VOL. LXXXIV.] PHILOSOPHICAL TRANSACTIONS. 421 



is self-evident, and if applied to any of the foregoing rales will be found to con- 

 firm the truth of the solution. 



Prob. 3. — To find the value of a given sum payable on the death of a and c, 

 provided b should survive one life in particular (a). After the analytical solution 

 of this problem, the author then adds : But the solution of this problem may 

 be obtained rather more easily by the assistance of the first problem in this 

 paper, and of the 2d problem which I communicated to the r. s. in the year 

 1788. For the value of a given sum payable on the death of a and c should b 

 survive a, is evidently " the difference between the value of that sum depending 

 on the contingency of b's surviving a, and the value of an annuity equal to the 

 interest of the given sum during the life of c after a, provided a should die 

 before b." The first of these is e, and if an annuity of ^1, by prob. 1, be 



denoted by q, the 2d will be = s. a. The required value therefore will be 



T — 1 



= E s. Q If the 3 lives be equal, the general theorem will be- 



come = — — s X (v — cc — c + c^), which may be derived from either of the 

 foregoing rules, or from the different series given above. 



Prob. 4. — To find the value of a given sum s, payable on the death of a 

 and c, should b die before one life in particular (a). After the general solution 

 of this problem, it is inferred, that the solution of this problem may also be 

 derived from the 2d problem in this paper, and the 3d problem in the paper 

 communicated in the year 1788. In other words, " the value of s in the pre- 

 sent case is equal to the difference between its value after the death of a and b, 

 provided b should die before a, and the value of an annuity equal to the interest 

 of s during the life of c after a, provided a should survive b." Let the first of 

 these values be denoted by w, and the second by x, then the required value will 



r — 1 



be = \v s X X. When the 3 lives are equal, the value of the reversion 



T — 1 



evidently Jaecomes = -—— s X (v — l), which expression may be easily derived 

 from either of the rules given above, or immediately from the series themselves. 

 And having given so many examples of the accuracy of the rules in the first and 

 second problems, it becomes unnecessary to add any further examples in regard 

 to the 2 foregoing problems, as the solutions of the latter are derived from those 

 of the former, and consequently are equally correct in all cases. 



Prob. 5. — To find the value of a given sum payable on the decease of b and 

 c, should their lives be the last that shall fail of the 3 lives a, b, and c. 



In the first year the given sum can be received only provided the 3 lives shall 

 have failed, and the life of a have been the first that became extinct. In the 2d 

 and following years it may be received provided either of 4 events shall have 

 happened : 1st, If ail the 3 lives shall have failed in that year, K dying first. 



