MATHEMATICAL COIsTRIBUTIOHS TO THE THEORY OF EVOLUTION. 
305 
mares, not identical with the series in (ii.)'“ and working ont their dams’ records most 
carefully, rejecting any cases in which the breeder was clearly sending the mare to 
the stallion long after it was obvious {'post facto) that she was sterile. In this case 
four coverings wmre retained as a minimum, and the results are given in Table XIA^. 
We find : 
= -OSGO, M,, = -6616, 
cr,„ = 'ISSS, a-a ■= T604, 
^rad = '0995. 
The coefficient of regression = ‘llGO. 
The probable error of the correlation is '0193, and of the regression ’0194 ; the 
correlation is accordingly more than five, and the regression more than six times Its 
probable error. We conclude that fecundity is most certainly inherited. The 
regression found is, however, only about two-fifths of what is recpiired by the law of 
ancestral heredity, 
(v.) It has been suggested that fertility or fecundity might alternate in tivo 
generations ; when the offspring are numerous their offspring might have less fertile 
or fecund offspring. I do not see how this wmuld be possible Muthout its exercising 
an influence on the correlation of two generations, for we must come to one fertile 
followed by an infertile generation. But I had made preparations in my alphabet of 
mares for testing the correlation between mares and their granddams, and I went on 
to the construction of a table, although the results for mares and their dams showed 
me that whatever result might be reached, it would be within the probable error of 
the observations. I reached this conclusion in the followino- manner : If we o’o back 
^ o 
one generation we introduce, owing to the nature of the record, so much fictitious 
correlation and so much in-and-in breeding that the coefficient of inheritance is 
reduced to two-fifths or less of what its value should be accordino’ to the law of 
o 
ancestral heredity. In going back two generations we come to fewer mai'es, to more 
in-and-in breeding, and to just the type of famous old mare, whose breeder kept her 
at the stud long after she was sterile. I expected accordingly a great and artificial 
fall in the fecundity of granddams and a double drop, something like | X |, in the 
value of the regression as indicated by the law of ancestral heredity. This would 
reduce the apparent regression to about f X f of T5, or to about ’025, say, a value 
about equal to the probable error of the table. The results actually reached are given 
in Table XV., and we find, if the subscript g refer to granddam : 
M„j = '0345, 
o-,„= -2040, 
> mg 
The coefficient of regression = ‘0204. 
* In the first series the mares’ names ran from A to G; in the second from G to M, with 300 
additions made to the A to G series, while I was completing my alphabet. 
VOL. CXCII.—A. 2 R 
M„ = -6232, 
o-y — ’1687, 
= -0169. 
