Mathematical Contributions to the Theory of Evolution. 165 



5. The brood-mare is for many causes, detailed at length in the paper, 

 a highly artificial product, and accordingly the record gives a consider- 

 able percentage of fictitious fecundities. The effect of a mixture of corre- 

 lated and uncorrelated material on correlation and variation is next 

 investigated, and it is shown that the former is more seriously affected 

 than the latter. Hence results based on variation are more likely to 

 be trustworthy than those which use correlation. Incidently the 

 problem of the mixture of heterogeneous materials uncorrelated in 

 themselves is investigated, and it is shown that a correlation will result 

 in the mixture. This spwrious correlation is of some importance for 

 the question of mixtures of classes in fertility problems, but it is also 

 significant of the general danger of heterogeneity in bio-statistical inves- 

 tigations, and further indicative of the possibility of creating correlation 

 between two characters by breeding between small heterogeneous groups 

 in which this correlation is zero. This illustration suffices to indicate how 

 correlation between characters does not necessarily indicate a causal 

 relationship. 



6. Part II of the memoir deals with the inheritance of fertility in 

 man. It is first shown by large numbers that fertility is undoubtedly 

 inherited from mother to daughter, but that if we include all types 

 of marriages the inheritance is largely screened by other factors. An 

 attempt is made to remove one by one these factors, and the more 

 stringently this is done the more nearly the regression of daughter on 

 mother moves up towards the value required by the law of ancestral 

 heredity. If we could take only marriages in which both daughter and 

 mother were married during the whole of their fecund period there 

 is little doubt that we should find inheritance according to the law 

 of ancestral heredity. The sparseness of homogeneous material hinders, 

 however, such an investigation. 



The inheritance of fertility from father to son is then considered ; 

 this is really rather an inheritance of sterility or tendency to sterility, 

 for the full fecundity of a man is not usually exhibited in monogamic 

 union. It is rather a problem of whether his fecundity lasts as long as 

 his wife's. We find definite inheritance of this sterile tendency from 

 father to son, although for the reason just given it falls below that 

 indicated by the law of ancestral heredity. 



Lastly, the inheritance of fertility in the woman through the male 

 line is dealt with, and it is shown that a woman's fertility is as highly 

 correlated with that of her paternal as with that of her maternal grand- 

 mother. In other words the latent character, fertility in the woman, is 

 transmitted through the male line, and with an intensity which ap- 

 proximates to that required by the law of ancestral heredity. Inci- 

 dentally the problem .of " heiresses " is dealt with. It is shown that 

 in the case of women who are chiefly " heiresses," there is at once a 

 considerable drop in the correlation between their fertility and that of 



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