Mathematical Contributions to the Theory of Evolution. 409 



Hence if r" be the uncle-nephew correlation and 1ST the total 

 number of pairs 



N<r V = iS(nW+JS(nk») 

 = i(p-St + r)N<r % , 

 since S(?i^o) = ^(ipi^i + ipz^-h . . . . ) 



= <x 2 N(J r ca 2 +|ca 4 + ....) 



Thus r" = 0*15, or is double the correlation of first cousins. 

 Here, as throughout, the variations of all generations, in this case 

 those of uncles and nephews, have been treated as equal. 



Returning to the correlation r" of second cousins we have 



+ *S(nfe'AJ + ^) HS(* } . 



Evaluating each of these terms we have — 



S(«&iV) = NrVo 2 ; SinhM) = Nr<r 2 ; S(?^^ 2 ) = Nr<r 2 . 



S[»i(/iiV + ^i'^2)] = product moment of all uncles and nephews 

 = 21SV V. 



S[n7c (h 1 -\-h 1 , )~\ = product moment of all offspring and the mid- 

 parental system of their grandfathers = 2S(wA;u/ii) for all 

 values of h x 



= 2S ^ h±^ n KfK'+mW' + hr)+ 

 = 2(Jr 2< 7 2 JSr + ir 3( T 2 ]Sr + ir 4 (r 2 K+ . . . . ) 



= 2N(7 4 XO-1. 



Similarly, S[w^(^2 + ^a0] = 2Nff 2 X0-2 as before (p. 408). 

 Thus finally : r'" = t V(^' + tV+tV+]K' + 3 X 0*2 + ^ x 0'4) 

 or r" = 0-0171875. 



More distant collateral relationships, which can be found in like 

 manner, and may be needed for the case of in-and-in breeding, say 



