Mathematical Contributions to the Theory of Evolution, 147 

 From (xv) 



„, 2 + . + 48 + 2«S^^^j^0 



where 



2(i + a+6S) 

 1 



e = 



Lastly from (xiv) y = ~ (xxi) 



Thus (xviii) and (xix) give 8 and c. (xx) then gives a, taking the 

 root less than unitij. Finally (xvii) and (xxi) give P and y, completing 

 the solution. 



(4) Comparison with Law of Ancestral Heredity. — -Now let us compare 

 these results with those I have obtained from the law of ancestral 

 heredity."^ On p. 390 of the memoir on that subject we have for the 

 3ith midparental correlation with the offspring pn = 2^%j, where is 

 the correlation of the offspring with the individual Tith great grand- 

 parent. By p. 394 pn = coL^. Hence 



Tn = cipcj J2y^ (xxii) 



or the correlations of the offspring with the ancestry follow a simple 

 geometrical progression. 



Comparing this with the result (xii) of this paper, or 



= + + (xii) 



where c is now a different constant, we see that the two cannot possibly 

 be in agreement, unless one of the terms of the latter result vanishes. 

 Thus there is in general a fundamental difference between the law of 

 ;incestral heredity and the law of reversion ; they give expressions 

 differing in character for the correlations between the offspring and 

 individual ancestors. Let us see when the two laws will agree. There 

 is unfortunately a bad slip in my memoir of 1898. The series at the 

 top of p. 403 leads to 



y^'l{l-P') = 1 and not as there given yP'{^~yP) = 1- 

 Thus we have ^'(1+7) = 1 /5' = 1/(1+7). 



- l/{ V2(l+r)} (xxiii) 



and therefore by (xii) a = 1/^2. 



Thus by (xxii) of this paper 



ni = ^(2)'" ' (xxiv) 



• ♦ Eoj. Soc. Proc.,' vol. 62, pp. 386—412. 



