56 REPORT 1891. 



only, the tliermodynamic potential is a minimum, is thus identical with 

 the principle of minimum action. For non-reversible processes the 

 thermodynamic potential tends to a minimum, and this fact expresses the 

 principle of degradation of energy involved in the Second Law, though as 

 yet the corresponding dynamical property has not been worked out. 



J. J. Thomson's applications of his ' theorem ' have no bearing on the 

 subject of this Report, as they do not depend to any extent on the 

 dynamical aspect of the question. 



Section II. — Hypotheses based on the Properties of Monocyclic Systejas. 



19. The peculiarity of the theories to be discussed in this section is 

 that they are not in themselves statistical. They do not therefore postu- 

 late the existence of an infinitely large number of molecules the motion 

 of which, taken individually, is uncontrollable. Instead of this, the funda- 

 mental hypotheses on which they are based have reference to the forms 

 of the kinetic and potential energy as functions of the coordinates of the 

 system. Thus the equations of motion of any finite system of rigid bodies 

 fulfilling the necessary qualifications will give rise to equations analogous 

 in form to those which represent the laws of Thermodynamics. 



Under the present category may be classed Rankine's very early 

 theories, already mentioned, Helmholtz's papers on the statics of Monocyclic 

 Systems,^ and the proof of the Second Law given by J. J. Thomson in 

 his ' Applications of Dynamics to Physics and Chemistiy.' Boltzmann has 

 endeavoured to show how a system satisfying the properties of a mono- 

 cyclic system may be derived from statistical considerations, but this 

 investigation naturally falls under Section III. of this Report. 



Rankine's hypotheses call for no comment, being only very special 

 cases of those of Helmholtz. 



20. H. L. F. von Helmholtz on the Principles of Statics of Monocyclic 

 Systems. — As no account of these papers has hitherto been given in Eng- 

 lish, we shall now consider them somewhat fully. The introductory por- 

 tion has already been noticed in §§ 2, 3. 



Helmholtz defines a polycyclic system as a dynamical system containing 

 one or more periodic or circulating motions. If there is only one such 

 motion, or if, owing to the existence of certain relations between the 

 velocities of the diflFerent parts of the system, the circulating motions can 

 all be defined by a single coordinate, the system is called monocyclic. 



As in other investigations the coordinates of the system fall under 

 two classes — those which, following the suggestion of J. J. Thomson, 

 we have called ' controUahle ' coordinates, and those defining the in- 

 ternal or circulating motions within the system, which that writer calls 

 ' unconstrainahle ' coordinates. In applying the results to Thermody- 

 namics, the latter coordinates are those which fix the positions of the 

 molecules, and thus define the thermal state of the body; they may, 

 therefore, be called ' molecular ' coordinates. 



A polycyclic or monocyclic system is assumed to possess the following 

 properties : — 



(i.) The kinetic and potential energies of the system do not involve 

 the actual values of the molecular coordinates which define the circulating 

 motions, but only depend on their generalised velocities or rates ol 

 change. 



' ' Principien der Statik monocyclischer Systeme,' Crelle, Journal, xcvii. pp. Ill, .S17. 



