168 Clausius's Theorem for Irreversible Cycles. 



density in some other body." The truth of this statement, 

 and the conclusions therefrom, I must directly contest. 

 There is nothing to hinder us from supposing that the gas- 

 holder together with the weights, which are acting against 

 the gas pressure, are placed in a vacuum. If, then, the 

 gas pressure lifts the weights, the gas expands without 

 producing a change of density in some other body. That 

 it is impossible to obtain an absolute vacuum, that abso- 

 lutely unchangeable weights do not in reality exist, and 

 that there may be other difficulties opposing the realization 

 of this process, does not of course affect the validity of 

 the proof. J am sorry to see from this misconception, that 

 the note, which Dr. Ogg on my advice added to that par- 

 ticular part of his translation of my Thermodynamics, has 

 not been sufficient to make the point clear. 



8. If a process, e. g. the flow of a gas into a vacuum, takes 

 place so violently that one can no longer define temperature and 

 density, then the usual definition of Entropy is inapplicable. 

 Of course, on this point I completely agree with Orr and 

 Bertrand. My observations with regard to the entropy 

 of a gas (p. 512) which is not in a state of equilibrium, refer 

 only to the case where we may speak of a temperature and a 

 density of the gas in all its parts. But, as is well known, 

 Herr L. Boltzmann has shown from the point of view T of the 

 Kinetic Theory of Gases, that, even in the case of violent 

 motion, a definition of Entropy, which includes the usual one 

 as a special case, may be deduced from the Theory of Proba- 

 bilities. The suppositions of the Kinetic Theory of Gases, 

 however, were purposely excluded from my Treatise on 

 Thermodynamics. 



4. Prof. Orr gives (p. 518) u A shorter proof of the principle 

 of Increase of Entropy, if the substance of Planck's definition 

 of Irreversibility be adopted." Closer examination shows 

 that this proof starts with the assumption expressed in Lord 

 Kelvin's version of the Second Law, and that, therefore, the 

 essential import of the thing to be proved is already assumed 

 to be true. I cannot, therefore, accept this proof as an 

 improvement on my own. 



With these remarks I shall conclude. I should not, how- 

 ever, have again given my views on these questions at such 

 length, had it not been that I wish to express my pleasure at 

 having found in Prof. Orr's paper much that was stimulating 

 and interesting to me. 



I am, Yours faithfully, 



Dr. Max Planck. 



