ON AN ENSEMBLE OF SYSTEMS. 159 



external, may be derived from a single force-function, which, 

 taken negatively, we shall call the potential energy of the 

 combined systems and denote by e 12 . But we suppose that 

 initially none of the systems of the two ensembles EI and 

 E% come within range of each other's action, so that the 

 potential energy of the combined system falls into two parts 

 relating separately to the systems which are combined. The 

 same is obviously true of the kinetic energy of the combined 

 compound system, and therefore of its total energy. This 

 may be expressed by the equation 



'=/ + ,', (458) 



which gives e 12 ' = i/ + e 2 '. (459) 



Let us now suppose that in the course of tune, owing to the 

 motion of the bodies represented by the coordinates called 

 external, the forces acting on the systems and consequently 

 their positions are so altered, that the systems of the ensembles 

 E l and E% are brought within range of each other's action, 

 and after such mutual influence has lasted for a time, by a 

 further change in the external coordinates, perhaps a return 

 to their original values, the systems of the two original en- 

 sembles are brought again out of range of each other's action. 

 Finally, then, at a time specified by double accents, we shall 

 have as at first 



" = e/' + i a ". (460) 



But for the indices of probability we must write * 



W + W ^ W' (461) 



The considerations adduced in the last chapter show that it 

 is safe to write 



W 5 W- (462) 



We have therefore 



5i" + i" < ^ + i', (463) 



which may be compared with the thermodynamic theorem that 

 * See Chapter XI, Theorem VII. 



