MATHEMATICAL CONTRIBUTIONS TO HE THEORY OF EVOLUTION. 307 
reproductive selection changes these quantities to *6337 in the case of the daughters, 
and to the apparent high fecundity of '6525 in the case of the dams. 
We can now find cr, to a second approximation by aid of (xxi.). In the small term 
multiplied by r, we put cti = 0-2 = cr",. Hence we find 
2 _ '/2 I ^.2 / '/2 f'2\ 
0-2 = O- 2 + r (o- 2 — o- 1), 
and deduce, on substituting the numerical values. 
a-, = '1896, 
or is scarcely different from a" 2 . We accordingly conclude that we may quite 
reasonably assume the variability of the mares to represent the variability of the 
mares without reproductive selection, but the effect of weighting the dams with 
their fertility is to reduce the variability of the dams from about '1896, if there be 
no secular change, to an apparent value as low as '1643. 
The same formulae applied to the slightly better results in (iv.) on p. 305 give us : 
M'l = -6205, and Mb = '6342. 
If we pass back from Mb and Mb to Mj and M 2 we find: 
First case. 
Second case. 
Ml . 
‘5460 
•5567 
M 2 . . . 
•6266 
•6278 
If these results be considered as valid, we notice a remarkable difference between 
the fecundity of the younger and elder generation. While the crude results on 
pp. 304 and 305 might lead us on first examination to suppose the elder generation 
more fecund than the younger, these results show us that it is distinctly less so. 
The greater part of the difference, however, is due, not to a secular change, but to 
the causes we have so often referred to as weakening the fecundity recorded for the 
older mares. At the same time the whole system of breeding is so artificial that we 
may well doubt whether our equations (i.) and (v.) can be legitimately applied. For 
the chance of a mare getting into the stud-book as a dam, i.e., having daughters at 
the stud, depends less on her fertility than on the degree of fashion in her stock. 
Thus the record weighting with fertility is hardly a probable hypothesis, and the 
values just given for Mi are, I suspect, much below what they should be. For the 
above reason 1 have not proceeded to consider the changes in variability connoted by 
(ii.) and (xxii.). As I have made no attempt to form a correlation table for mares 
and dams in which the dam would have only one daughter to her record, I cannot 
make any plausible guess at the real magnitude of the cubic summation term in 
2 R 2 
