SCHWEITZER 1950 Problem 10

Consider an arc of a planar curve such that the total curvature of the arc is less than $\pi$ . Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.