144 



Prof. Karl Pearson. 



number of correlated pairs to the total number of pairs.* Thus the 

 correlation of parent and offspring = 1 x piN/N = pi. 



It thus follows that pi, p2, ps , Pn are the correlation 



coefficients to be expected between offspring and parent, grandparent, 

 ...... Tith great parent, &c. Here we have assumed equal potency for 



both sexes and all lines of descent, otherwise these coefficients must be 

 looked upon as mean values of the correlations for different genera- 

 tions of ancestry. 



Lastly, it seems to me that reversion may not be the proper word to 

 apply to those who directly follow their parents, and that these may 

 be fairly considered direct inheritors and distinguished from reverters. 

 I shall accordingly assume no a priori relation between these two 

 classes, certainly not that direct inheritors and reverters are equally 

 numerous, i.e., \ and J, as in Mr. Galton's Law. As for reversion 

 itself I will only suppose it to diminish in geometrical progression as 

 we step backward to more and more distant ancestry. I shall 

 accordingly take /?N offspring to follow either parent and yaN, ya^N, 

 ya^N, &c., to follow grandparent, great grandparent, great great 

 grandparent, &c. With these preliminaries arranged we can now pro- 

 ceed with the analysis. 



(3) The Generalised Law of Reversion. — The total number of off- 

 spring N is clearly the sum of all those that follow all the successive 

 ancestors, i.e., 



1^ = 2im-\-iyoLl^ + Syam+l^yam+ 



or 1 = 2/3 + 47a/(l-2a) (i) 



Now consider how the number of offspring " like " or absolutely corre- 

 lated with one parent are made up : they are /oiN in number ; they 

 consist first of /5N, the number directly inheriting from this parent ; 

 also there will be yaN like each of the parent's parents, and the parent 

 will be like one or other of the parent's parents in pi proportion of cases ; 

 similarly there will be ya^N like each of the parents' grandparents, 

 and the parent is like each of the parents' grandparents in p2 cases ; and 

 so on. Thus we have 



piN = /3N -f 2ya/)iN + iyoC^pi^ + Sya^sN + 



or = /? -h 2ya(/0i 4- 2a/)2 + ioc^pz + ) (ii) 



Now note how the p^^ like any one grandparent is made up. We have 

 directly yaN reverting to this grandparent, ya^N to each of the grand- 

 parents' parents, and in each case /oiya^N like the grandparent; 

 similarly out of those ya^N reverting to any grandparents' grand- 

 * In this case there is every reason for supposing = a. 2 = ffi = Cj', and 

 mi = Wj, rrti = ^3'. Thus S = 2' = <r/, and since r = 1, R = %/N. 



