30G PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
If q > 3 and < p we have 
IR, = 
V 
1 
0 
r 
V 
0 
0 
r~ 
r 3 
0 
7 ” 
0 
0 
0 
0 . . . 
1 
0 
1 
0 
0 
V 
r 
0 
0 
0 
0 
q 
r* 
q 
r* 
q 
7 ” 
q 
r* . . . 
o 
r - 1 
r 
0 
1 
0 
0 
0 
r 
r 
0 
0 
0 
0 
0 
0 . . . 
0 
V 
V 
0 
0 
1 
0 
0 
0 
0 
r 
r 
0 
0 
0 
0 . . . 
Q 
7 W 
0 
r 
0 
0 
1 
0 
0 
0 
0 
0 
r 
f 
0 
0 . . . 
o 
V 
0 
r 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
r 
r . . . 
r 3 
r 3 
0 
r 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 . . . 
' 
r‘ 
q 
V 
0 
r 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 . . . 
Whence, adding the s 
econd and 
third 
columns 
multiplying by ; 
. . . . 
and subtracting 
from 
the first, v 
r e ha 
ve 
= 0. 
Lastly, R 12 = 
= r 
R,i 
and R 13 
— 
rb 
Tl 1 
for R, 12 
is 
of the form 
of 
Ri ? 
above 
wifchou 
t the 
second column. Divide the first column by r, and subtract the second column (the 
third of R K/ above) and it becomes the first of R n , the remainder is identical in R 12 
and R n . Hence, R 12 = rR u . Similarly we find R 13 . Thus we conclude that 
i = v " 1 h, + 
- h,. 
a 1 = o-f (1 
the formulae for biparental inheritance with equal parental correlation, and no assorta- 
tive mating. The analysis for unequal parental correlation and assortative mating 
follows the same lines, is far more lengthy, but leads to the same result, i.e., no gain 
by selection of the same amount, oft repeated. 
(d.) Secular Natural Selection and Steady Selection. Focus of Regression. —We 
thus see that, on the theory with which we are concerned, a knowledge of the ancestry 
beyond the parents in no way alters our judgment as to the size of organ or degree of 
characteristic probable in the offspring, nor its variability.* An exceptional father 
is as likely to have exceptional children if he comes of a mediocre stock as if he comes 
of an exceptional stock. The value of £ will be no greater nor the value of less if 
the parents have been selected for p generations than if they have been selected for 
one only. This result seems to me somewhat surprising, but I cannot see how it is 
to be escaped so long as we assume the normal distribution of frequency, which 
appears in so many cases to be a close approximation to fact. It is of course possible 
* This seems specially noteworthy; it would seem natural to suppose that the offspring of a long 
selected stock would be less variable than those of one just started—that the offspring of a gradually 
created variety would be more stable than those so to speak of a sport. It appears not. 
