178 



APPENDIX I — DIVERGENT EVOLUTION. 



Table II is a preliminary formula for showing the proportion of 

 half-breeds to pure-breeds. 



Let i? = 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 Umits of the species and one- 

 tenth of the unions are with an allied species i? = 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 (RM) ' = 2d generation. 



A (RM) ^ = 3d generation. 



AiRM)"^ = 4th generation. 

 Substituting (1 — c) for R in the 2d 

 generation, we have A (M — 

 Mc) ^ = 2d generation. 



1st generation = Ac7n. 



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

 2d generation = (AcmR — Ac-inc)m + Acw(RM). 

 2d generation — Acm(R — c)m -t- Acm(RM). 



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

 2d generation = Acm(l — 2c)m + Acm(M — Mc). 



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 i st, 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{RR'[)c — Acme, form 

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



