INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 73 



A Q ^^ 1) 



Hence : Deviation of offspring = -^ ^ but the deviation of m" 1 great-grand- 

 parent = s n. 



Thus we have 



Deviation of offspring l 1 



Deviation of m ttt great-grandparent ~" ^ 'l m 



This result is independent of \ and of n. 



Thus we conclude : 



(i.) The regression of offspring on any individual ancestor is linear; 

 (ii.) The correlation coefficient is halved at each stage in ancestry ; 

 (iii.) The result is perfectly independent of the number of couplets introduced into 

 the formula. 



The first two results are very familiar to biometric workers in heredity. 



The actual numerical values of the grandparental, great -grandparental, great-great- 

 grandparental correlations are y, -j^-, -/j, &c. 



These are distinctly less than the values so far readied fur ancestral correlation, 

 the grandparental correlations, for instance, Iving between "2 and "'>. 



The results show, however, that a general theory of the pure gamete, embracing 

 the simpler forms of the Mendelian principle, leads us directly to a series of ancestral 

 correlations decreasing in a geometrical progression. Thus, when we suppose a 

 population arising from hybridisation to cross at random, we find that it obeys the 

 second fundamental assumption of the biometric theory of heredity.* In other words, 

 ancestry is of the utmost importance, and the population follows laws identical in 

 form with those propounded in the biometrical theory on the basis of a linear 

 regression multiple correlation. Only the values of the constants deduced for the 

 law of ancestral heredity from the present theory of the pure gamete (which appears 

 to cover the bulk of Mendelian formulae hitherto propounded) are sensibly too small 

 to satisfy the best recent observations on inheritance. 



It is of interest to find "Mendelian Principles" when given a wide analytical 

 expression leading up to the very laws of linear regression, of distribution of 

 frequency, and of ancestral inheritance in populations, which have been called into 

 question as exhibiting only a blurred and confused picture of what actually takes place. 



It would be an immense advantage if we could accept such a theory of the pure 

 gamete as has been here analysed as a physiological basis for the theory of heredity. 

 We should then have a physiological origin for the ideas of regression and of ancestral 

 inheritance which statistics of heredity in populations have made familiar to 

 biometric workers. Unfortunately, even such a general pure gamete theory as we 

 have here dealt with, while leading to results which form a special case of the law of 

 ancestral heredity, is not sufficiently elastic to cover the observed facts. The lesson 



* ' Biometrika,' vol. 2, p. 220. 

 VOL. COIII. A. L 



