LENGTH OF THE INTERNODES 43 



senting the diameters of 5.3, 6.3 and 7.3 micra respectively are not 

 parallel, but show a slight divergence (see also Takahashi '08, 

 tables 4 to 8). The amount of divergence is very slight, but is 

 regularly greater for the line corresponding to the greater diameter. 

 This* relation agrees with that found to exist between the two 

 characters in the frogs of different sizes. We feel justified there- 

 fore in concluding that our hypothesis that the relation between 

 the diameter and the length of the internode is exponential, even 

 when applied to the length of the internodes from the different 

 segments of the leg of the same frog, is correct. 



On the basis of the preceding argument, we present the follow- 

 ing final conclusion: — The exponential equation 



h x 



y = Ae 



expresses the relation between the length of the internode and 

 its diameter either in different segments of the leg of the same 

 frog, or in frogs of different sizes. This law seems applicable to 

 both Rana pipiens and Rana temporaria (fusca). 



Thus far we have merely demonstrated that when {he data 

 are examined, the relation existing between the two characters 

 under consideration is adequately expressed by an exponential 

 formula, but we do not make any inference as to the immediate 

 factors which bring the two characters during the growth period 

 into such exponential relation. As one of the factors, Takahashi 

 counts the segmental influence, by which I understand the elonga- 

 tion of the internodes of Ranvier depend on the elongation of the 

 segments of the limb in which they are found. 



If this view of Takahashi is correct, we shall be justified in con- 

 cluding that the growth of the segments in Rana pipiens follows 

 also an exponential formula. Although unfortunately I have not- 

 sufficient data to test this point just mentioned in either Rana 

 pipiens or Rana temporaria, I find it to be true at least in the case 

 of the leg of the toad as shown by the recent investigation of 

 Kellicott on Bufo lentiginosus (1907) in which the rate of incre- 

 ment in the length of the segments is proportional to the length 

 of the segment itself. 



