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L. A Problem in Age-Distribution. 

 By F. R, Sharps, Ph.D., and A. J. Lotka, M.A.* 



PI^r/IE age- distribution in a population is more or less 

 JL variable. Its possible fluctuations are not, however, 

 unlimited. Certain age-distributions will practically never 

 occur ; and even if we were by arbitrary interference to 

 impress some extremely unusual form upon the age-distri- 

 bution of an isolated population, iu time the "irregularities " 

 would no doubt become smoothed over. It seems therefore 

 that there must be a limiting " stable " type about which the 

 actual distribution varies, and towards which it tends to 

 return if through any agency disturbed therefrom. It was 

 shown on a former occasion f how to calculate the " fixed " 

 age-distribution, which, if once established, will (under con- 

 stant conditions) maintain itself. 



It remains to be determined whether this " fixed " form is 

 also the " stable " distribution: that is to say, whether a given 

 (isolated) population will spontaneously return to this" fixed" 

 age-distribution after a small displacement therefrom. 



To answer this question we will proceed first of all to 

 establish the equations for a more general problem, which 

 may be stated as follows: — 



" Given the age-distribution in an isolated population at 

 any instant of time, the f life curve' (life table), the rate 

 of procreation at every age in life, and the ratio of male to 

 female births, to find the age-distribution at any subsequent 

 instant." 



1. Let the number of males whose ages at time t lie between 

 the limits a and a-\-da be F(<2, t)da, where F is an unknown 

 function of a and t. 



Let p(a) denote the probability J at birth that a male shall 

 reach the age a, so that p(0) = 1. 



Further, let the male birth-rate (i. e. the total number of 

 males born per unit of time) at time t be B(£). 



Now the F(a, t)da males whose age at time t lies between 

 a and a-\-da are the survivors of the B(£ — a)da males born 

 a units of time previously, during an interval of time da. 

 Hence 



F(a, t)da = B(f — a)p(a)da 



F(a,t)=p(a)B(t-a) (1) 



* Communicated by the Authors. 



t A. J. Lotka, Am. Journ. Science, 1907, xxiv. pp. 199, 375; ' Science,' 

 1907, xxvi. p. *J1. 



t As read from the life table. 



