46 VARIATION AND CORRELATION IN THE CRAYFISH. 



SUMMARY OF RESULTS AND CONCLUSIONS. 



This study, dealing with eleven characters of the body and appendages 

 of the crayfish, had for its primary purpose the determination of the rela- 

 tion of variation and correlation to the morphological factors, differentia- 

 tion and homology. The results and the conclusions drawn therefrom 

 may be summarily stated as follows: 



(1) Variation in all the characters studied is skew rather than sym- 

 metrical in its distribution. The skewness is in all cases positive, or the 

 mean lies above the mode. In respect to the degree of flat-toppedness or 

 kurtosis the variation curves all deviate from the mesokurtic condition of 

 the normal curve. In general, we conclude that the variation in the char- 

 acters of the crayfish studied can not possibly be adequately described by 

 the normal or Gaussian curve of errors. 



(2) The correlations between the different characters studied are gen- 

 erally of an unusually high order of magnitude. The coefficients are in 

 general of about the same magnitude as those which have been found for 

 the correlation between bilaterally homologous organs in other animals. 

 The regressions are linear throughout. 



(3 ) It is found that the cheliped, which is the most differentiated leg, 

 is more variable in all the joints studied than is either the first or the 

 second ambulatory appendage. This result is obtained whether we measure 

 the variation in the absolute size of the organs or in their relative propor- 

 tions when referred to some other dimension of the body as a standard 

 base. 



(4) The most variable and the most differentiated and specialized 

 single part of all those studied is the great chela. 



(5) The frequency distributions for the different joints of the cheliped 

 have, on the average, the greatest skewness of any of the characters 

 studied. Degree of skewness and degree of relative variability appear to 

 run parallel in the variation of the characters we have considered. 



(6) The correlation between the homologous segments of two legs is 

 higher when these legs are contiguous than when they are separated by 

 an intervening leg. In so far, the crayfish furnishes evidence in favor of 

 a "rule of neighborhood" in correlation such as has been found by 

 Lewenz and Whiteley in the correlations of the bones of the human hand. 



(7) When the correlations of the non-homologous joints of the dif- 

 ferent legs are considered such a "rule of neighborhood" is not found to 

 hold uniformly. 



