PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 305 
Hence £ = r — h 2 , and a ' 2 — oq 3 (1 — r 3 ), 
O’ o 
or precisely the results we should have obtained by selecting only the immediate 
parent. 
To simplify the analysis for biparental selection, assume that the correlation 
coefficients of both parents are equal, and that there is no assortative mating. 
We have the scheme for the correlation-coefficients subscripts, —>• marking a 
generation : 
r-» 4 - 
r—- 8 — 
1 — 
ffi— 
-- 9 — 
— 10 — 
—IT— 
— 12 — 
— 13 —- 
—> 14 —- 
— 15 — 
and so on. 
Thus r mn is at once expressible as zero, or a power of r, the simple coefficient of 
correlation for parent and 
direct descent. 
Hence we find 
offspring, according 
as 
to and 
n 
do not 
or 
do 
lie in the 
R = 
1 
v 
r 
^2 
c, 
T" 
9 
r " 
r 3 
r 3 
r 3 
r 3 
r 3 
r 3 
r 3 
r 3 
r 3 . . . 
V 
1 
0 
r 
r 
0 
0 
r 2 
r 2 
9 
o 
0 
0 
0 
0 . . . 
r 
0 
1 
0 
0 
/■ 
r 
0 
0 
0 
0 
9 
r v 
0 
r v 
0 
)- 
0 
/y$J 
r 
0 
1 
0 
0 
0 
r 
r 
0 
0 
0 
0 
0 
0 . . . 
0 
r* 
r 
0 
0 
1 
0 
0 
0 
0 
r 
r 
0 
0 
0 
0 . . . 
O 
r 4 
0 
r 
0 
0 
1 
0 
0 
0 
0 
0 
r 
r 
0 
0 . . . 
9 
r 
0 
r 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
r 
r . . . 
r 3 
o 
r" 
0 
r 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 . . . 
r 3 
o 
i w 
0 
r 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 . . . 
r 3 
T" 
0 
0 
V 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 . . . 
r 3 
0 
r~ 
0 
0 
V 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 . . . 
r* 
0 
0 
rj 
0 
0 
V 
0 
0 
0 
0 
0 
1 
0 
0 
0 . . . 
r 3 
0 
9 
0 
0 
r 
0 
0 
0 
0 
0 
0 
1 
0 
0 . . . 
r 3 
0 
r 2 
0 
0 
0 
r 
0 
0 
0 
0 
0 
0 
1 
0 . . . 
r 3 
0 
9 
r w 
0 
0 
0 
r 
0 
0 
0 
0 
0 
0 
0 
1 ... 
Add the second and third rows, multiply them by r and subtract from the first, 
and we find : 
R = (1 — 2r 3 ) E n . 
9 p 
Li Lv 
MUCCCXCVI.—A. 
