152 



Prof. Karl Pearson. 



Thus the law of reversion fits no better than the law of blended 

 inheritance the data to which I have referred (in § 6) when we adopt 

 the 1/4, 1/16, 1/64 hypothesis, i.e., the original form of Mr. Galton's 

 statement.* 



(iii) Let us suppose the parental correlation to be 0-5, a value not 

 very far from what I have found for eye-colour in man and coat-colour 

 in horses. Then by (xvi) 



^ = h P2 = h 

 Putting e = 00 in (xx) we deduce : 



a^-m^'^ + i = 0, or a2-fa + i = 0, 



which gives us a = 1 or J. 



But remembering the value of S we have, using (xxi), 

 ocy = -{oL-^Y and oL + fS = 1. 



The first equation shows us that a = 1 is impossible, for it gives 

 7 negative. Accordingly we conclude that a = J and ^ = J, while 

 7 = 0. Thus reversion is totally excluded and one-half the offspring take 

 after each parent. In this case the grandparental correlation, p2, is 

 0'25, the great grandparental 0*125, and so on. The ancestry beyond 

 the parents have no direct influence on the offspring, beyond the fact 

 that they have determined the parents. We are dealing indeed with 

 a case like that investigated in my memoir on " Regression, Heredity, 

 and Panmixia."! So far our theory of exclusive inheritance with 

 parental correlation = 0'5 agrees with that of blended inheritance 

 with the same value of the parental correlation. But we have seen 

 that the latter leads to an impossible value for fraternal correlation, 

 i.e., one which does not fit the facts. Does perfect fraternal correla- 

 tion necessarily flow from exclusive inheritance without reversion ' 

 Certainly not, for this would connote that all the off'spring of a given 

 set of parents would be alike, or one parent in each family be abso- 

 lutely prepotent. This is of course not the fact. 



Supposing all families to consist of n members, and that both 



n[n ,\ 



parents were equipotent in the family, there would be ^ " ) P^^^^ 

 of brethren alike, out of a total of pairs, or the fraternal 



correlation would be l)/(%- 1). The average size of a human 



* See * Natural Inheritance,' chapter viii, p 149, &c. Mr. G-alton there uses the 

 correlation coefficients corresponding to blended inheritance for eve-colour, an 

 exclusive inheritance. But, directly investigated, such values are far from holding 

 for eye-colour. 



t ' Phil. Trans.,' vol. 187, p. 303. 



