304 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
whence 
\ = 1 “ 2r 3 , v 18 = r (1 - 2r 2 ), „ 13 = ^ = 0, 1 / K = 1 - 8t* + 2r±, 
or, 
A = r & =-- / 3 o = 0, ^ = o-! v/1 - r 2 . 
Co 
Thus we see that in both cases the grandparents are quite indifferent, when the 
immediate parent has been selected. 
These theorems can be at once generalised by means of Edgeworth’s theorem. 
Suppose we select a complete parentage for p generations in the case of partheno- 
genetic reproduction, or a parentage of one sex, say males, in the case of sexual 
reproduction, then in either case our scheme of subscripts of the correlation-coefficients, 
—> marking a generation, is 
and 
2 
4 
5 —> 
r 
„3 
r 3 . . 
. r p ~ l 
1 
r 
0 
r* . . 
r V-2 
r 
1 
r . . 
r p ~ 3 
r p ~ 2 
r p-z 
. I 
Multiply the second line by r, and subtract from the first, and we have 
R = (l - >-) R n . 
Take E. I? (q < p), and we have 
II 
(—1 
r 
1 
r 
0 
. 
. r 2-3 , r 2 ~ 1 . . 
y p- 2 
0 
r 
1 
v . 
. . r 2 ~ 4 , r 2-2 . . 
pp — 3 
r p ~ ] 
r p -\ 
+ - j -P—'! 
. . 1 
Multiply the second column by 
Ei 2 = 0 if q > 2. 
If q — 2 , we have 
R w = | r 
v, and subtract from the first, and we have 
r 
1 
? w . . . r l 
r ... 7 
,v -3 
I r p ~ 3 . . . . 1 
or. dividing the first column by r, K 12 = ER n . 
