how is it possible that equation 407 is correct? v' is in the coordinates of the rotating reference frame. simply swapping the x and y axes of the rotating reference frame would change v' which would imply that dr/dt=v changes in the inertial reference frame, which is not right. you can't add a position in one coordinate system to a position in another coordinate system and expect a sensible result in the second coordinate system. what is it that i am misunderstanding?