178 APPENDIX I—DIVERGENT EVOLUTION. 
Table II is a preliminary formula for showing the proportion of 
half-breeds to pure-breeds. 
Let R = 1 —c = the ratio of pure breeding, 7. ¢., the segregation. 
Let c = the ratio of cross-breeding, 7. 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 =0.1. R will always 
equal 1 —c. 
Let / = 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) = Ist generation. | Ist generation = Acm. 
A(RM)? = 2d generation. | 2d generation = (AcmR + A(RM)c — Acmc) X m. 
A(RM)* = 3d generation. | 2d generation = (AcmR — Acmc)m + Acm(RM). 
A(RM)* = 4th generation. | 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 — | 24 generation = Acm(1 — 2c)m + Acm(M — Mc). 
| 
Mc)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(M)c represents the total number 
of pure-breeds of the 1st generation that form mixed unions; of these Acmc 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. 
