On Hamilton's Dynamic Principle in Thermodynamics. 427 



it is indeed true that it is not very conceivable how any con- 

 nexion capable of being clearly indicated can exist between that 

 equation and the second proposition of the mechanical theory of 

 heat. 



Clausius then quotes the original form of Hamilton's equation 

 used by me, and proves that Boltzmann's equation (23 a), rightly 

 interpreted, perfectly corresponds with Hamilton's, is indeed 

 identical with it. But what he concedes in relation to Boltz- 

 mann's equation, that he decidedly questions in reference to his 

 own; nay, he declares without hesitation the correspondence 

 impossible. Thus the same question is again raised which has 

 been discussed in the Annalen, between Clausius and Boltz- 

 mann*, on the occasion of the claiming of priority by the latter, 

 namely :— Does there actually exist a difference, and, if this is 

 the case, what is the difference, between the equation of Clausius 

 and that of Boltzmann ? I believe that I am now entitled — nay, 

 after the reply of Clausius, am bound to enter into the discussion 

 of the question, at least in regard to Hamilton's equation. 



I will first discuss the equation which Clausius claims as his 

 own. 



For the more convenient elucidation of the matter, we will 

 first of all, as Clausius does, confine ourselves to the considera- 

 tion of the periodic motion of a single point. Given, therefore, 

 a material point which with the original motion describes a closed 

 path, or moves between two given points. The only forces act- 

 ing on the movable point shall be such as possess a force-func- 

 tion (or, as Clausius is accustomed to express it, an ergal) — that 

 is, forces of which the components can be represented by the 

 partial differential quotients, taken negatively, of a function of 

 the coordinates of the point. Let us now suppose that this mo- 

 tion undergoes an infinitesimal alteration. With the changed 

 motion let the point likewise describe a closed path, or move 

 between two points, which latter may be either identical with 

 those previously given, or, if not so, fulfil the condition that the 

 quantity 



dx ~ ( dy 5, , dz ~ 



has the same value at the terminal point of the motion as at the 

 initial point. Then let i denote the period of the motion, T the 

 vis viva, U the force-function ; and let us agree to indicate the 

 mean value of the variables by putting a horizontal stroke over 



* Pogg. Ann. vol. cxliii. p. 211 : "Zur Prioritat der Auffindung der 

 Beziehung zwischen dem zweiten Hauptsatze der mechanischen Warme- 

 theorie und dem Principe der kleinsten Wirkung; von L. Boltzmann." 

 Ibid. vol. cxliv. p. 265: " Bemerkungen zu der Prioritatsreclamation des 

 Hrn. Boltzmann; von. R. Clausius." 



