Mathematical Contributions to the Theory of Evolution. 143 



individual. If this be true, then his law, or possibly some generalisa- 

 tion of it, is very comprehensive ; it embraces the two distinct types of 

 heritage, blended and exclusive. But I think it most desirable to keep 

 the two ideas quite separate, and speak of the one dealing with blended 

 inheritance as the Law of Ancestral Heredity ; the second, dealing with 

 exclusive inheritance, as the Law of Reversion. If this be done, we 

 shall, I venture to think, keep not only our minds, but our points for 

 observation, clearer ; and further, the failure of Mr. Galton's statement 

 in the one case will not in the least affect its validity in the other. 



(2) The Law of Reversion. — Let us examine first what I take to be 

 Mr. Galton's view of this law. Out of an array of N offspring, 1/4 N 

 will follow each parent, 1/16 N follow each of the four grandparents, 



and follow each of the 2^^ nth great parents. In this manner 



the total offspring N is distributed by reversion among the ancestry. 



Now I want to draw attention to one or two points here. 1 /4: N will 

 not be all the children like, say, their father ; for out of the 1 /4 N 

 who are like members of his ancestry, those who are like ancestors 

 like him — and these ancestors will occur in certain proportions — will 

 thus also be like him. This holds for each individual ancestor ; the 

 number like any ancestor will be considerably greater than the number 

 who " follow " that special ancestor. Now let piN, /)2N, p^, p^N, 

 and pn^--' be the number of the offspring like a parent, a grand- 

 parent, a great grandparent, and nth. parent, &c. 



This brings me to my second point. A special meaning is here 

 given to the word like. p^N is not in the usual sense of the word all 

 the number like the father. If the offspring had the same distribution 

 of character as wc find in the general population, then undoubtedly 

 some would have the same quantity or quality of the character as he 

 has — some, for instance, would be blue-eyed if he were blue-eyed — but 

 this is a random likeness and not like in the special sense in which we 

 are using the word. pjN are like the father owing to the laws of 

 heredity, the remainder have a random distribution so far as he is 

 concerned, and we exclude any random likeness from our considera- 

 tion. 



How then are we in actual observation to distinguish hereditary from 

 random likeness 1 * The answer is simple ; piN out of N pairs of 

 parent and offspring will be absolutely correlated, i.e., have a correlation 

 equal to unity, but the remaining (1 - pi) x N pairs will have zero 

 correlation, although there may be random likenesses. Hence, by the 

 theorem given by me in the ' Phil. Trans.,' vol. 192, p. 276, the actual 

 correlation will be perfect correlation reduced in the ratio of the 



* I exclude for the present the influence of assortative mating. A likeness to 

 the mother, otherwise random so far as the father is concerned, may thus become 

 a real likeness to the father. 



N 2 



