21 
SELECTION ON THE VAIHAEILITY AND COKRELATION OE ORGANS. 
say, and find the “ centre ” of the quadric of the remaining xs, the co-ordinates of 
this centre, expressed in terms of the h’s, are tiie regression equations. Now it will bo 
clear, that if we })ut all the selected x’s equal to Iis, the dilferentials of the (juadric 
with regard to the remaininp’ or non-selected a;’s can contain no o’s or the coefficients 
of the regression equations thus found will not he modified by selection. 
Further, we might have given not only the selected organs, but any number of the 
non-selected organs constant values, and the resulting regression equations would 
involve only the c’s and not the c’s. 
Hence we have the following general theorems : — 
(i.) If an organ has been modified only by indirect selection, then its y)Cf'>'tial 
regression coefficients on any complex of other organs, hoivever large or small,provided 
it includes all the directly selected organs, will remain unchanged by the selection. 
(ii.) The same organ in two different local races which have been derived from a 
common stock by the selection of two cornghexes of organs, some of ivhich may or may 
not be common, will, if it has not been directly selected in either case, give the same 
partial regression coefficients for any group of organs ivhich includes the members of 
both complexes and, any number of non-directly selected, organs besides. 
If the partial regression etpiatlons liave changed coefficients, then we cannot at 
once determine whether— 
(a) We are dealing with a non-directly selected (jrgan, and have not included all 
the directly selected organs in the gi'oup upon which we are calculating the I'egres- 
sion ; or 
(b) We are really dealing with a directly selected organ. In this case, we have 
also certainly not included at least one directly selected organ in the regression 
group. 
Theoretically, however, ( 1 .) and (ii.) suffice to find out which, if any, are tlie non- 
directly selected organs in the diffierentiation of local races. Practically, however, 
the number of organs and characters may be so great, and our ignorance of those 
probably selected so complete, that the arithmetic of determining so extensive a 
series of partial regression coefficients may be quite beyond our ]jowers. Still, where 
the divergence between local races is not too great, and the source of the differentia¬ 
tion not too obscure, it is pr(d)ahle that the aljove tlieorems will lead to results of 
great interest.* 
Without laying too great weight on these theorems, I would still venture to 
suggest that if the criterion of a species be tlie discovery of any numerical constant 
* Mr. L. Bramley-Moore has working with this end in view at the long-bones in man. But 
even here the direct selection of jjarts of the vertebral column—for which, at jjresent, we have no correla¬ 
tion values either among themselves or with the long-bones—and of the hand and foot, which 
Dr. W. R. Macuonelt. has just shown, are very highly correlated with the long-bones, may render 
nugatory all attempt to ascertain which, if any, long-bone has been'only indirectly, or, at any rate, least 
directly selected. 
