24 



VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



the observed. It is not, however, distributed on the two sides of the mode 

 in the same manner. The theoretical range overestimates the observed 

 below and underestimates it above the mode. Thus the two whorls 

 with 12 leaves are theoretically excluded. It is altogether probable 

 that with proper corrective terms for the moments we should get a still 

 better estimate of the range. In the case of plant 2, Series IV, the 

 observed range is between 3 and 11 leaves, inclusive, or a total of 8. 

 The theoretical curve has a range between 1.14 and 10.96, or a total of 

 9.82 leaves. No better result could be desired for the upper end, but 



Leaf number 

 Fig. 4— Frequency histogram and fitted curve for variation in Ceratophyllum. All 

 unmutilated whorls on all plants of Series I, II, and III. 



the lower end is considerably overestimated. Thus, the occurrence of 

 2-leaved whorls is theoretically possible. There are three things which 

 should be kept in mind, however, in passing judgment on this result. 

 In the first place, the undue extension of the lower end of the theoretical 

 curve is in a large measure due to the single observation of a 3-leaved 

 whorl. Now, this whorl was almost certainly to be regarded as an 

 abnormality. It was a very small whorl in respect to size as well as 

 number of leaves. It presented the appearance, to borrow zoological 

 terminology, of being an intercalated, abnormal whorl between two 

 normal ones. What its origin was it is impossible to say, but it cer- 



