Mathematical Contributions to the Theory of Evolution. 407 



If we suppose two individual parents with no assortative mating, 

 we have 



r = 2(r 1 2 +r a 2 ) (xix), 



where o\ and r 2 are the male and female parental correlations. With 

 Galton's law r x = r 3 = 0'3, and r again = 0*36. Assuming the value 

 r y = r 2 = i adopted by Mr. Galton in his ' Natural Inheritance ' 

 (p. 133) for parental regression, the fraternal regression deduced 

 from this ought to have been -§- = 0'44, and not 0*67 as obtained 

 by Mr. Galton.* The mean of the sister-sister, brother-brother, 

 brother-sister correlations that I found in 1895,| duly weighted for 

 the number of pairs in each case, is exactly O4000. The value as it 

 might have been a priori predicted from Galton's law = 0*4000, with 

 a rise to 0*4402, if we " tax " up to 10 per cent. 



I conclude therefore that this law of ancestral heredity is at least 

 to a first approximation in agreement as complete as could possibly 

 be expected with the facts we as yet know as to collateral heredity. 

 It confirms the view I took in 1895, that fraternal heredity cannot be 

 taken greater than 0"5. I think the high value (about 0*6) obtained 

 from Mr. Galton's " special data " must be explained by my suggested 

 causey (a) i.e., unconscious selection of approximately equal heights 

 in brothers who join Volunteer regiments ; for the explanation (b) is 

 taken away if we accept Galton's law without a modified 7. 



(11) Turning now to the inheritance of cousins, we notice that 

 their regression may be represented by 



%i = ihi + ^ko +iV + iV+® r j 



*% = W+ih .... +ih" , +ih"+x". 



Here hi and h" are children of the same parents and have fraternal 

 correlation ; hi and h{" are their other parents, and without a 

 double cousin marriage have no correlation with each other, or 

 neglecting sexual selection with hi or hi" ; h is the mid-parental 

 system§ of h u and therefore of hi"; Jc ' and Jc " the mid-parental 

 systems of hi and hi", and accordingly, if there be no in-and-in 

 breeding, uncorrelated with each other or with k . 

 Summing first for the array corresponding to hi, hi", 



S(xi% 2 ) = n{^- 6 hh" +^h (hi + hi")+^h 2 }, 



* Mr. G-alton took r = 2r Y ; this is part of what, I think, the erroneous theory 

 of regression developed in ' Natural Inheritance,' a theory which is inconsistent 

 with the law of ancestral heredity given in the same work. 



t ' Phil. Trans.' A, vol. 187, p. 281. 



% ' Phil. Trans.,' A, vol. 187, p. 284. 



§ This means that k = + ±7c 3 + \7c A + .... where Jc q is the common mid-^th 

 parent of the two cousins. 



