206 A. J. Lotka — Mode of Growth of Material Aggregates. 

 Introducing the relation (c) into (4), we have : 



D, = kN t I c(a)da. 

 From the nature of c(a) it is evident that 



c(a)da = 1. 



/ 



o 

 Hence D^ = k'Nf, 



which, of course, is not a new result. 



The form of e(a) follows from the relation : 



C *(«) = Y ? 1 , («) (3) 



fr'N' 

 = e -JcaE*_™ (by c)m 



= <r~ ka {l _ e - {k+k , )t) (by a and b). (g) 



When t =ao i.e., for equilibrium : 



c m {a)=ke-*«. (A) 



"We cannot determine c'{a) exactly, as we do not know the 

 ages of the N/ molecules originally present at time t = 0. By 

 simply neglecting these N/ molecules, we can, however, find 

 an approximate expression which will give c'(a) more and more 

 nearly as time advances, and the number of survivors of the 

 original N/ molecules diminishes. 



We thus obtain : 



c t '(a) = ke~ ka -A- — n — ,..,,,, ; . (t) 



v ; k + k'e —(*+*')* v 7 



^(a) = /fcV-*'«. (#) 



Equation (a) holds good (approximately) when t is sufficiently 

 large, for values of a small as compared with t. 



Equation (k) holds good for all finite values of a. 



Equations (A) and (Is) correspond to equations (19) and (21) 

 developed for the general case. 



The above conclusions still hold if other substances enter 

 into the reaction beside A and A', provided that their concen- 

 tration is practically constant, (e.g., if they are present in large 

 excess). 



As a concrete example, we may, for instance, quote the case 

 of the reaction : 



HCOOH + CJTOH ±zr HCOOC„tL + H„0 



