32 



VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



of whorls on the main stems of Ceratophyllum plants, the chance that 

 a whorl chosen at random will have 10 leaves is, on the basis of the 

 combined Series I, II, and III, tVoV- Or, in other words, the chance 

 in favor of the 10-leaved whorl is greater than the combined chance of 

 all other whorls together. This means, of course, very low variability. 

 In order to gain a more precise idea of the nature of this differentia- 

 tion I propose to treat the distributions analytically. In doing this I 

 shall, as before, combine the frequency distributions for Series I, II, 

 and III. These series are so slightly differentiated from one another 

 that the error made by combining them is altogether negligible. In 

 fact it is practically certain that we shall come nearer to the true facts 

 by using the combined data for these three series from the same locality 

 than by taking any one of them separately. For the same reasons as 

 before I have used the raw moments in determining the constants. 

 The analytical results for 'main-stem" and "all-branch" whorls are 

 given in table 10. The combined distributions are, for the main-stem 

 whorls, No. 180 in table 5, and for the whorls on branches. No. 181 of 

 table 7. 



Table 10. — Analytical constants for variation in leaf-number. Series I, II, and III 



combined. 



CJonstant. 



Total frequency 



M2 



^3 



/*4 



^i-^- 



l//3i 



^2 



^2—3 



Main-stem 



whorls (Dis 



tribution 



No. 180). 



374 



.8798 



— .9013 

 4.0691 

 1.1928 

 1.0921 

 5.2568 

 2.2568 



+ .9354 



+1.2455 



Whorls on 



all branches 



(Distribution 



No. 181). 



1,954 



1.5350 



— .8739 

 5.8052 



.2112 



.4595 



2.4637 



— .5363 

 —1.7060 



Constant. 



Skewness —0.4270 



Modal divergence — .4005 



Standard devia- 

 tion 9380 



Mean 9.5348 



Mode 9.9353 



Range 



Lower end of 

 range 



Upper end of \ 

 range 13.2096 



Main-stem 

 whorls (Dis- 

 tribution 

 No. 180). 



Whorls on 



all branches 



(Distribution 



No. 181). 



—0.6119 



— .7581 



1.2390 

 8.4765 

 9.2345 

 6.0426 



4.5435 



10.5861 



This table brings out clearly the essential differences between ' 'main- 

 stem" and "all-branch" whorls in respect to variation in leaf-number. 

 We note at once that the main-stem distribution is markedly less skew 

 than is the branch distribution, the distance from mean to mode being 

 in the former case only a little more than half what it is in the latter. 

 The direction of the skewness is the same however, in both cases, namely 

 negative, or the mode is larger than the mean. The range for the 

 "all-branch" distribution is, as was to be expected, less than what it is 

 for the plant as a whole (cf . table 4) . 



