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PROFESSOR KARL PEARSOX AXD MR, LESLIE BRAMLEY-MOORE 
(xxii.). Apart, however, from the numerical application of these variation formulEe 
to a somewhat doubtful case, we see in these formulse the theoretical basis for the 
observed fact that the fecundity of mothers is far less variable than that of daughters. 
It is really only an apparent divergence, due to the fact that the mothers have been 
v/eighted wuth their fertility ; this, while it increases the apparent mean of their 
fecundity, reduces its apparent variability, 
(17.) On the Inheritance of Fecundity in the Brood-rnare through the Male Line. 
For the thoroughbred horse this problem is fairly easily answered by investigating 
whether mares related to the same stallion have any correlation between their 
fecundities. The two cases I have selected are : (i.) “ Sisters,” daughtei’s of the same 
sire, but in general not of the same mare ; and then (ii.) “Nieces” and “Aunts,” or 
daughters of a sire and the daughters of his sire. As we have only 760 sires and 
nearly 5000 mares, the daughters or aunts fall into rather large arrays, and we are 
compelled to use the methods discussed in Proposition lY., A and B. Even so the 
arithmetical work for a correlation based on the index of sires was far more laborious 
than for one based on the index of mares. 
(i.) To find the Correlation between Hcdfi-Sisters, Daughters of the same Sire^ 
Here we have to use formulae (xxiii.), (xxv.), and (xxvi.) of pp, 272-273, In order 
to do this a table was formed of the mean fecundity M of the array of sisters due to 
each sire, and of ^n{n — 1), the number of pairs of sisters in each array. Then the 
products hn (n — 1) M and ^ n {n — 1)M“ were formed, and the numerator of (xxiii.), 
or al, calculated by adding up for all the 760 sires. The result gave : 
al = -6655167, 
v/here the unit is the fecundity group element of 1/15. The number of pairs of 
sisters dealt with was 54,305. The denominator cro(l — p') + crl is not so easily 
ascertained, o-q is the standard deviation of all the series of mares who are sisters 
without weighting; o-qs/{\ — p") is the standard deviation of an array of sisters, or 
if the regression be not linear, the mean of such standard deviations for all arrays, 
or rather its square is the mean of the squares of such standard deviations ; p is the 
correlation between a patent character in the daughter and a purely latent character 
in the sire, and cannot therefore he found directly. 
In order to get an appreciation of the standard deviation of an array of sisters— 
it being practically impossible to work out these quantities for 760 arrays—-I selected 
twenty sires having fairly large arrays of daughters, and reached the following- 
results : 
