282 
STATISTICS: PEARL AND REED 
Proc. N. a. S. 
X = — CO y = 0 
Relations (x) and (xi) define the asymptotes. 
The point of inflection is given by 1 — = 0, or 
X = — - log c 
a 
y = 
2c 
(xi) 
(xii) 
The slope at the point of inflection is — . 
Ac 
Expressing the first derivative of (xi) in terms of y, we have 
dy _ ay{h — cy) 
dx b 
Putting the equation in this form shows at once that it is identical 
(xiii) 
FIG. 2 
General form of curve given by equation (ix). 
with that describing an autocatalyzed chemical reaction, a point to which 
we shall return later. 
The general form of the curve is shown in figure 2. 
The question now is how well does (ix) represent the known historical 
facts as to the growth in population of the United States, and to what 
legitimate deductions as to the future course of population in this country 
does it lead ? 
It is obvious that equation (ix) as it stands cannot be fitted to ob- 
servational data by the method of least squares. It is possible to write 
momental equations and fit by the method of moments, but at this tim 
we do not care to develop that method because, as will presently appear, 
