40 
PROFESSOR K. PEARSON ON THE INFLUENCE OF NATURAL 
in correlation produced by the stringent selection of one parent, but we must 
remember the complexity of the factors—the variety of other influences at work in 
selecting and modifying selection—before we lay much stress on this source of 
altei-ation in parental correlation. 
Now let us deal with the case of both parents selected, and suppose their selection 
given l)y sjcr^ = /Xo, and the change of their coefficient of assortative 
mating from to We have from formnlse (Ixviii.) and (Ixx.) by a little 
rearranging 
—3^ — '^.3^ {1 (l /^i”) p/) 2 (^’la ^ 13 /^ 23 } 
^ 3^13 “ ^3 [^'13 Pi) /^13 ~ (^’12 PllP-l) .(Ixxii.), 
where yS^g = (^'^g ~ / (1 — /3og = (v’gg — / (1 — Now let us take 
special cases to bring out points. Let us suppose r^o = oi’ assortative mating to 
exist, and let us enquire what change would be made in j^arental correlation if we 
selected parents who had assortatively mated, without altering their variability, he., 
let us take 1 x 1 = 1, p., = 1, Ave have at once 
V’ = (1 + ^di 2 ’‘i 3 ^‘ 23 ).(Ixxiii.), 
— 3^13 ~ ^3 ('^’13 fl“ PllP'is) .(Ixxiv. ), 
AA'hence we find for r^g = = ’4 and 
P 12 — ’L 
%s/o-s= 1-0U9, 
1 jg — 4335, 
P 12 = 
= 1-0315, 
= -4653, 
P 12 — '3’ 
— 1-0469, 
= -4967, 
Pl-2 = 
= 1-0621, 
= •5271, 
P 12 ^ 
il 
1 —' 
0 
0 
= - 5571 , 
P 12 = L 
= 1-1489, 
= -6963. 
Thus the general effect of assortative pairing of j^arents is to increase the correlation 
between parent and offspring sensibly, hut not to very rapidly increase the variability 
of the offspring. Thus marriages within a class Avonld, if heredity statistics were 
collected for a class, tend to shoAA^ increased parental correlation. Very high assorta¬ 
tive mating no doubt occurs Avith some forms of breeding, and we may Avell find in 
such cases higher values of the parental heredity than Ave should obtain for a popula¬ 
tion of the same species Avith random mating. I think this may he an effective factor 
in the raising of the parental correlation in the case of coat-colour in thoroughbred 
horses. 
