Mr. Morgan on Survivorships. 
*44 
X AT. If A be the oldest of the three lives, let z, C', B'C', 
and <p denote the same quantities as in the second part of 
prob. I, and let F'C' be the value of the joint lives of F and C 
for % years, it will then be evident that the value of the rever- 
* S * ^ 1 1 " 
sion for the first % years will be = E -f- — * C '— * A'C r 1 • 
B'C' — ABC — 
s.g 
zb 
x F'C' — AFC -J — — 
. P' 
T' — AT, and its value after this term = V — C=; 
C* being the value of an annuity on a life z years older than 
C, and k the number of persons living at the age of C 2 . . . If B 
be the oldest of the three lives, the v alue, by proceeding as abo ve, 
may be easily found = E + ~ x C' — A'C' — r— 1 . BC — AdC 
-^xFC_AFC + ^x 
S.d jM.Pf — APT 
T-A'T+7 
S.r-i 
HT x v C x ; C, A'C', 7 r and x denoting the same quan- 
tities as in the third part of prob. I, C* the value of an an- 
nuity on a life x years older than C, q the number of persons 
living at the age of C* . . and T' and A'T' the values of annui- 
ties on the single life of T, and on the joint lives of A and T 
for x years. 
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 second problem which I communicated to the Royal 
Society in the year 1788* For the value of a given sum pay- 
able on the death of A and C should B survive A, is evidently 
W the difference between the value of that sum depending on 
« the contingency of B's surviving A, and the value of an 
* Phil. Trans. Vol, LXXVIII. p. 3+1. 
