56 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



constants had been increasing, they now begin to decrease, and vice 

 versa. The nature of these changes in the character of the leaf- 

 number variation in different portions of the plant may best be grasped 

 as a whole from a graphical representation, such as is given in fig. 10. 

 This diagram shows for four groups (viz, Series I, II, and III combined, 

 Series IV, Series V, and Series VI) the changes as we pass from central 

 to peripheral divisions in the following constants— mean, standard devi- 

 ation, coefficient of variation, and skewness. The different constants 

 are plotted to different vertical scales, which are in each case given at 

 the side of the diagram. 



This diagram makes plain at once the law which the variation in leaf- 

 number in different sets of whorls follows. It may be stated in the 

 following way: The mean number of leaves per whorl is highest in the 

 whorls on the most central division of the plant (the main stem) , and 

 decreases regularly in the peripheral divisions. The whorls on the main 

 stem are the least variable, and the variation increases regularly in the 

 more peripheral divisions, till a maximum is reached in secondary-branch 

 whorls. The variation then tends to diminish in the whorls on higher- 

 order branches. Now, from the method of growth of Ceratophyllum, it 

 is clear that as a class the main-stem whorls are the oldest, the primary- 

 branch whorls as a class stand next in age, secondary and tertiary 

 branch whorls next, while quaternary-branch whorls will on the whole 

 be youngest. Of course these distinctions are not absolute for every 

 whorl; there may, for example, be individual whorls on the main stem 

 which are younger than individual whorls on any branch, but on the 

 average it is evident that the more peripheral parts will be the younger. 

 So, then, we find that as a general rule the older the portion of the plant 

 the greater will he the average number of leaves to the whorl. Further, 

 the variation in leaf-number is least in the oldest portion of the plant and 

 increases in the younger portions, but reaches a maximum one or tiuo divi- 

 sions short of the youngest. 



Besides the changes in type and variability, there are marked differ- 

 ences in other respects between whorls in different parts of the plant. 

 Thus for example, the skewness appears to be greater in the variation 

 of whorls on the younger portions of the plant, though on account of 

 paucity of material we can not go farther than secondary branches, 

 with the analytical constants. 



It is perfectly clear from the results which have been presented that 

 in a general way at least the number of leaves in a whorl and the posi- 

 tion of the whorl on the plant are related. But how close is the relation? 

 Does it hold within an axial division (say primary branches) that the 



