178 APPENDIX I — DIVERGENT EVOLUTION. 



Tabic II is a preliminary formula for showing the proportion of 

 half breeds to pure-breeds. 



Let R = 1 — 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. 1. ./v will always 

 equal 1 — c. 



Let .1/ = 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 I , T . , ....... . , , „ 



the nure form umber of individuals representing the half-breeds. 



A = Initial number. 



A(RM) = 1st generation. 

 A(RM)" = 2d generation. 

 A(RM) 3 — 3d generation. 



1st generation = Acm. 



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

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

 A (RM) * = 4th generation. I 2d generation — Acm(R — c)m + Acm(RM). 



Substituting (1 — c) for R in the 2d . Substituting in this (1 — c) for R, we have 



generation, we have A(M — ! 2d generation = Acm(l — 2c)m + Acm(AI — Ale 



■J- 



Me) 2 — 2d generation. 



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 1st 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. 



