40 BULLETIN 1121, U. S. DEPARTMENT OF AGEICULTTJEE. 



Aa + y^aa where x and y are the relative proportions of dominant and 

 recessive genes. The proportion of heterozygosis in the random-bred 

 stock is thus 2xy. Let p be the percentage of heterozygosis after a 



certain amount of inbreeding. Then (x-^]AA + pAa + (y-^)aa 

 represents the composition of the population. The correlation be- 

 tween uniting gametes (/) comes out I-2 ' ^PP^^^^S ^^® product- 

 moment method to the above formula. Thus p = 2xy{l—f). In the 

 calculations (Wright, 1921) on which the percentages in Figure 24 are 



based, the formula p=o( 1— /) was used, which, as stated, applies to 



the case in which x = y^:^- The formula* for / under the various 



systems of inbreeding applies to any composition of the population. 

 Thus the decline in the percentage of heterozygosis shown in the figure 

 applies to any population provided merely that the scale is changed 

 so that the percentage under random mating is 2xy instead of 50 per 

 cent. 



As regards the rate of decline in vigor (if any), it is easy to show 

 that it is proportional to the decline in heterozygosis, regardless of 

 the relative number of dominant and recessive genes, and regardless 



of the degree of dominance. In the population \x — *^\ AA + p Aa + 

 (y — ^) aa the mean deviation from the dominant type is p (Aa) + 



( 2/— o ) (aa), where (Aa) represents the deviation of the heterozygotes 



(zero if dominance is perfect) and (aa) represents that of the recessives. 

 The deviation in the ultimate inbred population, xAA + yaa, is y (aa) . 

 Thus the deviation at any time from the ultimate level is the differ- 

 ence p [^(aa) — (Aa)]. This is proportional to p, the percentage of 

 heterozygosis regardless of the values of x and y, or of the degree of 

 dominance. Thus Figure 24 should represent the rate at which vigor 

 declines, relative to the ultimate level, under any conditions under 

 the various systems of mating. The absolute rate of decline, if any, 

 depends of course on the factors in the particular case. 



In comparing the theoretical with the actual rate of decline, it must 

 of course be borne in mind that the character which is being studied 

 must be measured on a scale such that unit differences at all parts of 

 the range are physiologically equivalent. A correction may be neces- 

 sary such as we have used in the case of percentage born alive and 

 similar cases. In other cases a, logarithmic scale may be the proper 

 one to use (Zeleny, 1920). 



