PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 303 
correlation coefficient of parent and offspring, and oq, <r 0 their standard-deviations in 
the unselectecl state. Thus we have 
01 
— h 
tt-,, 
o 
If the parent and offspring are of the same sex and there be no reproductive 
selection, cr 0 = o-j, and we have 
s = * = v (i - >oi 2 ) + i-oiV- 
Cor. 2.—If a bi-parentage be selected with parental means /q, h 2 , standard- 
deviations s v s 2 , and coefficient of assortative mating p 12 , then 
{■ 1 0] ' 12' 02 7 , i 1 0-2 1 2' 01 a 0 
b — 2 *1 "T -| o 'O 
1 ?i2 0"i 1 7 13 °"3 
^1^1 4 ~ ^2^2 J 
VC 1 - 
01 
r 02 2 Tis 2 ~b ffi’odWis) d" /^LW r 
/3oV + 2 ^1^2P12 S 1 5 2 
Let us use these results to investigate how the offspring of a selected parentage or 
bi-parentage degenerate. At first sight, it would appear that with our general 
proposition the discussion of the effect of p selections would be perfectly straight¬ 
forward. So it is, but the conclusion which follows, although it might have been 
foreseen, is remarkable in its consequences. We have only to calculate out the yd’s 
for p selected ancestors, and we obtain the regression £ in the descendant by putting 
in the values 7 q, h 2 , h 3 , ... of the means of the selected ancestors. For example, 
suppose now a parent, a grandparent, and a great-grandparent to have been selected. 
We can find the yd’s at once from the results on p. 294 . If 1, 2, 3 , 4 denote the suc¬ 
cessive generations, and r the correlation coefficient of parent and offspring, we find 
y»^ - ry /r* - /)•» - syO /y - 
r > T 1S — r ) — r > ; ’-23 — r ’ V -2i — V ~> r 43 — r > 
whence we deduce at once 
X x = 1 — 2r 3 -f r 4 , v l2 = r (1 — 2r' 3 + r b ), 
v u = v u — °> 
i/x = (i - ^ 2 ) 3 > 
or 
ft = r A ft = ft = 0. 2, = <r, Pi - rK 
Similarly, if we take offspring (l), parent (2), and maternal and paternal of the 
same sex, grandparents (3 and 4 ), we have : 
— r > r l3 — 1 
r u = r 
23 
— r, r u = r, r 34 = 0, 
