1904] 



serve to Test various Theories of Inheritance. 



277 



+ ^(/*ii 2 + ^i 2 ) + ^(rf + Pi 2 2 ) (xxii). 



To simplify still further, assume no distinction in the totals of 

 "mothers' offspring" and "fathers' offspring," or take = m 2 = 

 o-j = cr 2 = cr, %i = ri2 = Jw. Hence 



p = i(rn + r 2 2) + ir 12 , 

 V = cr2 {1 - J (r n 2 + r 22 2) - 1 (1 - r 12 2)} 

 + i {(ni - rf 2 + (r 22 - />) 2 + 2 (r 12 - a* 



Hence we see that if the variability of the arrays of brethren is to 

 be constant, it is absolutely necessary that r n = r 22 = r 12 = p, or the 

 degree of likeness between brothers whether they belong to " mothers' 

 offspring " or " fathers' offspring " must be identical. If this be not 

 true, the variability of the array of brethren of a brother of given 

 character must obey a hyperbolic law, being least for a brother of 

 mean character. 



If we adopt Dr. Boas's theory of complete alternate inheritance we 

 have 



7*11 = r 22 and r 12 =0. 



Whence 



fli = 2/>, , 



and 



= o-2(l- 2p2) + p W (xxiii). 



This is a very easy result to test. On the whole, however, it is 

 better to ask the general question : Are the variability of arrays of 

 brethren hyperbolically distributed 1 



(11.) I propose to answer this by appeal to my data for nearly 

 2000 pairs of brothers measured for their cephalic index. Selecting 

 the arrays of 20 brothers and upwards the results plotted in 

 Diagram II. were reached. Here p = 0*4861 and cr == 3*314 and the 

 mean cephalic index = 78*92. Hence the " Law of Ancestral Heredity " 

 gives for the mean value of 2 X , 2 = 2*896. On Dr. Boas's hypothesis 



X, 2 = 5*7924 + 0-2363Z 2 . 



These results are shown in the same manner as in the first diagram , 

 the broken lines marking the limits of twice the probable error of 2 

 and the actually observed 2^. 



We may, I think, safely conclude from this ' result that the 

 VOL, LXXIII. X 



