 9 \times \text{L. } 0{\cdot}952381}{181 \times 11}$$: therefore, if the present value of an annuity of L. 1 on the joint continuance of two lives, now aged 90 and 100 years respectively, be L. 0·950; by reasoning as in No. 41, it will be found that the present value of an annuity on the joint continuance of two lives, of the respective ages of 89 and 99 years, will be worth $$\frac{142 \times 9 \times 0{\cdot}952381}{181 \times 11} \times 1{\cdot}950 = 1{\cdot}192$$ years’ purchase.

45. In this manner, commencing with the two oldest lives in the table that differ in age by ten years, and proceeding according to No. 43, the values of annuities on all the other combinations of two lives of the same difference of age, may be determined.

The method is exemplified in the following specimen:

46. Hence, and by what has been advanced in the 39th number of this article, it is sufficiently evident, how a table may be computed of the values of annuities on the joint continuance of the lives in every combination of three, or any greater number; the differences between the ages of the lives in each combination remaining always the same in the same series of operations, while the calculation proceeds back from the combination in which the oldest life is the oldest in the table, to that in which the youngest is a child just born.

47. But, when there are more than two lives in each combination, the calculations are so very laborious, on account, principally, of the great number 1em