340 STATISTICS: A. J. LOTKA Proc. N. A. S. 
where N{t,ai,a2) denotes the number of individuals Hving at time t and 
.comprised within the age Hmits ai and a^. We may speak of c{t,a) as 
the coefficient of age distribution. It is, of course, in general a function 
of /, only in the special case of the fixed or self-perpetuating age distri- 
bution is c{a) independent of the time. 
Now, without assuming anything regarding the stability of the self- 
perpetuating age distribution, it is easy to show^ that its form must be 
f e''p{a)da ^ ^ 
where r is the real root of the equation 
1 = pe-'''p{a)^(a)da (4) 
In this equation r is the natural rate of increase of the population, i.e., 
the difference r = b — d between the birthrate per head b and the death- 
rate per head d, and p(a) is the probability, at birth, that a random indi- 
vidual will live to age a (in other words, it is the principal function tabu- 
lated in life tables, and there commonly denoted by 1^). The limits a,- 
and Gj of the integral are the lower and upper age limits of the reproductive 
period. The factor i8(a), which might be termed the procreation factor, 
or more briefly the birth factor, is the average number of births contri- 
buted per annum by a parent of age a. (In a population of mixed sexes 
it is, of course, immaterial, numerically, to what parent each birth is 
credited. It will simplify the reasoning, however, if we think of each 
birth as credited to the female parent only.) 
The factor jS(a) will in general itself depend on the prevailing age dis- 
tribution. This is most easily seen in the case of extremes, as for example 
in a population which should consist exclusively of males under one year 
of age and females over 45. But, except in such extreme cases, ^{a) will 
not vary greatly with changes in the age constitution of the population, 
and we shall first develop our argument on the supposition that ^(a) 
is independent of the age distribution. We shall then extend our reason- 
ing to the more general case of /3(a) variable with c{t,a). 
Referring now again to fig. 1, let two auxiliary curves be drawn, a minor 
tangent curve and a major tangent curve 
y, = K^e-'^'p^a) (5) y, = K.e-'^'pia) (6) 
the constants i^i, being so chosen that the minor tangent curve lies 
wholly beneath the giv^en arbitrary curve, except where it is tangent 
thereto, while the major tangent curve lies wholly above the given curve, 
except where it is tangent thereto. 
