MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 243 



It will thus be seen that when r, 2 is small the absolute and partial probable errors 

 are both large and nearly equal ; that when r l2 is large the absolute and partial 

 probable errors differ more widely, but, as both are small in this case, their difference 

 is not of much importance. Bearing these facts in mind, it will be found that the 

 reasoning based on the partial error in previous memoirs remains valid, even if the 

 partial error be replaced, as it generally should be, by the somewhat larger absolute 



error.* 



(/) If we know definitely the variability of any organ, and we take a definite 

 group of the general population to find the variability of a second correlated organ, 

 then there will be correlation between the deviation of the variation of this group 

 with regard to the first organ from the variation of the first organ in the general 

 population and the deviation of the variation of the group with regard to its second 

 organ from that of the general population's variation for the second organ. This 

 correlation is measured by r? 2 , and thus, if the organs be slightly correlated, it is 

 small ; but if the organs be closely correlated, it is large. Suppose, for example, we 

 know the variability of the tibia, and require to find that of the ulna from a com- 

 paratively few specimens. Let cr, -}- Ao^ be the variability of the tibia in the 

 specimens for which the ulna can be measured, and *, the correlation observed 

 between the ulna and tibia in these specimens ; then, the variability of the ulna being 

 observed as cr/ = cr 2 -f- ACT,, the most probable variability of the ulna in the general 

 population is 



a-.,' /''f^Ao-i/X, 

 or 



/ 4"i . 



cr, - Ao-,, 

 <TI 



or, since the second term is small, we may write cr./ for cr 2 , and the above expression 



equals 



,/', ., A 



= cr, 1 r]., - 



' 



For the long bones, r,., = '9 roughly, and therefore we have the ratio of variability of 

 the ulna in the general population to the variability observed in the group 

 = 1 -8 Ao-j/cr,. 



It is clear that this expression also measures the change in the variability of the 

 ulna due to a random selection of tibia. 



(8) Although the correlation between deviations in the variability of two organs from 

 their mean variabilities only varies as the square of their correlation, the correlation 

 between the deviations in the variability of an organ and in its correlation with a second 

 organ varies as the first power of the correlation of the two organs. In other words, 



* See, for example, the reasoning as to the non-constancy of the correlation coefficient for local races 

 in ' Phil. Trans.,' A, vol. 187, pp. 267 and 278. 



2 I 2 



