7 8 



APPENDIX I DIVERGENT EVOLUTION. 



Table II is a preliminary formula for showing the proportion of 

 half-breeds to pure-breeds. 



Let R = i c = the ratio of pure breeding, i. e., the segregation. 



Let c = the ratio of cross-breeding, i. e., the segregation viewed from the other 

 side. 



Ex. When nine-tenths of the unions are within the limits of the species and one- 

 tenth of the unions are with an allied species R = 0.9, c = o. i. R will always 

 equal i c. 



Let M = the ratio of fertility in each generation for those that breed with their 

 own kind. 



Let m = the ratio of fertility in each generation for the cross-unions and for 

 the hybrids when breeding together. 



Let A = the initial number of individuals representing the pure species when 

 the computation commences. 



TABLE II. 



Number of individuals representing 

 the pure form. 



Number of individuals representing the half-breeds. 



A Initial number. 



A(RM) = 1st generation. 



A(RMY 2d generation. 



A(RM) 3 = 3d generation. 



A(RM)* = 4th generation. 

 Substituting (1 c) for R in the 2d 

 generation, we have A(M 

 Me) 2 2d generation. 



1st generation - A cm. 



2d generation = (AcmR + A(RM)c Acme) X m. 

 2d generation = (AcmR Acmc)m + Acm(RM). 

 2d generation Acm(R c~)m + Acm(RM'). 



Substituting in this (1 c) for R, we have 

 2d generation = Acm(\ 2c)m + Acm(M Me}. 



EXPLANATION OF TABLE II. 



The term AcmR represents the number of half-breeds that form unions among 

 themselves, the offspring being half-breeds; A(RM)c represents the total number 

 of pure-breeds of the ist generation that form mixed unions; of these Acme form 

 unions with an equal number of half-breeds, and their offspring being three- 

 quarter breeds must be rejected; the remainder, namely, A(RM)c Acme, form 

 unions with the other race, and their offspring are half-breeds of the 2d generation. 



