MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 309 



children. In the same manner a selection of shorter children is more variable and 

 less normal than the distribution of the general population of eight years of age, i.e., 

 tends to reproduce the characteristics of a younger population. Generally, a random 

 selection, which increases variability, very sensibly increases skevvness and decreases 

 stature. What, perhaps, would hardly be expected, is that increase of skewness as 

 well as increase of interval from mean to mode, i.e., greater divergence from normality, 

 increases the frequency of the mean stature. 



It will be clear that by aid of this table we are able to predict the probable 

 changes in all the other physical characters of the distribution when any sub-class 

 has been selected at random from the general population with a difference of one 

 character. If two or more characters differ in the sub -class, the probable changes 

 in the other characters can be found by the principles of multiple correlation from 

 the correlation table on page 307. 



(22.) Conclusion. 



This study of the probable errors and error correlations shows us that these 

 quantities can be determined for the most complex system of organs in the case 

 of normal correlation, and in the case of either normal or skew variation with con- 

 siderable ease. It is only in the case of skew variation that the arithmetic becomes 

 at all laborious. But numerical examples suffice to show that the errors here 

 made are of the same order as in the case of normal variation, if we confine our 

 attention to the characteristic features of the frequency, e.y., the mean or modal 

 frequency, the standard deviation, the skewness, &c. Certain constants of the 

 algebraic form of the frequency curves have large probable errors, but these errors 

 are so highly correlated, that their existence does not suffice to substantially modify 

 either the form of the curve, or the " physical " characteristics of the distribution 

 calculated from such values. 



For the theory of evolution certain very important principles flow, beyond the 

 mere advantage of knowing the probable errors made in the measurement of racial 

 or organic characters. Above all we note the importance of a random selection in 

 altering in a systematic manner all racial constants. In most cases even size cannot 

 be altered without alteration of the size, variation and correlation of all correlated 

 organs. This principle is developed more at length in a memoir, nearly completed, 

 on the influence of directed selection, which covers as a special case that of random 

 selection. 



Later, we hope to apply the general theorem from which our memoir starts to 

 determine the probable errors in the constants of the components into which 

 a heterogeneous frequency distribution may be resolved by the method of the first 

 memoir of this series.* It applies equally to such an investigation. 



* The importance of such a determination was emphasized by Professor GEORGE DARVCIN in the 

 discussion which took place at the reading of that memoir. 



