The Co-Efficient of Humidity. 141 
whose tangent is the mean value of the coefficient of humidity so 
as to satisfy the equation 
y = 11 ix. 
Using the values m—2'3 for the peats, and w=3 f 0 for the sub¬ 
peats lines are obtained that show the individual points almost 
equally distributed on either side of the lines. This should be the 
case and in the case of the peats no further analysis seems necessary 
and the equation y=mx may be accepted provisionally as a satis¬ 
factory solution. 
VII. — Reduction by the Method of Least Squares. 
But in the case of the sub-peats there is an obvious objection 
to the acceptance of the same equation even when m has a different 
value. For, in any case, when x—o , according to the equation y=o, 
but as a matter of fact, when the humus in the soil is reduced to 
zero there is still some water held by the soil particles. So that 
for these soils, with humus-content below 10 or 15% it is necessary 
to fall back on the equation 
y = b + nix 
where b represents the residual water present in the soil indepen¬ 
dently of any humus. 
It is impracticable to find the value of b by the usual method 
of making x=o, and hence b=y, because no soil is met with entirely 
devoid of humus. But the most probable value of both b and m 
may be calculated from the experimental results by the method of 
least squares—a method of greater refinement than the occasion 
requires, but the only one available. If the linear equation is 
written in the form 
b -f- nix — y = 0 
the values of y calculated from the equation will differ from the 
experimental values yi,y. 2 ,y 3 , corresponding to experimental values 
x lf * 2 » *V Let these differences be S lt S 2 , S 3 , Then 
b + mx 1 — y r = S x 
b + mx 2 — y 2 = S 2 
The method of least squares shows that when the sum of the 
squares of these differences is a minimum the calculated values of 
the constants b and m are the best that can be deduced from the 
experimental results, supposing all the observations are of equal 
accuracy and may therefore be given equal weight. 
To obtain the values of b and m in the linear equation 
y =J> + nix 
