Mathematical Contributions to the Theory of Evolution. 395- 



all the total mid-parental correlations would be perfect, and, there- 

 fore, any one mid-parent would suffice to fully determine any other 

 and the offspring. The individual parental correlations would then be 



1111 



v 7 ^' 2' iya' 4 



for parent, grandparent, great-grandparent, &c, with offspring.* 



(6) More generally, suppose any values of 7 and ft which lead to 

 c == 1, then 



L-/3 2 (l + 7) 



C== 1 = 



^[l-/3 2 (l + 2 7 )] 



whence we find /3(^-\-l) = 1, that is, a, — 1 ; or again, all mid- 

 parental correlations are perfect. Thus, as in case (i), the individual 

 parental correlations could be represented by 



These are the values I took in my memoir of 1895. f I took these- 

 values then because they seemed to express Mr. Galton's method of 

 passing from individual parental to individual grand-parental total 

 regression. J I had not perceived that there was any antinomy 

 between Mr. Galton's theory of regression and his law of ancestral 

 heredity. Had I done so I should certainly, at that date, have given 

 the preference to the former, and rejected his law of partial Coeffi- 

 cients of regression in favour of the values, based 011 numerical 

 observation, of his total regression coefficients. 



(c) Put 7 = 1, /3 = 1 ; this is Mr. Galton's form of the law. 

 2 y/ 2 



We find at once 



a, — 1 — 3 — 0'6 



" Vi' 5 



Hence we have for the successive mid- parental correlations p h p 2 ,. 

 0'6 0-3 . 



Vi' °' 3 ' 775' &c - 



and for the individual mean parental correlations, r u r 2 , r 3 , &c. 

 0*3, 0-15, 0*075, &c. 



* This is what, I think, must follow from any theory of the " continuity of the 

 germ plasma," and of its exact quantitative addition and bisection on sexual repro- 

 duction. 



f < Phil. Trans.,' A, vol. 187, pp. 303-5. 



X See ' Natural Inheritance,' p. 133. Mr. Gralton puts r = ± for a parent, 

 r- = i for a grandparent, and so on. 



