DIVERGENT EVOLUTION THROUGH SEGREGATION. 321 
TABLE Il.—Preliminary Formula for showing the Proportion of Half-breeds to Pure- 
breeds. 
* Let R=the ratio of pure breeding, i. ¢., the segregation. 
Let ¢ =the ratio of cross-breeding. 
Ex. When ,*, of the unions are within the limits of the species and 41, of the unions 
are with an allied species R=), c= ')._ R will always equal 1—e. 
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, 
Number of individuals representing the pure form.) Number of individuals representing the half-breeds. 
A =Initial number. 
A(RM) SU PONOLAMLON setter cis neem cine ea Ist generation—= Aem. 
A(RM)? =z2nd' ceneration..:---.-.---..-.-.- --) 2nd generation= (AemR+A(RM)ce—Acme) x m.* 
A(RM)* Sani lean ean ise oembacEmeneemcabe 2nd generation = (AcmR— Acme)m-|+Acm(RM). 
A(RM)# ——AGh PEHOLAWON=- 2-2 <- =< 6/2 = as | 2nd generation— Acm(R—c)m--Aem(RM). 
Substituting (1—c) for R in the 2nd gen., we have 
Substituting in this (1—ec) for R, we have 
A(M— Me)?=2nd generation. 2nd generation = Acm(1— 2c)m+Acm(M—Mo). 
*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 first generation 
that formed mixed unions; of these Aeme form unions with an equal number of half-breeds, and their 
offspring being three-quarter-breeds must be rejected; the remainder, namely A(RM)ce— Aeme, form 
unions with the other race, and their offspring are half-breeds of the second generation. 
TABLE III.—Developed Formula for Segregation and Segregate Fecundity, giving the pro- 
portion of Half-Breeds to Puwre-breeds. 
Half-breeds. 
Pure-breeds. 
A == Initial number. 
A(M— Me) =1st generation. 
A(M--Mc)?=2nd generation .| 
A(M—Mce)3=3rd generation - 
| Ist generation=Ame. 
2nd generation =A me(1—2¢)im4-Acm(M — Me). 
3rd generation—A me((1—2e)m)?-++ Aem(M — Me) (1— 2e)m-+ 
| Aem(M—Me)?. 
4th generation =A me((1 —2c)m)* 4 Acim(M — Me) ((1—2e)m)?4 
} Aem(M —Mce)?(1—2e)in+Acem(M — Me)3. 
((1—2¢)m)? | ((1—2e)m)? 
M—Mes + ( 
A(M—Me)*= 4th generation . 
(= 2eym 
(M— Me) 
veneration —/ e(M— Me)3 = 
4th generation —Ame(M Me)*( (M— Me)? + 
(M =) 
(M—Me)3}° 
nth weneration =A me(M—Me)"—!x (( y+ )+( y+( y+ 
(1—2e)m\1 1). 
.M—Me 
First rule.—The pure-breeds of any generation are found by multiplying the pre- 
vious generation of pure-breeds by M—M¢e, and the half-breeds of any generation are 
found by multiplying the previous generation of half-breeds by (1—2c)m and adding 
the previous generation of pure-breeds multiplied by cm, 
H. Mis. 334, pt. 1——21 
A(M—Me)"—nth generation | 
