INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 



75 



This gives a total of 25Gx (34^= 16 X 4^ (4^ 1) pairs of brothers, as it should, 

 every brother in 16 families having 4^ 1 brethren, and there being 4^ in each 

 family. Clearly we can divide by 4y, and we have the simplified form : 



where the subscripts 1 and 2 refer to the first and second brothers. 



Let us simplify this by considering only the allogenic elements, 17 denoting either a 



Then we have : 



heterogenic or a protogenic element. 



"l- 



9 x -4 



'H- 



X 

 41 X -12 



Thus the distribution of pairs of brothers in the case of the character being fixed 

 by a single couplet is 



- 4) 



(41 x - '-) 



This is to be read as follows : there are ( J^ 4 cases of both brothers being 

 allogenic, to 41^ 12 cases of neither brother allogenic, to ?x + 7x cases of one onlv 

 of the two brothers being allogenic. 



Now when we pass from a character fixed by one couplet to a character fixed by 

 two, the above distribution can occur in either couplet, and every possible pair of 

 brothers will be got by squaring the above expression. Proceeding in this way to 

 ft couplet characters, we have the following symbolic expression for the distribution 

 of brothers 



(4 X )" X {9 X - 4) 



(41 X - 



the omitted factor 4 X being restored. 



This, I think, represents the distribution of a character measured by allogenic 

 couplets in a population of pairs of brothers, i.e., is the correlation table for brothers 

 in the population any term involving // representing when we put 77! and 77., 

 equal to unity the number of pairs in which the first brother has p and the second / 

 allogenic couplets in their constitutions. 



Hence if we find the coefficient of uf in terms of u. it we shall have the array or 



L 2 



