C 68 3 
Demonstr at ion. 
For, let z reprefent- any indeterminate portion of 
71. Now the probability of the life’s attaining the 
end of the interval z } and then failing, is to be 
expreffed by * , (as fhewn in page 77» edit. i. and 
n 
in page 1 1 5, edit. 2. of my book of annuities upon 
lives) upon the fuppofition of a perpetual and uni- 
form decrement of life. 
But it is well known, that if an annuity cer- 
tain, of i /. be paid during the time Zj its prefent 
-I* 
value will be or — — — •== — • 
r — i r — I r—lXr l 
And, by the laws of the do&rine of chances, the 
expectation of fuch a life, upon the precife interval 
z> will be exprefled by — %=. 4 =.; which may 
«X ) — i nr z Xt — x 
be taken for the ordinate of a curve, whofe area 
is as the value of the life required. 
In order to find the area of this curve, let 
p~ n x r l j and then the ordinate will become 
K jc 
j — } a much more commodious exprefUon. 
Now it is plain, that the fluent of the firft part 
is _ : but as the fluent of the fecond part is not fo 
P 
readily difeover’d, it will not be improper,, in this 
place, to fhew by what artifice l found it ; for 1 do 
not know, whether the fame method has been made 
ul'e of by others: all that 1 can fay, is, that I never 
had 
