MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 299 
a moment to expect inheritance in the full intensity of the Galtonian law to be 
exhibited by such material. 
(15.) But there is another point of very considerable importance for the weakening 
of correlation, namel}^ the effect of in-and-in breeding’. To get correlation we must 
have a diversity of parents producing a diversity of offspring, but when the parents 
become more and more identical, we get larger and larger arrays between which and 
the parents the correlation is weakened. For example : suppose the correlation found 
between all parents and offspring in the general population, and now select only all 
the brothers in a large array and find the correlation between them and their 
oflspring, we shall find that the correlation is lower than in the ])revious case.'"" It 
would be impossible to apply theory to the present case, however, because we can 
only roughly appreciate the extent of such in-and-in breeding. That it is great the 
following statistics will show. 
Of the more than 1000 sires in my sire alphabet, only 7G0 were sires of mares 
which had been covered at least four times. These 760 sires had upwards of 
5000 offspring, of whom I had the fecundity recorded, but when mares with 
alternative sires were excluded, there remained only 4677 available niares.t These 
mares were distributed as follows :— 
Daughters .... 
I 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
Sires. 
280 
113 
78 
43 
29 
22 
20 
21 
22 
14 
10 
10 
Daughters .... 
13 
14 
15 
16 
17 
18 
19 
20 
Above 20 
Sires . 
11 
11 
8 
0 
2 
6 
4 
4 
46 
Here the second line gives the number of sires having the number of daughters 
in the first line in the 4677 cases, which I take to be a fair sample. 
Thus over a third of the sires had only one mare. Two-thirds of the sires had 
together only one-fifth of the mares. Seventy-six of the sires were fathers of 
about half the mares, and 46 sires alone produced 1801 mares, almost as many as 
642 sires did. We are here dealing with the fairly long period of 30 years, but 
even making due allowance for young stallions commencing and old stallions con¬ 
cluding their stud career, it will be manifest that our sample shows that the great 
bulk of mares for the period in question were the offspring of comparatively few sires. 
But let us look at the problem from the standpoint of the sires. My 760 sires 
* Tlie theory of such cases is fully developed in a memoir on the influence of selection on correlation 
not yet published. 
t Some other cases were also excluded for diverse reasons. 
2 Q 2 
