342 
STATISTICS: A. J. LOTKA 
Proc. N. a. S, 
tact occur at age ai, so that yt{a^ = Kie~^^^p{ai). At time f the sur- 
vivors of the individuals comprised in this elementary strip will be of 
age (ai + f — so that they will then be represented by a strip of 
width da and of altitude 
From this it is seen that the elementary strip of population which at time 
f contacts with the minor tangent curve (8) is built up of the survivors 
of the strip which at time t contacted with the minor tangent curve (5). 
In other words, if we follow up, by identity of individuals, the element 
of the population which at any instant contacts with the minor tangent 
curve, this element (so long as any part of it survives) continues in contact 
with that curve. (It must be remembered, however, that the tangent 
curve itself changes with time according to (5), (8).) 
Similarly it follows that the element of population which at any instant 
contacts with the major tangent curve continues in this contact so long 
as any part of it survives. 
And again, considering any element of the population which does not 
contact with the minor or the major tangent curves, but has its upper 
extremity at some point within the area enclosed between these curves,, 
it can be shown by precisely similar reasoning that such element continues 
in such intermediate position. 
Turning now from the consideration of the survivors of the original 
population, and taking in view the new population added by births since 
the time t = t, we note that if the original population had been that repre- 
sented by the shaded area in fig. 1, i.e., by the area under the minor tangent 
curve, then the birthrate would at all times have been such as continuously 
to reproduce a population represented by the minor tangent curve (5),, 
In point of fact, provided that contact with the minor tangent curve 
is not continuous over a range of ages equal to or greater than aj — a^, it 
is easily seen that the total birthrate 
is always greater (equality is here excluded) than that which would result 
from and in turn reproduce the age distribution represented by the minor 
tangent curve. 
Similarly the total birthrate is always less than that which would result 
from a population and age distribution represented by the major tangent 
curve. 
(8). 
