Theorem for Irreversible Cycles. Ras! 
state into any other in which the sum of the entropies is 2¢’, 
and accordingly into a state in which the first n—1 gases 
have respectively the entropies 1, de, . - - dn—1, and the nth in 
consequence the entropy $,' + de’ +.. -¢n' —¢1—ge—- - » bri, 
which by supposition is less than gx. The first n—1 gases 
may now be restored to their original states by rever- 
sible adiabatic processes. The entropy of the nth gas has, 
therefcre, been diminished without leaving changes in other 
bodies, which has been shown to be impossible. The pro- 
position thus follows. 
4, From these theorems it then follows that if a gas-system 
undergoes an “irreversible” (“irreversibel”) process in 
which no change is left in other bedies the sum of the 
entropies is increased * ; for it cannot be decreased according 
to the last proposition, nor can it be unaltered, as then, 
according to the proposition of Art. 2 above, the process could 
be “reversed ” (in Planck’s sense). 
The definition of entropy and the above propositions 
relating to it are then extended to bodies other than perfect 
gases. 
Planck gives one Definition of “ Reversibility,” but uses 
another. 
5. Now if the phrase “ without leaving changes in other 
bodies ’’is to be interpreted literally,it is obvious that the proofs 
of the above propositions break down and that the propositions 
themselves are as a matter of fact untrue. In the case of the 
theorem of Art. 2 above, for instance, changes of volume are 
described in which work is done on or by bodies external to the 
system ; and it may be necessary to allow the aggregate volume 
of the system to increase in order that it may pass from one state 
to the other, thereby altering the volume, density, and possibly 
the temperature, of some other body. This has been noticed by 
the English translator ; he indicates f that the work to be done 
in such an expansion may be performed by raising weights, 
and that changes of position of such weights are not to be 
considered but that of course changes of density are. Under 
ordinary circumstances, however, no body can expand with- 
out producing a change of density in some other body. It 
appears, then, that the enunciations of the propositions should 
be amended by changing the phrase “ without leaving changes 
in other bodies ”’ into “ without interchanging heat with other 
bodies,” and that there should be a corresponding change in 
* This is in substance part of the theorem of Art. 126, loc. cit. 
jis eos 
