Mathematical Contributions to the Theory of Evolution. 331 



earlier memoir of this series the effect of random selection, or what it 

 is better to term random sampling, on the characters of a population. 

 Isolation of a few individuals who form a random sample may produce 

 very sensible modifications of race characters, but it is to directed 

 selection that we must look for changes on the largest scale. The 

 subject is a very broad and complex one no less than the total 

 effect upon a population containing individuals at all ages of a selec- 

 tive death-rate applied for a long period and a function not only of 

 the organs of each individual, but of the relationship of these organs 

 to each other, and of the stage of growth of the individual. In 

 its complete form the problem presents very considerable difficulties ; 

 but if we confine our attention to one class of the population, 

 namely, individuals in the same stage of growth, we are able to 

 trace fairly well the effect upon such a class of selection, however 

 complex may be the relation between the organic characters and 

 the death-rate. Thus we can measure the death-rate which would 

 convert one race into a second by a cataclysmal action on the mean 

 standard deviations and correlations of p out of n possible organs 

 in mature individuals. New complexities arise if the individuals 

 are reproducing themselves during the process of selection, which is 

 then assumed to be continuous and not sudden. 



At this point a very definite distinction is reached, namely, that 

 between directly and indirectly selected organs. It may be said that, 

 although it is possible for the recruiting sergeant to select stature, and 

 in so doing differentiate the arm-length of his troop from that of the 

 general population, yet that in natural selection we are given only the 

 modified organs, and so we cannot tell which of them have been directly 

 and which indirectly selected. Both are changed ; how discover which 

 was the source of the change 1 The answer is : In the same manner as 

 we could distinguish between two recruiting sergeants, one of whom 

 selected his troop from the general population by stature, and the 

 other by cubit ; in either case the stature and cubit would be both 

 modified, but the mathematical theory of regression would enable us 

 to distinguish between the methods of operating of the two men, and 

 even between them and one who selected by both stature and cubit at 

 once. The mathematical theory as developed in this paper shows us 

 that, although the whole complex of characters may have been changed, 

 still, if direct selection have only occurred in p out of n possible cases, 

 there will be certain of the partial regression coefficients which remain 

 unmodified and which will theoretically enable us to distinguish among 

 the whole group of differentiated organs, between those directly 

 selected and those modified only because they happen to be correlated 

 with the directly selected organs. Thus the distinction becomes one 

 of singular importance, for though the selection of a few organs 

 modifies the means, variabilities and correlations, possibly of the whole 



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