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