CALCULATIONS OF POPULATION IN JUNE, 1900 409 



ually increase (thus showing always larger additions per decade) 

 and approximate to an arithmetical progression with common 



difference equal to -. for very large values of p if g were zero ; 

 that without any such supposition J p increases in value as long 



as p is less than* / - , but decreases whenp exceeds-* / - ; and that 



the effect of the constant g is to make the population, when it be- 

 comes very large, nearly proportional to the square root of the time 

 elapsed. If A p is taken as a differential coefficient, it is easy to 

 deduce a value of the time in terms of the population, involving 

 the logarithm of p as well as its first and second powers ; but a 

 statement of p in terms of the time seems to require a series to 

 express it. To treat Ap as a difference instead of a differential 

 only introduces further complexities, so that it will not be worth 

 while to go further into the mathematical discussion of the 

 formula. 



For convenience the constants /and g are made to apply to a 

 population in millions, a million inhabitants being taken as a 

 unit in the calculation. The table to be shown is constructed 

 accordingly. To apply to natural units, /would have to have 

 six ciphers prefixed, while g would require twelve. 



Table of Results. — The results of four calculations will appear 

 in the table. In the first all the coefficients — e, / and g — are de- 

 termined. The census figure for 1870 is rejected, but the law of 

 natural increase is supposed to operate undisturbed from 1860 

 to 1870, as in the decades before and after. This gives the re- 

 sults headed A. , 



In the second calculation g is taken as zero, and, as before, no 

 break is supposed between 1860 and 1870. This calculation is 

 denoted B. 



The third calculation, C, differs from B by supposing that the 

 law of increase, which applies from 1820 to 1860 and after. 1870, 

 is not true for the decade of the civil war, and that a new start 

 must be made from the latter date, the difference between the 

 new value and that calculated from the 1860 figure denoting the 

 effect of the extraordinary losses by wounds, disease, etc., during 

 that decade. 



The fourth calculation, D, agrees with C in recognizing a break 

 after 1860, but it takes /= 0, and so determines e and g. Two 

 lines are given in the table for the date 1870, where the calcula- 

 tion assumes a break, the first showing a normal increase from 



