40 BULLETIN 1121, U. S. DEPARTMENT OF AGRICULTURE. 
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 2zy. Let p be the percentage of heterozygosis after a 
certain amount of mbreeding. Then (xB) aA +pdat(y—$)aa 
represents the composition of the population. The correlation be- 
tween uniting gametes (f) comes out eters applying the product- 
moment method to the above formula. Thus p=2zy(1—f). In the 
calculations (Wright, 1921) on which the percentages in Figure 24 are 
based, the formula p= (1 ~ f) was used, which, as stated, applies to 
the case in which r=y= 5° The formula for f 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 2ry 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 (7-2) WabsbeRYiedl iy=5 
(v-2) aa the mean deviation from the dominant type is p (Aa) + 
(y-8) (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, xA A+ yaa, is y (aa). 
Thus the deviation at any time from the ultimate level is the differ- 
ence p [4(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). 
