INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 01 



10VR a _ 1; and if we add a couplet of form A,,A,,, the array will he of the form 

 Now start with a father of one couplet ; this must lie a l n l , or a, A,, or A,A], or in our 

 symholic notation u, r, or w ; the offspring array arc respectively 8,a, + S^A] or 

 4 1 a 1 + S^A! + ^A^ or Sa^ + SAjAj, i.e., 1GU, 16V, or 1GW. These, therefore, 

 are the possihle values of P^. Hence, hy the principle just developed above, the 

 array of offspring due to a father of type 



u n ~f~i 

 is 



or remembering that such fathers occur with a frequency of 2>'c Mt! , ? , we have for the 

 total distribution of offspring of all fathers of type 



u-p-i ,/> ic'i, 

 the symbolic result 



4" X 4". c,,,,,, /[?'-/'-? (2V)'' W?. 



Substituting, the following expression would give all offspring of fathers of the 

 type u"~P~iv r> w' ! , i.e., with n p q allogenic, p heterogenic, and q protogenic 

 couplets 



4" X 4" . c a , M ($u + -\vy-f-i ('- + v + ^''Y Q-'.' + -I-" 1 ) 7 - 



Therefore, given n and given p and <j, it is merely a matter of expansion to find the 

 array of offspring due to any special class of father. 



Corollary (i.). So far we have supposed our special class of father to lie defined by 

 the exact couplet distribution constitutional to him. But it is of interest to consider 

 the array of offspring we get supposing only the allogenic couplets fixed in number, 

 for example, in a generalised Mendelian theory if the number of recessive couplets be 

 fixed, but the heterogenic and dominant, as both exhibiting dominant characters, be 

 considered as indifferent. Let s = number of allogenic couplets, then we have to 



sum all arrays like 



4" X 4". c, ]]iV/ U'(2V)nV?, 



subject to the condition that p + q = n .s. 

 The result is clearly 



4" X 4.c^. ie 2{c, + ,, Pie (2V)>W'} 

 = 4" X 4" . c Bi . i0 U' (2V + W)"-' 

 = 4 X 4" . C B>1 , , (u + i-r)' (i + ft' + w) 

 = 4" X 4" . c,, .,.( + Jv)' {($u + %v) + (v + u<)}"-'. 

 This, we note, is not a pure binomial., or the arrays of offspring of a father with a 



