6o6 TRAXS. ST. LOUIS ACAD. SCIENCE. 



VALUE OF THE FORMULA FOR PREDICTION. 



How closely formula (i) will continue to represent the growth 

 of population during future decades depends, of course, upon the 

 continuance of the same conditions of growth. A decided change 

 in the birth-rate, or rate of immigration, or a destructive war, 

 would bring out a large discrepancy between the computed and 

 observed values. A fair test of*the formula is found by com- 

 puting the population for 1S90. According to the formula we 

 should expect in 1S90 a population of 62,677,280, The Census 

 Bureau has within the last few weeks finished its count of the 

 population in 1S90, obtaining the result 62,622,380. The agree- 

 ment between these two results is all that could be desired, the 

 difference of 55,000 being within the limit of error of both the 

 formula and the census count. , 



The general law governing the increase of population, as usu- 

 ally stated, is that, when not disturbed by extraneous causes, such 

 as wars, pestilences, immigration, emigration, <&.c., the increase 

 of population goes on at a constantly diminishing rate. By this 

 it is meant that the percentage of increase from decade to decade 

 diminishes. The law of growth expressed by equation (i) in- 

 volves such a decrease in the percentage of growth. 



Differentiating equation (i) we have 



dt B -f- 2C^ -h3D^2 



A 4- B/- 4- C/2 -f D i-a 



which diminishes as t increases, and approaches zero as t ap- 

 proaches infinity. In 1790 the percentage of increase per decade 

 was 33 per cent. ; in iSSo 24 per cent. ; in 1990 will be 13 per 

 cent., and in i.ooo years will have sunk to a little less than 3 

 per cent. 



In order to include all available data, I have re-solved for A 



