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it be deflected very slightly from motion in that surface, so that it will strike against the inner guide-surface, we mav be quite ready to learn, that the energy of knocking about between the two surfaces, will grow up from something very small in the beginning, till, in the long run, its time-integral is comparable with the time-integral of the energy of component motion parallel to the tangent plane of either surface. But will its ultimate value be exactly half that of the tangential energy, as the doctrine tells us it would be ? We are, however, now back to Class I,; we should have kept to (.'lass IL, by making the normal force on the particle always Unite, however great. § 45. Very interesting cases of Class II., § 28, occur to us readily in connexion with the cases of Class I. worked out in §§ 38,41,42,43. § 46. Let the radius of the large circle in § 38 become infinitely great: we have now a plane F (floor) with semicircular cylindric hollows, or semicircular hollows as we shall say for brevity; the motion being confined to one plane perpendicular to F, and to the edges of the hollows. For definiteness we shall take for F the plane of the edges of the hollows. Instead now of a particle after collision flying along the chord of the circle of § 38, it would go on for ever in a straight line. To bring it back to the plane F, let it be acted on either (a) by a force towards the plane in simple proportion to the distance, or (/3) by a constant force. This latter supposition (/3) presents to us the very interesting case of an elastic ball bouncing from a corrugated floor, and describing gravitational parabolas in its successive flights, the durations of the different flights being in simple proportion to the component of velocity perpendicular to the plane F. The supposition (a) is purely ideal; but it is interesting because it gives a half curve of sines for each flight, and makes the times of flight from F after a collision and back again to F the same for all the flights, whatever be the inclination on leaving the floor and returning to it. The supposition (/3) is illustrated in fig. 8 § 47. Let the rotator of §§ 41 to 43, instead of bouncing to and fro between two parallel planes, impinge only on one plane F, and let it be brought back by a force through its centre of inertia, either (a) varying in simple proportion to the distance of the centre of inertia from F, or (/?) constant. Here, as in § 40, the times of flight in case (a) are all the same, | iind in (/3) they are in simple proportion to the velocity of its centre of inertia when it leaves F or returns to it. Fig:. 8. § 48. In the cases of §§ 46, 47, we have to consider the time-integral for each flight of the kinetic energy of the component velocity of the particle perpendicular to F, and of the whole velocity of the centre of inertia of the rotator, |