MATHEMATICAL COHTRIBUTIOHS TO THE THEORY OF EVOLUTION. 311 
are not “ aunts ” and “ nieces ” in the human sense, for the auuts are only half- 
sisters of the sire. By a process similar to that on pp. 408 and 409 of m 3 - paper 
on the “Law of Ancestral Heredity,”I deduce that the correlation between a sire’s 
sisters and daughters ought to be ‘05, and not *15 as in the case of Man. If this be 
weakened down to the f of previous results, we should not expect a result differing 
much from ' 02 . As the variability of the elder generation is always less than that 
of the younger, we ought to exj^ect a coefficient of regression of about this value. 
The theory used will be that of p. 273 of the theoretical part of this paper. The 
weighted mean fecunditj’ found for the arrays of aunts and nieces was as follows :— 
Without grouping. 
With grouping. 
AiTays of aunts. 
Arrays of nieces .... 
•6195 
•6346 
•6199 
•6338 
The grouping was done in fecundity units of i.e., 1/30 change in fecundity. The 
agreement may accordingly be considered very good. The “ aunts ” are the daughters 
of the older sires, who owing to in-and-in breeding form a comparatively small group, 
and are the sires of mares belonging to the older period, whose fecundity is much 
weakened by causes already referred to. Their mean fecundity is slightly less than 
that of granddams, given on p. 305, while the mean fecundity of their nieces agrees 
well with that for the corresponding group of mares. 
The method of grouping being adopted, a correlation table was formed for the 
mean fecundities of arrays of mares, daughters of a sire, and of arra}-s of mares, 
daughters of his sire. This is Table XVI. Here each mean is weighted with the 
number of pairs of aunts and nieces in the two arrays, i.e., the extent of the data on 
which it is based. It represents accordingly 138,424 pairs of aunts and nieces. 
The following results were obtained, corresponding to 687 pairs of sires :— 
Sire’s Sire. Sire. 
M„=-6199. M,, = -6338. 
cr„ = ‘04344. cr„, = ‘07609. 
E = ‘1174. 
It will be at once noticed how much more variable are the array-means for the 
sire than for the sire’s sire. The means of many of the sire’s arrays are based u|)on 
small numbers, which would have been selected out, if we had gone to another 
generation as in the case of the sire’s sire. 
It will clearly not be legitimate in this case to put cr'„ = cr^ as suggested onp. 274. 
There is probably no secular change of importance here, but the sire’s sire requiring 
* ‘ Roy. Soc. Proc.,’ vol. 62. 
