Yoh. 8, 1922 
STATISTICS: A. J. LOTKA 
339 
THE STABILITY OF THE NORMAL AGE DISTRIBUTION' 
By Alfred J. Lotka 
ScHooi. OF Hygiene and Pubi^ic Heai^th, Johns Hopkins University 
Communicated, September 13, 1922 
There is a unique age distribution which, in certain circumstances, ^ 
has the property of perpetuating itself when once set up in a population. 
This fact is easily established,^ as is also the analytical form of this unique 
fixed or normal age distribution. 
More difficult is the demonstration that this age distribution is stable, 
that a population will spontaneously revert to it after displacement there- 
from, or will converge to it from an arbitrary initial age distribution. Such 
a demonstration has hitherto been offered only for the case of small dis- 
placements,^ by a method making use of integral equations. The pur- 
pose of the present communication is 
to offer a proof of stability which em- 
ploys only elementar}^ analytical opera- 
tions, and which is readily extended to 
cover also the case of large displace- 
ments. This method presents the fur- 
ther advantage that it is molded in 
more immediate and clearly recogniz- 
able relation to the physical causes that 
operate to bring about the normal age 
distribution. 
Consider a population which, at time 
t has a given age distribution such as ^ 
that represented by the heavily drawn j^ff^^ ^ 
curve in fig. 1, in which the abscissae 
represent ages a (in years, say), while the ordinates y are such that 
the area comprised between two ordinates erected at ai and ao, re- 
spectively, represents the number of individuals between the ages ai 
and a2. 
If we denote by N{t) the total population at time t, and if the ordinates 
of our curve are 
y = N{t) c{t,a) (1) 
we have, evidently, 
J-»a2 /*a2 
y da = N(t) 1 c{t,a)da = N(t,ai,a2) 
(2) 
