Method of Least Squares. 365 
sections on each side of the origin, and that the space on each 
side of the origin is divided by ordinates (one at the origin, 
one at infinity, and two intermediate) into three compartments 
such that for each compartment the area bounded by the 
abscissa, ordinates, and primary curve is equal to the area 
bounded by the same right lines and the average curve. 
Suppose, further, that throughout the first and third compart- 
ments the disutility-function is almost level, the disadvantage 
of a particular error only just increases with the extent of error, 
while in the second compartment the same function rises steeply. 
Then the appropriate mathematico-psychical reasoning will 
show that it is better to abide by a single observation than 
to take an average. 
The two preceding examples, to which it would be easy to 
add others similar, are not put forward as practically impor- 
tant, but rather (like the imaginary cases put by Hume in 
his inquiry concerning moral sentiments) as assisting us to 
distinguish the general principle from the particular rule, and 
in the act of doing so to discern the supremacy of the prin- 
ciple of utility. 
We have next to review some typical instances of the 
problems solved by the method of least squares. Let us take 
as the principle of a rough classification, complexity. As 
a first division we may demarcate those cases in which there 
is but one measurable; a single quantity x, whereof x x , x 2 , &c. 
are values, or more generally functions, given. This class 
may be subdivided according as the facility-curves which 
generate x x , x 2 , &c. are (I.) or are not (II.) symmetrical. In 
the simpler cases we need not take the trouble of distinguishing 
between the most probable and most advantageous values, since 
they are coincident. But the distinction soon emerges. 
Subclass I. may be subdivided according as the symmetrical 
facility-curves under which the observations are ranged are 
(A) or are not (B) Probability-cuvves. 
I. A (1) The following is one of the simplest problems 
which our subject presents : — Given a set of observations x 1} 
x 2 , &c, and given that they have been generated by diver- 
gence from an unknown point according to one given law of 
error, a probability-curve of given modulus, to find the most 
probable (and advantageous) value for the unknown point. 
By a familiar application of the. differential calculus the sought 
value is the mean of the observations. 
A variant of this problem is when the observations are dis- 
tributed into groups, each of which diverges from one and the 
same unknown point according to different given probability- 
curves. The solution is of course the weighted mean. 
