GIBBS' PAPERS I AND II 31 



with respect to coexistent homogeneous phases such as a sub- 

 stance part liquid and part vapor which has been under discus- 

 sion; we should add the entropies as well as the volumes and 

 energies of the two parts. It is, however, necessary to proceed 

 with some caution because entropy and energy have arbitrary 

 origins and it is essential that the entropy and energy in one 

 phase should be consistent with those in any other phase into 

 which the substance may go or from which it may come. Sup- 

 pose we have a substance in various phases, and not necessarily 

 all in one working unit. Suppose the substance receives 

 amounts Qi, Q2, • • • • of heat at temperature ^1, iz, . . . . , negative 

 values of Q meaning that heat is returned to the reservoir. Also 

 a certain amount of work is done by the substance or on it. The 

 number of temperatures ti, ^2, . • • • of the reservoirs from which 

 the substance receives heat may be infinite. Let the substance 

 work on a cyclic process or on cyclic processes which may or may 

 not be reversible. With this entire system we combine a per- 

 fect (reversible) thermodynamic engine or a number of such 

 engines to take the quantities of heat Q2, .... all to a reservoir 

 of the given temperature ti. The quantities may be sche- 

 matized as follows : 



Reservoir tempera- 

 tures tl, tzi tzf ti, .... 



Heat absorbed by 



system Qi, Q2, Qs, Qi, 



Heat used by engines — Q2, —Qs, —Qi, 



Heat yielded by en- 



gmes - Q2, 7 Qh -Qi, 



ti tz ti 



Work done by engines — - — Q2, — - — Qs, — - — Qi, 



t2 t3 ti 



Work done by system Qi, +Q2, +Q3, +Q4, 



