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the same age. Under these conditions there are three
different cases for investigation; namely, Ist, that in which
the births exceed the deaths and the population consequently
increases; 2nd, that in which the births and deaths are equal,
and the population consequently undergoes no change of
number; and 3rd, that in which the deaths exceed the
births, and the population therefore progressively diminishes.
In the first case (that in which the births exceed the
deaths), the population, the births and the deaths, all increase
from year to year in the same ratio by geometrical progression.
And if the number of the population from which
we start be represented as a unit, and the letters b and d
represent respectively the birth-rate and death-rate of this
unit, and r its increase for one year, or what is the same
thing, the difference between b and d, the following three
series will represent the annual growth of the population, of
the births and of the deaths respectively :
Population, Births. Deaths.
1st year 1 b d
2nd year (1+r) b(1+ r ) d(1+r)
3rd year (1+ r)2 - b(l+r) 2 - d(1+r)*
4th year (1+ r)3 b(1+r) 3 d (1+r) 3
5th year (1+ r ) 4 (1+ r ) 4 (1+r)4
* * * *
nthyear (1+r)n-i b(1+r)n-i d(1+r)n-i
Now, according to the hypothesis with which we
started, all the persons born in any one year, attaining as
they do exactly the same age, will die together in the
coarse of some future year. Thus, the births represented by
the letter b, occurring in the first year of the series, will die
in a lump, so to speak, at the end of some definite period,
and those of the next year represented by b(1+r) will die
similarly in the course of the year next following, and so on