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
