38 



VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



we compare the first four series. Tliis is so evident from the distribu- 

 tions themselves and from the values of the constants which follow 

 that it does not appear to be necessary to figure separately the polygons 

 of variation in the primary branches. The general character of the 

 distributions is shown graphically in fig. 8, p. 44, in which primary and 

 secondary branches are compared in respect to variation in leaf-number. 

 From distribution 84 it is seen that the single 3-leaved whorl observed 

 in the course of the work was situated on a primary branch. In Series 

 VI, the single 12-leaved whorl of that series was on a primary branch. 

 Thus it would appear that the whole range of variation in leaf-number 

 in Ceratophyllum may be shown by primary-branch whorls. 



The chief physical constants for the distributions of table 14 are 



given in table 15. 



As we should expect, the values of the constants for primary branches 

 throughout are of the same order as those for the "all-branch" distri- 



