200 A. J. Lotka — Mode of Groiotli of Material Aggregates. 



We will for the present restrict our considerations to cases 

 in which : 



1. The individuals are either all of one class as regards their 

 general properties, and especially as regards those which affect 

 the character of the limitation of their "life period" ; or, if they 

 belong to a number of different classes (e.g., males and females 

 of a community of living organisms, etc.) then the relative 

 proportion of individuals of each class among those formed 

 during any element of time is constant. 



2. The ''length of life" of each individual is independent 

 of the total number of individuals in the ao-ffreo-ate, and of the 

 distribution of ages among them. 



3. The general conditions of the system, in so far as they 

 affect the "length of life" of the individuals (see §2), are, on 

 an average, uniform and constant throughout. 



4. The variations in the conditions of the system are of such 

 fixed type that, when conditions 1, 2 and 3 are satisfied, the 

 number of individuals surviving age a out of any large number 

 Cl counted at the moment of their formation and picked out at 

 random, can be expressed in the form flp(a), where p(a) is -a 

 (univalent) function containing only a. 



Then, if c(a) is such a factor that out of the total N f the 

 number of individuals whose age lies between the age limits 

 a and (a + da), is given by N t c(a)da, it readily follows that 



<•(«) = %fi>(«). m 



^=^f\ a) ^sm da . (4) 



"We may substitute these values in (2) : 



ST t = N o + f\dt + C N,/* *c(a) d ] °S p ( a K u da. (6) 



^'o *-^ o o 



= K a +jT t B t dt+£ t J^Br_ a ^§j£*t da. (7) 



Lastly, given that the average mass of one individual at age 

 a is m{a), we have for the total mass M f of the aggregate at 

 the instant t : 



M t = N A c(a) m(a) da. (8) 



=.J B t _ a p(a) m (a) da. ( 9) 



