168 THERMODYNAMIC ANALOGIES. 



the modulus (@), the external coordinates (a l9 etc.), and the 

 average values in the ensemble of the energy (e), the index 

 of probability (?;), and the external forces (A 19 etc.) exerted 

 by the systems, the following differential equation will hold : 



cfe = dj A l da-L JT 2 da 2 etc. (483) 



This equation, if we neglect the sign of averages, is identical 

 in form with the thermodynamic equation (482), the modulus 

 () corresponding to temperature, and the index of probabil- 

 ity of phase with its sign reversed corresponding to entropy.* 



We have also shown that the average square of the anoma- 

 lies of e, that is, of the deviations of the individual values from 

 the average, is in general of the same order of magnitude as 

 the reciprocal of the number of degrees of freedom, and there- 

 fore to human observation the individual values are indistin- 

 guishable from the average values when the number of degrees 

 of freedom is very great. f In this case also the anomalies of q 

 are practically insensible. The same is true of the anomalies of 

 the external forces (A^ , etc.), so far as these are the result of 

 the anomalies of energy, so that when these forces are sensibly 

 determined by the energy and the external coordinates, and 

 the number of degrees of freedom is very great, the anomalies 

 of these forces are insensible. 



The mathematical operations by which the finite equation 

 between e, 77, and a x , etc., is deduced from that which gives 

 the energy (e) of a system in terms of the momenta (j) l . . . .p n ) 

 and coordinates both internal (^ . . . <?) and external (a x , etc.), 

 are indicated by the equation 



$ all 



\ e~ & =f. . .e~dq, . . . dq n dp, . . . dp n , (484) 



phases 



where ^ = rj + e. 



We have also shown that when systems of different ensem- 

 bles are brought into conditions analogous to thermal contact, 

 the average result is a passage of energy from the ensemble 



* See Chapter IV, pages 44, 45. t See Chapter VII, pages 73-75. 



