Dynamical Theory of Heat and Light. 31 



it be deflected very slightly from motion in that surface, 

 so that it will strike against the inner guide-surface, we may 

 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 com- 

 ponent motion parallel to the tangent plane of either surface. 

 But will its ultimate value be exactly half that of the tan- 

 gential energy, as the doctrine tells us it would be ? We are, 

 however, now back to Class I,; we should have kept to 

 Class II., by making the normal force on the particle always 

 finite, 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 : w r e have now a plane F (floor) with semi- 

 circular cylindric hollows, or semicircular hollows as we shall 

 say for brevity; the motion being confined to one plane per- 

 pendicular to F, and to the edges of the hollows. For defi- 

 niteness 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 (ft) by a constant force. This 

 latter supposition (ft) presents to us the very interesting case 

 of an elastic ball bouncing from a corrugated floor, and 

 describing gravitational parabolas in its successive Mights, the 

 durations of the different flights being in simple proportion to 

 the component of velocity perpendicular to thephine 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 (ft) is 

 illustrated in fig. 8, with only the variation that the corru- 

 gations are convex instead of concave, and that two vertical 

 planes are fixed to reflect back the particle, instead of allowing 

 it to travel indefinitely, either to right or to left. 



§ 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 (ft) constant, 

 Here, as in § 4<>, the times of flight in case (a.) are all the same, 



