66 PEOFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



or 



Now 



and by differentiating 



2 ( 7i _ 



Repeating the process 



4 ( - s) (n - * - 1) (1 + 2ic)"-'~ 2 = 2c B _ fi ,., 2 ; i (i - 1) a 1 '- 2 . 



Hence, putting a; = 1, we have the required expressions on the right of the above 

 result, or 



0.-.-S - Q,-, J" i ( n ~ *) 2 _i_ ( Ji _- *) ( n -.Sj-J) _. 2_( ? 1 

 U 9 9 9 



Therefore 



a-, 2 = ^i i ( n s) = 3 1 ,; (5w, + 4s). 



Now the standard deviation of the whole population, as far as allogenic units are 

 concerned, is 



a- = v/ni f = X/I-G". 

 Thus 



n 



This result is of singular interest. The variability of an array of offspring 

 corresponding to a father of given allogenic constitution is not independent of the 

 father, but increases steadily from a minimum of \/jfan t when there are no allogenic 

 couplets in the father, to a maximum of \/\n when there are only allogenic couplets. 

 In other words, fixing our attention on the same character, let the offspring of the 

 hybrids inter se be crossed first with one pure race, and then with the second pure 

 race, i.e., first with pure allogenic and then with pure protogenic individuals, there 

 ought to be a marked difference in the variability of the resulting offspring in the 

 two cases. 



Corollary (i.). In the theory of linear regression as apart from the theory of 

 normal correlation on the basis of the Gauss-Laplacian distribution,* if <r be the 

 standard deviation of any character, and r its correlation with a second character, 

 then 



o- v/1 _ r 2 



* YULE, 'Journal of Royal Statistical Society,' vol. 60, December, 1897, 



