146 ./. W. Glbbs — Equilibriion of IIete)'0(jeneous SubsUinces. 



When it is not possible to bring the system from one to the other 

 of the states to which //'' and '/'" relate by a reversible process without 

 altering the temperature, it will be observed that it is not necessary 

 for the validity of (107)-(109) that the temperature of the system 

 should remain constant during the reversible process to which TTand 

 Q relate, provided that the only source of heat or cold used has the 

 same temperature as the system in its initial or final state. Any 

 external bodies may be used in the process in any Avay not affect- 

 ing the condition of reversibility, if restored to their original con- 

 dition at the close of the process ; nor does the limitation in regard 

 to the use of heat apply to such heat as may be restored to the 

 source from which it has been taken. 



It may be interesting to show directly the equivalence of the condi- 

 tions (111) and (2) when applied to a system of which the temperature 

 in the given state is uniform throughout. 



If there are any variations in the state of such a system which do 

 not satisfy (2), then for these variations 



6e<Q and 6}] = Q. 



If the temperature of the system in its varied state is not uniform, 

 we may evidently increase its entropy without altering its energy 

 by supposing heat to pass from the warmer to the cooler parts. 

 And the state having the greatest entropy for the energy f -|- (Je will 

 necessarily be a state of uniform temperature. For this state (regarded 

 as a variation from the original state) 



dE<Q and 6i]>Q. 



Hence, as we may diminish both the energy and the entropy by cool- 

 ino- the system, there must be a state of uniform temperature for 

 which (regarded as a variation of the original state) 



rff < and (J// = 0. 



From this we may conclude that for systems of initially uniform tem- 

 perature condition (2) will not be altered if we limit the variations 

 to such as do not disturb the uniformity of temperature. 



Confining our attention, then, to states of uniform temperature, we 

 have by differentiation of (105) 



6s - tdi}=dil^-\-})dt. (112) 



Now there are evidently changes in the system (produced by heating 

 or cooling) for which 



de - t (h/ = and therefore Si/^ -[-7jdt=:0, (113) 



