108 RKPORT— 1891. 



that tbe geometrical equations which that author has investigated 

 will any longer hold good ; the same may also be said with regard to the 

 alternative hypothesis underlying J. J. Thomson's investigation. More- 

 over, even if the latter hypothesis be assumed to hold good for an 

 unequally heated body, the function which plays the part of tempera- 

 ture will be the whole molecular kinetic energy, so that instead of the 

 entropy we shall obtain an expi'ession which does not alter in value as 

 the temperatures of the various parts become equalised. Another 

 hypothesis, which does not seem to me to be unreasonable, is that 

 possibly irreversible changes may take place when any portion of the 

 potential energy of the system depends partly on the molecular as well 

 as on the controllable coordinates of the system, so that this portion of 

 potential energy, as well as the kinetic, is uncontrollable. But then 

 thei'e appear to be no grounds, except from statistical considerations, for 

 supposing that this enei'gy will all be rendered kinetic by the action of 

 the intermolecular forces. Such would certainly not be the case in a 

 system possessing only one or two degrees of freedom. 



The consideration of dissipative forces, such as friction, is of course 

 precluded by the conditions of the problem, for their presence would be 

 a violation of the principle of Conservation of Energy. And as we are 

 thus left with a dynamical system which is pei'fectly reversible (provided 

 that the system is complete and all the velocities are reversed), it seems 

 necessary to accept the principle of degradation of energy as a statistical 

 property and not as a dynamical principle. We shall consider the matter 

 more fully in the third section of this Report. 



35. Dr. Ludwig BoUzmann on the MecJianical Representation of Mono - 

 cycles. — In his paper on the properties of monocyclic systems, already 

 referred to,^ Dr. Boltzmann discussed at great length a mechanical model 

 illustrative of a system in which it appeared not only that dQ/T was not 

 a perfect diflerential, but that clQ did not possess any integrating factor 

 whatever. 



In a volume only just published ^ Boltzmann has again taken up the 

 representation of monocyclic systems by means of mechanical models, and 

 has slightly elaborated ideas suggested in Helmholtz's papers. On 

 account of their greater simplicity we will consider the latter represen- 

 tations before the former. ■ 



As a simple example of a monocyclic system Boltzmann takes a 

 vertical revolving shaft having attached to it a horizontal spoke along 

 which a bead can slide without friction. A string, which is attached to 

 the bead, passes over a small pulley close to the shaft, and hangs freely, 

 •can-ying a scale-pan, on which varying weights can be placed. The 

 arrangement may be illusti-ated by the shaft C D and the spoke carrying 

 the mass E in the figure of § 38. 



If we suppose the shaft and spoke to be without mass, and if m be 

 the mass of the bead, r its distance from the shaft, w the angular velocity, 

 T the kinetic energy of the system, and clQ the amount of work performed 

 by turning a handle attached to the shaft, we have 



'^^=dlog(r*u.^-) .... (70) 



' Crelle, Journal, xcviii. p. 88. 



= Vorlcsungen ilher Maxirell's Thcorie der Electricitdt und des Lichtes, I. Theil 

 ^Leipzig: Johaun Ambrosius Barth, 1891), pp. 8-23. 



