498 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



5 = P — — = c v log T + k log v + Const., . . . (4) 



c p and c v are, respectively, the specific heats at constant pressure 

 and at constant volume; k is a constant for an}' particular gas. 

 Utilizing equation (3) and remembering that 



e = p and k= (c p - c v ) 



C v 



we get 



k 

 s — c v l°g + (e — 1) log — + Const. 



Po 



or 



s = c p log + Const (5) 



This gives us the relation sought between potential temperature 

 and entropy. Since c p is invariably a positive quantity, it follows at 

 once that for any process the potential temperature varies in pre- 

 cisely the same direction as the entropy. If the entropy is increased 

 as it invariably is for irreversible processes in accordance with the 

 second law of thermodynamics, then is the potential temperature 

 likewise increased. When the entropy remains constant, as for 

 reversible processes, e.g., a strict adiabatic process, then the poten- 

 tial temperature likewise remains constant. In other words as far 

 as perfect gases are concerned it is possible to express the entropy 

 function in its simplest form by means of a quantity — the potential 

 temperature — not only readily interpretable but also easy of direct 

 graphical representation. 



Owing to the intimate relationship between entropy and potential 

 temperature the term "entropic temperature" might appear as 

 being possibly a more suggestive one for von Helmholtz's "waer- 

 megehalt" than that of "potential temperature," but it may 

 hardly seem advisable now since von Bezold's extensive use of the 

 latter term to recommend a change. 



Cyclical process. — By turning back to the diagram, it will be 

 seen that the change in potential temperature in going from a to b 

 is precisely the same as from a' to b' , i. e., the same as for a simple 

 expansion process under constant pressure. Hence, in carrying 

 out the cyclical process abb' a' a, it will readily be seen that the sum 

 total of the potential temperature changes is zero, just as in the 

 case of the sum total of the entropy changes. 



