104 EEPOET 1891. 



cliaractorise temperature as the criterion of tbermal equilibrium between 

 two or more bodies. 



It has also been deduced that </>! and ij/^ are integi-ating divisors for 

 the two respective systems, so that tliey satisfy the definition given by 

 Carnot's laws. 



The only other property of heat — namely, the principle of limited 

 availability — follows at once on the hj^pothesis of § 26 as to the uncon- 

 strainable nature of the gyrostatic coordinates of the system, and the 

 analogue is therefore complete. 



30. Helmholtz is almost the only writer who has made any attempt 

 at a complete mechanical theory of heat. The other writers have simply 

 endeavoured to show that an equation of the form (2) can be deduced 

 from dynamical considerations by assuming that the kinetic energy due 

 to the uncontrollable motion of the system takes the place of temperature. 

 This assumption is not necessary in Helmholtz's investigations — a great 

 advantage considering our uncertainty as to the nature of temperature. 



Although the properties of temperature are explained by means of 

 monocyclic systems, it cannot be said that they are ])roved. on these 

 hypotheses. Thus, it would be very easy to couple a monocyclic system 

 with two other systems in such a manner that the two latter could not 

 also be coupled together — as, for example, in the case of revolving wheels 

 connected together by cogs. What Helmholtz has done is to show the 

 possibility of dynamical analogues and the conditions they must satisfy, 

 rather than to establish an analogy between all dynamical systems and 

 heated bodies. 



The omission of the work done by intermolecular forces also intro- 

 duces certain restrictions on the generality of the proof. In the vortex 

 atom theory of matter no difficulty of any kind presents itself, because 

 the vortex atoms are essentially monocyclic in character ; but on Bosco- 

 vich's hypotheses there will be difficulties, although these difficulties do 

 not appear insuperable. There seems, for example, no reason why the 

 molecular potential energy should not be controllable, in which case the 

 work done by the intermolecular forces would be of the nature of 

 available energy — available, that is, through the controllable coordinates 

 of the body. Thus, for example, if we suppose a number of molecules 

 enclosed in an envelope at rest under their mutual repulsions, and if we 

 imagine the envelope to expand so that the distances between the mole- 

 cules are increased, the intermolecular forces do work in expanding the 

 envelope, and the whole of this work will be available. Thus there is 

 nothing impossible in such an hypothesis. But it cannot be regarded as 

 axiomatic, and can only be justified if it is found to accord with observed 

 phenomena, among which must be included the Second Law itself. In 

 fact, it must not be forgotten that the object of all such investigations is 

 to discover theories which will account for facts, and not to prove facts 

 by means of theories. 



31. Professor J. J. Thomson's Proof of the Second Law. — The investi- 

 gation now to be considered is one which in its principle and fundamental 

 hypotheses is intimately related to Helmholtz's researches, although the 

 method of proof is somewhat different. I refer to the proof of the Second 

 Law given by Prof. J. J. Thomson in his ' Applications of Dynamics to 

 Physics and Chemistry,' chap. vi. §§ 46-49. It is in connection with 

 this investigation that the author introduces the terms ttnconstrainahle 

 and controllahle, which he uses to distinguish coordinates defining the 



