366 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(viii) Put a, = © in (ii.), 
y = yp (w/a, — 1)-™ e™. 
y 
O--a@> zr 
<———_ 60 
This is an asymmetrical frequency curve, with an ordinate varying from @, to « 
along an infinite range. 
All eight of the above types are included in the single form 
Y = Yo (1 + %/a,)™ (1 — 2/ay)’, 
or 
y= yx (1 — 2/c), 
if we give positive, negative, or limiting values to the constants. But to do this we 
require to give values to 7 and 7 in the expressions for 8,, 8,, and 3, which are not 
easily intelligible, if we rigidly adhere to our example of drawing a definite quantity 
of sand from a limited mixture of two kinds of sand. The last type of curve given 
is, however, the frequency curve for @ priori probabilities,* and readily admits of a 
direct interpretation of the following kind. 
Given a line of length /, and suppose 7 + 1 points placed on it at random; what is 
the frequency with which the point pr from one end and qi from the other of the 
series of 7 + 1 points falls on the element Sz of the line? 
The answer is clearly 
rela Oa 
pr |qr\ 1 l U 
or, we have a frequency curve of the type 

y = ye?" (1 — al)”. 
We may express the problem a little differently. Take 7 + 1 cards and slip them 
at random between the pages of a book, the frequency of the page succeeding the 
pr + 1™ card is given by the above curve.t 
* See Crorron, ‘“ Probability,” § 17, ‘ Encycl. Brit.’ 
+ The important point to be noticed here is that we are dealing with a distribution in which 
contributory causes are inter-dependent, 
