VOL. XC.] PHILOSOPHICAL TRANSACTIONS. 577 



blems; and, having succeeded in the solution of them, I hope the following will 

 not be considered as an improper addition to my former communications. 



Problem 1. To determine the value of a given sum, payable on the death of a 

 or b, should either of them be the J st or 2d that fails, of the 3 lives, a, b, and c. 

 — Solution. In this case, the payment of the given sum must certainly take place 

 on the extinction of the joint lives of a and b, independent of c, and therefore the 



value of the reversion will be = ' ^ "*" — — .* 



r 



Prob. 2. To determine the value of a given sum, payable on the decease of a 

 or b, should either of them be the 2d or 3d that shall fail, of the 3 lives, a, b, 

 and c. — From the analytical solution of this problem, which is of considerable 

 extent, it is at length deduced, that the value required will be generally denoted by 



• , r— 1 ,, A + B + C, . AC x. 



s into— X (v + ABC - ) + - - ab — - x (ak + bk- 2abk) + 



s x (af - apc) + _ x (i + ap + d. — r - ). 



But, it is observed that, unless a and b are very nearly of the same age, and 

 both older than c, this rule will not be sufficiently accurate. If b be the oldest of 

 the 3 lives, the annuities a, ac, and ak, should be continued only for as many 

 years, x } as are equal to the difference between the age of b and that of the oldest 

 life in the table of observations. Let those annuities be respectively denoted by a', 

 ac', and a'k'; also let p denote the probability that c survives b, q the number of 

 persons living opposite to the age of a at the end of x years, then will the value of 



s, after x years, be = -~ X - , and the whole value of the reversion 



•11 i • i r ~~ 1 v* / ■ A' + B + BC\ Kj , a'c' k , , 



will be = s into —^ X (v + abc 2i_ ) X -|- — ab — — x (a V-f BK 



_ 2 bk) + A X (A ; _ AFC) + £ x (1 + AP + 1. azftj. % x <^£-, 



a* denoting the value of an annuity on a life x years older than a. 



If c be the oldest of the 3 lives, let a — s, s — t, t — u, &c. be substituted 

 for their equals a, a", a"', &c, and c — c',c— (c + c"), c — (c -f c" -f- c"), &c. 

 for their equals e,f, g, &c. then will the value of the reversion be found = s into 



^ X (V-*(A + B) + ABC) + ^1 - AB + id»=™> + i^«3 



If the lives be all equal, the value, according to the first rule, will be = s into 



r — 1 / \ cc . * , \ . d 



— X (v - c - -lcc + ccc) + - - cc + - x (kcc - kc) + — x (1 + ct) 



-| X (tt — ctt); and, according to the 2d rule, it will be = s into ^~ x 



/ i \ i cc i *.(ck — cck) d v ^ i' . . dd 

 X (v - c + ccc) + - - cc + — i— -' - - X (1 + ct) + — x (1 + 



ctt). If these expressions be resolved into their respective series, the value in 



* The same symbols are uniformly retained in this, as in my last 2 papers on the subject. See Phil, 

 Trans., vol. 81, p. 247.— Orig. 



VOL. XVIII. 4 E 



