202 A. J. Loika — Mode of Growth of Material Aggregates. 



'-- if^* (u) 



I 



da 



f 



(15) 



l p{a) da. 



Substituting (13) in (11) we have : 



c («)=" (16) 



--.00 \ ' 





For the mass of the aggregate we have : 

 M t = N t / c ( a ) w»(«) <^«- 

 If m(a) as well as c(a) is independent of £ 



/CO 

 c(a) w?(a) «fa = constant = m say; (1 



then M t = mlS t 



= M n e 



j-t 



(18) 



The above is the type toward which tends, for instance, a 

 population in which the influence of emigration and immigra- 

 tion is negligible, and general conditions are approximately 

 constant. Equations (10) and (18) then give the number* and 

 mass of the population as a function of the time (geometric 

 progression) ; equations (13), (14) and (15) are three different 

 forms of the same relation between the birth-rate per head and 

 the death-rate per head, while equation (16) gives the " age 

 scale." 



A comparison was made between some observed values and 

 the corresponding figures calculated according to the above 

 formulae. Below are given the results for England and Wales, 

 1871-1880, which, in spite of very considerable emigration and 

 immigration, show comparatively close agreement. For 1881- 

 1890 the divergence is greater. 

 * Compare M. Block, Traite theorique et pratique de statistique, 1886, p. 209. 



