INHERITANCE, WITH SPECIAL ItEFEKENCE TO MENDEL'S LAWS. M 



(8.) PROPOSITION VI. To find the Array of Offspring due to a Grandfather of 

 s-allogenic Couplets, supposing Complete Random Mating in the Population. 



The general distribution of the population is 



The fathers with s-allogenic units in their correlation are given by 



f,, i . so ?/,*(2r + w)"~\ 



The array of offspring due to these fathers is simply obtained by writing U ior u, 

 V for r, and W for iv, and multiplying by 4" X 4". This is a general rule for getting 

 the offspring from any father if lie mates at random. It gives us, as on p. 61, for the 

 offspring distribution 



4" X 4" X f,, ; ,,,_,!? (2V + W)"- 

 - 4" X 4" X ,,, (. + ')' {( + ir) + (r -f w )} . 



To get the offspring of this array treated as fathers and mating at random, we 

 have only to repeat the process, and we find 



Offspring of grandfather of s-allogeuic couplets 



= 4" X 4" X 4" X 4"e, M , u (i U + \)> {^U + 2V + W}- 



= 4 4 "c.,,.,.(* + |c)'(s" +^+ H''+ ^))- J X (*)-, 

 where 



e^Sr + lu', 



and is equal to unity if we identify -o and w as something not allogenic. This can be 

 dealt with exactly as in Proposition IV. we dealt with the array of offspring due to 

 a father of s-allogenic couplets, i.e., by analysing the array into the sum of a number 

 of weighted binomials ; in this case all. skew. 



Writing as before, i? = (r + iv), we have to expand 



The general term is 



This has a total frequency O B _,, ,, (!)' X / ]s and its mean is at a distance i+ 1 -H("~ *') 

 from the origin which is taken at (n + 1) allogeuic couplets. 



