estimation of the value of life contingencies. 293 
&c. from the present time after the failure, provided it takes 
CT 
place between the intervals n —p and m. And P 
m — p 
. r +nlP . M„ +ip ) or its equal r~^ • Q — — N„ . r\ M n _^ p 
is the value if it is to be paid immediately after the failure, 
provided it takes place between the intervals n and m* It is 
necessary to add, that the method pointed out in Art. 4 of this 
Scholium, for solving the problems of Section 4, will gene- 
rally produce results which do not agree to absolute mathe- 
matical equality, with the results of that Section, except the 
interval p be infinitely small ; but they will agree with each 
other as far as the first power of p is concerned ; which when 
p is taken, the smallest interval of the tables will be as near 
the truth as any method should be considered to reach, as 
long as the real function of life is not known ; except indeed 
there be sufficient regularity in the tables to induce the belief 
that we may approach nearer by interpolation, as hinted in 
Art. 4, Section 3 ; but if the interval p be not greater than 
one year, this will give, I think, sufficient accuracy for any 
useful purpose, except perhaps in very rare cases, and in 
which our tables (from more minute observation), should, I 
imagine, be divided into less periods than yearly interval, and 
then the same method would still apply by taking p smaller. 
The same observation will apply if a comparison is made with 
what is done in the present Article 6 of this Scholium, with the 
other articles of the Scholium ; for instance, with Article 3. 
As an annuity secured by land, only differs from common 
annuities, in as much as in case of death of the lives on which 
the annuity is determined, during the portion of a year, 
mdcccxx. Q q 
