116 LIFE ANNUITIES. 
To check the additions in the last example 
l l0i r m — .01692512 
l m r l03 = .05280636 
l m r m - .09153125 
/ 101 r 101 = .13326950 
/ 100 r loo = .17820036 
.47273259 — sum of the products above the age of 99, 
l 99 r" = .2265124 agreeing with the result obtained before. 
l 9n r 98 = .2998201 
l 97 r 97 = .4009023 
l 9 , r 96 = .5327548 
1.9327222 = sum of products above the age of 95, as before. 
115. Mr. Griffith Davies was the first who computed tables of the 
values of annuities on the above plan, some of which he has published 
in a tract, in which are given formulae for computing various cases of 
Annuities and Assurances on Single Lives. 
116. Tables have been inserted at the end to show the application of 
some of these formulae, the notation varying hut slightly from Mr. Da 
vies’s. The number in column D opposite any age is the product obtained 
by multiplying the number living opposite that age in Table 1. by the 
present value of £l due the same number of years as the age; thus, at 
the age of 30 the number living by Table 1, is 5642, and the present value 
of £l due at the end of 30 years is by Table 4, Part I. = .30831867. 
The product of these two numbers =1739.53393, which is the number 
in Table 13, under column D, opposite the age of 30. Having found in 
this manner the numbers in column D at all ages from birth to the extre 
mity of life, those in column N are found by beginning at the oldest 
age, and taking the successive sums of the numbers in column D, as in 
Art. 113, the number in column N at any age being the sum of the 
numbers in column D at all the ages above the given one. Column M 
is formed by multiplying the decrements opposite each age in Table 1 
by the present value of £l due the same numbers of years as the age 
increased by unity, and taking the successive sums from the extremity 
of life, as in the formation of column N from the numbers in column D. 
Column S is the sum of the number at any given age, and at all ages 
above in column N ; and column R is the sum of all the numbers in 
column M at any given age and above. 
D m , N m , M m , S m , R,„, represent the numbers opposite any age m 
in the respective columns so marked, 
D m i, N m _,, M m _!, S m _!, R m _!, opposite an age one year younger 
than m. 
D m+ ., N m+< , M m+( , S m+; R.„ +f , . . i years older than m, 
is *, M^, n _i)_)_f, i)+i j R(m—1;+( j i years older than a 
life one vear younger than m.