1904.] serve to Test various Theories of Inheritance. 



275 



of the " Ancestral Law," but it does not equal the latter law in its 

 description of the facts, because it shifts the mean variability of the 

 array out of its proper place owing to its absolute rigidity in the value 

 of parental correlation. I think we may say that for the three 

 characters here considered the " Law of Ancestral Heredity " is the only 

 one of the three theories which clearly describes the facts, and the 

 facts at no point differ significantly from its statements. 



(10) The theory of alternative inheritance, which differentiates the 

 offspring into " father's offspring " and " mother's offspring," permits 

 of being tested in another manner, namely, by considering the 

 variability of the array of brothers who have a brother of given 

 character x. Let us first look at the problem generally. A man 

 of extreme character value will have a group of brethren, say, 

 " father's offspring," who allowing for regression are like himself, but 

 the other group of his brethren will be far less like himself, and on 

 the average of all mothers near the population mean. Thus, the 

 array of brethren of a man of extreme character will be made up of two 

 components, one tending to be like himself, the other like the general 

 population. Hence on the whole the array of brothers corresponding to 

 men of a given type must become more and more variable, the more 

 marked the deviation of the given type. I will now give the results of 

 the analysis. Let p be the general correlation of brothers, m their mean, 

 and o- their standard deviation. Let r n be the correlation of brothers 

 who are " father's offspring," mi their mean, and v their standard 

 deviation ; let r 2 2j m 2 and cr 2 be the corresponding constants for the 

 brethren who are "mother's offspring." Let v Y and v 2 be the number 

 of brethren of each class. Then if an individual have a deviation x 

 from the mean of all brethren, he will if a " father's offspring " have 

 a co-fraternity of standard deviation J (I - r n 2 ) and at distance 

 from the mean of offspring given by 



= (rn -p)x + (mi - m) (1 - rn). 



He will have v 1 - 1 of such brethren, but he will also have v 2 brethren 

 of mother's offspring type with a standard deviation <r 2 J (I - r 12 2 ) and 

 at distance from mean of offspring given by 



Similarly, if the brother of deviation x were a " mother's offspring," 

 his brethren would be made up of . v 2 - 1 of standard deviation 

 (T. 2 J (I - r-22 2 ) and mean /x 22 , and of v\ brethren of standard deviation 

 °i "JO- - n 2 2 ) and mean /x 2i where 



p, u = m 1 + ?*n -l(x + m- mi) - m - px 



Pi2 = m 2 + ?*i2 — (x + m - m 2 ) - m - px 



