526 Mr. Gompertz on the nature of the function 
the result for the age of 70 agreeing nearly with the living 
corresponding to the age 71 ; and the result for the age 90, 
agreeing nearly with the living at the age 89 of the Carlisle 
tables. 
Art. 1 1 . Lemma. If according to a certain table of mor- 
tality, out of dj persons of the age of 10, there will arrive 
c, d, &c. to the age 20, 30, 40, &c.; and if according to the 
tables of mortality, gathered from the experience of a parti- 
cular society, the decrements of life between the intervals 
10 and 20, 20 and 30, 30 and 40, &c. is to the decrements in 
the aforesaid table between the same ages, proportioned to 
the number of living at the commencement of those intervals 
respectively, as 1 to w, 1 to n' , i to n ", &c. it is required to 
construct a table of mortality of that society, or such as will 
give the above data. 
Solution. According to the first table, the decrements of 
life from 10 to 20, 20 to 30, 30 to 40, &c. respectively, will 
be found by multiplying the number of living at the com- 
mencement of each period by &c., and 
therefore, in the Society proposed, the corresponding decre- 
ments will be found by multiplying the number of living at 
those ages by n ; n' ; n" &c.; and the number of 
persons who will arrive at the ages 20, 30, 40, &c. will be 
the numbers respectively living at the ages 10, 20, 30, &c. 
multiplied respectively by j ^ — > 
&c.; hence out of the number a, living at the age 10, there 
will arrive at the age 10, 20, 30, 40, 50, &c. the numbers 
. a-\- 7 ib ; 1 ^n. nb y. ^ ~7— 1 — /z .a-\-nb x 
1 — n' .b n c ^ 1 — n". n". d . numbers for 
b c 
the intermediate ages must be found by interpolation. 
