estimation of the value of life contingencies. 
22 5 
Art. 8. If one pound is to be received at the expiration of 
the first of the equal periods /, after the expiration of the 
time ii — p, which shall happen after the failure of any of the 
above conditions of joint or single existence of the persons in 
question, provided that that failure should take place between 
the intervals n — p and m ; the present value of this sum will 
be the present value discounted for the period/ of the income 
on the contingency of the conditions not failing, the first pay- 
ment to be made at the period n — /, and the last at the period 
m — /, less the present value of the income on the contin- 
gency of the conditions not failing, the first payment to be 
made in the time n, and the last in the time m. Thus suppose 
the income, payable at every interval/, of one pound on the 
contingency of certain lives remaining, the first payment to 
be made at the time n, and the last at the time m> be repre* 
sented by n 
m 
•, and therefore a similar regulated income of 
which the first payment is to be made in the time n — •/, and 
the last in the time m — -/, will be n—p 
m — p 
and the present 
value of one pound to be received at the first period / which 
shall happen after the failure of the condition, &c. is 
rf x n-p 
m—p 
Because 
is the value of one 
pound to be received at the period 7 r in the contingency of the 
conditions not having failed, on or before that time ; «• 
7T 
is the chance of its not having failed on or before that time ; 
