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 2 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 

 A a in general is no more variable than the homozygous conditions A A 

 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, F 2 frequency distributions 

 arising from extreme variants of the Fi population, should be practically 

 identical, since in this case all FI variation should be due to external 

 conditions. 



This follows because all the Fi 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 Fz population from such crosses should be 

 much greater than that of the Fi 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 2 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 F 2 . 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 2 decreases, so that with 

 a very large number of factors the limiting value is that of the Fi 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 2 individuals are available, the 

 grandparental types should be recovered. 



Simply a restatement of the consequences of factor recombination. 



