530 
Mr. Gompertz on the nature of the function 
CHAPTER II. 
Article i. The near proximity to the geometrical pro- 
gression of the series expressing the number of persons living 
at equal small successive intervals of time during short 
periods, out of a given number of persons living at the com- 
mencement of those intervals, affords a very convenient 
mode of calculating values connected with life contingencies, 
for short limited periods ; by offering a manner of forming 
general tables, applicable (by means of small auxiliary tables 
of the particular mortalities) to calculations for any parti- 
cular mortality; and by easy repetition, to calculate the 
values for any length of period for any table of mortality we 
please. 
If, for instance, it were required to find the value of an annuity 
of an unit for p years, on three lives of the age 5, c, d, the rate 
of interest being such that the present value of an unit to be 
received at the expiration of one year, be equal to r, then the 
value of the first payment would be - ^ x x - x r; 
L, 
and of the payment the present value would be — x 
“l — ^ T X ^ ;butifL 
c d 
b+p 
(^,y 
whether p be 
1, 2, 3, &c. which will be the case when L,, L, , L, 
' ’ ’ o’ b-i-l ’ b + z, 
&c. form a geometrical progression, and similarly, if = 
C+I 
and also, L , , = L , x 
’ d + p a 
d + I 
, the pre- 
