50 AVERAGE VALUES IN A CANONICAL 



for the moment of these forces, we have for the period of their 

 action by equation (3) 



* =- ( ^- d ^ + F l = - + F l 



dqi dqi dqi 



The work done by the force F may be evaluated as follows : 



r r d * 



= I Pi dq t -f I ydqit 

 J J dq^ 



where the last term may be cancelled because the configuration 

 does not vary sensibly during the application of the forces. 

 (It will be observed that the other terms contain factors which 

 increase as the tune of the action of the forces is diminished.) 

 We have therefore, 



f* f* n f* 



\ dqi = I pi 1 dt = I qi dp t =. I Pi dpi . (131) 



For since the p's are linear functions of the q's (with coeffi- 

 cients involving the #'s) the supposed constancy of the <?'s and 

 of the ratios of the <?'s will make the ratio fa/Pi constant. 

 The last integral is evidently to be taken between the limits 

 zero and the value of p 1 in the phase originally considered, 

 and the quantities before the integral sign may be taken as 

 relating to that phase. We have therefore 



i = i pl ^L t (132) 



That is: the several parts into which the kinetic energy is 

 divided in equation (128) represent the amounts of energy 

 communicated to the system by the several forces F l , . . . F n 

 under the conditions mentioned. 



The following transformation will not only give the value 

 of the average kinetic energy, but will also serve to separate 

 the distribution of the ensemble in configuration from its dis- 

 tribution in velocity. 



Since 2 e p is a homogeneous quadratic function of the jo's, 

 which is incapable of a negative value, it can always be ex- 

 pressed (and in more than one way) as a sum of squares of 



