266 0)1 the Ej^ect of a Di^ereutial Fertility on Degeneracy 
It is clear therefore that the fertility rises to a maximum at -SlSlo-a; and then 
falls again, so that the lowest values of the character have a lowered fertility, but 
are still in excess of the corresponding high values. Thus m^, the mean of the 
effective parentages, is given by 
= ka^^l{a," + a^') = •2169o-^, 
measured below the old mean, and 
2i = standard deviation of effective parentages 
or, tlie effective parentages have 16 per cent, less variability than individual 
parentages, i.e. parentages unweighted with their offspring. We can now find 
the average fertility of the lowest decile of the population : 
Nf ^^-rrJ n 1, 
Nf ^/27^J a X 
V2 
27ri n' 
1-2434. 
where q' = q/'^^ = ^ 
_ 1-2816 --2169 
■8563 
Hence A = 1 A = -10686, 
Nf 10 5 
or, = 5-34 ; 
that is to say, that while the average value of the uppermost decile's fertility is 
only 2, that of the lowermost decile is 5-34, i.e. is still in excess of the average 
fertility, 5, of the population. Indeed for only about 6 per cent, of the population 
with the very lowest values of the character does the fertility fall below some 
4 per cent. 
If we take N as before for the number of pairs of parents, then the distribution 
of effective parentages is given by 
1 - ^ AT--2169.r, \2 
w = -j= e 2V -soeso-^ ) . 
V27r -SoeSo-^ 
We have now to consider the distribution of offspring from these effective 
parentages. We have to consider p of p. 260 in relation to and a^. Four 
hypotheses are possible : 
(i) We may suppose that the character is equally influential in the case 
of the fertility of both sexes ; thus brain capacity and resulting intellectual energy 
might mark equal reduction of fertility in both sexes. In this case the proper 
value for a; is* 
Here as later it must be borne in mind that a constant factor is of no importance. 
