HEREDITY 



533 



2. The correlation of sons with respect to mid-parents (1?) is 

 equal to that of sons with respect to fathers multiplied by V^.* 



We have answered the question as to the value of h z . It 

 remains to answer the question as to the value of 2, the vari- 

 ability (standard deviation) of the array of offspring from the 

 particular parents of deviations h-^ and h^. If we assume as 

 before that ^ = r 2 , then 



The fuller treatment of the meaning of this formula will be 

 taken up in a succeeding section on " Selection." 



We could now proceed to form a mid-grandparent in the same 

 way by transmuting female values into their male equivalents 

 by multiplying by the ratio of male to female standard devia- 

 tions. Having four grandparents, we take the mean of the four 

 values thus obtained for our mid-grandparent. Similarly this 

 could be carried back to any number of generations, and we 

 should thus derive a mid-parent for the first, second, third, etc., 

 generations of ancestry. These can be conveniently referred to 

 as the first, second, third, etc., mid-parents of an offspring. 



Formula for ancestral heredity. In terms of these mid-parents 

 and their variabilities Pearson has stated, in modified and 

 generalized form, Galton's law of ancestral heredity as follows : 



/, IO "7v_i_ IO "w_L IO "*7_i 1<r / 



h = -- //i H --- //a + T: ~~ -"a 4- ---- h H n -r - , 

 2 <TI 4 <r 2 00-3 !1.~ <r, t 



in which h is the deviation from the mean of offspring in general 

 to be expected in offspring of mid-parents of successive genera- 

 tions backward whose deviations were H^ H^, H^ , H n \ cr is 

 the standard deviation of offspring in general [the cr 3 of formulas 

 (i) to (4) ] ; and <r lt <r 2 , cr 3 , . . ., cr w , etc., are the standard deviations 

 of the mid-parents of successive generations of ancestry. 



It may be noted from this formula that if we take no 

 account of differences of variability in successive generations 

 (<r = a l <r 2 = ...), and make the deviations of successive mid- 

 parents equal (ff^ H^ = H^ = }, we obtain Galton's series, 



* 



* That is, in K = , if r a be disregarded, the formula becomes 



VI 



