ENSEMBLE OF SYSTEMS. 49 



where e might be written for e p in the differential coefficients 

 without affecting the signification. The average value of the 

 first of these parts, for any given configuration, is expressed 

 by the quotient 



/+ f+ de ^r . 



/ i*l ~fo 6 d Pl ' ' d Pn 



_oo J oo api 



-=r- (129) 



e dpi . . . dp n 

 Now we have by integration by parts 



ty-C 



r PI <^~^- d Pl = r 4 



,/ _oo api j _ 



By substitution of this value, the above quotient reduces to 



, which is therefore the average value of \P\ for the 

 2 dpi 



given configuration. Since this value is independent of the 

 configuration, it must also be the average for the whole 

 ensemble, as might easily be proved directly. (To make 

 the preceding proof apply directly to the whole ensemble, we 

 have only to write dp 1 . . . dq n for dp . . . dp n in the multiple 

 integrals.) This gives J n for the average value of the 

 whole kinetic energy for any given configuration, or for 

 the whole ensemble, as has already been proved in the case of 

 material points. 



The mechanical significance of the several parts into which 

 the kinetic energy is divided in equation (128) will be appar- 

 ent if we imagine that by the application of suitable forces 

 (different from those derived from e q and so much greater 

 that the latter may be neglected in comparison) the system 

 was brought from rest to the state of motion considered, so 

 rapidly that the configuration was not sensibly altered during 

 the process, and in such a manner also that the ratios of the 

 component velocities were constant in the process. If we 

 write 



