MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 263 



If we put 4=1 in this result and remember that r,, = 1, we find, after some 

 reductions, 



T> i 1 ~ ? "Ti ~ 9*18 ~ ? 'li + 2?V/B)?'a 1 



'''""" (l-ry(l-^ 3 ) 



which 'agrees with (xxxvii.), and may be taken as a verification of this result.* 



(12.) We may draw several conclusions from the results (xxxvi.), (xxxvii.), and (xl.). 

 (a.) While errors in the correlations of a first organ with a second and a third 

 have a correlation themselves of the first order, errors in the variation of a first 

 organ and the correlation of two others, or in the correlation of two organs and in 

 the correlation of a second two, have only correlation of the second order. Thus a 

 selection of the correlation between two organs modifies the variation of all organs 

 correlated with one or other or both of the first, but only in the second degree. 

 Again, a selection of the correlation between two organs modifies the correlation of 

 every other pair of organs, one or both of which are correlated with one or both of 

 the first pair ; but this is only in the second degree. 



(j8.) If two organs be entirely uncorrelated a random selection of the variation of 

 a third organ correlated with both of them will tend to generate correlation between 

 the hitherto uncorrelated organs, i.e., put ?v, = in (xxxvi.), and we have 



If a variation Ao^ be made in cr l the probable value of r, :; is 



A<7, 



An, -- * - 



which may clearly be of sensible magnitude. Thus correlation may be generated 

 by selection of variation, and vice versa. 



(y.) If two organs be each entirely uncorrelated with a third, yet a random 

 selection, which produces a correlation between one of these organs and the third, will 

 produce a correlation of the first order between the other of these organs and the 

 third, i.e., put r, 2 = r 13 = in (xxxvii.), we have 



a correlation of the first order between the probable changes. 



(8.) Consider four organs of which the first is alone correlated with the third and 

 the second with the fourth, the third and fourth being themselves uncorrelated. 

 Then any random selection which produces a correlation between the first and 



* The probable error of a partial regression coefficient for p organs has not been worked owing 

 to the labour involved, but judging by the cases on pp. 245 and 260, it may safely be taken as 

 67 - 449 v/H/^w ( R 12 ), where R is now the determinant of they" 1 degree. 



