MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 301 
nmsfc use the formula (i.) for the mean value of an index given in my memoir on 
spurious correlation.’’- We shall then obtain an approximate value to the mean 
number of coverings of each mare. Formulae (hi.) of the same paper will then give 
the standard deviation for the number of coverings. In our present notation : 
and therefore 
We find 
and ; 
iW = 
M, 
1 + 
too 
^ r'th 
VJl\ 
CTr 
- V(im) + (foo) 
_ 2r 
10,000 
. 100 / 
M,/M^ = 10-2196, 
M, = 10-2196 X 1-007 = 10-2911, 
(Tc = 4-4455, 
V. = 43-20. 
To the same degree of approximation we can further ascertain the correlations 
between the number of coverings and the apparent fertility and fecundit}^, i.e., 
and A short investigation similar to those in the memoir on spurious correlation 
just cited shows us that: 
’V = i'^j - 
Th ese lead to the numerical results : 
= *8259, = — -0572. 
The conclusions to be drawn from these results are all of some interest. In the first 
place we may ask : How does agree with its value found from other and more 
complete series? For 4677 mares—my complete series without mares with alternative 
sires—the average fecundity was -6373. A better agreement could not have been 
hoped for. In a group of 1509 mares dealt with for variation only and entered as 
“ daughters ” on the cards—so that they had not been selected by the fact that their 
daughters must have recorded olfspring, as is the case with “ dam ” entries—I found 
the following results 
Variation in Fecundity of 1509 Brood-mares (Four Coverings). 
Eecundity. 
a. 
b. 
c. 
J. 
e. 
/• 
17- 
h. 
li 
j- 
h. 
1. 
1)1. 
11. 
P- 
2- 
Frequency 
9 
3 
11 
26 
46 
43-5 
85 
122-5 
154-5 
232-5 
194 
223 
146 
100 
23 
90 
Total 1509. M.^ = ’6345. = T965. 
* ‘ Rcy. Soc. Proo.,’ vol. GO, p. 492. 
