16 
Research Bulletin No. 2 
If three independent size factors instead of two were involved 
in this cross, the F, individuals would fall in the same class as 
before, but the F 2 classes would be seven in number and the 
grandparental sizes would each be recovered only once out of 
sixty-four times. For four factors there would be nine classes of 
F 2 individuals, and the grandparental types would each occur 
only once out of 256 times; while with only eight factors, the 
forms of the grandparents would each appear only once out of 
65,536 times, and it would be quite remarkable if they were ever 
recovered from an ordinary cross. 
The entire scheme of this type of inheritance can be expressed 
in mathematical form just like ordinary Mendelian inheritance 
with full dominance. Let us recall that the F 2 Mendelian ex- 
pression for n allelomorphi c pairs when dominance is complete is 
the expanded binomial. 
(3 + 1)" or (| + 
7i = l (3 + l) 1 = 3 + l 
n=2 (3 + 1) 2 = 3 2 + 3 + 3 + 1 = 9 + 3 + 3 + 1 
» = 3 (3 + l) s = 3 8 + 3(3) 2 +"3(3) +1 — 
27 + 9 + 9 + 9 + 3 + 3 + 3-1 
Likewise the expanded binomial (l + ^) 2n or (1 + l) 2w gives 
the numerical relationships when dominance is absent and /? 
represents the number of allelomorphic pairs. The expression 
(i + i) 2n instead of (| + J) n is used because it is supposed that 
the presence of any allelomorphic pair in the heterozygous condi- 
tion produces one-half the visible effect on tlie character that is 
produced when the genes are present in the homozygous condition. 
When n is very large the frequencies with which the different 
classes occur form a regular curve called the normal curve of 
error. This is the curve that is produced when the errors in any 
physical measurement are similarly plotted using as classes any 
constant deviation from the average, as a. 2a, 3a, etc. This same 
curve is also produced when one plots the fluctuations of any 
organic character produced by the infinite complexity of external 
conditions. 
If no non-heritable fluctuations intervened to obscure the 
class to which any particular zygote belongs, therefore, one 
should expect the following classes in F 2 when parents of different 
sizes differing in n allelomorphic pairs are crossed. The extremes 
represent the grandparental types in each case, and the inter- 
mediate classes theoretically divide the difference between the 
parents into aliquot parts. It should be noted, however, that 
this is theory only: in reality the influence of one factor might be 
somewhat different from that of another factor. 
