FACTOR RELATIONS IN QUANTITATIVE INHERITANCE 185 
indicated limits. This would show that it would be necessary to 
grow about 5,000 individuals in order to recover the parental form. 
The values differ strikingly but they give a rough idea of the ratio of 
occurence of parental forms in F, populations of this kind. 
East has pointed out that there are about eight mathematical re- 
quirements many of them independent which must be fulfilled in order 
to establish the validity of the multiple factor theory of size inheritance. 
1. Crosses between individuals which from long-continued  self- 
fertilization or other close inbreeding approach a homozygous condition 
should give F; populations comparable to the parental races in uniformity. 
Continued self-fertilization tends very quickly to reduce a race to a 
condition in which the individuals are nearly all homozygous. The 
assumption, therefore, here involved is that the heterozygous condition 
Aa in general is no more variable than the homozygous conditions AA 
and aa. There are many exceptions to this rule, so that it can be said 
fairly that a slight increase in variability need not be taken to invalidate 
this first condition. 
2. In all cases where the parent individuals may reasonably be pre- 
sumed to approach complete homozygosis, F2: frequency distributions 
arising from extreme variants of the F; population, should be practically 
identical, since in this case all Ff; variation should be due to external 
conditions. 
This follows because all the F; individuals in such a case presumably 
belong to the same genotype. The student should compare this state- 
ment with the ideas developed in the chapter on pure lines. 
3. The variability of the F. population from such crosses should be 
much greater than that of the 1 population. 
This proposition is merely a statement to the effect that segregation 
for size factors takes place in the same manner as segregation for other 
factors. Accordingly in F. a series of forms is obtained depending upon 
the recombination of size factors. Furthermore, it may be stated that 
with a given range, the less the number of size factors involved the 
greater will be the increase in variability in fs. The maximum vari- 
ability as measured by the coefficient of variability would be attained 
by a single factor difference. As the number of factors for the given 
range increases the coefficient of variability of F, decreases, so that with 
a very large number of factors the limiting value is that of the F; popula- 
tion. It is therefore possible to have size differences which give inter- 
mediate forms which appear to breed true in subsequent generations 
unless a large number of individuals be grown. 
4. When a sufficient number of F,: individuals are available, the 
grandparental types should be recovered. 
Simply a restatement of the consequences of factor recombination, 
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