104 



THE AMERICAN NATURALIST 



[Vol. XLV 



ered as a limiting case of Mendel's law of alternative 

 inheritance, thus saving- the original name, is a matter 

 for later decision, though the latter name seems to me 

 more appropriate to retain owing to the fact that the 

 phenomenon of segregation, most important of all, had 

 been first stated by Mendel. 



Let us now consider a limiting case of our formula (2) 

 when the values of. n (number of allelomorphic pairs of 

 characters) increase. In the typical Mendelian ratio, 

 the relative frequency of the various zygotes with re- 

 spect to any given visible character is proportional to an 

 expansion of (1 + 3)" which is the same as (1/4 + 3/4 ) n 

 if we consider the relative values of the frequencies. 

 Thus in all known cases of the inheritance, we have to 

 deal with an expansion of (r -f 5)" where r + s = 1. A 

 concise mathematical formula which represents a limit- 

 ing case of the binomial series arising from an expansion 

 of (r -f s) n will be very useful, especially when we are 

 dealing with a quantitative measurement such as weight, 

 length, area, volume, etc., since in these cases the values 

 of the variates will be graded. Further, the theoretical 

 frequency corresponding to eaeli variate when the value 

 of n becomes very large, can best be determined from 

 such a mathematical expression which represents a limit- 

 ing case. 



Without going into any detail of the mathematical 

 treatment, it will be seen that we obtain two forms of 

 expression according as r=sorr + s. The former will 

 be represented by the normal probability curve and the 

 latter by a limiting case of a skew binomial curve. For 

 representing a skew binomial curve we can best use 

 DeForest's formula (Professor Pearson's curve of type 

 3). It may be useful to the reader to know that De- 

 Forest's formula degenerates into the normal probability 

 curve as its simplest form, as will be seen below. 



DeForest's formula (Hatai: '10) is usually written in 

 the following form : 



