Miscellanea Curiosa.

a Sum of Money, to be paid certainly after so many Years, to the present Value of the same Sum to be paid, provided one of those Three Persons be living at the Expiration of that Term. Which Proportion being yearly repeated, the Sum of all those present Values will be the Value of an Annuity granted for three such Lives. But to explain this, together with all the Cases of Survivance in Three Lives: Let N be the Number in the Table for the younger Age, n for the second, and ν for the elder Age; let Y be those dead of the younger Age in the Term proposed, y those dead of the second Age, and υ those of the elder Age; and let R be the Remainder of the younger Age, r that of the middle Age, and ρ the Remainder of the elder Age. Then shall R + Y be equal to N , r + y to n , and ρ + υ to ν, and the continual Product of the three Numbers N , n , ν, shall be equal to the continual Product of R + Y × r + y × ρ + υ, which being the whole Number of Chances for three Lives, is compounded of the eight Products following. (1) Rr ρ, which is the Number of Chances that all three of the Persons are living. (2) r ρ Y , which is the Number of Chances that the two elder Persons are living, and the younger dead. (3) R ρ y the Number of Chances that the middle Age is dead, and the younger and elder living. (4) Rr υ being the Chances that the two younger are living, and the elder dead. (5) ρ Yy the Chances that the two younger are dead, and the elder living. (6) rY υ the Chances that the younger and elder are dead, and the middle Age living. (7) Ry υ, which are the Chances that the younger is living, and the two