THE MECHANISM OF HEREDITY 73 



the Fa Mendelian expression for N allelomorpMo pairs 

 when dominance is complete is the expanded bionominal: 



(3 + 1)" or (3/4 + 1/4)" 

 N = l (3 + 1)1 = 3 + 1 



JV = 2 (3 + 1)2 = 32+3 + 3 + 1 = 9+3 + 3 + 1 

 N = 3 (3 + 1)3 = 33 + 3(32)2+3 (3) + 1 = 27 + 9 + 9 + 9 

 + 3+3+3 + 1 



Likewise, the expanded bionomial (1/2 + %)^" gives the 

 numerical relationships when dominance is absent and N 

 represents the number of aUelomorphic pairs. The ex- 

 pression is (% + %)*" instead of (% + %)" because it is 

 supposed that the presence of any aUelomorphic pair in 

 the heterozygous condition produces one-half the visible 

 effect on the character that is produced when the heredi- 

 tary factors are present in the homozygous condition. 

 When N is very large the frequencies with which the dif- 

 ferent classes occur form a regular curve called the nor- 

 mal curve of error. This is the curve that is produced 

 when the errors in any physical measurement are sim- 

 ilarly 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 many 

 organic characters produced by the infinite complexity of 

 external conditions. 



If no non-heritable fluctuations intervened to obscure 

 the class to which any particular zygote belongs, there- 

 fore, one should expect the following classes in Fg when 

 parents of different sizes differing in N aUelomorphic 

 pairs are crossed. The extremes represent the grand- 

 parental types in each case, and the intermediate classes 

 theoretically divide the difference between the parents 

 into aliquot parts. It should be noted, however, that this 



