270 
PROFESSOR KARL PEARSON, MATHEMATICAL 
S (C) vanish. Thus all the values given for M'l, M'2, (r\, and cr'2 remain the same, if 
their results be interpreted in the sense of fecundity and not fertility. If p he the 
correlation between fecundity and fertility, and cti, 0-3 the standard deviations of 
these quantities, then Xj = po'zjcri ; but we have seen that it is also the ratio of mean 
fertility to mean fecundity. It follows accordingly that p is the ratio of the coefficient 
of variation in fecundity to the coefficient of variation in fertility. If we may judge 
by the cases of man and horse, so far as I know the only cases in which fertility and 
fecundity have yet been examined, a coefficient of variation in fecundity amounts to 
about 30 per cent., while one in fertility is something like 50 per cent. Thus the 
correlation of fertility with fecundity would be about '6. We should expect it to 
have a high value, perhaps even a higher value than this. In the case of thorough¬ 
bred horses, p will be the correlation between fecundity and apparent fertility. By 
direct investigation in the case of 1000 brood mares I find its value to be ‘ 5152 . 
Passing now to the correlations r, r, r", I observe that the proof given for fertility 
is valid with but few modifications, if these be fecundity correlations (see p. 266 ), for 
the proof involves no expansion of the factor {X^x -f- 0 ^- Hence we conclude that the 
regression coefficient for the inheritance of fecundity will not be modified by the 
nature of the record or the weighting of individuals with their fertility. 
When we come to the last series of constants, M"i, M"2, o-"i, or''25 we find that these 
will be modified, owing to the presence of the square factor (XjX + although 
{ is not correlated with x. The term now comes in, and S (^“) will give the 
standard-deviation of an array of fertilities corresponding to a given fecundity, i.e., 
S ({“) = 0-3(1 — p~) X number in the array. 
I find after some reductions that M'b and M'j are given by 
M'b = 
Af I .. °'iy2 / 1 I _ ^ 
• • 
(xviii.). 
^ ^ 1 + 
(xix.), 
the correlation of fertility and fecundity being now introduced into the results. 
Clearly the result (xvii.) 
M", — M', 
^- T-- = coefficient of regression.(xx.) 
M 1 — jM , ^ ’ 
still remains true. 
For the remaining two constants a".3 and cr"i, I find, after some rather long 
analysis in the second case, which it seems unnecessary to reproduce,! 
* Sliould the regressiun not be linear, {\. — /<-) is the mean of the standard-deviations of the arrays, 
t In the course of the work the squared standard-deviation of a fertility array is assumed to be the 
same foi' all arrays o-] (I — p"), and Xj is given its value rffs/cri. See, however, the previous footnote. 
