Vol. 8, 1922 
STATISTICS: A. J. LOTKA 
341 
The given arbitrary curve representing the age constitution of the popu- 
lation at time t then Hes wholly within the strip or area enclosed between 
the minor and the major tangent curves. 
Now consider the state of affairs at some subsequent instant f. Had 
the population at time t consisted solely of the individuals represented by 
the lightly shaded area in fig. 1, i.e. the area under the minor tangent 
curve, then at time the population would be represented by the lower 
curve of fig. 2, whose equation is 
(7) 
(8) 
Vor the age distribution (5) is of the fixed form (3), and therefore persists 
in (7); on the other hand, given such 
:fixed age distribution, the population as 
a whole increases in geometric progres- 
sion, » so that K\ = Ki/^^'~^\ In point 
of fact, we have left out of reckoning 
that portion of the population which in 
fig. 1 is represented by the dotted area. 
Hence, in addition to the population un- 
der the lower curve of fig. 2, there will, 
at time t\ be living a body of popula- 
tion which for our present purposes it 
is not necessary to determine numeri- 
cally. We need only know that it is some 
positive number, so that the curve re- 
presenting the actual population at time 
f must lie wholly above or in contact 
with the curve (8). 
By precisely similar reasoning it is read- 
ily shown that at time f the actual curve 
lies wholly beneath or in contact with 
the curve 
= K2/^''-''^e-"'p{a) (9) 
Hence at time f the actual curve lies 
Fig. 2 
The curves shown are intended to 
wholly Wlthm the strip comprised be- interpreted only in a qualitative 
tween the two curves (8) and (9). sense; so, for example, the increase in 
Consider now an elementary strip, of the ordinates in passing from fig. 1 to 
width da, of the original population ^ ^^^y. "^^^^ exaggerated, to 
(shown heavily shaded in fig. 1), which ^^^^^^ obvious to the eye. 
at time t is in contact with the minor tangent ciurve. Let this con- 
