VARIATION AND DIFFERENTIATION. 17 



From these results we conclude that the appendage which is most 

 highly developed and specialized morphologically is also relatively the 

 most variable. This may be taken as a confirmation of Darwin's law that 

 highly developed parts tend to be more variable than corresponding parts 

 of ordinary size. It is in agreement with Yerkes' (1901) results on 

 Gelasimus. 



So far we have been considering the variability in proportion to the 

 absolute size. Another equally important point to be considered is the 

 variation, not in the absolute size of the parts, but in the proportions of 

 the body. Thus it is conceivable that the great chela might be relatively 

 the most variable part of the body in its absolute size, and yet that the 

 proportionate length of this organ in relation, say, to the length of the 

 cephalothorax would be one of the least variable dimensions. Having 

 determined the relative variability of the parts in respect to absolute size, 

 we must next study the variability in respect to proportions. This we 

 may of course do by expressing each dimension in the form of an index. 

 Thus, we may express the length of each joint of the legs as a percentage 

 of the cephalothorax length, instead of in terms of millimeters or other 

 absolute units. The variation of these indices will then measure the 

 extent to which the proportions, as distinct from the size of the parts 

 of the body are varying. In order to determine the variation shown 

 by these indices we may resort to the theorem given by Pearson (1897). 

 He has shown that if we let 1,3 denote the mean value of an index, 

 x^/x^; 2,3 the standard deviation of this index; and v^ and v^ the coeffi- 

 cients of variation of j^j and x^ and r,, the coefficient of correlation between 

 the same characters, then 



^13 = -- (1 -{- v-i^ — ruViVs) (i) 



7/Z3 



where w, and m^ are the means of the characters x^ and x^, and 



Si3 = ii3 t/(Vi2 + V3^ — 2ri3 V1V3) (11) 



In the present case it seems best to use the length of the cephalothorax 

 as the basis of reference in determining the relative proportions of the 

 different joints of the legs. That is, in the index fraction xjx^, x^ will 

 denote in every case the length of the cephalothorax, while for x^ will be 

 put successively the different joints of the different legs. Proceeding in 

 this way, and putting the result in each case in the form of a percentage, 

 we shall have the length of each joint of the legs measured in hundredths 

 of the cephalothorax length. 



