500 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



and b thermally, whereupon in accordance with nature's law heat 

 will flow from the hotter body to the colder until the two are of 

 the same temperature. It will be found that here again the potential 

 temperature of the entire system at the end of the process is 

 greater than at the beginning. This may be proven most readily 

 thus: For a perfect gas we have from (3), when the volume remains 

 constant; 



6 = k" T Ve 

 hence 



k" 1 

 dd = — . --dT 

 e 7> 



where ft = 1 — i/e = positive quantity, since i/e < 1. Conse- 

 quently, the change in potential temperature for a given change in 

 absolute temperature, the volume remaining constant, decreases with 

 absolute temperature. Hence, although the two bodies, a and b, 

 under the conditions imposed, change in absolute temperature by 

 the same amount, the first losing, the second gaining, because of 

 the law just stated, the potential temperature of the colder body, b, 

 suffers a greater increase than the decrease in potential temperature 

 experienced by the warmer body, a, which was to be proven. 



So also for imperfect gases the law of increase of potential tem- 

 perature for natural processes can be established independently 

 of the entropy principle. It is merely necessary to show that in 

 general the adiabat is steeper than the isotherm or that the change 

 in potential temperature varies inversely with the absolute tempera- 

 ture, when the volume remains constant. 



Thus far it has appeared as though the potential temperature law 

 might suffice equally as well as the entropy law. However, in all 

 thermodynamic problems where the element of mass enters, the 

 former law necessarily fails to give as complete a representation of 

 the second law of thermodynamics as the entropy law. The entropy 

 function is not alone a function of pressure and volume but also of 

 mass, whereas the potential temperature is independent of the 

 latter. Equation (8) shows likewise that the substitution of the 

 obviously more convenient function — potential temperature — for 

 entropy cannot be made in general. There are doubtless, how- 

 ever, a number of thermodynamic problems, aswassho.vn by von 

 Bezold, as also in this paper, where the application of the potential 

 temperature law may be found convenient. The main purpose of 

 this paper, as above stated, has been to show the precise relation- 

 ship between the two functions. 



