Theorem of the Mechanical Theory of Heat. 41 5 



that is impenetrable by heat; and the extensive remodelling 

 which the theory of the steam-engine, and the theory of thermo- 

 dynamic machines in general, has undergone, a change wherein 

 the second fundamental theorem has played at least as important 

 a part as the rirst. In like manner I might quote from other 

 departments of physics many additional and not unimportant 

 results which have been arrived at as consequences of this theo- 

 rem ; and without doubt the number of such results will become 

 continually greater when the theorem has been more extensively 

 applied. The points of connexion between the two theorems, 

 in the investigations that have been carried out by help of the 

 mechanical theory of heat, are so numerous and so intimate that 

 but few of these investigations are intelligible without a know- 

 ledge of the second theorem. 



It has been said above that the two fundamental theorems of 

 the mechanical theory of heat are very similar to each other. I 

 must now, however, direct attention to an essential difference, 

 the existence of which indicates a very remarkable characteristic 

 of all natural processes. 



In the foregoing considerations from which we deduced the 

 second fundamental theorem, it was made a condition through- 

 out that all the changes that occurred should be reversible — that 

 is, that they should take place in such a way that the reverse 

 changes should be capable of occurring under the same condi- 

 tions. We must now put to ourselves the question, what results 

 should we arrive at if this condition were abandoned ? 



Let us examine, in the first place, the alteration of the dis- 

 gregafcion of a body, connecting our considerations, as before, 

 with the case of a perfect gas which undergoes a change of 

 volume. 



If a gas expands and at the same time overcomes at every 

 instant the greatest possible external pressure that its expansive 

 force enables it to overcome, so that its pressure and the resist- 

 ance are always equal to each other, or at least so nearly equal 

 that the excess of one over the other may be disregarded, then, 

 by the application of the same external force as the gas overcame 

 during its expansion, it can be compressed again to its original 

 volume, all the phenomena taking place during the compression 

 in the inverse direction from what they did during the expansion, 

 but otherwise in the same way. This kind of expansion of a gas 

 is, accordingly, reversible. 



The expansion of a gas may, however, take place in a different 

 manner. Let us suppose a vessel in which the gas is contained, 

 and let us assume that this vessel is suddenly put into commu- 





