Karl Pearson 
259 
(2) We will suppose that some character — e.g. grade of intelligence — influences 
the fertility. We have then to consider whether this character in the male and 
female influences equally or unequally the fertility of the pair. It has been 
asserted by some that academic training in the woman lessens her power of child- 
bearing. In this case the more intelligent women would have fewer offspring, 
but owing to assortative mating such women marry the more intellectual men, and 
whether intelligence was or was not associated with a lesser grade of fertility in 
the male, assortative mating would handicap the fertility of the intelligent male, 
and the more intelligent males would be practically less fertile. 
Let ^1 and be the deviations from their respective sex means of male and 
female for any character, cti and 0-3 the standard deviations with respect to this 
character in the two sexes, represented by the subscripts 1 and 2. Let ?'io be the 
correlation of |i and fa, i.e. the intensity of assortative mating. 
Then it is reasonable to assume that the fertility of the pair is some function 
of X, where « is a linear function of ^1 and or : 
a; = Ci^' + c, (i). 
0"i (To 
The mean of 00 is clearly zero, and we can free ourselves from the influence of 
either parent by putting Ci or Cj zero. If fertility were related to « by a simple 
linear correlation then we should obtain, it is well known, the highest correlation 
of fertility (y) and x by taking (the subscript 3 denoting fertility) : 
or proportional to these quantities. We need not at present however assume any 
special values for Cj and c^. Further, it is desirable to suppose that the curve of 
mean fertility for each value of x is not necessarily linear, but of a more general 
type, allowing us to make fertility a maximum at other grades than the extreme 
values of the character. 
(3) This leads us to the next point. What law of fertility seems reasonable ? 
We want a law of fertility which will allow closely of the fertility (a) increasing 
nearly uniformly and at any given rate with the character, (6) decreasing nearly 
uniformly and at any given rate with the character, and (c) being concentrated 
with any degree of intensity round any grade of the character, and falling away 
on both sides of this maximum. All these conditions are fulfilled if we assume 
the mean fertility y for any grade of the character x, not to be given by a line 
but by a normal curve, e.g. 

Here by a proper choice of o-q and h, we can make the distribution of fertility 
fall or rise with the character (i.e. h positive or negative and both h and o-q very 
large, — h so taken that the centre of fertility lies outside the region of the range 
of values of x), or concentrate the fertility on any grade of the character (i.e. make 
