36 SHINKISHI HATAI 



Let us first examine the curves (see chart 2). Although the 

 curves are irregular at the ends where the number of observations 

 is not large, yet each can be seen to follow a characteristic course 

 when the middle portion is examined. We notice, as Boycott 

 found, that the length of the internode varies proportionately 

 more than the diameter, since the length of the internode becomes 

 longer for a given diameter as the frogs become larger. In addi- 

 tion these curves are not parallel to one another but diverge 

 more from the curve of group 1 as the frog becomes larger. This 

 divergence in the course of the curves would of course mean that 

 the relative amount of the increment to the internode following 

 to the increase in diameter is not equal, but is greater for the larger 

 frog than for the smaller frog. Since the smaller frog has a 

 shorter internode for a given diameter than in the larger frog, 

 it follows that the longer internode gains in increment propor- 

 tionately more than the shorter internode. This relation suggests 

 that the rate of increment in the length of the internode may be 

 proportional to the length of the internode itself. 



In mathematical terms this corresponds to the expression 



— = hy 

 dx 



where x and y represent the diameter and the length of the inter- 

 node respectively, and h is a constant. 



In order to test this hypothesis the equation has been solved in 

 the following manner: — 



— = hy, — =h dx 

 dx y 



The integration of both terms gives at once the required formula. 



f^ = /if dx + k orlogy =hx + k or finally y= A e hx (1) 

 where A = e k . 



Therefore if our hypothesis is correct, the foregoing exponential 

 equation should adequately express the relation between the 



