Mr. Morgan on Survivorships. 
*55 
d ■ PT _ APT 
— PC — APC 
g • FC - AFC . C' — A'C' 
6b 2 r 
BT _ ABT 
1' — A' 1 ' + - ' r ~ 1 x V — C*. 
1 cr ~ + 
When A is the oldest and B the youngest of the three lives-, 
let the symbols be changed which denote the decrements and 
probabilities of life of A and B ; let z be the difference between 
the age of A, and the oldest person in the table, and the whole 
value of the given sum during the life of A will be = — * 
db" . eb 1 " . , . 
S db , 
X h 
r 1 
b'+ b 
zbcr 
db‘ 
eb" 
~ it * T + F + 4 + • • • • ( z ) — 
eb 
zbcr 
7 + 
+ . . . . (*> 
/• v+ v . ' 
(*) 
3 a be 
acb' 
_j_ sdV 
db" , te.b'" . q 
- + — + 
a d . h 
4 &c. 4 
1 tf-b'+b" , 0 _ S sob' , td.b" 
+ r &C. — Tabc X — + — 
ue . b“' . Q S 
— 7T- + &c. + 
6abc 
dsb' . et . b" 
x— + - 1 - 
/« • 
+ &c. Let £, 4,. e, &c. denote the 
number of persons living opposite the age of C in the table at 
the end of 2 , % 4 1 , % 4 &c. years, /S'; /3", &c. the de- 
crements of life opposite the age of B at the end of those years 
respectively, tt the probability that B dies after A, and C* the 
value of an annuity on a life z years older than C ; then will 
the value of S in the % 4 1 st y ear (depending on the contin- 
gency of C’s dying after B in that year, or of C's dying in 
that year, B having died after A in either of the foregoing %■ 
years) be expressed by 44 + 444 In the z + 2d > 
* 4 3 d , &C. years, the payment of S will depend upon either 
