DIVERGENT EVOLUTION THROUGH SEGREGATION. 323 
the sum of the infinite progression is $2=18. And M—Mce=18, which 
makes the half-breeds=the pure forms xem; and em= 5. Let M=2, 
m=1, c= ;%;; then half-breeds=pure fone Kanes) et M— 2° m—1, 
e=4; then the infinite progression=1, M—Mc=1, and the pure forms 
in each generation will equal A, and the half-breeds A x4. Therefore 
Half-breeds=Pure-breeds x $. 
Let M=3, m=2,c=3; then the sum of the infinite progression=1, 
and the. Half breeds=4 x 2x A(M—Me)n—1, and the Pure-breeds=14 x A 
(M—Me)»—1; therefore Half-breeds=Pure-breeds x 3. 
Let M=3, m=2, c=4; then Half-breeds= Pure breeds x 2. 
Let M=3, m=2, c=4; then Half-breeds= Pure-breeds x 2 
Let M=3, m=2, c=}; then Half-breeds=Pure-breeds x 2. 
Let M=3, m=2, c= 145 y ; then Half-breeds= Pure-breeds x ;2;. 
Let M=3, m=2, c then Half-breeds=Pure-breeds x ;3,. 
ae oe 
TABLE IV.—Simplified Formulas for the Proportions in which Half-breeds and Three- 
quarter-breeds stand to Pure-breeds when all are equally vigorous. 
From Table III we learn that 
eee Ue 1 (1—2c)m ) 
p=M_MeX(t+ Wroate eae 
When (1—2c)m is less than M—Me, the series within the brackets is a decreasing 
geometrical progression, and we may obtain the value of the whole series by the 
a 
formula S=j—¢, Applying this formula, we have 
H___me a eee __M—Me — me = 
P M—Me (1—2¢e) ~ M—Me**M—Mc—m-+2me M—m--(2m—M)e -- (1) 
-\- Me 
SOS TS) a LL a a a RE Nhe fs (2) 
M—m-+-(2m—M)e 
If m’ — the ratio of fertility for the Three-quarter-breeds, then according to the 
reasoning given in Tables VII and VIII, 
fe 2m'e 
Sree opie tat sate: Ue eae (3) 
H M—m’ +-(2m’ =Mye? 
AU Sable 
and = == tS SEE DSS Sess Goes sss och oss 4 
VC Pop xy (4) 
The following solutions, as well as those given in Table V, are ob- 
tained by substituting values for M, m, and ¢ in formula (2): 
When M = 4, m=3, then if 
c= 4, half-breeds = pure-breeds xX #, 
c=}, half-breeds = pure-breeds x 3, 
c= 4, half-breeds = pure-breeds X 2, 
c= ji, half-breeds = pure-breeds xX #, 
e=1, half-breeds = pure-breeds x 2, 
c=}, halt-breeds = pure-breeds X 3, 
c=t, half-breeds = pure-breeds X 4's, 
c=4, half-breeds = pure-breeds xX +';, 
c= 75, half-breeds = pure-breeds xX +%, 
€= rho, half-breeds = pure-breeds X yz 
