364 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
or, if 
1/v = Bs (a — 2), 
y= yo(e# —m)-™" (x — 4,)" 
= Yo (1 — a/a,)- (1 — w/a) 
by changing constants. 
Assuming that yo, v, a, and a, can take any sign whatever, we see that there are 
three fundamental subtypes of this’ frequency curve, 
(.) y= yo (1 + w/o) (1 — e/a). 
— a 0 a 
This is an asymmetrical curve with limited range and maximum towards mediocrity. 
Asa rule va, and va are fractional and the curve becomes imaginary beyond the 
limits v= — a, and «= a,. 
(i1.) Y= Yo (2/a,— De (1 as 2/9)". 
<—a,— 
= aes LSS — 
Here the ordinate between 2 =a, and « = a, varies from infinity to zero, and 
resembles the frequency curves given by “ wealth” distribution or infant mortality. 
(ils) yy = yy (1 — e/a.) (1 + w/a.) 
<——_ a4, —>0<——_ a z 
This is an asymmetrical curve with limited range, mediocrity being in a minimum. 
The disappearance of mediocrity is not a very uncommon feature of statistics ; the 
