THE BIOMETRICAL ANALYSIS 



and a the frequency v^ ( =: i — uj, etc., we find that the con- 

 tribution of the gene ^ — d to D is 4 u,v^ [^a~r hgCv^— uj]^ 

 and to H is 16 u^^^^-h^- so that the general fornmlac are 



D = S|4uv[d -f h(v - u)]2} and H = S(i6u2v2h2) 



These reduce to the original forms when u = v = o • 5 for all genes. 

 The special virtue of equal gene frequencies for analysis now becomes 

 clear. Unless u = v = 0*5 for all genes, the full effects of all the 

 dominance relations is not represented by H; some of them appear 

 inD. 



Using these general forms of D and H, we can represent the 

 relations between parents and offspring of any randomly breeding 

 group or population in terms of the variance, hT) + |-H + E, and 

 the parent-offspring covariance, ^D. 



The parent-offspring correlation is given by r^^^ = ^ :. We 



can now see that Wp/„ = JD, while V^ = V^ = lD+ JH + E, 

 so that: — 



iD 



'p/o 



ID+IH+E 



D, H and E are quadratic quantities and so must always be positive. 

 The maximum correlation which parents and offspring can show 

 for simple genetic reasons within a randomly breeding population 

 must, therefore, be r= 0*5 when H= E=: O. 



We can find the covariances between other pairs of relatives, of 

 which the most useful is that between pairs of siblings. This, too, can 

 be represented in terms of our three components of variation as: — 



Then the fraternal correlation will be: — 



^s/s 



4D + 1H+E 



This, too, will have a maximum of o • 5 when H =E = O. But when 

 dominance is present, so that H > O, r^/^ must exceed rp^^ because 

 of the additional term -^H in its numerator. 



90 



