Mathematical Contributions to the Theory of Evolution. 151 



These give y and ^ when the parental and fraternal correlations are 

 Imown. 



Now, since r is < 1, /^'^ will be imaginary, if r be not >4ri2. Hence 

 we should again need perfect fraternal correlation for fi to be as large 

 as 0-5. 



Thus with blended inheritance and little or no assortative mating 

 we cannot get a parental correlation as high as the value 0'5, which 

 actually does occur in my data for both men and horses. 



We must now consider how the problem will be affected, if we 

 suppose exclusive and not blended inheritance. 



(7) Illustrations of the Law of Reversion in Exclusive Inheritance. — 

 (i) Let us first consider what happens if we take the chief feature of 

 Mr. Galton's view, i.e.^ that the likeness to the parent is the beginning, 

 so to speak, of the reversion series. Then y = /3 (in the notation of 

 the present memoir). It follows from Equation (i) that : 



1 - 2(a + ^) = 0, or S = 0. Thus by (xviii) 



Pi = 2/)2, 



and generally pn = 2/)„_i. 



Equation (xx) to find a now becomes 



a2-(l+l.)a + l = 0, 

 while y = ^ - a. 



Thus as soon as we know pi, we can find all the ancestral correlations 

 and the whole series of reversions. For example : ii pi = 0*4 we 

 should have p2 = 0-2, e = 5, and a^- 3-5 a + 0-5 = 0. a = 0-149 

 and y = 0*351 = j3. Thus in this case 35 per cent, of the oft'spring 

 take after each parent, and 30 per cent, revert to higher ancestry. 

 Of this 30 per cent. 100 x 0*35 x 0-149, or 5-23 per cent., revert to 

 each of the four grandparents, leaving 9 per cent., about, to revert to 

 great grandparents and higher ancestry still. 



(ii) Next suppose Mr. Galton's full view to be correct, and that 1/4 

 of the offspring follow each parent, 1/16 each grandparent, 1/64 each 

 great grandparent, and so on. Then we have — 



a = r = ^ = i. 



Hence from - (1 + Jg) a + J = 0, we find 



e = 2'0 and pi = 0-3. 



Thus we should have: pi = 0*3, p2 = O'lS, pz = 0-075, &c., or, pre- 

 Hsely the same ancestral correlations in the case of exclusive that we have in 

 the case of blended inheritance by the law of ancestral heredity for the special 

 case of y = \ (see table, p. 149), 



