132 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



Finally, in closing, I desire to emphasize the great importance in any 

 study of variation of the analysis of the intra-individual variability of 

 the characters which are to form the basis of intra-racial investigations. 

 We have seen in a repeated character of the same individual a whole 

 series of variation constants appearing in an orderly manner and repre- 

 senting degrees of variation from practically zero up, degrees of skewness 

 from positive through symmetry to negative, etc. It is obvious that on 

 any problems of intra-racial variation we could with Ceratophyllum get 

 widely divergent results by taking whorls from different parts of the 

 plant. Ordinarily when an investigation of a problem of geographical 

 variation, or of natural selection, or of almost any phase of intra-racial 

 variation is made on the basis of characters like the leaf- whorls of Cera- 

 tophyllum, only a few are taken from each individual. Either no attention 

 at all is paid to differentiation in the characters, or at most only the 

 characters in the immediate neighborhood of those chosen are examined, 

 and if there is no differentiation easily detectable among them, the 

 conclusion is reached that any factor of this kind may be neglected. It 

 seems to me that the results of the present paper show that the question 

 of whether or not there is a differentiation within the character group 

 chosen for investigation is not necessarily the important thing at all. 

 If one takes the 10 most distal whorls on Ceratophyllum plants, there is 

 substantially no differentiation in respect to leaf-number. Yet, to con- 

 clude, when this had been ascertained by superficial examination, as it 

 could be, that these whorls, since they were undifferentiated and there- 

 fore could be considered homogeneous material, might be taken to 

 represent the conditions of the individual as a whole in an intra-racial 

 investigation, would lead to absolutely fallacious results. It is true 

 that there is no marked differentiation among these distal whorls, but 

 the very reason that there is not is that a perfectly definite and orderly 

 process of differential development has led as an end result to the con- 

 tinued production of substantially the same type of whorl, regardless of 

 environment or other influences. These whorls are not greatly differ- 

 entiated, but neither are they truly representative of the individual. 

 If characters of this kind are to be used as the basis of investigations of 

 problems of intra-racial variability, it clearly is absolutely necessary 

 that the laws according to which the characters are differentiated during 

 the development of the individual must first be ascertained if valid 

 results are to be obtained. When this has been done we may turn to 

 intra- and inter-racial problems, and, using the methods which have 

 been developed by Pearson for dealing with differentiated characters, 

 hope to reach definite conclusions. 



