AVERAGES IN A CANONICAL ENSEMBLE. 69 

 or, since \f/ q = 7 g + ^ g , (182) 



and <fy c = < g 4- ^ g rf + <fy a , (183) 



ckg = cfyg ^etai J 2 ^2 etc. (184) 



It appears from this equation that the differential relations 

 subsisting between the average potential energy in an ensem- 

 ble of systems canonically distributed, the modulus of distri- 

 bution, the average index of probability of configuration, taken 

 negatively, and the average forces exerted on external bodies, 

 are equivalent to those enunciated by Clausius for the potential 

 energy of a body, its temperature, a quantity which he called 

 the disgregation, and the forces exerted on external bodies.* 



For the index of probability of velocity, in the case of ca- 

 nonical distribution, we have by comparison of (144) and (163), 

 or of (145) and (164), 



(185) 



which gives ^ = Yp ~ * p ; (186) 



we have also ^, = n , (187) 



and by (140), fa = - \ n log (2ir0). (188) 

 From these equations we get by differentiation 



<%=^d, (189) 



and <, = d^. (190) 



The differential relation expressed in this equation between 

 the average kinetic energy, the modulus, and the average index 

 of probability of velocity, taken negatively, is identical with 

 that given by Clausius locis citatis for the kinetic energy of a 

 body, the temperature, and a quantity which he called the 

 transformation-value of the kinetic energy, f The relations 



* Pogg. Ann., Bd. CXVI, S. 73, (1862) ; ibid., Bd. CXXV, S. 353, (1865), 

 See also Boltzmann, Sitzb. der Wiener.Akad., Bd. LXIII, S. 728, (1871). 

 t Verwandlungswerth des Warmeinhaltes. 



