ON OUR KNOWLEDGE OF THERMODYNAMICS. 121 



Conclusion. 



51. Although many of the researches mentioned in this report are not 

 unfrequently called dynamical proofs of the Second Law, yet to prove 

 the Second Law, about which we know something, by means of mole- 

 cules, about which we know much less, would not be in consonance with 

 the sentiments expressed at the end of the last paragraph. The most 

 conclusive evidence for regarding Carnot's principle as a theorem in mole- 

 cular dynamics lies in the remarkable agreement between the results 

 obtained by the methods described in the three different sections of this 

 report, all of which are based on different fundamental hypotheses. It 

 is worthy of note that the method of Clausius alone is independent of 

 any assumptions regarding the nature of the intermolecular forces. 



It has been proved, on each of the various hypotheses, that when a 

 system of molecules undergoes transformations analogous to reversible 

 processes in thermodynamics the molecular kinetic energy T is an inte- 

 grating divisor of the work c^Q communicated to the system through the 

 molecular coordinates. Thus any quantity proportioned to T satisfies 

 the definition of temperature afforded by (2), § 2. The evidence that 

 such a quantity possesses the properties mentioned in § 3 is far less 

 conclusive. These properties have never been investigated by the 

 methods of the first section, while, if the statistical method be adopted, 

 the evidence is confined to the very limited cases in which Maxwell's 

 theorem is valid. The methods of the kinetic theory of gases do not 

 afford a direct proof of any relation between the molecular kinetic 

 energies of two substances which are in thermal contact, but which do 

 not mingle. 



In the volume already alluded to in this Report, Prof. J. J. Thomson 

 claims to have deduced certain thermal properties of matter from the 

 generalised equations of dynamics without the use of the Second Law of 

 Thermodynamics, and he further claims that the results thus obtained 

 afford evidence of the connection between the Second Law and the 

 Hamiltonian principle. It would seem, however, that the novelty of this 

 point of view is not fundamentally very great, for the molecular assump- 

 tions involved in the proofs are identical with those required in order ta 

 deduce the Second Law from dynamical principles. And, moreover, 

 properties of temperature are assumed which, as we have just seen, 

 have not hitherto been satisfactorily deduced from dynamical principles. 



If, on the other hand, we decide, for the present at any rate, to regard 

 Carnot's Principle (like Newton's Laws of Motion) as an axiom based 

 on experience, the researches which we have considered show how this 

 principle may be reduced to a theorem in molecular dynamics by making 

 suitable assumptions as to the nature and motion of molecules. In this 

 way the reversible thermal properties of matter may be represented by 

 means of monocyclic or other dynamical systems, and the fundamental 

 equations of thermodynamics may be replaced by particular cases of the 

 ordinary dynamical equations. This is the point of view adopted by 

 Helmholtz in his valuable paper on the physical meaning of the Principle 

 of Least Action.' 



In conclusion we may reasonably hope that future researches in the 

 domain of molecular science will still further strengthen the bond of 



' Crelle, Journal, c. 



