106 REPORT— 1891. 



while Helmholtz's inYestigations involve the assumptions of (21), namely, 

 that 



opi, ou oil on 



assumptions which characterise the molecular coordinates as gyrostatic 

 or speed coordinates. 



With the above assumptions it is shown that 



^^=;^8 1og/(<^) + 81ogT„ . . . (65) 



an equation analogous to the Second Law (2). Also 



/^9Q^ =-T,/.^^) . . . (66) 



V Vf^J T„ constant VO i „/ * constant 



where <& is the generalised component of external force corresponding to 

 the coordinate ^. This relation is analogous to the well-known thermo- 

 dynamical relation 



(r) =^(l) • • • (^^) 



\0V J B constmit \UV J v constant 



32. J. J. Thomson also mentions the case in which V, the potential 

 energy of the system, is a function of the molecular as well as of the 

 controllable coordinates. But here he tacitly assumes that the molecular 

 coordinates only enter into V in the form of the temperature, an assump- 

 tion quite unjustifiable from dynamical considerations, for no dynamical 

 meaning can be attached to temperature until the Second Law has been 

 completely (vide §§ 2, 3) established by dynamical principles. 



On the hypothesis that T„ is the quantity which is analogous to tem- 

 perature in the dynamical system, the assumption takes the form 



^'Z''=tTf- ■ ■ ■ ■ («8) 



and unless this condition is satisfied the relation (66) will not be true, as 

 J. J. Thomson asserts, when the potential energy is a function of the 

 molecular as well as of the controllable coordinates. 



Concerning the physical aspect of equation (68) Mr. C. V. Burton has 

 suggested to me the following argument : — If we consider a vessel of 

 unalterable volume containing ice, water, and steam at tlie triple point 

 it is evident that heat may be communicated to the system isothermally, 

 the effect being to decrease the quantity of ice and to increase the quantity 

 of water and of steam without altering the pressure or volume. In this 

 case the molecular potential energy would in all probability be increased 

 without any concomitant change in the temperature or in the potential 

 energy of the controllable coordinates. 



33. H. FoincarS on the Applicability of Monocyclic Systems to Irreversible 

 Processes. — The question whether Helmholtz's monocyclic systems can 

 be employed to illustrate irreversible processes has been considered by 

 Mons. H. Poincare,' and answered by him in the negative ; but his inves- 

 tigation is far from satisfactory. 



In the first place, he points out that an irreversible process is only 



' Comptes Rendus, cviii. (1889), p. 550. 



