Mr. Morgan on Survivorships. 249 
t^alues as in the third part of the first problem, and let A'T' 
denote the value of an annuity on the joint lives of A and T 
for x years, P' the value of an annuity on the life of P for the 
same term, C the value of an annuity on a life x years older 
than C, and k the number of persons living in the table at 
that age, then will the value of the given sum be in this case 
= S into - 
■ BU- ABC 
3 r 
+ F* 
13 . FK— AFK 
— BK— ABK 
/8 ■ FC — AFC 
3b 
-f JL. 
~ 2 br 
4-d . PT — APT 
3 C 
PC — APC 
3 
BT— A BT 
3 
+ T'— A'T' -f 
— P' — AT' — 
+ 
B + C — AB-j-A'C' , <rr.k.r i 
cr xJ e l ~ x 
V — O. f 
If Abe the ol d est of the three lives, it will be necessary to 
substitute a 5, 5 — t, t — «, &c. for their equals a', a", a!". See. 
and b', b", b"\ &c. for their equals b 
■m, m—n, n — o, &c. 
In this case let C be supposed the oldest of B and C, and the 
series expressing the value of the reversion during the life of 
s 
A will become x 4- 
7 abc r I 
ft-b'+b" 
+ See. -f 
dsb' 
3abc 
+ ^ + = + &c ._ • 
fub 
6abc 
^ + ^ + - +&c ._ 2 - x ^zp T ^ : _ 
csb' 
dtb" 
30.be 
4- 411- _L eub I , S 
-r- r * T — + &C. + -r X 
s 
2 ab 
+ f + % + &c. + 
2 b 
S 
2 be 
+ 
r 1 I 
+ &C. 
** + ? +- + 8CC. 
S db , eb" fb'" e *T , , -=~==~— 
^ x r + ? + - + &c. +^ x ^ + f^A±£: +&c . 
Let y represent the difference between the ages of A and of 
the oldest person in the table, let K', C', B', T', B'C' B'K f 
MDCCXCIV. K k J 
