lotka: a natural population norm 



243 



In these formulae b is the birthrate per head per annum, d 

 the corresponding deathrate, r is the ''natural rate of increase" 

 of the population, i.e., r = b — d; p{a) is the probability at birth 

 that a given individual picked at random (at birth) shall reach age 

 a, and is the function tabulated in the so-called life tables, where it 

 is commonly denoted ?x; p(«) is the derivative of p(a) with regard 

 to a. Lastly, c{a) is the coefficient of age-distribution, which is 



Fig. 1 . Age-distribution in the living population, England and Wales 1871-1880. 

 Areas represent numbers of persons. Males. 



so defined, that, out of the total number iV of the population, 

 the number comprised between the age-limits a and' (a + da) 

 is given by Nc{a)da. 



Now the formulae, as given above, are not in the form best 

 adapted for certain numerical computations. They were there- 

 fore developed in series by expanding the exponential function 



