Vol.. 8, 1922 
STATISTICS: A. J. LOTKA 
343 
From this it is clear that, after the original population has once died 
out, the representative curve can never again contact with the original 
minor and major tangent curves, but must henceforth be separated from 
them everywhere by a finite margin (except, of course, where the several 
curves terminate upon the axis of a). 
We may then begin afresh by drawing a new pair of tangent curves, 
lying within the original pair, and so on indefinitely, until the minor and 
major tangent curves coincide, and with them also coincides the actual 
curve of age distribution, which is then of the form 
In this argument we have expressly excluded the case that the original 
age distribution curve contacts continuously with one of its tangent curves 
over a range greater than the reproductive period {aj — a^) = A. If 
such continuous contact occurs in the 0th generation over a range nA , 
where ay/A > n > 1, then a simple reflection shows that in the next generation 
(the first) contact will occur over a range (n — 1)^4, in the second genera- 
tion over {n — 2) A, etc. A time is therefore reached (in practice very 
soon), when contact is over a period less than the period of reproduction 
A. After that the argument set forth above applies. 
If the curve representing the original age distribution contacts with 
one of the tangent curves all the way from a = 0 to a = aj, so that 
aj/A ^ n, then, of course, the fixed age distribution is practically estab- 
lished ah initio, or at any rate from the moment the original population 
above reproducing age Oj has died out. 
It remains to consider the effect of variability in the form of |8(a) with 
changes in c{t,a). Some such variability undoubtedly exists owing to 
the influence of the ages of the male and female constituents of the popu- 
lation upon the frequency of matings. We may, nevertheless, in this 
case also, define a minor and a major tangent curve (5), (6) in terms of 
the value of r given by equation (4) ; in order, however, to make this value 
determinate, it is now necessary to make some definite disposition re- 
garding the form of (3{a), which is now variable. Merely for purposes 
of defining r, we shall suppose that the function /3(a) under the integral 
sign has that particular form which corresponds to the fixed age distribu- 
tion.^ 
We cannot, however, now reason, as before, that the portion of the popu- 
lation represented by the shaded area in fig. 1 will, by itself, reproduce 
N{t) c{a) = Ke''e-''p{a) 
which we recognize as the fixed age distribution (3), with 
0 
