312 THE MATHEMATICAL PRINCIPLES [BJOK 11. sisting medium will be as the ordinate of the circle or ellipsis described upon the diameter Ba ; and therefore the figure BKVTa will be nearly ac ellipsis. Since the resistance is supposed proportional to the velocity, le\ OV be the exponent of the resistance in the middle point O ; and an ellip sis BRVSa described with the centre O, and the semi-axes OB, OV, will be nearly equal to the figure BKVTa, and to its equal the rectangle Act X BO. Therefore Aa X BO is to OV X BO as the area of this ellipsis to OV X BO; that is, Aa is to OV as the area of the semi-circle to the square of the radius, or as 1 1 to 7 nearly ; and, therefore, T7T Aa is to the length of the pendulum as the resistance of the oscillating body in O to its gravity. Now if the resistance DK be in the duplicate ratio of the velocity, the figure BKVTa will be almost a parabola having V for its vertex arid OV for its axis, and therefore will be nearly equal to the rectangle under f Ba and OV. Therefore the rectangle under