1900.] on the Dynamical Theory of Heat and Light. 389 



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 arc, 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 connection 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 

 holloics, 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 dis- 

 tance, or (/3) by a constant force. This latter supposition (/?) presents 

 to us the very interesting case of an elastic ball bouncing from a cor- 

 rugated floor, and describing gravitational parabolas in its successive 

 flights, the durations of the different flights being in simjde proportion 

 to the component of velocity perpendicular to the plane F. The sup- 

 position (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, with only the variation 

 that tho corrugations 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 (/?) constant. Here, as in § 46, the times of flight in case 

 (a) are all the same, and in (/3) they are in simple proportion to the 

 velocity of its centre of inertia when it leaves F or returns to it. 



§ 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, which is itself perpendicular to F. 

 If q denotes the velocity perpendicular to F of the particle, or of the 

 centre of inertia of the rotator, at the instants of crossing F at 

 the beginning and end of the flight, and if 2 denotes the mass of the 

 particle or of the rotator so that the kinetic energy is the same as 

 the square of the velocity, the time-integral is in case (a) \ q 2 T and in 

 case (/3) % q 2 T, the time of the flight being denoted in each case by T. 



