366 
STATISTICS: PEARL AND REED 
Proc. N. a. S. 
or 
dy 
^ , -^=-a'. (iii) 
y{k-y) 
If y be the variable changing with time x (in our case population) 
equation (iii) amounts to the assumption that the time rate of change of y 
varies directly as y and as (k — y), the constant k being the upper limit of 
growth, or, in other words, the value of the growing variable y at infinite 
time. Now since the rate of growth of y is dependent upon factors that 
vary with time, we may generalize (iii) by using / (x) in place of — a', 
f (x) being some as yet undefined function of time. 
Then 
dy 
y{k-y) 
= f{x)dx. 
whence 
and 
my 
where 
F{x) = —kf f{x)dx. 
Then the assumption that the rate of growth of the dependent variable 
varies as (a) that variable, (b) a constant minus that variable, and (^7) an 
arbitrary function of time, leads to equation (iv), which is of the same form 
as (i), except that ax has been replaced by F(x). If now we assume that 
f{x) may be represented by a Taylor series, we have 
y = — (v) 
X _|_ + ^^2^^ + 03^c3 + + anXn ^ ^ 
If 
a2 = as = = Gn = 0 
then (v) becomes the same as (i). 
If m becomes negative the curve becomes discontinuous at finite time. 
Since this cannot occur in the case of the growth of the organism or of 
populations, nor indeed so far as we are able to see, for any phenomenal 
changes with time, we shall restrict our further consideration of the equa- 
tion to positive value only of m. Also since negative values of k would 
