308 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
indicated in Mr. Galton’s idea of organic stability. In either case panmixia would 
only carry back the mean to the current focus of regression, and so be a very minute 
reversal of natural selection. 
What our theory really shows is a regression of the offspring of selected parents 
towards the mean of general offspring. This latter mean, supposing no secular 
natural selection, can, it seems to me, only be determined by experiment. It can 
hardly agree with the general parental mean, if the parents themselves are the 
product of natural selection. On the other hand, the statistics actually obtained for 
stable, or sensibly stable, populations seem to mark a focus of regression close to the 
mean of the current population, and, therefore, a progression of the focus due to past 
selection." Meanwhile, till experiment has settled how continuous selection affects 
the focus of regression, we may see whither extreme hypotheses lead us. Such are : 
(1.) The focus of regression remains stable during selection. 
(2.) The focus of regression is the mean of the population from which parents have 
been selected. 
(e,) Focus of Regression Stable during Selection. 
(i.) Steady Selection cannot be Secular or Produce Truer Breeding .—We have 
seen that on this hypothesis ancestry, as distinct from immediate parentage, is indif¬ 
ferent. Thus, in the case of parthenogenetic reproduction, or of sexual reproduction 
with one parent selected, we have seen that one selection leads to the distribution 
(Cor. 1, p 301 ) : 
r — r 0-0 1, V2 _ _ s /i _ r 2\ i r 2 Cl 2 
t — ' oi nr n a - 1 — G 1 oi ) i ' oi _ 2 *1 > 
o-1 
and if out of this we again select a parentage, defined by h x and s l5 we shall obtain 
the same distribution of offspring, and this however often the process be repeated. 
We must increase the divergence ( 7 q) of the selected from the general population or 
its concentration (l/hq) or both, if we require any progressive effect from continual 
selection. The same remarks apply to bi-parental selection (wi 1} on. 2 , s z , s s , p l2 ). 
Persistent selection only suffices to keep the mean and variation at a definite distance 
from those of the general population. Or, on the hypothesis of a stationary focus of 
regression, we conclude that steady selection, however long it persists, can only be 
periodic and not secular. 
This point seems of such importance that it may be best to illustrate it by an 
example drawn from our Table I. and Table III. The mean height of fathers being 
about 69 "' 2 , the regression of the average sons of fathers of 6' in height is about 
l"’ 25 , or the average height of sons of a 6' fatherhood = 70 ,,- 45 . Hence, if we 
select fathers forming a normal distribution of any standard round 6', we shall have a 
normal distribution of sons round 70 "‘ 4 -, 5 . If we select a second parentage from 
* The determination of the focus of regression for some organ in selected domestic ducks for several 
generations and comparison with, the means for wild and general domestic ducks would seem a 
possibility. 
