and therefore
We have also
If, therefore, we set, according to a well known theorem,
then we have
therefore
or, since
The ambiguous sign in the last formula might at first seem out of place, but upon closer consideration it is found to be quite in order. In fact, since this expression depends simply upon the partial differentials of
and since the function
itself merely defines the nature of the curve without at the same time fixing the sense in which it is supposed to be described, the question, whether the curve is convex toward the right or left, must remain undetermined until the sense is determined by some other means. The case is similar in the determination of
by means of the tangent, to single values of which correspond two angles differing by
The sense in which the curve is described can be specified in the following different ways.
I. By means of the sign of the change in
If
increases, then
must be positive. Hence the upper signs will hold if
has a negative value, and the lower signs if
has a positive value. When
decreases, the contrary is true.
II. By means of the sign of the change in
If
increases, the upper signs must be taken when
is positive, the lower when
is negative. The contrary is true when
decreases.
III. By means of the sign of the value which the function
takes for points not on the curve. Let
be the variations of
when we go out from the