426 ON INCREASE IN SIZE [pt. iii 



curve, which falls so markedly during the same time. Thus the 

 growth-rate expressed in this manner also falls off as time goes on, or 

 rather rises less and less rapidly, becoming eventually asymptotic to 

 the mature value. 



This curve is undoubtedly a great improvement on Minot's, for 

 it involves no arbitrary time period and depends on the differential 

 calculus, which has as its special province the evaluation of instan- 

 taneous change. An infinitesimally small period dt, and an infini- 

 tesimally small increase of weight dW, are the basis of its operation. 

 Then all the infinitesimal differences can be added together (i.e. 

 integrated). Thus: .j^ 



"^ _ 



becomes, if the number of dfs, and dW's is infinitely large [n) , when 

 integrated, W = Ae^*, for 



I + ^y = e^\ 



A being the weight at the beginning of the whole period. If this is 



turned into logarithms 



log W =\og A ^ kt, 



, _\ogW - log A 

 or k , 



and, as where growth is being considered A is, to all intents and 



purposes, o, ^ ^ 



k = -^ — . 

 t 



The instantaneous relative growth-rate for a unit time is the sum 

 of all the instantaneous rates during the given unit of time, and may 

 therefore be multiplied or divided, according to the time-unit in 

 which it is desired to express it. 



The log. weight/age graph is, therefore, a measure of the instan- 

 taneous growth-rate, and the value of the constant k which can be 

 calculated from the last equation will give the slope of the straight, or 

 approximately straight, line. The log. weight/age graph could, of 

 course, have been plotted by Minot, but Brody's use of the differential 

 calculus was required to show that the slope of the curve gave an 

 instantaneous growth-constant. Thus, in the example given, Lamson 

 & Edmond's data, the constant is 56 from the 5th to the 8th day of 



