343 
FOUNDATIONS OF 
THEORETICAL STATISTICS. 
The first of these forms may be rewritten 
df oc l - 
— 
a 5 
r + 2 
— i-tanh *— , 
e a dx, 
r being negative, showing its affinity with the second class. 
In order that these expressions may represent frequency curves, it is necessary that 
the integral over the whole range of the curve should be finite ; this restriction acts in 
two ways :— 
(1) When the curve terminates at a finite value of x, say x = a,, the power to which 
a 2 — a? is raised must be greater than — 1. 
(2) When the curve extends to infinity, the ordinate, when x is large, must diminish 
more rapidly than — ; 
x 
In Fig. 2 is shown a conspectus of all possible frequency curves of the Pearsonian type ; 
A 
