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continuously; and it is obvious, therefore, that if we can
ascertain the number of terms of the series which intervene
between these respective epochs of birth and death, we shall
thereby ascertain the mean duration of life. Now, since the
index of (1+r) is 1 at the expiration of one year from the
t ime at which the births, b, occurred, 2 at the expiration of
two years,3 at the expiration of three years, and n at the
expiration of n years, it follows that the index of that term of
the third of the above series which equals b represents the
mean duration of life in years.
Let x be that unknown index the value of which it
is required to ascertain. Then it is obvious that,
1:(1+r)x::d::d(1+r)x
but by the hypothesis
d(1+r)x=b;
so that the proportion becomes
1:(1+r)x::d:b;
and, consequently,
log.b/d
(1 +r)x=b/d, or x=/.
log. (1+r)
a formula from which the value of x can be readily deduced.
For the second case (that in which the birth-rates and
death-rates are exactly equal) the above formula is useless;
for here b and d being equal to one another,b/d-=1, r=0,
and the equation consequently becomes x=log.1/log.1=o-,o
the value of which is indeterminate. The mean duration of
life, however, is readily ascertainable by another process. In
this case the whole number of persons supposed to be living