VARIATION— GENERAL RESULTS. 19 



smooth and regular polygons. But these polygons represent composites. 

 While they show the condition of the plants as a whole with respect to 

 variation, they tell us nothing about how this condition comes to be. 

 Our purpose is to analyze, piece by piece, these gross frequency polygons 

 and try to find out the different factors which determine the proportionate 

 frequencies of whorls of different types. In other words, we must by 

 statistical analysis dissect these polygons, and in that way, if possible, 

 gain some idea of the physiological factors to which they fundamentally 

 owe their origin. By proceeding in this way it should be possible to 

 obtain results of real significance. 



In the continuation of our preliminary survey we may next examine 

 the values of the chief physical constants, means, standard deviations, 

 and coefficients of variation of the distributions given in table 1. These 

 constants are collected in table 2. 



This table brings out a number of points of interest. In the first 

 place, we note again that in respect to all the constants Series I, II, and 

 III are very much alike, whether we compare the series as wholes or 

 take single plants. Series IV is significantly but not widely different 

 from Series I, II, and III. It has a somewhat higher mean number of 

 leaves to the whorl, and a lower variability in this character. Series V 

 differs from the others in having a lower mean and higher variability. 



If, in order to get a general idea of the values of the constants for 

 variation in the Ceratophyllum of this region, we take the means of the 

 constants for the six series, weighting them in each case with the 

 number of whorls they include, we have the following results: 



Mean of means 8.655 



Mean of standard deviations 1.206 



Mean of coefficients of variation 13.966 



It may, then, be fairly concluded that the Ceratophyllum of this 

 region has an average number of leaves to the whorl of about 8.7, a 

 standard deviation of about li leaves, and a relative variability of approx- 

 imately 14 per cent. 



It is of some interest to compare these results with the statements 

 of botanists who have investigated Ceratophyllum from other stand- 

 points. The first thoroughly scientific investigation of the biology of 

 Ceratophyllum was that of Schleiden ('37). Considering its date this 

 memoir is a most remarkable one, both for the wealth and the accuracy 

 of its observations. Regarding the number of leaves to the whorl 

 Schleiden says (loc. cit., p. 516): "Der Stengel der Pflanze ist stielrund 

 mit vollstandigen Knoten versehen, an denen sich die Blatter in 6-10 

 theiligen Wirteln befinden. Smith giebt falschlich 8 als Kegel (cf . Eng- 



