268 



Prof. K. Pearson. On a Criterion which may [Mar. 4, 



Or, we see that unless 



ri/o-cjo-f = T.jcrjo-f, 



i.e., the paternal influence be the same in both groups, or there be no 

 question of individual dominance of the parents in " father's offspring " 

 and " mother's offspring " respectively, the arrays of offspring due to 

 different classes of fathers will not be equally variable. It is clear 

 that if the standard deviation of the array be plotted to the size of 

 the father's character, the resulting curve will be a hyperbola with 

 real axis vertical, and in the two directions across the correlation table 

 taken from the parental value 



the variability of the arrays will rapidly increase from a minimum. 



As there can hardly be a sensible distinction between mi and m- 2 , 

 for it would mean bimodality in all the characters dealt with, which 

 is contrary to experience, we may say that the variability %f the 

 offspring arrays increases in both directions with the deviation of the 

 father (or mother) from the mean. 



If we put mi = m 2 we have : 



IV = ^cHl-r 1 ; 2 ) + n 2 o- c ^(l-r,/) + n V j,, / ^A 2 ^ _ . , ( xii \ 



11 ll 2 \ " ay ay/ 



which shows clearly how the variability of the array increases 

 hyperbolically with the deviation of the father from the mean. 



(5) Now it is as well to take one or two numerical cases to 

 appreciate the degree of curvature of this hyperbola, for if it were 

 a very flat hyperbola, it possibly could not be readily distinguished 

 from a horizontal straight line (Ancestral Law) or from a parabola 

 (Mendel's Law). I take the hypothesis suggested by Dr. Boas, i.e., a 

 negligible influence of the father on " mother's offspring," and a 

 negligible influence of the mother on "father's offspring." 



We have r 2 / = rim = 0- Further, if we suppose the two groups of 

 offspring to be equal in number and equally variable, %i - iu and 

 <r Cj = o- C2 = o- c . From (ii) it follows that 



Pfc = - — 1 Tif = J Tj f. 

 11 (T c 



Therefore from (xii) it follows that 



y 2 _ 

 z, x — 



I ay- j 



