PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 315 
the proposition that inheritance gains strength simply through long continuance, 
but I doubt whether it can be proved. In one sense the proposition is little better 
than a truism; if any character has remained constant during many generations, it 
will be likely to continue so, if the conditions of life remain the same. So again 
in improving the breed, if care be taken for a length of time to exclude all inferior 
individuals, the breed will obviously tend to become truer, as it will not have been 
crossed during many generations by an inferior animal.” (“Animals and Plants 
under Domestication,” vol. 2 , p. 37.) 
Down to the words “ if the conditions of life remain the same,” all is consistent 
with the extreme theory of panmixia, but making a breed truer by selection for 
many generations is only consistent with belief in a progression of the focus of 
regression, or in a change towards unity in the coefficient of regression with continued 
selection. The latter alternative would, I think, be quite inconsistent with our whole 
theory of heredity as applied to a practically stable population. As we cannot 
mathematically deal with a theory of progression of the focus of regression without 
some hypothesis of the nature of progression with continued selection, we will 
assume an extreme case, and suppose the focus to progress very rapidly, i.e., that 
offspring regress to the mean of the population from which their parents have been 
immediately selected. This will at least offer some explanation of animals breeding 
truer with persistent selection, if at the same time it leads to results inconsistent 
with the extreme theory of panmixia. 
(ii.) Panmixia and Bi-parental Selection. —Let /q, s- ± be the paternal, h 2 , s 2 the 
maternal distribution at each selection. Then with assortative mating after p genera¬ 
tions, the standard-deviations of the male and female populations will be of the same 
form as after one generationand be given by the e 1? r/ 1 of p. 310. Now this result is not 
like the stable focus of regression out of accord, I think, with experience. It is note¬ 
worthy how comparatively little difference there is in the variation constants of the 
different races of man, although in many cases pretty severe selection may have been 
supposed to have been in progress for many generations. For example, the mean 
cephalic index varies from 70 to 83, but the probable deviation from this mean only 
varies from about 2 to 2 ’7, so that even very primitive races (where the variation is 
small and we may suppose the selection has been severe, or the strain is very pure), 
do not “ breed much truer ” than highly civilised races with a far less mortality. 
The difference between the variation of the most and least variable races is probably 
not more than the /3-terms in the values of q and rj l (p. 310) may be able to account 
for. 
Turning now to the alteration of the male and female means in jp-generations of 
selection, let as before /3 3 , /3' 2 , /3' 3 , be the regression coefficients and u n , v n , the 
distances from m 2 , m 3 , of the means of the male and female populations out of which 
the nth bi-parentage (m 2 , m 3 , s 2 , s 3 , p) is selected. 
Hence : u„ — (/3 2 u n + fi 3 v n ) and v n — (/3' 2 u H -f- ft ? v,) are the distances from m 2 , m 3 
2 s 2 
