Vol. 6, 1920 STATISTICS: PEARL AND REED 283 
we do not regard equation (ix) as the final development of this type of 
equation for representing population, and we have no desire to encumber 
the literature with a mathematical discussion which we expect later to 
discard. 
For present purposes it will be sufficient to fit (ix) to the observations 
by passing it through three points. Given three equally spaced ordinates, 
yi, y2 and ys, the necessary equations are : 
b ^ 2yiy2y3-y2%yi + ^3) ^^^^^ 
c yiyz — yi^ 
a = logio — r h logio e (xv) 
y{- - n) 
where h is the abscissal distance in years between yi and ^2, or y^ and y^. 
c = -i-fi^- - (xvi) 
y,-y\e-"' ^ ' 
where a is the abscissal distance in years from the origin to yi. 
Putting xi at 1790, at 1850, and xzot 1910, and taking origin at 1780 
we have 
y, = 3,929^ 
« = 10 
y2 = 23,192 
h = 60 
ys = 91,972 
and taking (ix) in the form 
b , ... 
y = , (xvn) 
y = .- 0313395. . (^^") 
we find these values for the constants ; 
2,93 0.3009 
+ 0.014854 
The closeness with which this curve fits the known facts is shown in 
table 3. 
The closeness of fit of this curve is shown graphically in figure 3. 
Though empirically arrived at this is a fairly good fit of theory to 
observations. The root-mean square error from the last column is 463,000, 
or slightly smaller than that from the logarithmic parabola in table 2. 
It must not be forgotten, however, that the root-mean square error is 
reduced in the present case by virtue of the fact that in three out of the 
13 ordinates theory and observation are made, by the procrustean method 
of fitting, to coincide exactly. The most that can be asserted is that 
