CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
271 
U') 
(7 2 
0 -^ 1 + r- 
i a 
(xxi.), 
//o 
cr I 
7q~T \ 
w\) 
+ 7 
1 - 
L 
S [(x - 
N'lM'i 
(xxii.), 
and y is the factor 
Ml - 1 ^ 5 ? 
Ml + 
or as we can write it 
If p be unity or near unity, i.e., fecundity very closely correlated with fertility, 
y = 1, the second term vanishes and (xxii.) becomes identical with the corresponding 
fertility formula (xvi.), just as (xxi.) is already identical with (xv.). 
Thus we see that the whole series of fecundity relations are strikingly like those 
for fertility, except that in certain of them—those for IVT'i, M'h, cr", and cr'k—the 
correlation p of fertility and fecundity is introduced. If p be considerable, all the 
remarks we have made on the fertility formulae may, mutatis mutandis, be applied to 
the measurement of fecundity. 
(5.) Proposition IV.—To deduce formula for finding the correlation between any 
grades of kindred from the means of arrays into ivhich the kindred may he grouped. 
This problem is of very great practical importance. In the case of Man, families 
are so small that there is comparatively small difficulty in forming all the possible 
pairs of brethren, say, for any family ; but when we come to animals or insects where 
the fertility may be extremely large, it is practically impossible to form a correlation 
table involving 50,000 to 100,000 entries."^ One thoroughbred sire may have 50 to 80 
daughters, and thus give us roughly 1200 to 3200 pairs of sisters to be entered in 
a correlation table. Still higher results occur in the case of aunts and nieces. It 
may be asked why we do not content ourselves with one or two pairs from each 
parent; the answer is simple : we have not {e.g., in the case of thoroughbred animals, 
pedigree moths, &c.) a great number of sires, and the sire with 50 offspring cannot, 
for accuracy of result, be put on the same footing as the sire with only 2 to 4. Our 
process is really an indirect weighting of our results. 
(A.) To find the coefficient of correlation hetiveen brethren from the means of 
the arrays. 
Let X be the measure of any character or organ in one brother (sister), and x that 
of a second brother (sister) ; let m be the mean of one set of brothers, and ni of the 
* Even with the reduction in labour, introduced by this proposition and by the use of mechanical 
calculators, Mr. Leslie Bkamley-Moore and I took practically a week, of eight-hour days, to deduce 
two coefficients of correlation, after the means of the arrays had already been found. 
