CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
177 
* 
indirect selection. The chanofes due to indirect selection are shown in the memoir 
referred to to be in many cases of considerable importance ; every mean, every 
standard deviation, every correlation may be altered ; but the following theorems 
govern the changes in the regression formulae :— 
(i.) The regression formula of a directly selected organ on any number of other 
organs, whether directly or indirectly selected, will change. 
(ii.) The regression formula of an indirectly selected organ on all the directly 
selected organs, and any number of the indirectly selected organs, does not change. 
(iii.) The regression formula of an indirectly selected organ on some, but not all 
the directly selected organs, will change, unless the selection happens to be one of 
size only, and not of variability and correlation at the same time, in which case the 
formula remains unchangfed. 
(iv.) Most local races show sensible but small diflPerences in both variability and 
correlation ; if we call these differences quantities of the first order of small quantities, 
then the changes in the regression formulge between two or more indirectly selected 
organs will be of this order of small quantities X the squares and products of corre¬ 
lations, quantities which are themselves less than unity, or what we may term a 
quantity of the third order ; further, the changes in the regression formulee between 
an indirectly selected organ and some but not all the directly selected organs will be 
of the first order of small quantities X the correlation, or what we may term a 
quantity of the second order. 
To sum up, then, it would appear that the I'egression formulse in general will 
change from local race to local race, but that a particular set (see (ii.) above) exist 
which would not be changed at all, while many others, supposing size* to be the chief 
character selected, would only be changed by quantities of the second or third order. 
It will be obvious then that a knowledge of a considerable series of regression 
formulae of two local races will enable us to ascertain to some extent the nature and 
amount of differentiation which has gone on from a common ancestral stock. Further, 
if we have not sufficient data for one local race to find the variabilities and correla¬ 
tions of its organs, but if we can find fairly closely the inean size of its organs, then 
the degree of consistency of the results obtained when these means are inserted in the 
regression formulse for the second local race is an indication of the amount of 
differentiation which has taken place. The larger the number of organs we include 
in a regression formula the more likely we are to embrace all the directly selected 
organs, and so to obtain a formula which remains unchanged for the two races. 
Thus we see that the extension of the stature regression formulse from one local 
race—say, modern French—to other races—say, palseolithic man—must be made 
with very great caution. The extension assumes (i.) that stature itself has not been 
* A selection of the mean sizes of two organs, which, wonld alter their relative proportions, does not 
of course involve a selection of correlation; in other words, selection of mean relationship does not 
necessarily connote a selection of differential relationship. 
VOL. CXCII.-A, 2 A 
