The change could now be written : 



AR = AA = Change due to migration 



AB 

 Ar := ^a = Change due to natural causes 



Ab 

 AR + Ar = Total change 1940-1950 



Using the partial derivatives, the equations were: 



Ar =J_Aa + (- a) Ab 



b b- 



AR =_1 AA + (- A) AB 



B B2 



Since small populations (county) were the objects of analysis and since 

 the sum of the partials is not likely to give true increments if the change is 

 large compared to the base, it was felt desirable to minimize this limitation 

 by taking the mean value of the two populations under consideration. Thus 



a = 1940 pop. + 1950 expected population 



2 

 A = 1950 expected pop. + 1950 population 



The use of this device so ordered the values that in all cases the sum 

 of all partials 1/bAa + (— aJ_Ab + 1/BaA + (—A) AB is within 



P B^ 



.002 of the difference between the measure for 1940 and 1950. 



In terms of the meaning 1/b Aa represents change in the numerator 

 of the ratio due to natural causes; — a Ab represents change in the de- 



b2 



nominator due to natural causes; 1/B AA represents change in the numer- 

 ator due to migration; and — A AB represents a similar change in the 

 denominator. ^T 



>9 



If the measure used has more than one term in the numerator, the 

 function can be written in the form a -|- t> ^'''d the equation will become 



c 



1/cAa -\- 1/cAb -f" ( — a-|-b) Ac and equivalently for the change due to 



migration. c^ 



There follows an example worked out for the dependency ratio, in- 

 cluding tabular computation form used and symbols. 



a = mean population 0-14 1940 and expected 1950 



b = mean population 65 and over 1940 and expected 1950 



c = mean population 20-64 1940 and expected 1950 



A ^ mean population 0-14 1950 expected and 1950 actual 



B = mean population 65 and over 1950 expected and 1950 actual 



C = mean population 20-64 1950 expected and 1950 actual 



19 



