1904] 



serve to Test various Theories of Inheritance. 



271 



(c) On the generalised theory of alternative inheritance which 

 divides the offspring into two groups more intimately associated with 

 one or other parent, the resulting curve is a hyperbola with vertical 

 real axis. 



(7) I have applied the criterion here developed to my measurements 

 on father and son in more than 1000 families. The three characters 

 stature, span and forearm are dealt with. The correlation tables for 

 the paternal inheritance of these characters will be found in a recent 

 memoir by Dr. Lee and myself on the " Inheritance of the Physical 

 Characters in Man."* We excluded all arrays with less than eight 

 individuals in them, deeming it absolutely untrustworthy to find a 

 mean and standard deviation from fewer than eight cases. The standard 

 deviation of each array for a given paternal character was found by 

 Dr. Alice Lee. These were then plotted to the paternal character by 

 Mr. W. L. Atcherley, and results are shown in the accompanying 

 Diagram 1. The zigzag polygons in each case give the plotted 

 variabilities of the arrays, the vertical numbers being the total on which 

 the variability is based. The horizontal line AA gives the mean value, 

 -cr c J(l - pf c 2 ), of the standard deviations of the arrays according to the 

 .statistical theory. The broken lines ccc and c'c'c' give cr c J(l -p 2 f e )± 

 twice the probable error of the deviation of an array from o- c J (I - p 2 f C ). 

 Thus if 2 = cr c ^/(l - p2f c } and 2^ = the standard deviation of an array 

 •of m individuals out of a total of n, we have plotted up and down from 

 .2 the quantity 



Now unless a difference is at least twice its probably error we certainly 

 cannot assert it to be significant. Now, if the diagram be examined, 

 it will be seen that almost without exception the zigzag polygons 

 fall well within the non-significant areas bounded by ccc and c'c'c' . 

 There are, indeed, three exceptions, but all three occur in arrays with 

 less than twenty individuals, or arrays where some eccentricity of 

 individual or measurement might easily make itself felt. But there is 

 really no need to appeal even to this explanation, we have thirty-nine 

 observations on arrays, and three of these only exceed the double of 

 their probable error • this is actually less than half the excesses we 

 might have expected on the theory of probability. 



There is clearly absolutely nothing in the observed results opposed 

 to that constancy of variability in the arrays which is suggested in the 

 usual treatment of the " Law of Ancestral Heredity." 



(8) I next look at the theory of alternative inheritance, or at the 

 hypothesis that some children follow the one, some the other parent. 

 An examination of the diagrams show that there is not the least 



* ' Biometrika,' vol. 2, pp. 415 — 417. 



