MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 367 
Until we know very much more definitely than we do at present, how the size of 
an organ in any individual, say, depends on the sizes of the same organ in its 
ancestors, or what are the nature of the causes which lead to the determination of 
prices, or of income, or of mortality at a given age, I do not see that we have any 
right to select as our sole frequency curve the normal type 
Y= hor 
in preference to the far more general 
y= yo (1 + win) (= aly 
which not only includes the former, but supplies the element of skewness which is 
undoubtedly present in many statistical frequency distributions. As we may look 
upon the former as a limit to a coin-tossing series, so the latter represents a limit to 
teetotum-spinning and card-drawing experiments. It is not easy to realise why 
nature or economics should, from the standpoint of chance, be more akin to tossing 
than to teetotum-spinning or card-dealing. At any rate, from purely utilitarian and 
prudent motives, we are justified so long as the analysis is manageable, in using the 
more general form. It will always give us a measure of the divergence of particular 
statistics from the normal type, and in many cases of skew frequency, it can be used 
when it would be the height of absurdity to apply the normal curve at all. 
Since Types I., II., III., and V. are all represented by the curve 
y=y (1 + v/a) (1 — w/a,)"* 
and Type IV. by the curve 
— Ta See —vtan-la/a 
Yaa Yo (1 a eel lae FP ’ 
we have only to deal with these two cases in general. We shall refer, in the 
course of our work, to special simplifications arising in particular sub-cases. After a 
description of the manner in which these generalised probability curves may be fitted 
to statistics, we shall indicate, by examples, their practical applications. 
(14.) On the Generalised Probability Curve. Type I. 
Y = Yo (1 + w/a) (1 — w/a). 
Let the range a +a,=b; let m=va,, m= vo, z= (a, + x)/(a, + a), 
whence x= —a,, 2=0 and z=a, 2=1., 
Further let 
1 = Yo (% + ay) ™*™/ay"049", 
= Yq (m+ My)™*"/m" My”, 
thus y = ne (1 poe 2); 
