256 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(2.) Definitions. 
It is necessary to give definitions to several current biological conceptions, in 
order to introduce them into our mathematical analysis. 
(a.) Variation. — If a curve be constructed, of which the ordinate y is such that 
y Sx measures the frequency with which an organ lying in size between x and x -f- Sx, 
occurs in a considerable population (500 to 1000 or more), the constants which, for 
any particular organ for any particular animal determine the form of this curve, are 
termed the constants of variation , or more briefly, the variation of the given organ. 
The assumption is made that the frequency is continuous, or that we really reach a 
curve. In the great majority of cases, where real statistical methods have been used, 
continuous curves (or, practically, polygons) have been found, and we shall assume this 
continuity to hold in all cases to which our formulse are applied. 
The size of the organ (x) which corresponds to the ordinate ( y ) through the 
centroid of the frequency curve, is termed the mean ; the size of the organ, which 
corresponds to the ordinate bisecting the area of the frequency curve, is termed the 
median ; the size of the organ corresponding to maximum frequency is termed the mode. 
We assume, what may be considered as fairly established, that variation curves in 
zoometrv, and more especially anthropometry, approximate closely to probability 
curves. When the variation curve has more than one mode, it may, as a rule, be resolved 
into simple probability curves, each with a single mode, and it may be even hetero¬ 
geneous and require resolution, when only one mode is apparent.* These probability 
curves may be skew, and in this case the treatment of the problem of heredity involves 
a discussion of skew-correlation,! but in a very great range of cases the frequency 
is sufficiently closely given by the normal probability curve. Here the variation is 
defined b} 7 a single constant,| the standard deviation cr, and the equation to the curve 
is given by 
y = 
N 
\/ 27TO- 
-*7(2<r2) 
5 
and we shall confine our attention to such variation in the present memoir. The 
following assumption, therefore, lies at the basis of our present treatment of heredity. 
The variation of any organ in a sufficiently large population—which may be selected 
in any manner other than by this organ itself from a still larger population— 
is closely defined by a normal probability curve. 
(b.) Correlation .—Two organs in the same individual, or in a connected pair of 
* On resolution and skew variation, see ‘ Contributions to the Mathematical Theory of Evolution,’ 
Memoirs I. and II., ‘ Phil. Trans.,’ vols. 185 and 186. 
t Dealt with in a memoir not yet published. 
f Inheritance can be treated by single-constant variation in the case of most organs in human adults, 
but it could not be dealt with in like manner in the of case pedigree buttercups, see De Vries : ‘ Berichte 
der Deutschen Botanischen Gesellschaft,’ 1894 and 1895. 
