1!l| "l Johnson: Quantitative study of Sal pa Chain. 169 



whether these relations change as the parts grow, these and many 

 other related questions can be answered only by accurate meas- 

 urement of carefully selected material. 



The two phenomena, the grand period of growth, and the 

 length period of the internodes, are regarded by Moll and some 

 other botanists as wholly different. Whether they are different 

 or not, it is not within my province to discuss. However, if they 

 are wholly different phenomena, since the blocks of zooids are 

 developing structures, they would seem to be more comparable 

 with the grand period of growth than with the length period of 

 the internodes. 



These considerations, while dealing with a different aspect of 

 the problem, suggest the law of differentiation with growth which 

 Dr. Pearl found in Ceratphyllum — i.e., that "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 leaf- 

 number diminish but also the rate of increase diminishes as the 

 ordinal number of the whorl, measured from a fixed point, in- 

 creases. It means, broadly speaking, that the form of any par- 

 ticular whorl of a Ceratophyllum plant is a function (in a mathe- 

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

 before it on the same axis." 



"The same law of growth holds (with appropriate changes of 

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

 primary, and secondary branches)." 



The second law of growth given by Dr. Pearl is as follows, 

 "As whorls are successively produced by a growing bud, they are 

 formed with ever increasing constancy to their type, the ultimate 

 limit toward which the process is tending being absolute con- 

 stancy. ' ' 



In other words, Dr. Pearl found that the "leaf -number" in 

 Ceratophyllum is a function of its place on the axis, and that 

 the leaf number becomes more constant as one goes out on the 

 stem. In our problem we are dealing with size rather than 

 with number, but we have found the length and the width of the 

 zooid to be a function of its position in the block. 



But when we try to correlate our results with Dr. Pearl's 

 second law a discrepancy appears. The mean leaf-number in- 



