THROUGH CFMULA.T1VE SE(JREGA.TlOJf. 



251 



crossing with eacli other, but also upon their Positive Segrega- 

 tion. We can further see (when considering an extreme case, like 

 either of the above-supposed cases) that Segregate Fecundity, 

 without the aid of Positive Segregation, must lead to extinction. 

 We have already seen that partial Segregation cannot by itself 

 prevent the fusion of species. It therefore follows that in order 

 to account for the continuance of divergent races we must sup- 

 pose either that the Segregation is complete, or that the divergent 

 evolution is strong enough to more than counterbalance the 

 influence of the occasional crossing, or that the partial Segre- 

 gation is aided by Segregate Pecundity or Segregate Vigour. 



Between the members of species belonging to different orders 

 we find not only complete Segregation, but complete sterility 

 when attempts at crossing are made; but hope of gaining an 

 explanation of how these characteristics have arisen is found, 

 not in the study of those cases in which the process has been 

 completed, but in the study of the relations to each other of 

 species and varieties that are characterized by partial Segrega- 

 tion and mutual sterility, that is not complete. Here, again, 

 mathematical analysis will help us in understanding the subject. 

 Though I have not succeeded in constructing a complete mathe- 

 matical representation of all the grades of intermingling that 

 will take place, I have found a general formula that gives a close 

 approximation to the proportion in w^hich two species will breed 

 pure as contrasted with the proportion of first crosses and their 

 descendants that will be produced, in any case in which the 

 degree of Segregation and the ratios of fertility for the pure and 

 crossed breeds are known. As my object is simply to show 

 under Avhat conditions the pure races will continue without 

 being swamped by crossing, it is not necessary that I should 

 follow the action and reaction between the three-quarter-breeds. 

 I wish, however, to call attention to the fact that when the 

 number of the pure forms and of the half-breeds is constantly 

 decreasing, without a general decrease in the sum of the de- 

 scendants, it is evident that the three-quarter-breeds and tlieir 

 descendants are increasing ; and when a three-quarter-breed on 

 one side crosses with a three-quarter-breed on the other side, the 

 offspring will usually be about intermediate between the two 

 species ; therefore, where the two species are equally numerous, 

 if we find that the pure forms will disappear through fusion, we 



