84 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



means of the array of offspring due to any pair of parents are marked in slim figures 

 along the Mendelian hyperbolic contours, and in heavy figures along the Galtonian 

 straight lines. We see comparatively small differences as long as we deal with 

 matings within the 99 per cent, loop, somewhat greater differences as we approach 

 the 9 9 '9 per cent, loop, and very marked differences as we go beyond the latter 

 boundary towards pure allogenic parentage. A study of the diagram illustrates at 

 once how the Mendelian theory exhibits for the bulk of the population Galtonian 

 regression, such regression becoming, however, less and less as we proceed to 

 individuals, the frequency of whose matings may be less than one in a million. 



It will be seen again that in this proposition we have no fundamental antagonism 

 between the Mendelian and biometric standpoints. We reach a single formula whicli 

 approaches more and more closely to the biometric standpoint when we deal with 

 characters depending on many allogenic couplets.* On the other hand, it gives the 

 absence of regression which is obtained when pure allogenic parents are mated. We 

 see, however, quite clearly that it is totally erroneous to argue from this single case 

 against regression in general. Such regression actually exists on " Mendelian 

 Principles" when any population breeding at random is taken, and involves in itself 

 the whole conception of ancestral correlation and the influence of ancestry. 



Of the formula for the midparent now reached, however, we can only say that on 

 the basis of our experience in populations the factor seems too inelastic to work. 

 Here again the data must be especially investigated from the standpoint of a 

 midparent given by 



before judgment can be final on this test. 



Theoretically, by assuming the midparent to be 



we should have a means of finding ^ by averages, and therefore n, the number of 

 couplets involved. 



(12.) General Conclusions. 



In this paper we have dealt with a general theory of the pure gamete possibly 

 not the widest that could be conceived but sufficiently wide to indicate the real 

 bearing of Mendelian formulas when applied to a population mating at random. We 

 see that under such circumstances : 



(i.) The^ population which results from the offspring of hybrids remains stable; 

 every variation which appears, appears with a certain definite and predicable 



* For n = 50 or 100 the population within the 1 in 1000 line sensibly obeys ordinary linear midparental 

 regression. 



