72 PEOFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 

 Hence we have to find the mean of the system 



(pu + Wdiu + Xc + vriY-: 



The i th component binomial of this sum of binomials is 



e,,_ I ,, > iyOnw + Xe) '. 

 It therefore has its mean at a distance 



i + 1 + X (n i) 



from (n + 1) allogenic couplets, and a frequency given by 



fi = c n _,,,- i/ft. 



The total frequency of the whole array = (1 -j- v}"~'f\. 



Hence, taking moments round the origin at n + 1 allogenic co\iplets, we have, if 

 m', be the mean of the array, 



)-/; x m f , = 



_ ,s) (-2 + X ( - 1)) 





>(8 + X(n-2)) + ... 



+ ^ H -^^- s - 2 ^;>-^- { + 1 )(i + i + x(,-^) + ...}. 



Summing and dividing by (1 -f- v)"~ ! we find 



m', = 1 + Xn + ( " " *^ ~ X) . 



Hence the mean number of allogenic couplets in the members of the array 

 = n + 1 - m ', = n (1 - X) - (n - s) - v - (l - X). 



Deviation of offspring from mean of general population 



v(l \) 



= s - - ' n 



We now substitute for v and X in terms of M,,, and W m , and find 



= - 1 - - - 1 n 4- M' } - l ~ 2M '" - 

 ~ ell M- - - 



