MATHEMATICAL COHTETBUTIOHS TO THE THEORY OF EVOLUTION. 
308 
Variation in Fecundity of 2000 Brood-mares (Eight Coverings), 
Fecundity 
1 
a. 
h. 
c. 
</. 
e. 
/• 
'J- 
h. ' L 
i- 
1. m. 
n. 
i q- 
1 
Frequency 
0 
2 
7'5 
11-5 
21'5 
55 
104-5 
182 : 271-5 
1 
315 1 337 
293-5 204 
127 
49 , 19 
I 
i 
Total, 2000. 
M,^ = 
= -6330. 
= -1568, 
Thus, making the minimum number of coverings 8 instead of 4, has removed 
the terminal humps, zero fecundity is now unknown, and perfect fecundity very rare. 
We have reached a smooth skew frequency distribution ; we see fecundity as a con¬ 
tinuous character obeying the usual laws of variation.* The mean fecundity in the 
two cases is sensibly the same, '633, but owing to the fact that we have made a 
selection of a limited group in the second case, the variability is considerably 
decreased. 
The average apparent fertility of brood-mares, 6'515, must not be confused with 
their average real fertility, for, as we have seen, we have in many cases not a com¬ 
plete record of their stud-life, or such a full record has not been used {e.g., in case of 
mares still at the stud, but having been already covered four or more times). Its 50 
per cent, variation shows that an apparent fertility of 9 to 12 is not infrequent. 
The average number of coverings being 10 and more, it will be seen that the records 
of between 50,000 and 60,000 coverings have been dealt with to form our mare and 
sire alphabets. The large variability in the number of coverings shows that 15 
to 20 coverings will not be infrequent, and cases of 26 actually occurred. Lastly, 
we have the correlation between fertility and the number of coverings, high as 
might be supposed, for a high apparent fertility could only be exhibited by many 
coverings. Although a low apparent fertility might correspond to any number of 
coverings, still, in practice a sterile mare will not be sent indefinitely to the sire. The 
correlation between the number of coverings and the fecundity is small and negative 
(— *0572). This follows from the principle that, fertility being the same, a high 
number of coverings reduces the fecundity, and this factor is more potent than the 
high correlation of fertility and the number of coverings. 
(ii.) Table XTI. exhibits the correlation of 1200 mares and their dams with regard 
to fecundity. Here the more fertile dams are weighted with their fertility, and at 
least four coverings were required of each mare. If the subscript m refers to mare, 
and d to dam, we find : 
* The actual equation to the curve referred to the mode '6.531 as origin, the axis of x being positive 
towards perfect fecundity, and the unit of x being 1/15 is: 
y = . 342'187 (1 + a’/47'1358)«2«2Ci (l _ .r/12'1106)2i-229i. 
The fit will be found to be very .satisfactory. 
