INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 59 





 16 possibilities 8a n a n + 8a,,A,,, or this is the same thing as multiplying the R array 



by the symbolic factor 8 (u + v) = 1GU, say. 



The next pair of fathers 2 (P + a u A n ) with the four mothers reproduce the array R 

 of offspring combined with 8 (a,,a n + 2a,,A + A,,A M ), or 32 possibilities. But this is 

 the same as multiplying the R-array by the symbolic factor 8 (u -\- 2v + w) = 32V, 

 say. Lastly, the P + A,,A M father with the four mothers gives 16 possibilities of the 

 form 8a,,A,, + 8A,,A B to be combined with the R-array of offspring, which is the same 

 thing as multiplying the R-array by the symbolic factor 3 (v + w) = 1GW, say, 



We have at once the symbolic relation among the operators ; 



U + 2V + W = u + 2v + w ; 



and, further, the important result that the array of offspring due to any pair P and Q 

 of n 1-couplet parents can be converted into the arrays of offspring due to the 16 

 pairs of parents formed by adding an additional couplet to P and Q, by multiplying 

 that array by the symbolic factor 



16U + 32V + 16W = 16 (u + 2r + w). 



We have thus by induction a means of finding the array of offspring due to a 

 population of parents of n couplets from the series of arrays due to a population of 

 n 1 couplets. Since all the arrays are to be multiplied by the same symbolic 

 factor, we can multiply their total by this factor. Or the distribution of offspring of 

 (n l)-couplet parents being J, that of H-couplet parents 



= 16 (u + 2v + 10) J = 4 X 4 X 4 . (M + |v f J. 



Now consider parents of one couplet, their distribution is given by aa + 2A + AA, 

 and they are to mate with the same series, an -\- 2aA + AA. 

 But 



art X aa = 4na, 



2 (aa X 2aA) = 2 (4aa + 4aA), 

 2 (aa X AA) = 2 (4A), 

 2aA X 2aA = 4aa -f SrtA + 4AA, 

 2 (2aA X AA) = 2 (4aA + 4AA), 

 AA x AA = 4AA, 



Total = IGaa + 32aA + 16AA. 



= 16 (u + 2v + w) = 4 X 4 X 4 (> + f v + w) 

 symbolically. 



I 2 



