curve toward the right, at right angles to the tangent, that is, in the direction
and let the length of this normal be
Then, evidently, we have
or
Since now, when
is infinitely small,
and since on the curve itself
vanishes, the upper signs will hold if
on passing through the curve from left to right, changes from positive to negative, and the contrary. If we combine this with what is said at the end of Art. 2, it follows that the curve is always convex toward that side on which
receives the same sign as
For example, if the curve is a circle, and if we set
then we have
and the curve will be convex toward that side for which
as it should be.
The side toward which the curve is convex, or, what is the same thing, the signs in the above formulæ, will remain unchanged by moving along the curve, so long as
does not change its sign. Since
is a continuous function, such a change can take place only when this ratio passes through the value zero. But this necessarily presupposes that
and
become zero at the same time. At such a point the radius