578 PHILOSOPHICAL TRANSACTIONS. [ANNO 1 80O. 



t— 1 



each case will be found = s . X (v — c — cc + ccc), which is known to be 



the true value, from self-evident principles. 



But the solution of this problem may be obtained by the assistance of the 1st 

 problem in my last paper, for the year 1794, supposing, instead of a given sum, 

 it were required to know the value of the reversion of a given estate. For since 

 the possession of this estate is an event which must certainly take place, and the 

 only point to be determined is the time in which it will probably happen, it is ob- 

 vious that no event can postpone the possession, but the contingency of c's being 

 the 2d that fails, of the 3 lives. If therefore the sum of the values of an annuity 

 on the life of b after a, provided a should die before c, and of an annuity on the 

 life of a after b, provided b should die before c, both found by the problem just 

 mentioned, be subtracted from the whole value of the reversion after the joint lives 

 of a and b, the remainder will be the value required. Let x and y respectively 

 denote the annuities found by problem 1st, Phil. Trans, vol. 84, then will the 

 general rule expressing the value of an estate be = v — ab — (x + y), and con- 

 sequently of a given sum = s — X (v — ab — (x + y)), which, when the 



s (r — l) 



lives are equal, may be reduced, as in the former cases, to — — x (v — c -— 



cc -f- ccc). 



Prob. 3. To determine the value of an estate, or of a given sum, after the 

 decease of a or b, should either of them be the first or last that shall fail, of the 

 3 lives, a, b, and c. 



Solution. The reversion of the estate in this problem, like that in the preceding 

 one, cannot be prevented ultimately from taking place; and there is only the single 

 contingency of c's being the first that fails, of the 3 lives, which can postpone the 

 possession of it after the extinction of the joint lives. The whole value therefore 

 of the reversion, after the joint lives of a and b, must in this case be lessened by 

 the sum of the values of an annuity on a's life after b, provided b should survive c; 

 and of an annuity on b's life after a, provided a should survive c, both found by 

 the 2d problem in my last paper, of the year 17Q4. Let these two values be res- 

 pectively denoted by w and z, then will the general rule expressing the value of an 

 estate be = v — ab — (w -|- z), and the value of a given sum = s ' (r ~ — x (v — 

 AB — (w -f- z)). When the lives are all equal, the value of the reversion, by sub- 

 stituting the values of w and z, becomes = v -f- cc — c — ccc, or s, ^ r ~ X 

 ( v _|_ cc — c — ccc), according as it consists of an estate, or a given sum. 



Prob. 4. To determine the value of a given sum, payable on the death of a, 

 should his life be the 1st or 2d that fails, and should b's life, if it fail, become 

 extinct before the life of c. 



Solution. In this case, the payment of the given sum can only be prevented by 

 the contingency of c's dying before a, and therefore its value is immediately found 



