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 of combinations, that no tables of that kind have yet been published, except three or four for three lives.

And, in the books that contain tables of the values of two joint lives, methods are given of approximating towards the values of such combinations of two and of three lives, as have not yet been calculated.

Therefore, assuming the values of annuities on single lives, and on the joint continuance of two or of three lives, to be given; we have next to show how the most useful problems respecting the values of any interests that depend upon the continuance or the failure of life, may be resolved by them.

48. Proposition 1. The value of an annuity on the survivor of two lives, A and B, is equal to the excess of the sum of the values of annuities on the two single lives, above the value of an annuity on their joint continuance.

49. Demonstration. If annuities on each of the two lives were granted to P, during their joint continuance, he would have two annuities; but if P were only to receive these upon condition that, during the joint lives of A and B, he should pay one annuity to Q; then, there would only remain one to be enjoyed by him, or his heirs or assigns, until the lives both of A and B were extinct; whence the truth of the proposition is manifest.

50. ''Prop. 2''. The value of an annuity on the joint continuance of the two last survivors out of three lives, A, B, and C, is equal to the excess of the sum of the values of annuities on the three combinations of two lives (A with B, A with C, and B with C) that can be formed out of them, above twice the value of an annuity on the joint continuance of all the three lives.

51. Demons. If one annuity were granted to P on the joint continuance of the two lives A and B, another on the joint continuance of A and C, and a third, on the joint continuance of B and C; during the joint continuance of all the lives he would have three annuities.

But if he were only to receive these upon condition that he should pay two annuities to Q, during the joint continuance of all the three lives; then, there would only remain to himself one annuity during the joint existence of the last two survivors out of the three lives. And the truth of the proposition is manifest.

52. ''Prop. 3''. The value of an annuity on the last survivor of three lives, A, B, and C, is equal to the excess of the sum of the values of annuities on each of the three single lives, together with the value of an annuity on the joint continuance of all the three, above the sum of the values of three other annuities; the first dependent upon the joint continuance of A and B, the second, on that of A and C, and the third, on B and C.

53. Demons. If annuities on each of the three single lives were granted to R; during the joint continuance of all the three, he would have three annuities, and from the time of the extinction of the first life that failed, till the extinction of the second, he would have two.

So that he would have two annuities during the joint existence of the two last survivors out of the three lives; and besides these, a third annuity during the joint continuance of all the three.

Therefore, if out of these, R were to pay one annuity to P during the joint continuance of the last two survivors out of the three lives, and another to Q during the joint continuance of all the three; he would only have left one annuity, which would be receivable during the life of the last survivor of the three.

But in the demonstration of the last proposition (51) it was shown, that the value of what he paid to P, would fall short of the sum of the values of annuities dependent respectively on the joint continuance of A and B, of A and C, and of B and C, by two annuities on the joint continuance of all the three lives. Whence it is evident, that the value of the annuities he paid both to P and Q, would fall short of the sum of these three values of joint lives, only by the value of one annuity on the joint continuance of all the three lives.

Wherefore, if from the sum of the values of all the three single lives, the sum of the values of the three combinations of two that can be formed out of them were taken; there would remain less than the value of an annuity on the last survivor, by that of an annuity on the joint continuance of the three lives.

But if, to the sum of the values of the three single lives A, B, and C, there be added that of an annuity on the joint continuance of the three, and from the sum of these four values, the sum of the values of the three combinations, A with B, A with C, and B with C be subtracted; then, the remainder will be the value of an annuity on the last survivor of the three lives. Which was to be demonstrated.

54. ''Prop. 4. Problem''. The law of mortality and the values of single lives at all ages being given; to determine the present value of an annuity on any proposed life, deferred for any assigned term.

55. Solution. Find the present value of an annuity on a life older than the proposed, by the number of years during which the other annuity is deferred; multiply this by the present value of L. 1 to be received upon the proposed life surviving the term, and the product will be the value sought.

56. Demons. Upon the proposed life surviving the term, the annuity dependent upon it will be worth the same sum, that an annuity on a life so much older is now worth; therefore, it is evident, that the deferred annuity is of the same present value as that sum to be received at the expiration of the term, provided the life survive it.

57. Corollary. In the same manner it appears, that the present value of an annuity on the joint continuance of any number of lives, deferred for a given term, may be found by multiplying the present value of an annuity on the joint continuance of the same number of lives, older respectively than the proposed, by the number of years in the term; by the present value of L. 1 to be received, upon the proposed lives all surviving it.

58. A Temporary Annuity on any single life, or on the joint continuance of any number of lives, that is, an annuity which is to be paid during a certain term, provided that the single life or the other lives jointly subsist so long; together with an annuity on the same life or lives deferred for the same term, are evidently equivalent to an annuity on the whole duration of the same life or lives.