88 REPORT— 1891. 



Clerk Maxwell's theorem, named after its discoverer, was tlie first 

 attempt at a kinetic analogue of thermic equilibrium. It was generalised 

 by Boltzmann, and afterwards further generalised by Maxwell himself ; 

 but the latter extensions are probably incorrect, as we shall see here- 

 after. 



Having thus briefly mentioned the earliest researches on the present 

 subject, let us turn to a consideration of the papers themselves, beginning 

 with the writings of Clausius and Szily. 



Section I. — The Hypothesis of Stationary or Quasi-Feriodic Motions. 



10. Clausius and Szily. — In 1870 Clausius showed that when a sys- 

 tem of particles is in stationary motion, the mean vis viva of the system 

 is equal to its virial.' About a year later he gave a proof of the Second 

 Law, based on the laws of motion, in a paper entitled ' On the Second 

 Axiom in the Mechanical Theory of Heat.' - The methods of proof 

 employed by Clausius in this paper are very laborious and complicated, 

 while his arguments are artificial and, in places, not very intelligible. 



Soon after Clausius' paper had appeared, Szily endeavoured to show 

 that ' what in the mechanical theory of heat is called the Second Law is 

 nothing other than Hamilton's Principle of Least Action.' ' The proofs 

 which Szily gave are, in many places, quite at variance, not only with 

 the principles of dynamics, but also even with the laws of Thermo- 

 dynamics themselves. Thus he repeatedly mistook f?E for cZQ, and tried 

 to show that fZE/T is a complete difierential (a result not in general 

 true) ; moreover, in endeavouring to account for the principle of degra- 

 dation of energy in a non-reversible cycle, he altogether ignored the First 

 Law, and supposed some of the molecular energy of the system to be 

 actually lost or annihilated by friction, viscosity, or imperfect elasticity 

 of the molecules, or by other similar resistances. In consequence he had 

 to employ methods of proof that were far from rigorous, and even, in 

 many instances, illogical. 



Szily's papers seem, however, to have had one good eSect — namely, 

 that of stimulating Clausius to remodel his investigations in a simpler 

 and more intelligible form. Those who care to examine the original 

 papers of these writers will find them translated in the volumes of the 

 ' Philosophical Magazine ' from 1871 to about 1876. Among them is a 

 paper by Szily,^ in which he claimed to have deduced the Second Law 

 from the First ' without any further hypothesis whatever.' Yet Szily 

 based this investigation on two hypotheses which are hardly more 

 axiomatic than Carnot's principle. 



11. Clausius' Methods. — It would be useless to enter into further criti- 

 cism. We now proceed to give a proof of the Second Law based on 

 the methods of Clausius, with the object of bringing into prominence the 

 more salient features of his investigations, and of presenting them in a 

 concise form. 



The assumptions which form the basis of Clausius' proof may be stated 

 as follows : — 



(i.) In the steady or undisturbed state of the system the motion of 

 the molecules shall be stationary or quasi-jjeriodic ; in other words, the 

 potential and kinetic energies of the molecules shall fluctuate rapidly 



' Phil. 3Iag. vol. xl. (1870), p. 122. - Ihid. vol. xlii. (1871) (September). 



» Hid. vol. xliii. (1872), p. 339. * Ihid. V. series, vol. i. (1876), p. 22. 



