18 



VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



explanation of this fact will be taken up later when certain other results 

 are in hand. In anticipation of these results it may be said that the 

 matter is related to the fact that Series II and III were collected at later 

 dates than Series I, and the change is a growth phenomenon. 



Examining the frequency polygons for Series V and VI, we see that 

 these are essentially different both from each other and from all the 

 other series. Series V, while it has as the most frequent whorls those 



Leaf number 



Fig. 3.— Frequency polygons for variation in leaf number. Totals for series. The abscisses give the 



number of leaves per whorl and the ordinates frequencies percent. Series I, • ; Series 



II, o ; Series III, m ; Series IV <t ; Series V, a ; Series 



VI, X.0— -o—o. 



with 9 leaves, yet differs from the other series in having a compara- 

 tively high frequency of whorls with 7 leaves. The polygon for Series 

 VI is quite smooth and regular, yet differs from all the others in having, 

 as the most frequent class, 10-leaved whorls. 



With these polygons before us the nature of our task becomes perhaps 

 plainer. We find that when we determine the number of leaves in 

 every whorl of each plant and plot up the results we get these very 



