88 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



SUMMARY OF SECTION. 



We may now put together in orderly form the chief results which 

 have been gained in this section of the paper. It has been shown that: 



(a) There is a positive correlation between the number of leaves to 

 the whorl and position of the whorl on the plant, or, since in a plant 

 growing in the manner in which Ceratophyllum does, position on the 

 plant is determined by the order of the formation of the organ or 

 character under consideration, there is a correlation between the number 

 of leaves to the whorl and the ordinal rank in the process of successive 

 whorl formation. 



(6) This correlation is considerable in amount. It is lowest for the 

 main-stem whorls, and increases steadily in the higher axial divisions 

 of the plant (primary, secondary, and tertiary branches). 



(c) The regression of leaf -number on position is not linear, but log- 

 arithmic. 



(d) The mean number of leaves to the whorl increases with succes- 

 sive whorl formation according to the equation 



y = A -{- Clog {x — a) 



where y denotes the mean number of leaves per whorl, x the position 

 of a whorl on an axial division of the plant, and^, C, and a'are constants. 

 Stated in words, the law of differentiation with growth in Ceratophyllum 

 is: The mean number of leaves per whorl increases with each successive 

 whorl, and in such a way that not only does the absolute increment in 

 each leaf-number diminish, but also the rate of increase diminishes as 

 the ordinal number\ofthe whorl, measured from a fixed point, increases. 

 This may be, for convenience, designated as i\\e first law of growth in 

 Ceratophyllum. It means, broadly speaking, that the form of any 

 particular whorl of a Ceratophyllum plant is a function (in the math- 

 ematical sense) of the number of whorls which have been produced 

 before it on the same axis. 



(e) The same law of growth holds (with appropriate changes of the 

 constants) for all axial divisions of the plant (main stem, primary, 

 secondary, and tertiary branches) . 



( f) The absolute size of the elements of the developing system given 

 by an axial division of a Ceratophyllum plant is modified by environ- 

 mental differences, but the proportional differentiation of the elements 

 is in accord with the same law in plants from different environments. 



{g) The absolute increment in mean leaf-number between similarly 

 situated successive whorls is least on the main-stem, and increases 

 regularly in each of the higher axial divisions (primary, secondary, and 



