82 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



data solely, we could get a reasonable graduation of Ann Arbor material. 

 Unfortunately the Series IV data for secondaries are so meager that the 

 results are rather irreg-ular. The number of plants is not large enough 

 to get rid of the disturbing influence of the individuality of a single 

 large specimen. This especially shows itself in the first whorls of sec- 

 ondary branches in this series, where we get an abnormally high mean, 

 due to the marked tendency for one of the plants in this series to have 

 exactly 7 leaves in these whorls. If, however, we change the posi- 

 tion of the growth-curve (equation II) by lowering it 0.1 of a leaf 

 on the y axis, it represents the data for Series IV, with the exception 

 of the first whorl, with very considerable accuracy. Thus, taking the 

 first 10 whorls (after this the observations are too few to give reliable 

 means) , I find the total deviation between observation and theory to be 

 +0.253, giving an average deviation per observation of 0.025. This is 

 a sufliciently close agreement to justify the contentions made that (a) 

 environmental differences affect chiefly the absolute size of the elements, 

 and (6) that the essential character of the differentiation with growth 

 is practically independent of environmental influences. We could of 

 course get a better graduation of the Series IV material by specially 

 determining all the constants of the curve anew from that data, but 

 what I have tried to show is that we get for all practical purposes a 

 sufficiently good fit by simply shifting slightly the position of the curve 

 deduced from the other series, this alteration being merely.to take into 

 account the difference in absolute size between the two sets of plants. 



Summing up, then, we see that the same law of differentiation of 

 whorls with growth holds jor secondary as for primary branches. The 

 chief difference in the two cases is that the change in successively 

 formed whorls is, at any given point, greater in secondary than in 

 primary branches. 



We have now to consider the question of whether the same law 

 which we have found for secondary and primary branches holds for the 

 other divisions of the plant (main stem and tertiary branches) . We 

 may take up first the main stem. It is of course to be expected that 

 the relations are not essentially different here. Unfortunately it is 

 impossible to test the matter directly in the way which has been used 

 for the other divisions, because of lack of material. A plant may have 

 50 or more primary branches, but of course it has only one main stem. 

 Therefore, to get satisfactory means for main-stem whorls in particular 

 positions it would be necessary to have a very considerable number of 

 plants. There is a further difliculty in the case of the main stem. 

 Ordinarily the plants have several inches of the lower (proximal) end 

 of the main stem embedded in the soft debris and mud of the bottom. 



