552 



THE AMERICAN NATURALIST [Vol. LV 



as large as a, it is possible to trace the curve in both 

 directions until it reaches zero values of dx/dt. The ac- 

 tual values of dx/df in the second cycle are much smaller 

 than those in the first cycle. 



Attention must be drawn to the way in which these two 

 curves overlap. It will be noted that the second cycle 

 of growth began shortly after the rate of the first cycle 

 began to decline and that the first cycle continued to 

 about the time of maximum rate in the second cycle. The 

 true curve of the growth rate of these animals is, there- 

 fore, the arithmetical sum of the values of dx/dt for the 

 various values of t. This is shoMTi in Fig. 6 and agrees 

 very well with the observed weekly increments. 



The curves in Fig. 6 were obtained by plotting dx/dt 

 as ordinate and t as abscissa. 



If we let dx/dt = s, we may write 



which when differontiatod becomes 



If the right-hand member of this equation be equated to 

 zero it will give the values of x for which z is either max- 

 imum or minimum. 



Let ak — 2kx — ; then 



Therefore the rate, z, is either maximum or minimum 

 when x^a'l. To find whether z is maximum or min- 

 iinuin it is only necessar^^ to get the second differential 

 of the above equation 



Sill,-,' <|unrility has a lu-alivc sign z i=dx/dt) is a 

 maxiimim when j a J. hi other words the rate of 

 growth is a maximum when the cycle has reached a stage 

 at which the weight of the animal is half the weight it 

 attains at the end of that qycle. When x<{a/2) the curve 

 rises and when x> (a/2) the curve falls. 



