THERMODYNAMIC ANALOGIES. 185 



where indeed the individual values of which the average is 

 taken would appear to human observation as identical. This 



gives 



d 2c v 



whence =' < 493) 



a value recognized by physicists as a constant independent of 

 the kind of monatomic gas considered. 



We may also express the value of K in a somewhat different 

 form, which corresponds to the indirect method by which 

 physicists are accustomed to determine the quantity c v . The 

 kinetic energy due to the motions of the centers of mass of 

 the molecules of a mass of gas sufficiently expanded is easily 

 shown to be equal to 



where p and v denote the pressure and volume. The average 

 value of the same energy in a canonical ensemble of such 

 a mass of gas is 



J0v, 



where v denotes the number of molecules in the gas. Equat- 

 ing these values, we have 



pv = v , (494) 



whence J~~T~^' ( 495 ) 



Now the laws of Boyle, Charles, and Avogadro may be ex- 

 pressed by the equation 



pv AvT, (496) 



where A is a constant depending only on the units hi which 

 energy and temperature are measured. 1 / K, therefore, might 

 be called the constant of the law of Boyle, Charles, and 

 Avogadro as expressed with reference to the true number of 

 molecules in a gaseous body. 



Since such numbers are unknown to us, it is more conven- 

 ient to express the law with reference to relative values. If 

 we denote by M the so-called molecular weight of a gas, that 



