[ 7 * ] 
Problem II. 
To find the 'value of an annuity for a limited inter- 
val of life, during which the decrements of 
life may be confidered as equal. 
Solution. 
L ET a and b reprefent the number of people liv- 
ing in the beginning and end oi the given 
interval of years. 
s reprefent that interval. 
P the value of an annuity certain for that 
interval. 
J^the value of an annuity for a life fuppofed to 
be neceffarily extinttin the time s 5 or (which 
is the fame thing) the value of an annuity for 
a life, of which the complement is s. 
Then ~ x ^ will exprefs the value re- 
quired. 
Demonstration. 
For, let the whole interval between a and b be 
fill’d up with arithmetical mean proportionals j there- 
fore the number of people living in the beginning 
and end of each year of the given interval s will 
be reprefented by the following ferics ,• viz. 
a. sa ~ a ~ s r h . sa ~ 2a + z b , 'a — t0 £ 
s s s s' 
Confequently, the probabilities of the life’s con- 
tinuing during 1, 2, 3, 4, 5, &c. years will be ex- 
prclfed by the feries, 
sa — a-\-b sa — za-\-zb sa—$a-\-$b sa— \a^-\b » b 
sa sa sa sa ’a’ 
Wherefore, 
