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



1 is evidently the " mean length of life " ; its reciprocal then, 

 by (20), is that value of 1) which just suffices to keep the 

 aggregate from decreasing under the conditions corresponding 

 to the particular form of p(a) by which I is given. 



Thus, in the case of England and Wales, while according to 

 the three life tables 1838-54, 1871-80, 1881-90, the mean 

 length of life has risen from 40*9 to 42"9 and 45*4, the 

 "equilibrium birth-rate per head" has been corresj^ondingly 

 reduced from -0245 to "0233 and -0220. 



II. 



Isothermal Monomolecular Reaction. 

 "We will consider a system undergoing the chemical change, 

 A' < > A 



which we will suppose to take place at constant volume and 

 temperature, and in a homogeneous system. 



Let N denote the total number of molecules of A. 



" >J' " " " " " " " A' 



Also, when t = 0, let N = 



N'.== n; 



Hence, N+N' = N '. 



Let D' denote the number of molecules of A' decomposed per min. 

 a Ty " " " " " " A " " " 



« B' " ' k " " " " A' formed " " 



cc "D Li « a tc « a A a a a 



The reaction is monomolecular in either sense. Hence, by 

 the law of mass action : 



(k and k' are the reaction coefficients.) 

 Then, for the aggregate of N molecules of A. we have : 

 B = D' = k'W = #(N Q '-N). 



Integrating 



, = B-D = *'(N i '-N)-»*N 



= k'X () '-{k + k , )?$. 





