Vol. 8, 1922 
STATISTICS: PEARL AND REED 
367 
give negative values of y, which in the case of population or individual 
growth are unthinkable, we shall limit k to positive values. 
With these limitations as to the values of m and k we have the follow- 
ing general facts as to the form of (v). y can never be negative, i.e., less 
than zero, nor greater than k. Thus the complete curve always falls 
between the x axis and a line parallel to it at a distance k above it. Further 
we have the following relations : 
If F(x)~ OO y = 0 
F{x) = — 00 y = k 
k 
F{x) = — 0 y= from below 
1 + m 
k 
F{x) == + 0 j= from below. 
1 + w 
dy 
The maximum and minimum points of (v) occur where — =0. 
dx 
But ^ = y{k-y) . F\x), 
dx 
therefore we have maximum and minimum points wherever F' {x) = 0. 
dy 
The fact that — = 0 when either y = Oory — k = 0 shows that the 
dx 
curve passes off to infinity asymptotic to the lines y = 0 and y = k. 
The points of inflection of (v) are determined by the intersections of (v) 
with the curve. 
k k F'\x) , 
T = - — ^ — „ (vi) 
2 2 \F'{x)f 
Since we are seldom justified in using over five arbitrary constants in 
any practical problem, we may limit equation (v) still further by stopping 
at the third power of x. This gives the equation 
(vii) 
axx + aix"^ + azx'^- 
1 -f me' 
If a„ is positive the curve of equation (v) is reversed and becomes 
asymptotic to a line A B,Sitx = — and to the x axis atx = + oo . Thus 
in equation (vii) negative is a case of growth, and a 3 positive is a case 
of decay. 
Equation (vii) has several special forms that are of interest, among them 
being a form similar in shape to the autocatalytic curve (i.e., with no 
maximum or minimum points and only one point of inflection) except 
that it is free from the two restrictive features mentioned in our first 
