124 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



the farther out on an axis we proceed, but the rate of increase varies 

 inversely as the distance from the proximal end of the system. We 

 thus get further evidence, from an entirely different source, of the 

 generality of our first law of growth in Ceratophyllum. The data fur- 

 nish positive evidence that branch production and leaf production by the 

 growing plant follow the same law. 



SUMMARY. 



A study of branch production leads to the conclusion that '*as the 

 plant grows it tends with ever-increasing certainty to produce a branch 

 at each node." In other words, as an axis of the plant grows longer 

 the morphogenetic processes concerned in the production of lateral 

 branches work, so to speak, more smoothly, and attain their results with 

 greater regularity and constancy. But this is precisely the same conclu- 

 sion to which we came with reference to the production of leaf-whorls 

 (p. 106, mpra). It was there shown that the leaf-number per whorl 

 approaches more and more closely to a constant value the longer an axis 

 grows. Putting both sets of results together, we now see that the plant 

 as it grows tends to produce one or more branches and a constant num- 

 ber of leaves at each node. When a bud begins its growth it does not do 

 either of these things with anything approaching certainty or regularity, 

 but the longer it grows the more regular do the results become, until 

 finally they are almost mathematically precise. Objectively it ''profits 

 by its experience" just as does an animal in its behavior. The results 

 which have been obtained in this section of the paper show that our two 

 laws of growth (pp. 88 and 106 supra) operate in branch production as well 

 as in leaf production. 



