60 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



* Hence, by the above proposition, the distribution of offspring- of parents of two 

 couplets is 



4 X 4 X 4 . (\u + \v + X 4 X 4 X 4 . (11 + f y + 



and, by induction, the distribution of offspring for the random mating of parents of 



n couplets is 



4" X 4" X 4" . (\u + f v + \w}"- 



This, except for the constant factor 4" X 4", is absolutely identical with the 

 distribution of the parental population, and accordingly if the next generation also 

 mates at random, the mixed race will continue to reproduce itself without change. 

 We therefore reach the following result : 



However many couplets we suppose the character under investigation to depend 

 upon, the offspring of the hybrids or the segregating generation if they breed at 

 random inter se, will not segregate further, but continue to reproduce, themselves in 

 the same proportions as a stable population. 



It is thus clear that the apparent want of stability in a Mendelian population, the 

 continued segregation and ultimate disappearance of the heterozygotes, is solely a 

 result of self-fertilisation ; with random cross fertilisation there is no disappearance 

 of any class whatever in the offspring of the hybrids, but each class continues to be 

 reproduced in the same proportions. Thus our generalised theory lends no countenance 

 to the appearance of any "mutations" within a hybrid population under random 

 mating; the only appearance of new constitutions is in the segregating generation, or 

 the first generation of hybrid offspring. Except at this stage, the appearance of the 

 unfamiliar is only the chance occurrence of a very rare normal variation. When we 

 recollect that a purely allogenic individual is only to be expected once in a population 

 of 4" individuals, or if there be ten couplets, once in more than a million individuals, 

 it will be clearly seen that the rarity of some of the more exceptional normal 

 constitutions may easily lead to their being looked upon as " mutations," even if they 

 appear in the offspring of a population many generations removed from hybridisation. 



(5.) PROPOSITION III. To find the Array of Offspring due to a Parent of given 

 Gametic Constitution mating at Random. 



This can be again deduced by the method of induction adopted in the last 

 proposition. 



Supposing a male P of n 1 couplets to mate with all possible females, and R,,_ 1 

 to be the array of offspring, then we have seen in the last proposition that if we add 

 an n ih couplet a a a u -io P, the array of offspring due to P + a H a Jt will be 16UK B _j ; if 

 we add a couplet cr,.A. H , the array of offspring due to fathers of type P + aA,, will be 



