Ill] OF PARTS OR ORGANS 97 



having the form of a surface of revolution generated by the same 

 curve. This then is a simple and not unimportant illustration of 

 the direct and easy passage from velocity to form. 



A kindred problem occurs when, instead of "' zones "' artificially marked out 

 in a stem, we deal with the rates of growth in successive actual "internodes" ; 

 and an interesting variation of this problem occurs when we consider, not the 

 actual growth of the internodes, but the varying number of leaves which they 

 successively produce. Where we have whorls of leaves at each node, as in 

 Equisetum and in many water-weeds, then the problem presents itself in a 

 simple form, and in one such case, namely in Ceratophyllum, it has been 

 carefully investigated by Mr Raymond Pearl*. 



It is found that the mean number of leaves per whorl increases with each 

 successive whorl; but that the rate of increment diminishes from whorl to 

 whorl, as we ascend the axis. In other words, the increase in the number of 

 leaves per whorl follows a logarithmic ratio; and if y be the mean number of 

 leaves per whorl, and x the successional number of the whorl from the root 

 or main stem upwards, then 



y — A + C log {x - a), 



where A, C, and a are certain specific constants, varying with the part of the 

 plant which we happen to be considering. On the main stem, the rate of 

 change in the number of leaves per whorl is very slow; when we come to the 

 small twigs, or "tertiary branches," it has become rapid, as we see from the 

 following abbreviated table : 



Number of leaves per whorl on the tertiary branches of Ceratophyllum. 



Position of whorl ... 1 2 3 4 5 6 



Mean number of leaves 6-55 8-07 9-00 9-20 9-75 10-00 

 Increment — 1-52 -93 -20 (-55) (-25) 



We have seen that a slow but definite change of form is a 

 common accompaniment of increasing age, and is brought about 

 as the simple and natural result of an altered ratio between the 

 rates of growth in different dimensions : or rather by the pro- 

 gressive change necessarily brought about by the difference in 

 their accelerations. There are many cases however in which 

 the change is all but imperceptible to ordinary measurement, 

 and many others in which some one dimension is easily measured, 

 but others are hard to measure with corresponding accuracy. 



* Variation and Differentiation in Ceratophyllum, Carneqie Inst. Publica- 

 tions, No. 58, Washington, 1907. 



