VOL. XVII.] PHILOSOPHICAL TRANSACTIONS. 489 



other dead ; whence jnay be cast up what value ought to be paid for the rever- 

 sion of one life after another, as in the case of providing for clergymen's 

 widows and others by such reversions. And as AD to AG or rY, so are all 

 the chances, to 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 itself would not have been so easy 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 an- 

 nuity 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 v for the elder age ; let Y be those dead 

 of the younger age in the term proposed, 7/ those dead of the second age, and 

 V those of the elder age ; and let R be the remainder of the younger age, r that 

 of the middle age, and j the remainder of the elder age. Then shall R -|- Y 

 be equal to N, r -{■ y to n, and j + u to v, and the continual product of the 

 three numbers N nv will be equal to the continual product ofR-f-Yx r -\- y 

 X f + u, which, being the whole number of chances for three lives, is com- 

 pounded of the eight products following, viz. l.Rrj, which is the number of 

 chances that all three of the persons are living; 2. r^Y, which is the number 

 of chances that the two elder persons are living, and the younger dead ; 3. Rf^, 

 the number of chances that the middle age is dead, and the younger and elder 

 living ; 4. R?u, being the chances that the two younger are living, and the elder 

 dead ; 5. ^Yy, the chances that the two younger are dead, and the elder living ; 

 6. rYu, the chances that the younger and elder are dead, and the middle age 

 living; 7. R^u, which are the chances that the younger is living, and the two 

 others dead ; 'and lastly and 8thly, Y^u, which are the chances that all three are 

 dead. Which latter subtracted from the whole number of chances N wn, leaves 

 Nwv — Yyv the sum of all the other seven products, in all of which one or 

 more of the three persons are surviving. 



To make this yet more evident, I have added fig. 3, pi, 12, where these 

 eight several products are at one view exhibited. Let the rectangled parallelo- 

 pipedon ABCDEFGHbe constituted of the sides AB, GH, &c. proportional 



VOL. III. 3 R 



