40 SHINKISHI HATAI 



Obviously no simple curve will represent the system of points 

 shown in group 4 and 5 better than the continuous lines obtained 

 by the exponential equation. There is no doubt that the diminish- 

 ing value of the ordinates following the increase in diameter, to 

 be seen at the right end of the curves in groups 3 and 4, must be 

 due to the small number of observations, as well as high irregu- 

 larity in the length of the node, since it is not only opposed to the 

 general results, but the three remaining curves show contrary 

 relations. We are justified in assuming then that the relation 

 between the length of the internode and diameter in different 

 sized frogs is exponential, as it was supposed to be, and the law 

 can now be stated as follows : 



The rate of increment in the length of the internode following 

 the increase in diameter is proportional to the length of the inter- 

 node itself or 



dx 



In this connection we have tested three more hypotheses 

 touching the rate of increment in the length of the internode. 



1. Namely that the rate of increment is proportional to the 

 diameter : 



— = hx 

 dx 



2. That the rate of increment is proportional to the product 

 of the length of the internode and diameter: 



dy 7 

 — - = nxy 

 dx 



3. That the rate of increment is proportional to the quotient 

 of the length of the internode when divided by the diameter: 



dy = h y 

 dx x 



When however these three equations were tested, it was found 

 that so far as the constants are to be determined from the obser- 

 vations, all these equations give a very poor fit to the observed 



