80 PROFESSOR K. PEAESON ON A GENERALISED THEORY OF ALTERNATIVE 



in each couplet, and each such distribution must be combined with every other couplet 

 distribution. We then reach, dropping unnecessary indices, the general symbolic 

 relation, 



X (Jt' + ur) + ' + 



= 4" X 4" X ( j (> +ji (e + 17) + />??)" 



Tims the array of offspring due to parents of zygotes with p and q allogeuic 

 couplets respectively i.e., 'to w''X ?'/'' is the coefficient of e'ty' on tne right-hand side, 



or in the expansion of 



4" X 4" X (./ +Ji (c + i?) +>?)" 

 This may be written 



4" X 4" X (./, +jfl)e+jo+jflY- 



Thus the coefficient of e'' is 



4" X 4" X (/, + j s r})"(Ju +jtfT~ P c n , p , . 



We require to pick the coefficient of T?" out of this in order to get the array ot 

 offspring due to fathers of j>, and to mothers of q, allogeuic couplets. But this is 

 clearly 



4" X 4" X r,. f ,.{J**Ji f ~ 9 J<r i ' c t,*, 



.>:'"' JT '''in'" ''f',,- rt r...t> 1 ,-r.. 



or, more briefly, 



A" V -I" V / 



X 4 X ',,,j,, 



We shall first find the mean and frequency corresponding to the /"' term as given 

 above of this series. What we have to deal with is 



We may write this 



[ X 'it 

 4" 



X ( 



where x = '' ail( l x' == i r ~l~ t w > an< ^ l' )O th may be put unity when we are merely 

 rinding the distribution of allogeuic couplets. 



Now the general term in the above expression is 



