MATERIAL AND METHODS. 11 



became difficult near the ends of the branches. There the whorls are 

 very small and closely packed together, and in order to make an accurate 

 count it was necessary to use a lens, and in some cases even the com- 

 pound microscope. For the sake of uniformity it was necessary to adopt 

 some arbitrary rule with reference to divided leaves. After examining 

 a considerable number it was decided to count only the proximal ends of 

 leaves. Thus the whorl shown in fig. 1, would be recorded as having 8 

 leaves. 



The number of leaves in every unbroken whorl on 

 the plant was determined for each plant studied. In 

 case a w^horl was so mutilated as to make the deter- 

 mination of the number of leaves doubtful, it was so 

 recorded. Unfortunately Ceratophyllum is rather 

 liable to mutilation because of the fact that its tissues 

 are brittle. In comparison with the total number of fig. i.-Diagram of 

 whorls on the plant, however, the number so mutila- a whon, showing 

 ted as to be undeterminable was very small. form of leaves. 



In recording the position of the whorls the plan followed was to give 

 each whorl on a particular portion of the plant a consecutive number, 

 beginning with the proximal. In doing this the different axial divisions 

 of the plant were treated separately. The first of these is the main 

 axis, or, as it will be called throughout the paper, the 'main stem." 

 Using the usual notation for branching, we have designated the lateral 

 branches arising from this main stem "primary branches;" those 

 arising from primaries, "secondary branches;" those arising from 

 secondaries, " tertiary branches, " and so on. "Quaternary branches" 

 were the highest lateral-branch elements found in any of our plants. 

 Of these divisions the main stem alone presented any practical difficulty 

 in making the records. It often happens in Ceratophyllum that the 

 main stem branches dichotomously at a distance from its proximal end 

 which varies in different cases. Are the two new axes to be considered 

 as continuations of this stem, or is one to be regarded as a continuation 

 of the main stem and the other as a large primary branch to be included 

 in the records with the lateral branches? If the axes really arise by 

 dichotomy of the main stem it is clear that they ought to be recorded as 

 parts of it. The only difficulty is the practical one of being certain in a 

 particular instance that we are not dealing with a case of unusually vig- 

 orous growth of a lateral branch which comes to rival the main stem in 

 size. As a matter of fact it was found very early in the work that 

 the first few whorls of lateral branches are so clearly differentiated 

 that it is always possible in a given case of branching to tell whether 



