EXTENSION-IN-PHASE. 11 



which we have called the extension-in-phase, is also constant 

 in time.* 



Since the system of coordinates employed in the foregoing 

 discussion is entirely arbitrary, the values of the coordinates 

 relating to any configuration and its immediate vicinity do 

 not impose any restriction upon the values relating to other 

 configurations. The fact that the quantity which we have 

 called density-in-phase is constant in time for any given sys- 

 tem, implies therefore that its value is independent of the 

 coordinates which are used in its evaluation. For let the 

 density-in-phase as evaluated for the same time and phase by 

 one system of coordinates be DI, and by another system -Z> 2 '. 

 A system which at that time has that phase will at another 

 time have another phase. Let the density as calculated for 

 this second time and phase by a third system of coordinates 

 be Zy. Now we may imagine a system of coordinates which 

 at and near the first configuration will coincide with the first 

 system of coordinates, and at and near the second configuration 

 will coincide with the third system of coordinates. This will 

 give Dj' ^Y'- Again we may imagine a system of coordi- 

 nates which at and near the first configuration will coincide 

 with the second system of coordinates, and at and near the 



* If we regard a phase as represented by a point in space of 2 n dimen- 

 sions, the changes which take place in the course of time in our ensemble of 

 systems will be represented by a current in such space. This current will 

 be steady so long as the external coordinates are not varied. In any case 

 the current will satisfy a law which in its various expressions is analogous 

 to the hydrodynamic law which may be expressed by the phrases conserva- 

 tion of volumes or conservation of density about a moving point, or by the equation 



The analogue in statistical mechanics of this equation, viz., 



may be derived directly from equations (3) or (6), and may suggest such 

 theorems as have been enunciated, if indeed it is not regarded as making 

 them intuitively evident. The somewhat lengthy demonstrations given 

 above will at least serve to give precision to the notions involved, and 

 familiarity with their use. 



