DE. FAEE ON THE CONSTEUCTION OF LIFE-TABLES. 
843 
they were exposed to certain dangers represented by successive discharges of musketry 
which at every discharge shot down one-half of the numbers remaining, they would be 
reduced successively from 1600 to 800, to 400, to 200, to 100, to 50, and so on ad inji- 
nitum, if a fraction of a living man could be conceived : the numbers living at eacli 
year of age in a Life-Table would not decrease at these rates ^ but they would decrease at 
a constant rate if the dangers at every stage of life remained constant and equally great. 
The numbers of the living at successive ages would be in geometrical progression, and 
would be represented by the ordinates of the logarithmic curve. 
The law of mortality can only be derived from observation, and it is found to be less 
simple than either of these hypotheses implies. It can, however, be represented nearly 
by equations at different periods of age. Upon inspecting Table A (p. 864), it will be 
seen that at the age 55 — 65, which may be represented by the exact age 60, the mortality 
is such, that 2162 women die in a year out of a number equal to 100,000 living a year ; 
and the mortality, which is the ratio of the dying to the living in a unit of time, here 
set down as a year, is therefore '02162. Again, the mortality at the age of 70 is 
•04992 ; at the age of 80 it is -11866, and at the age of 90 it is -26711. The mortality 
increases rapidly, and is more than doubled every ten years. The four numbers differ 
little from the terms of a geometrical progression, the logarithms of which have a con- 
stant difference. Let the rate at which the mortality increases be r, and r’®=2-3116, 
and the first term {m) be -02177,- then a series of numbers will be formed differing 
little from those which express the value of m at decennial intervals of age. 
Values of m at the precise age x. — Females. 
Age (x). 60. 70. 80. 90. 
By observation . . . -02162 -04992 -11866 -26711 
By hypothesis . . . -02177 -05033 -11633 -26891 
Xote. — It may be assumed that m at 60 is the mean value of m in its range from 
to and so in other cases. 
The annual rate of the increase of m from the age of 55 to 95 is r= 1-0874; and if 
m is the mortality at any age after 55, then m^=mr^ =■ the mortality at z years after the 
age at which m is taken. The common logarithm of r is -03639. 
The mortality (m) of males at corresponding ages is higher than the mortality of 
females ; but the rate of increase as age advances is nearly the same. 
The value of m for females at the age of 20 is -00765, and the mortality increases at 
the rate of nearly one-seventh part every ten years. The exact value of r is 1-0149, 
and Xr= -006423. 
Values of m. — Females. 
Age. 20. 30. 40. 50. 
By observation . . . -00765 -00894 -00998 -01192 
By hypothesis . . . -00760 -00882 -01022 -01185 
By these observations in the healthy districts the mortality (m) of men at the ages 
15 to 45 is lower than the mortality of women at the same ages; yet during that period 
