72 



VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



has been seen in all our material, 30 whorls is the maximum for primary- 

 branch length. We must conclude, then, I think, that under natural 

 conditions the mean leaf-number per whorl becomes practically constant 

 after the formation of from 15 to 20 whorls. This number is 10. To 

 reach a mean leaf-number of 11 the branches must have more than 175 

 whorls, while to raise the mean to 12 would require the occurrence of 

 primary branches with more than 900 whorls each! Such an occur- 

 rence is of course far out of the bounds of reasonable expectation. 



Table 36. — Observed and calculated mean leaf-number in successive whorls on 

 primary branches. Series IV. 



Another matter which demands consideration is this: We have 

 reached our generalized law of growth from a study of plants collected 

 at one place, namely. Carp Lake. Do plants from other localities follow 

 the same law of growth? I think it is evident on general grounds that 

 this is altogether likely to be the case, and table 34 and fig. 11 show 

 very clearly that it is so. The law of the change in mean leaf-number 

 with successive whorl formation must be the same to lead to such par- 

 allel results as we find on comparing any two series, as for example. 

 Series I with Series IV or Series VI, etc. I propose to show in another 

 way, however, that the law of growth which we have deduced is general 

 for Ceratophyllum. A glance at fig. 11 indicates that the most marked 

 difference between Series IV and Series I, II, or III in respect to the 

 regression of leaf-number on position is that for corresponding positions 

 the mean leaf -number is in general higher in Series IV than in the 

 other three series. Now, suppose that without in the slightest changing 

 the shape of our growth-curve, we simply change its position by moving 

 it up on the y axis only 0.3 leaf. This will of course be done by simply 

 increasing the value of the constant A by that amount. The result is 

 shown in table 36 and graphically in fig. 12. The first column in the 

 table and the zigzag line give the actually observed values for the mean 



