44 BULLETIN 1121, U. S. DICPAETMENT OF AGRICULTITRE. 



It is not difficult to find the percentage of heterozyo;osis under 

 random breeding from stock derived from any given number of 

 families. Assume that there are n families. Symbolize each family 



by a letter P, Q, R, S, etc. There are -^ (n— 1) different first crosses 



(PQ, PR, PS, QR, etc.). On commencing random breeding there is 



one chance in ^ (n— 1) of making a mating in whicli both families 



are used twice (like PQ x PQ) and which is thus equivalent to a 



mating of experiment Cl. There are 2 (/i— 2) chances in ^ ('i— 1) 



of making matings in which one family only is used twice, (like 



11/ 



PQxQR). There are ^(n— 2) (n—3) chances in ^ {n— 1) of making 



matings in which neither family is used twice, (like PQ X RS) and 

 which are equivalent to Experiment CC. In the last case, as we have 

 seen, there is complete recovery, on the average, of the heterozygosis 

 of the original random-bred stock. In the first case (PQ X PQ) there 

 is only half ^recovery. One might expect to find an exactly intermedi- 

 ate result in the case (PQ X QR) , and this can easily be shown to be 

 true by the use of path coefficients (Wright, 1921). 



Since P, Q, R, and S are assumed to be completely homozygous 

 inbred families, the constitution of the germ cells is completely 

 determined. Thus the path coefficient from zygote to germ cell 

 {h') is 1.0 in the first generation. As there is assumed to be no 

 correlation between the families and hence none between their germ 

 cells (/' = 0), the coefficient for the degree of determination of the 

 progeny (first cross) by germ cells {a'") equals ^. For the cor- 

 relation between mated individuals of the next generation we have 

 in consequence m = 1 in the case PQ X PQ ; m=^ in the case PQ X QR, 

 and m = in the case PQxRS. For the path coefficient, zygote to 

 germ cell, second generation, we have h = >/^, by the formula h^ = 

 ^(1-1-/0. For the correlation between uniting germ cells of this 

 generation we have /= h^m.. Finally, by the formula for percentage 

 of heterozygosis, p = 2xy (1—/), we have p = xy in the case PQxPQ, 

 2? = 3/2 xy in the case PQxQR, and p = 2xy in the case PQxRS. 

 Thus the second case is exactly intermediate between the others, as 

 we set out to prove. 



Multiplying the number of matings of each kind by the correspond- 

 ing percentage of heterozygosis and adding, we find the total per- 

 centage of heterozygosis in the new random-bred stock to, be 



100 X 2 (n-l) = 100 X —^ where 100 X 2xy is the percentage 



in the original random-bred stock. Since each pair of factors comes 

 to equilibrium after one generation of random mating, this level of 



