260 On the Effect of a Differential Fertilitf/ on Degeneracy 
k equal that value, and take o-q small). It appears therefore that the distribution 
indicated, if not ideal, will still serve as a first approximation to the type of cases 
likely to occur. 
There is, however, an argument of still greater weight for the suggestion that 
a fertility curve of this kind cannot be widely removed from the actual state of the 
case; namely, when we measure almost any human character (i) in parents and 
(ii) in their offspring we find both distributions very approximately normal. This 
could not possibly be the case unless the fertility curve were itself approximately 
normal. We include of course under this the case in which is indefinitely great, 
or the fertility is not associated with the character in question, which is probably 
what we may describe as the organic condition of affairs. Of course in actual 
working we have to regard many special features. For example, in the matter of 
intelligence, it is fairly obvious that the abler men and women have fewer children ; 
but is it true that the fertility falls again towards the extremely dull ? This is a 
point which needs careful consideration ; I think it probable, especially when we deal 
with net and not with gross fertility. Extreme mental defect leads to neglect of 
offspring, to life in institutions and limited marriage. For example, pauper 
fertility is higher than criminal fertility. I shall illustrate later this point. 
(4) Let us now turn to the direct problem. ■ A parentage has the deviation x 
of the character and is the variability of this deviation ; the deviation in the 
offspring is z with a variability cr^ and a correlation p. Then, if there were no 
weighting with a differential fertility, the frequency surface would be 
u = u^e ^^-P'Vx^ ^.<^e <V (iv) 
in the usual way. But we must multiply this by the average fertility of each 
grade to obtain the frequency distribution. Tiiis will be 
v = Vi,e '^u" xe ^i-zj-V^'^ o-^o-^ a;-) (^^-^ 
1 1 j 2/ 1 •2xk[\-pi) ^2pxz_ z^i 
= v^e ~ '-^ "^"'•^ X e ~ ^ r Vx' ^ " <^o^ <'xo; ^ <^;4 . . .(v) bis 
_ 1 _ 2 {x -mi) {z-m.,) R (^-m,)a| 
= v:xe 21-iJM Si^ ^ f (y-^^ 
where V(,' is a fresh constant, i.e. independent of x and z, and and are the 
means of effective parentages (i.e. weighted parentages) and of actual sons, 2i and 
Sa their standard deviations and R their correlation. We have by finding' the 
centre of the elliptic contour of (v) bis in the usual way : 
vii = kax^o'o + o-x'} (vii), 
in., = vi^pa^jcr^ (viii). 
The first of these equations shows the shifting of the parental mean, the change 
being due to differential fertility ; the second equation shows that the shift of the 
