These observation equations follow:
Normal equations for each constant are formed from these observation equations by multiplying each equation by the coefficient of the constant concerned in the equation and adding. This gives us three equations containing three unknown quantities. These unknown quantities are determined by any method and substituted in the general formula for and respectively. For example, in 1900, before the census returns for that year were available, the process above outlined yielded the following equations:
When these equations are solved, it is found that
If we substitute these in the formula, we get
which is the forecast for 1900.
(It should be observed that in this work the year 1790 was considered 1, and 1800 was taken as the origin.)
This estimate proved somewhat low, as the census returns reported 76.3 millions for 1900. This indicates that the population of the country is growing a little more rapidly than would be indicated from its past history.
While the government authorities are at work on the census for 1910, it will be interesting to try this method of forecasting, and to see how well our results will compare with those to be announced later on. I have made a number of equations which are supposed to represent empirically the growth of the population of our country. These have been made in various ways, but all depend upon the parabolic formula, and the method outlined above.
The equations yield the following values for the census of 1910:
