56 PKOFESSOK K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



or, each couplet will give rise to four possibilities, representing, however, only three 

 constitutional differences, expressed by 



a r a r + 2a r A. r + A r A,. 



Since these four possibilities may occur with each of the n couplets, we shall have 

 when two perfect heterozygotes cross, 4" resulting possibilities. These form the 

 resulting population of the second generation. Our first problem will be to find the 

 distribution of this population. This, according to MENDEL, is the segregating 

 generation. We must inquire into the frequency of each constitutional difference in 

 this segregating generation. 



We now reach our second limiting hypothesis, which is needful if we are to apply 

 our theory to sexual reproduction. We suppose : 



There to be an absence of homogamy (including self- fertilisation), and the members 

 of the second generation to cross absolutely at random and with equal fertility.* 

 We have then to ask what is the distribution of constitutional differences in the 

 third generation. Does the process of segregation begun in the second generation 

 continue to the third, or does the population now remain stable ? Is the continual 

 segregation into pure protogenic and allogenic individuals a necessary result of any 

 pure gamete theory, or does the belief in such necessity depend upon the first 

 Mendelian experimenters working only with self-fertilising individuals ? 



(3.) PROBLEM I. To find the Distribution of the Offspring of the Perfect 



Heterozygotes. 



We shall here use a symbolic form of analysis. Let u stand for act, v for A, and 

 \v for AA ; then any corresponding couplets will give rise to 



u 



and any one of these constitutions may be associated with one of the similar 

 constitutions in any of the remaining n 1 couplets. Hence the general distribution 

 of the population will be given by the terms of the multinomial 



(u + 2v + to)". 

 This equals t 



'< 4- 7m"- 1 (2r + w) + ^L"- 1 ) u"~~ (2v + w)- + ... 



1 . _ 



* If one is to study heredity in populations with a view to the problem of evolution, the conditions as 

 to fertilisation should approach as far as possible the conditions we suppose them to be under in a 

 natural state; we must fix our attention on the mass relations between successive generations of the 

 population. 



t Throughout this memoir the symbol c n , v , ,, is used for the expression \n / { \n-p-q [g \q). 



