62 



in the next generation the numbers will 

 be as 



(p + qy:2(p + q)(q + r):(q + r)^, 



or as pi'.Iqi :ri, say. 



The interesting question is— in what 

 circumstances will this distribution be 

 the same as that in the generation be- 

 fore? It is easy to see that the condition 

 for this is q^ = pr. And since ^i^ = 

 piTi, whatever the values of p, q and r 

 may be, the distribution will in any 

 case continue unchanged after the 

 second generation. 



Suppose, to take a definite instance, 

 that A is brachydactyly, and that we 

 start from a population of pure brachy- 

 dactylous and pure normal persons, 

 say in the ratio of 1:10,000. Then 

 p = 1, ^ = 0, r = 10,000 and pi = 1, 

 qi = 10,000, n = 100,000,000. If 

 brachydactyly is dominant, the pro- 

 portion of brachydactylous persons in 

 the second generation is 20,001:100,- 

 020,001, or practically 2:10,000, twice 

 that in the first generation; and this 

 proportion will afterwards have no 

 tendency whatever to increase. If, on 

 the other hand, brachydactyly were 

 recessive, the proportion in the second 

 generation would be 1 : 100,020,001, or 

 practically 1:100,000,000, and this 

 proportion would afterwards have no 

 tendency to decrease. 



In a word, there is not the slightest 

 foundation for the idea that a domi- 

 nant character should show a tendency 

 to spread over a whole population, or 

 that a recessive should tend to die out. 



I ought perhaps to add a few words 

 on the effect of the small devia- 

 tions from the theoretical proportions 



HARDY 



which will, of course, occur in every 

 generation. Such a distribution as 

 pi:2<7i:ri, which satisfies the condi- 

 tion qi^ = piTi, we may call a stable 

 distribution. In actual fact we shall 

 obtain in the second generation not 

 pi:2qi:ri, but a slightly different dis- 

 tribution pi:2qi:ri, which is not 

 "stable." This should, according to 

 theory, give us in the third generation 

 a "stable" distribution p2-2q2'-r2, also 

 differing slightly from pi:2qi:ri; and 

 so on. The sense in which the distribu- 

 tion pi:2qi:ri is "stable" is this, that if 

 we allow for the effect of casual de- 

 viations in any subsequent generation, 

 we should, according to theory, obtain 

 at the next generation a new "stable" 

 distribution differing but slightly from 

 the original distribution. 



I have, of course, considered only 

 the very simplest hypotheses possible. 

 Hypotheses other than that of purely 

 random mating will give different re- 

 sults, and, of course, if, as appears to 

 be the case sometimes, the character 

 is not independent of that of sex, or 

 has an influence on fertility, the whole 

 question may be greatly complicated. 

 But such complications seem to be 

 irrelevant to the simple issue raised 

 by Mr. Yule's remarks. 



P.S. I understand from Mr. Pun- 

 nett that he has submitted the sub- 

 stance of what I have said above to 

 Mr. Yule, and that the latter would 

 accept it as a satisfactory answer to 

 the difficulty that he raised. The "sta- 

 bility" of the particular ratio 1:2:1 is 

 recognized by Professor Karl Pearson 

 \Phil. Trans' Roy. Soc. (A), vol. 203, 

 p. 60]. 



