1904] 



serve to Test various Theories of Inheritance. 



265 



heredity summed up in the " Law of Ancestral Heredity " would assert 

 that the offspring of all parents of a given type would have a constant 

 variability, whatever that type might be. 



(3) Mendel's Theory. — If we take as a fair sample of this theory the 

 generalised Mendelian theory, discussed by me in a recent communica- 

 tion to the Society, and now published in the ' Phil. Trans.,' we find that 

 this constancy of the standard deviation of the array is no longer true. 

 It only becomes true if the number of Mendelian couplets on which 

 the character depends is indefinitely great. In other cases, while 

 °"c - p/e 2 ) is still the mean of the standard deviations of the arrays, 

 the actual value of the standard deviation alters sensibly and con- 

 tinuously as we cross the correlation table, always tending to increase 

 in one direction and decrease in the other. Clearly we have, as I have 

 pointed out in the paper referred to, an excellent criterion here between 

 the two theories.* 



(4) Lastly, let us turn to the theory of individual parental domi- 

 nance. I will give the analysis for this case, extending and generalising 

 Dr. Boas's formulae. I suppose the total offspring n of a pair of parents 

 to be divided into two groups rti and n 2 in number. In the first groups 

 with a mean nil the fathers are considered as predominant without the 

 mothers being supposed at present entirely without influence ; in the 

 second group with mean m 2 , the mothers are supposed to have the 

 predominating influence. We may speak of these two groups, for 

 convenience only, as "father's offspring" and "mother's offspring." 

 Let or Cl and cr C2 be the standard deviations of "father's offspring" and 

 "mother's offspring" for a given characters; let oy, <r m , o- c be the 

 standard deviations of the fathers for the same character, of the 

 mothers, and of the offspring as a whole. The mean m of the 

 offspring as a whole will be given by m = (n x mi + n 2 m 2 )/n. Further 

 let rjf, r 2 / be the paternal offspring correlations for "father's offspring" 

 and "mother's offspring," and r lm , r 2m the maternal offspring corre- 

 lations for the same two groups respectively ; r fm shall be the coefficient 

 of assortative mating between parents, x, y, z are the characters in 

 father, mother, and child, x and y being measured from the parental 

 means and z from some other origin. Then, if Si stands for a summa- 

 ration of all the offspring of the first and S 2 of the second class, we 

 have 



n<r c 2 = ni \a- Ci 2 + (mi - m) 2 } + n 2 {o- C2 2 + (m 2 - m)' 2 } , 

 or o- c 2 = (?h°"ci 2 + n 2 a C2 2 )/n + n Y n 2 {m ± - m 2 ) 2 /n (i). 



Let pf C and p me be the total paternal and maternal correlations ; we 

 have 



p/c = [Si {x (z - m)} + S 2 \x (z - m)} ]l(no-jo- c ) 

 = (n 1 (T f a c {r lf +n 2 o-f(r C2 r 2f )/(n(r f cr c ), 



* 'Phil. Trans.,' A, vol. 203, p. 66. 



