36 VARIATION AND CORRELATION IN THE CRAYFISH. 



femur and humerus are "distinctly more closely correlated" than are the 

 femur and the radius. The tibia and radius appear to be "slightly more 

 correlated" than the tibia and humerus. His general conclusion is that 

 "serially homologous bones tend to be more closely correlated than non- 

 homologous bones." Similar results have been obtained by Lewenz and 

 Whiteley (loc. cit.). In their study of the intercorrelations of the bones 

 of the hand they found that homologous bones from two digits tend to 

 be more closely correlated than contiguous bones of the same digit. 

 Their statement (p. 350) is: "The next highest correlations* are between 

 lateral and not between longitudinal neighbours, each bone being on the 

 average more nearly related to the corresponding bone on the next digit, 

 than to the adjacent bone on the same digit." 



It may be said, then, that the evidence at present available indicates 

 that the morphological relationship implied in the homology of parts is 

 probably a real factor in influencing the degree of correlation in the 

 variation of these parts, but that this influence is nowhere a marked one. 



PARTIAL CORRELATIONS. 



We have so far been discussing the gross correlations between differ- 

 ent pairs of characters. It is of importance now to examine the "net" 

 or "partial" correlations of the joints of the legs with one another. The 

 nature and properties of the coefficients measuring partial correlation have 

 been fully discussed by Pearson (1902 and earlier papers), and it will not 

 be necessary here to discuss them in detail. We may, however, note 

 briefly certain fundamental points in the mathematical theory of multiple 

 correlation. 



Let #,, x 2 , x 3 , be any three characters of a population of organisms 

 varying about their respective means with standard deviations <r n <r 2 and 

 <r 3 , and organically correlated together to the degree indicated by coeffici- 

 ents r, 2 , r, 3 , r., 3 . Then suppose a group of individuals to be selected 

 from the population with reference to the character x l , so that after the 

 selection the variability of this character will be that indicated by a stand- 

 ard deviation s r It has been shown by Pearson (loc. cit.) that in this 

 selected group of individuals the correlation between x 2 and x, 3 will be given 

 by a coefficient 



' 



* The highest correlations were found to be between corresponding bones of the 

 right and left hands. 



