490 



THE GRAMMAR OF SCIENCE 



of perfectly correlated material. It will follow^ that the 

 actual correlation observed will be -g- unity or .5. Simi- 

 larly, taking the four grandparents, we should expect one- 

 quarter of the grandchildren to be perfectly correlated 

 with one of these, and three-quarters to be a purely 

 random or uncorrected group, consequently the grand- 

 parental correlation should come out .25. Next take 

 the case of brethren, and suppose the average number in 

 the family to be ;/ ; then ^71 will follow the father, ^n 

 the mother. These two groups will be perfectly corre- 

 lated among themselves, and have no correlation with 

 each other. Hence, taking the possible pairs out of n 

 brethren, or ^n{n — i ) pairs, we find twice ^i^fi)i-^n — i ) 

 or ^n{^n— i) are perfectly correlated, and the remainder 

 ■^n^n uncorrelated. Thus the total correlation will be 



^n{^n— i)l^n{n — 1) = ^^^, and this will vary according 

 to the average size n of families. For five in a family 

 it equals .375 ; for six in a family .4, and so on. Lastly, 

 turning to the avuncular relation, ^n — i of the brethren 

 of the father, say, would be like him, and ^n unlike him, 

 while ^Ji would be the proportion of resemblance to him 

 among his offspring, hence we have {^n — 1)^71 j n{n — i) 

 = -^{^n— i)l{n— i)=h2i\{ the fraternal correlation. We 

 can now compare these intensities of heredity for what we 

 may term ideal exclusive inheritance with the theoretical 

 values for blended inheritance under pangamy. 



Table of Intensities of Inheritance 



1 I have dealt with this point more at length. Philosophical Transactions, 

 vol. cxcii. p. 274. 



