INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 81 



and s must be taken from o to n p r. The frequency of this term is 



Q-S+r 



_ _ nn-i>-r- 



4 ^-p-r,t,<it 



and its mean is 7 r + 3 (p </ + 2r + .s) allogenic couplets. 

 The total frequency of the ?"' term is therefore 



Q-3 + i- 30.1 -j.-/ 



/i _i_ 2V'~''-'' 

 4" 4 ' 



which of course must ultimately be multiplied by the factorials in r omitted above. 

 If m r be the mean number of allogenic couplets in the r"' term, we have 



7 + )' 



- 9''-i'- r -v 'a r 4- ' / /) o 4- >; 4- V- 



An " c -J-r,,oia ' I 3 VJP 71^"' T */t 



4t 



= <-- + i(p-Sf+ 2r)} 3 "^'" + * (n -jp - r) ?^-) - 



Thus : 



ni. ( . = gr r + 1 ( i> </ + 2r) + -, 1 , ( - p r), 



= 9^ + 37 + I? >' 



This is the mean of the /"' term, and its total frequency is 



:T-" -*-'/ 



4. C-. ft rXC A , 



Hence, if 



F = 4" x.3 3 "~^~ S' 



>'= 



_ 4 3 a,-,-, x ^ ; 

 we shall have 



/X V/ = S l(a yi + ii'/ + f.^) - u' 1 } ( '^,r.. X 



where j/; yjf is the mean of the arrav of offspring due to fathers of p and n. others of </ 

 allogenic couplets. 



The only difficulty here is summing the series 



But this may be written 



r-I=-l 



S (n X c, _,,,,_, 



11=0 



or multij)led by 4"~ 1 3 2 ("~ ] )~/-"'^~ 1) /)? it represents the total offspring of fathers of p and 

 VOL. com. A M 



