INHERIT ANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 71 



or 



M,- +1 = f M, + |M', , 



M'< + i = iM, + fM',-. 



Therefore 



M,- +1 + M',- +1 = M; + M', = M + M' = 1 + = 1. 

 Hence - 



M /+1 = 'M ; + }, W i+l = pi', + , 



M, +1 - i = HM; - i) = A (Mo - ) = A- , 



1 1 fAP. . . J \ 1 /"VF . . l\ 



. M . -- m. n f ; = - . + 



Hence, finally, 



Thus the result of ??i changes on ^(n 4- v) is known. 

 Similarly the result of TO changes on ^(v + w) is 



We can now write down the array of offspring due to an ?/i th great-grandparent of 

 s-allogenic couplets. It is 



/ , ,.\ -Kf <-<' ~T~ V I 117 < ~ 



(4 x 4)'V n ,,, M,,,- ^ + M' ffl _ } 



\ A -J 



X |(M W + 2M' W ) n -+ - + (M' w + 2M W ) " + 



We must now find the mean of this array. For brevity let us write 

 MJ(M + v) + M' W -J('' + ') = PI + Xe 



(M, + 2M'.) 2 + (M '. + 2M.) ^ = M " + 2M - (^ + Xc 



where 



1 M' 



' l+M', 



, _ r _i f i w 



i~i Tl fc ^ 1 I AT/ W ' 



2_(M,,, 2 - M'^) 2 (M. - M / M ) 

 M M + 2M' M 1 + M' 



