262 On the Effect of a Differential Fertility on Degeneracy 
(5) We may next inquire into the mean fertility and variability of fertility 
in the population. 
The fertility of the parentage x is 
y = y.e ^ (in), 
and the frequency of such parentages is given by 
V27r(7^ 
where N is the total number of such parentages. 
Hence, if / = the mean fertility : 
accordingly : 
dx. 
f=~JML^ e 2<ro2 + <r/ (xii). 
This is one equation linking up the indeterminate quantities y^, o-q, k with 
observable quantities like f the mean size of family in the population. We can 
now find the standard deviation So of the size of families. Clearly : 
2,r = Sum(y=)/i\r_/, 
V27ro-3, 
1 A:2 
before. Hence : 
IfaJ 
Vf = = """JL^L e + + _ J 
Vf is an observable quantity and this equation may be used to find h if has 
been otherwise determined. The previous equation will then give y^. We 
should have to go to the third moment coefficient to determine y^, o-j and k 
from the fertility distribution, and it may be doubted whether the result would 
be of much value ; it seems better to endeavour to determine and k from 
observing the average fertilities of those exceeding certain grades of the character. 
(6) Determination of the constants of the fertility distribution from the 
average fertilities of those possessing more than a certain grade of the character. 
