Mathematical Contributions to the Theory of Evolution. 389 



Now, if n be >s, any Ti s will only (neglecting terms of order e 3 ) be 

 correlated with its own particular (n— s)th parents, male and female, 

 and there will be ^(2 n ~ s ) such male parents and J(2*~*) such female 

 parents. Hence 

 N 



+ m<r^' w 2 s - ] S 3 (r 5tt ) + mV^'^SiW } , 

 J$ff s ff n J Si(r^) + m — * S 2 (r,») + m— S 3 (r sn ) +m 2 ^^ S 4 (r,») 



(Hi). 



Here S L (/,$«) is the sum of all r m which begin and end with male 

 in the descent, S 2 (r,»), of those which begin with female and end with 

 male, S 3 (r$ M ), of those which begin with a male and end with a female, 

 and S 4 (r sw ) of those which begin and end with a female. Now, as 

 before, put m — a s \a \ = o n \a n . We thus have, supposing the varia- 

 bility of each generation to be constant, 



_ *2*<»-> S, (r Stt ) + S a (r,») + S 8 (r w ) + S 4 fa») 

 p ™ - I+i7 2^ ~ 



2i(«— *) 



l + e s 



where r s>i now stands for the mean value of all the correlation co- 

 efficients of an individual and its individual (n — s)th parents. It 

 may be written r n „ s , as it depends only on the difference of the gene- 

 rations. Hence supposing sexual selection to remain constant, if it 

 exists, for all generations, we see that p ns depends only on the differ- 

 ence of generations, and may be written p n -s, or : 



Pll - s = 2^-^r n _ s l(l + e)* 



Now if there be no selective breeding, e appears, at any rate for 

 man, to be small. Hence we have the important proposition : 



The correlation between two mid-parents, p generations apart, is 

 equal to the product of 2& and the mean of the coefficients 

 of correlation between an individual and its individual pth 

 parents, when they are taken for all possible combinations 

 of sex. 



When no allowance is made for reproductive selection, it has been 

 shown by Miss Alice Lee and myself that the four possible r's 



* The importance of this result is that it reduces the — — - correlation coeffi. 



1 "2 



cients between n mid -parents of different orders to n coefficients only. 



