THE FUNCTION 0. 101 



its most common or most probable value, and which is 

 determined by the equation 



d(f> 1 



de * ^ ' 



This value of d(f>/de is also, when n > 2, its average value 

 in the ensemble. For we have identically, by integration by 

 parts, 



'''= ! +4> r ~ 



v'=o v=o 



If n > 2, the expression in the brackets, which multiplied by N 

 would represent the density-in-energy, vanishes at the limits, 

 and we have by (269) and (318) 



It appears, therefore, that for systems of more tfyan two degrees 

 of freedom, the average value of d$/de in an eiis^ri^y canpni- / 

 cally distributed is identical with the value of the same, 

 ential coefficient as calculated for the most .eoavrooi'. < 

 in the ensemble, both values being reciprocals of the modulus. 

 Hitherto, in our consideration of the quantities F", V# V p , <, 

 </V 4>pi we have regarded the external coordinates as constant. 

 It is evident, however, from their definitions that V and < are 

 in general functions of the external coordinates and the energy 

 (e), that V q and $ g are in general functions of the external 

 coordinates and the potential energy (e g ). V p and <f> p we have 

 found to be functions of the kinetic energy (e p ) alone. In the 

 equation 



-/ 



de, (322) 



by which -vfr may be determined, O and the external coordinates 

 (contained implicitly in <) are constant in the integration. 

 The equation shows that i|r is a function of these constants. 



