90 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



From this we may conclude that when a system has a uniform 

 temperature throughout, the additional conditions which are necessary 

 and sufficient for equilibrium may be expressed by 



(W,so. (in) 



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

 of the states to which \f/ and \/r" 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 

 W and 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 way not 

 affecting the condition of reversibility, if restored to their original 

 condition 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 

 conditions (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 



&?<0 and &/ = 0. 



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 -\-Se will 

 necessarily be a state of uniform temperature. For this state 

 (regarded as a variation from the original state) 



Se<0 and cty>0. 

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



* This general condition of equilibrium might be used instead of (2) in such problems 

 of equilibrium as we have considered and others which we shall consider hereafter 

 with evident advantage in respect to the brevity of the formulae, as the limitation 

 expressed by the subscript t in (111) applies to every part of the system taken 

 separately, and diminishes by one the number of independent variations in the state 

 of these parts which we have to consider. The more cumbersome course adopted in 

 this paper has been chosen, among other reasons, for the sake of deducing all the 

 particular conditions of equilibrium from one general condition, and of having the 

 quantities mentioned in this general condition such as are most generally used and 

 most simply defined ; and because in the longer formulae as given, the reader will 

 easily see in each case the form which they would take if we should adopt (111) as 

 the general condition of equilibrium, which would be in effect to take the thermal 

 condition of equilibrium for granted, and to seek only the remaining conditions. For 

 example, in the problem treated on pages 63 ff., we would obtain from (111) by (88) 

 a condition precisely like (15), except that the terms td-rj', tdrj", etc., would be wanting. 



