64 BOTANICAL GAZETTE 



[JANUARY 



not at ail the size of parts relative to position, it is shown: (i) that A is the environ- 

 mental factor, and (2) that environment affects size but not at all the progress of 

 differentiation represented by the logarithmic curv-e. It is found that the cur\-e 

 for the main stem is most nearly horizontal and that for each successive grade of 

 branches is steeper and steeper, a change in the curve dependent upon the increas- 

 ing value of the constant C, which is therefore regarded as the constant which 

 expresses the action of internal formative factors. 



On any axial division the first whorls are the most variable in number of leaves, 

 and this variability decreases at first rapidly, then less and less rapidly as more 

 and more distal positions are reached, the theoretical limit attained being perfect 

 constancy. This constitutes the "second law of growth" in CeratophyUum. It 



is shown that these two laws are true in branch production as well as leaf produc- 

 tion in this species. 



Although few other studies have been made which can be definitely cited to 

 demonstrate the general validity of these two laws of growth, the essence of the 

 second law was recognized by the reviewer^^ in the case of Shim cicutaejolitim, 

 and other published observations are mentioned' which appear to indicate that 

 these are general laws. It is believed that this second law is the same which in the 

 physiological field has been stated by Jexxings^^ thus: "An action performed 

 or a physiological state reached is performed or reached more rapidly after one or 

 more repetitions, so that in time it becomes ^habitual' " It is shown that in psv- 

 diolog}- It is probably the same law that Ues at the basis of memorj', namely, the 

 ease, as measured by rapidity and accuracy, with which a state once reached is 

 reached again. It seems probable, at any rate, that suitable analysis of aU these 

 cases will show that they are governed by logarithmic laws, and though no sugges- 

 tion has been made as to the nature of the common causes upon which such general 

 validity of the logarithmic law must rest, their nature will doubtless be brought 

 to hght very soon when once it is recognized that they exist. 



The necessity of more studies of a similar nature is apparent. It is important 

 to know how these laws are affected by more complex environmental conditions. 

 The ease and conclusiveness with which they were demonstrated in CeratophvUum 

 probably depended upon the fact that in this species environmental modifications 

 are reduced to their lowest terms. It is to be hoped that the aims and methods 

 of this^ paper may serv-e as guides for many biometric studies, for the successful 

 analysis of ^the laws of differentiation within the individual must precede studies 

 which mvolve repeated structures in different individuals. Studies of the latter 

 type are msecure untH the laws of intra-indi vidua! differentiation and variation 

 are known for the particular species of plant or animal under investigation 

 Geo. H. Shull. ^ 



^8 Shitll, G. H., Stages in the d. 

 Institution of Washington, PubL no. 30. 



^ojEXNixGS, H. S., The methods 

 Jour. Exp. Zool. 2:473-49^ ^^5- 



Carnegi' 



