310 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
tions yet taken; it demands a mortality due to natural selection, which its 
propounders have hardly appreciated. 
(iii.) Panmixia and Bi-parental Regression .—The process by which corresponding- 
results may be deduced for bi-parental selection may now be briefly indicated. 
We suppose both natural selection and assortative mating to have gone on in any 
manner for any number of generations, the final effect, however, if the focus of 
regression be not changed, will be : 
Mean of males = / 3 .gn 2 4 - / 3 3 ?n 3 = g l5 say, 
Mean of females = ft H~ ftph —do sa y, 
(S.D. of males) 2 = o- 2 + + fi?? s 3 + 2 ; 8 2 /S 3 s 3 .s 3 p = e/, 
(S.D. of females) 3 = cr' 2 4 - ft dsg + ftds 3 2 + 2 ft. 2 ft 3 s. 2 s s p = gp, 
where m 2 , m 3 , s 2 , s 3 , p defines the last step of the natural and sexual selections, and 
are the regression-coefficients for females. 
Now, selection of all sorts ceasing, we must use for the regression-coefficients no 
longer their values modified by sexual selection, but simply : 
o 
cr~ = 
ft -2 — ^3 °"]/°"2> 
ftz = r s <r\/<r' 2 . 
fti = r 2 cr l /o" 3 , 
ft 3 = r ' 2 o-'i/o-'g, 
o-~ = erf (1 — r — rf), 
obtained from the general values, p. 287 , by putting r x = 0. Here r\ and r ' 3 are 
respectively the maternal and paternal correlation coefficients for inheritance in the 
female line. Further, we have very closely cq = oq = and c t \ = o -' 3 = cr 3 . Tf 
dp* dp g' ve the male and female means, e p , rj p the male and female standard-deviations, 
after p generations in which natural and sexual selection have both been suspended, 
we have: 
dr = 
dp — 
4 = 
v = 
fizdp-i + fizdp-i> 
fi zdp -1 ft 3d p-u 
°" + 
cr' 3 + ft ^ P -\ ft ft s \J. 
Solving the equations for the means first, we have : 
dp — -A-! y p \ + 
d p 
' - A Vl .-1 . A 7»-- 
- A 1 a y 1 + A 3 « 
ft 
yy 
w 
1 1 ere 
