July 10, 1908] 



SCIENCE 



49 



School of Economics and Political Science, to 

 which he was appointed in 1903, retains the 

 readership in geography, to which, under its 

 then title, he was appointed in 1902. 



DISCUSSION AND CORRESPONDENCE 

 MENDELIAN PROPOETIONS IN A MIXED POPULATION 



To THE Editor of Science: I am reluctant 

 to intrude in a discussion concerning matters 

 of which I have no expert knowledge, and I 

 should have expected the very simple point 

 which I wish to make to have heen familiar 

 to biologists. However, some remarks of Mr. 

 Fdny Tule, to which Mr. R C. Punnett has 

 called my attention, suggest that it may still 

 be worth making. 



In the Proceedings of the Royal Society of 

 Medicine (Vol. I., p. 165) Mr. Yule is re- 

 ported to have suggested, as a criticism of the 

 Mendelian position, that if brachydactyly is 

 dominant " in the course of time one would 

 expect, in the absence of counteracting 

 factors, to get three brachydactylous persons 

 to one normal." 



It is not difficult to prove, however, that 

 such an expectation would be quite ground- 

 less. Suppose that Aa is a pair of Mendelian 

 characters, A being dominant, and that in any 

 given generation the numbers of pure domi- 

 nants {AA), heterozygotes {Aa), and pure 

 recessives {aa) are as p : 2g : r. Finally, sup- 

 pose that the numbers are fairly large, so that 

 the mating may be regarded as random, that 

 the sexes are evenly distributed among the 

 three varieties, and that all are equally fertile. 

 A little mathematics of the multiplication- 

 table type is enough to show that in the next 

 generation the numbers will be as 



(p + qy:2{p + q)(q + r): {q + r]', 



or as Pj : 2q^ : r^, say. 



The interesting question is — in what cir- 

 cumstances will this distribution be the same 

 as that in the generation before? It is easy 

 to see that the condition for this is q'^pr. 

 And since q^^='Pi^v 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 brachydactylous and pure 

 normal persons, say in the ratio of 1 : 10,000. 

 Then p = l, g = 0, r = 10,000 and p^ = l, 

 g, = 10,000, r, = 100,000,000. If brachy- 

 dactyly is dominant, the proportion of brachy- 

 dactylous 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 pro- 

 portion would afterwards have no tendency to 

 decrease. 



In a word, there is not the slightest founda- 

 tion for the idea that a dominant 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 deviations from the 

 theoretical proportions which will, of course, 

 occur in every generation. Such a distribu- 

 tion as Pi : 2q^ : r^, which satisfies the condi- 

 tion qi<=p^T^, we may call a stable distribu- 

 tion. In actual fact we shall obtain in the 

 second generation not p^ : 2g, : r^ but a slightly 

 different distribution p/ : 2g/ : r/, which is not 

 "stable." This should, according to theory, 

 give us in the third generation a " stable " 

 distribution p^ : 2gj : r^ also differing slightly 

 from Pi : ^q^ : r^ ; and so on. The sense in 

 which the distribution p^ : 2^^ : r^ is " stable " 

 is this, that if we allow for the effect of casual 

 deviations in any subsequent generation, we 

 should, according to theory, obtain at the next 

 generation a new "stable" distribution dif- 

 fering but slightly from the original distribu- 

 tion. 



I have, of course, considered only the 

 very simplest hypotheses possible. Hypotheses 

 other that that of purely random mating will 

 give different results, and, of course, if, as 

 appears to be the case sometimes, the char- 

 acter is not independent of that of sex, or 



