322 GENETICS IN RELATION TO AGRICULTURE 
matical relations are much more complex than those given above. We 
may consider a simple case, however, and show the relations in that case. 
If we start out with a population consisting of equal numbers AA and aa 
forms, and assume that a given percentage of crossing occurs, then an 
equilibrium will be reached when the number of homozygotes produced 
by the heterozygotes in the population is equal to the number of hetero- 
zygotes produced by spontaneous crossing. Thus, if we assume 10 
per cent. of spontaneous crossing in such a population, in the first gen- 
eration of the 10 per cent. of AA which cross with other plants, half will 
be fertilized by other AA plants and half by aa. The latter will give 
heterozygotes, consequently the proportions of different genotypes 
produced by the AA plants will be 0.954A:0.05Aa. Similarly aa plants 
produce 0.05Aa: 0.95aa, so that in the first generation the ratio is 0.954 A: 
0.10Aa :0.95aa. Now in the next following generation if we assume 
that random mating occurs among the 10 per cent. of plants which 
cross with other plants, then one-third of the plants in each genotype will 
mate with the same genotype, one-third with one of the other two geno- 
types, and one-third with the remaining genotype. That is, of the 
0.954A one-tenth or 0.095 cross, as follows: 444A XK AA = 0.3244, 
AA X aa = 0.032Aa and 14AA X Aa = 0.016AA :0.016Aa. Simi- 
larly, of the 0.95aa, 0.095 cross: gaa X aa = 0.032aa, gaa K AA = 
0.032Aa and laa X Aa = 0.016Aa:0.01l6aa. Also of the 0.10Aa, 
one-tenth or 0.01 cross: 144A4a X AA = 0.0016AA: 0.0016Aa, 14Aa X 
aa = 0.0016Aa:0.0016aa and 4Aa X Aa = 0.0008AA :0.0016Aa: 
0.0008aa. Summating like genotypes we have 0.05AA :0.10Aa : 0.05aa. 
The 90 per cent. of AA and aa plants which are self-fertilized produce 
0.8554 A and 0.855aa respectively, while the 0.09Aa plants which are 
self-fertilized produce 0.0225AA : 0.045Aa : 0.0225aa. Combining 
these with the results of cross-fertilization we have the ratio for the 
second generation, 0.92844 :0.146Aa:0.928aa. Now the ratio of 
the proportion of homozygotes to the population in the first generation 
is of course 0.95 and in the second generation it becomes, 
0.928 + 0.928 
0.988 + 0.146 + 0.028 ~ °-92". 
The composition of the third, fourth and fifth generations and the ratio 
of the proportion of homozygotes to total population for each are shown 
in Table XLVII. It is evident that, under the conditions assumed in 
this case, the rate of change in the ratio of homozygotes to the total 
population becomes very gradual after the first three generations, so 
that for practical purposes the population has reached a state of 
equilibrium in the fourth generation. In this generation the ratio of 
heterozygous dominants to the sum of the heterozygous and homozygous 
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