25 4 Mr. Morgan on Survivorships. 
gdly, A must die in the first or second year, and C die after 
B in the third year. The value therefore of S for. this year 
S . ~o ■ ~f. a 1 " , S ~ . S .n-o.c-f.a- \-<f 
Will be — tuber 3 ' 2 abcr 3 ' zubcr 3 
By pursuing the same steps during C’s life, the whole value 
s 
may be found 
6 abc 
a!bc . a"ind , 
x — + — -h 
-fi &C. — 
6abc 
„ S nd-a ■ 
+ — -T- x — + 
+ "+ 
^+^+ 9 L + &c - + 6 ^ + 
4- &C. — 
6 abc 
a ne 
r z 
4- + &C. + 
s 
■zabcr 
ne .a! , of ■ a -f 
ne .a , 
X — + 
S d — e . a' . e—f. 
+ & c - + = * — + — 
+ 
&c. = S into A x 
e . fk-afk 
bk — abk + 
g. fc — AF> 
b b 
r i . BC — ABC 
6 r 
+ + 
+ 
When B is the oldest of the three lives , it is evident that none 
of the foregoing series ought to be continued beyond the ex- 
tinction of B’s life, and that after this period the payment of 
the given sum will depend simply upon the failure of C's life 
in each of the remaining years, A having previously been sur- 
vived by B. Let the difference between the age of B and of 
the oldest person in the table of observations be denoted by x, 
the probability that B dies after A by ». the value of an an- 
nuity on the life of a person x years older than C by C-, the 
number of living at this age by k, and the values of annuities 
on the single and joint lives of A, C, and T for xyears by C'.T, 
A'C' and AT, then will the required value in this case be — 
S into £ x 
g. fk-afk bk. — ABK + - 
. BC — ABC 
6r 
JL — 
+ 3 br 
