76 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



brothers to be found associated with a brother of p allogenic couplets. The above 

 expression may be written 



(4 X )" [{(9 X 4) u* + 7 X ?7, } MJ + (7 X M 2 + (41 X 12) y^ 773]". 



The term involving uf is 



(4 X ) H ((9x - 4). + rxntf {7 X w 2 + (4i x - 12) 7,!}"-*%-^,,,.. 



Neglecting the constant factor, the distribution in u. z is given by 

 {(9 X - 4)7*. + 7^}" [7 x u, + (41 X - 



We require to find the mean of this array. 

 Put 



X= 9 *- 4 - 7 X_ , = 



16 X -4' 16 X ~4' 7 X 



Then again, but for a factor independent of the power of u. 2 , the array may be 

 read 



The eneral term is therefore 



and we must sum from s = o, to .s- = n p. 



The general term has therefore its mean at the distance 1 + /* Q> + s) from 

 p + * + 1 allogenic couplets, or its mean 



= (1 p.) (/> + - s ') 5 allogenic couplets, 



and its total frequency = v n ~ p ~'c n _ j>iai0 . 



This gives a total frequency of the array proportional to (1 + v) n ~ p . 

 Hence, taking moments, we have for the mean m of the array given by 



1 hereiore 



m = l -^n + v -^-rtp. 

 1 + v I + v * 



Hence we see that 



(i.) The regression between brothers is linear. 



(ii.) The fraternal correlation which is equal to the regression 



_ 

 3(4 X -1)' 



