262 
PROFESSOR KARL PEARSON, MATHE:^rATlCAL 
(6.) fecundity of an individual shall be defined as tlie ratio of the total number 
of actual offspring to the total number of offspring which might have come into 
existence under the circumstances. 
These definitions ai’e not intended to give ^^recise statistical measures at this stage 
of our investigations. They are merely meant to convey a general sense of the 
words, which will be more precisely limited when they are applied to any given species. 
Fertility and fecundity, as we have thus defined them, leave out of account individual 
conditions and definite conditions of period, age and environment, which must be 
fully stated before numerical measures can be made in any special case. When the 
words are used in this theoretical section the reader mu.st suppose the phrase, “under 
definite individual and environmental conditions,” to be always inserted. 
Let Ml the mean fertility of parents of one sex ; jM'i = the mean fertility of 
parents of one sex weighted with their fertility Ni the number of parents con¬ 
sidered in the first case, ISfi, the apparent number dealt with in the second case; 
let cTi and mi be the standard deviations in the two cases, and let x represent the 
fertility of an individual parent and 2 its frequency among Ni parents. Let S 
denote summation for Ni parents. Then, without any assumption os to the type of 
frequency, Nj = S {kxz) = XMiNj, where \ is a constant such that \x is the weight 
of a parent of fertility x. This follows at once, since : 
Ni = S ( 2 ), M, = S ixz)/^ (z). 
Further, 
M'. = S (X.r X 
_ S {(a; - Mfz + 2Mi (xz) - 
MiNi 
Nicri + 2MfNi - MfNi 
MiNi 
by the definition of standard-deviation. Hence, finally : 
Further : 
0-1 = 
M'. = ^ ■ + M. 
S {\cc {x - U\fz} S {{x - Ml + M,) {x - Ml + - M'd's} 
(i.). 
N'l 
MiNi 
Hence, multiplying out, we find after some reductions : 
/2 
O- 1 
+ 
S {(x - Mi)s z] 
MiNi 
(ii.). 
At first sight it might seem a comparatively easy matter to avoid weighting parents 
with their fertility, but practically it is almost impossible. For example, if records 
* i.e., if / be tbe fertility of a parent, each parent is repeated A/times, where A is a constant. 
