100 THE AMERICAN NATURALIST [Vol. XLV 



determinants being designated p, and the gamete of 

 another parent also contains after the reducing division 

 n determinants, the whole group being designated q. 

 Then in the first hybrid zygote (F x ) there will be con- 

 tained at the time of the union of the gametes 2n determ- 

 inants. As we know, rearrangement takes place during 

 the maturation of the germ cells and we assume this 

 rearrangement to involve a random sampling by which 

 n determinants arc taken from the group of 2n. From 

 the theory of probabilities we find that n, n — 1, n — 2 

 . . . determinants of either parent contained in the 

 gametes of F y are proportional to the successive terms 

 of the following series : 



(i) 



The same phenomenon happens in the gametes of the 

 other hybrid parent (FJ and since the gametic consti- 

 tution of the two hybrid parents is assumed to be identi- 

 cal with respect to the distribution of determinants (1), 

 the frequency of the various combinations of the de- 

 terminants in the second hybrid offspring (F 2 ) will be 

 given by the square of (1) or 



( p . + ^-. 9+ ?fc -iy v + * - 35= V v+ • ■ •)' (2) 



(p 2 + 2pq + q 2 )\ 



This series, or the square of the binomial series, is 

 then the most general expression for the gametic com- 

 position of any hybrid arising from a combination of p 

 and q determinants and may therefore be considered as 

 the underlying principle of any law of inheritance where 

 the idea of determinants is used. 



It is evident that since the somatic characters in ques- 

 tion depend entirely on the behavior of the determinants, 

 the relative frequency of various zygotes, as well as the 

 character of the zygotes, depends on whether p or q de- 



