110 KEPOKT 1891. 



horizontal disc H, the edge of which is in contact with the /ace of the disc 

 G. The motion of the horizontal shaft is transmitted to the vertical 

 shaft B by means of the friction at the point of contact of the two discs 

 G, H. The disc H is capable of being- raised or lowered on the shaft B, 

 and in this way the ratio of the angular velocities of the two shafts A and 

 B can be varied. Lastly, the system is set in motion by turning a handle 

 attached to the shaft A. 



Let m, jn be the masses of the beads on the spokes attached to the 

 shafts A, B ; let r, p be their distances from the axes, iv, u the angular 

 velocities of the shafts, a the adjustable height of the horizontal disc H 

 above the axis of the horizontal shaft C. Boltzmann assumes the disc H 

 to be of unit radius, and the radii of the bevelled cog-wheels connecting 

 A, C to be equal, so that the angular velocities of the shafts A, B are 

 connected by the relation 



oj = aio. 



If, with Boltzmann, we neglect the inertia of everything except the 

 sliding beads, and supposing that r, p, a only vary very slowly, the 

 kinetic energy is evidently 



T = -^ (^mr^iv'^ + /Ao-o)-) = ^ (inr- + /ip-a"^) w^. 



The system has four generalised coordinates, namely, r, p, a, and the 

 angular coordinate corresponding to the angular velocity w. The latter 

 is the only speed coordinate of the system, for the kinetic energy does 

 not involve the rates of change of the other coordinates. 



Hence if we follow Helmholtz's assumptions (i.), (ii.) of § 20, the 

 coordinates r, p, a must be regarded as controllable, and the system is 

 monocyclic. We have, in fact, 



3T 



s = ■■>— = (?)i9-'^ + ii.p-a^)w, T ^ ^ifs, 



and 



dq=^wds = 'Ylcl (2 logs), 



so that T is an integrating divisor of cZQ. 



This result is quite at variance with that found by Boltzmann. The 

 reason is that he has not regarded r, p, a as controllable, but has included 

 in dQ the woi'k brought into the system through these coordinates. 

 This work properly belongs to —dW of equation (i.), § 2, and not to cZQ. 



In varying the height a there would, in the natural course of events, 

 be a loss of energy through friction, as the edge of the horizontal disc 

 H would have to slip up or down in contact with the face of the vertical 

 disc G. This slipping may be avoided by shifting the vertical shaft B 

 slightly to one side or the other of the vertical plane through the 

 horizontal shaft C. The friction between the rotating discs will then 

 cause H slowly to rise or fall (as the case may be) automatically and 

 without slipping. 



This simple device obviates a difficulty which in Boltzmann's original 

 paper requires several pages of explanation. 



38. Simple Mechanical Model of Garnofs Beversihle Heat-Enr/ine. — The 

 following model appears to be new. It may be of interest as furnishing 

 a mechanical representation of the properties of the source and 

 refrigerator of a perfect heat engine, although to do this it is necessary 



