INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 



79 



Still keeping n = 5, take p = (/ = \n = 4. Then we have the chance of such a 



mating 



= 1/4215 

 still extremely improbable. 



Thus when n is even moderately large, pure allogeriic matings are so rare that they 

 have vanishingly small influence in the population at large. Even if n were 2, the 

 chance of a pure allogenic mating is only -j-^. These points must be borne in mind 

 in what follows. 



Consider first a father and a mother of one couplet each, their zygotes are either u, 

 i' or w, involving a gametic constitution , -f- > + A or A -j- A. We have the 

 following scheme : 



Zygotc of fathe 



II 

 IV 



II- 



V 



Zygotu of mother. Number of mating. 



2 (!'+') 

 2 (' + /') 



Hence it there be 



1 allogenic couplet in lather and I in mother, offspring 



4", 



4(f + 2") 

 = 4(2 + u) 

 = 4n + 4r 



1 .......... i) 



.......... I 



.......... i) 



Let us write 



l(i/ = 1G ($n + o + /r), 



Then consider tlie relation 



( M o e u + n / e .) x (,,/y + ,,/ IT? I) = 4 o ( yo iy + y a ^i^, + ^^^ +|/ ., e y), 



where e and r; are mere symbols, and 0, 1, etc., denote their powers. , u' refer 

 respectively to father and mother, and their powers denote the number of allogenic 

 couplets in the zygotes of father and mother. Then the above is a symbolical relation 

 which gives, by equating any power or product of e and r) on either side, the offspring 

 of a pair of parents of definite constitution. 



Now suppose the parents not to consist of a single omplet, but of n couplets, then 

 the total distribution of offspring that we have given above for any couplet may occur 



