those that the elder survives the younger. I have been the more particular, and perhaps tedious, in this Matter, because it is the Key to the Case of Three Lives, which of it self would not have been so easie to comprehend. VII. If Three Lives are proposed, to find the Value of an Annuity during the continuance of any of those three Lives; the Rule is, _As the Product of the continual Multiplication of the Three Numbers, in the Table, answering to the Ages proposed, is to the difference of that Product, and of the Product of the Three Numbers of the Deceased of those Ages, in any given Term of Years: So is the present Value of 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) _Rrρ_, which is the Number of Chances that all three of the Persons are living. (2) _rρY_, which is the Number of Chances that the two elder Persons are living, and the younger dead. (3) _Rρy_ the Number of Chances that the middle Age is dead, and the younger and elder living. (4) _Rrυ_ being the Chances that the two younger are living, and the elder dead. (5) _ρYy_ the Chances that the two younger are dead, and the elder living. (6) _rYυ_ the Chances that the younger and elder are dead, and the middle Age living. (7) _Ryυ_, which are the Chances that the younger is living, and the two other dead. And Lastly and Eighthly, _Yyυ_, which are the Chances that all three are dead. Which latter subtracted from the whole Number of Chances _Nnν_, leaves _Nnν - Yyυ_ the Sum of all the other seven Products; in all of which one or more of the three Persons are surviving.