A. J. Lotka — Mode of Growth of Material Aggregates. 205 



The total mass of the aggregate follows immediately from 

 the relation : 



where m = absolute molecular weight of A. 



Similarly, for the aggregate of N' molecules of A' we have : 



f ° k + Jc' v ' 



M'i = m'N/. 



Both l$ t and jS"/ approach towards a limiting value as t 

 approaches oo , viz : 



£L = N ' 



N'_ ' = N ' 



A + A' 

 A' 



& + &'' 



These limiting values, of course, represent a state of equilib- 

 rium, for when they are reached, further increase in t pro- 

 duces no further increase in N. 



In this particular case evidently no knowledge of the form 



of p(a) and c(a) is required in order to determine — - — , and 



Ct v 



therefore ~N t . We can, however, conversely deduce the form 

 of jpici) and c(a\ 



For, since D depends only upon N, and not at all upon the 

 previous history of the system,* it is evident that the stability 

 of the molecule is independent of its age. (Otherwise D would 

 depend on the distribution of ages in the aggregate, and hence 

 on its past history.) 



Hence, if we pick out a large number n out of the total 

 number N of molecules, such that all these n molecules have 

 (nearly) the same age, then the fractional rate of decrease 

 among these will be the same as for the whole aggregate. 



„ . . dn dn ' 



VV e nave then — = =— = kn 



at da 



Therefore n = n a e— ka , 



i.e. p(a) = e~ ka (c) 



In this case we have for the mean length of life : 



l=\ (d) 



Similarly p'{a) = e— k ' a (<?) 



* Except in reactions which have a "period of induction." 



