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at any one time is made up of a series of successive annual
increments, which are due to the successive annual births and
are equal to one another; and it is obvious that the term of
the last added of these increments to die will arrive only after
all the other increments have been successively removed.
And hence the mean duration of life will be the quotient
obtained by dividing the population, by the annual increment
of births.
In the third case (that, namely, in which the annual
deaths exceed the annual births, and the population has a
consequent tendency to die out) the formula which was made
use of for the first case is again applicable; only since b is
now less than d, b—d or r becomes a minus quantity, and the
ratio less than unity, so that the series (whether of population,
births or deaths) which is a diminishing one is infinite,
and the population, though constantly decreasing, never
comes to an end.
In Table 1., I have arranged the calculations which I
have made (in accordance with the principles above enunciated)
in regard to the combined relation of the death-rate and
birth-rate to the duration of life. I have in eight successive
vertical columns, in which it is assumed that the mortality
is .5, 1, 1.5, 2, 2.5, 3, 3.5 and 4 respectively per cent.,
arranged the mean durations of life which result from a combination
of these several mortuary rates and of birth-rates
varying from a little over 100 per cent. to zero. The series
of birth-rates which corresponds to each death-rate is placed
in a vertical column to the left-hand side of the death-rate
column; and all the birth-rates are so arranged that the
differences between the birth-rates and death-rates on the
same horizontal line of the general table are in all cases