THERMODYNAMIC ANALOGIES. 177 



To make the problem definite, let us consider a system con- 

 sisting of the original system together with another having 

 the coordinates a^ , a 2 , etc., and forces AJ, A<, etc., tending 

 to increase those coordinates. These are in addition to the 

 forces A v A v etc., exerted by the original system, and are de- 

 rived from a force-function ( e g ') by the equations 



^;_ &J A,- _^L etc 



Al ' ~d^> da 2 ' 



For the energy of the whole system we may write 

 E = e + ej + Jm 1 a' 1 2 + im 2 a 2 2 + etc., 



and for the extension-in-phase of the whole system within any 

 limits 



I ... I dpi . . . dq n da,i mi da da z m z da 2 . . . 



or I . . . I e$ de da-^ m 1 da^ da z m 2 da 2 . . . , 



or again I . . . / e^ d& da t m x dai da 2 m 2 da 2 . . . , 



since de = c?E, when a x , a x , a 2 , 2 , etc., are constant. If the 

 limits are expressed by E and E + c?E, a^ and a-^ + da^ , a 1 and 

 j + da-^ , etc., the integral reduces to 



The values of ^ , a x , 2 , <z 2 , etc., which make this expression 

 a maximum for constant values of the energy of the whole 

 system and of the differentials dE, da 19 da l9 etc., are what may 

 be called the most probable values of a x , a^ , etc., in an ensem- 

 ble in which the whole system is distributed microcanonieally. 

 To determine these values we have 



de* = 0, 

 when d(e + e q ' + i m of + i m 2 2 2 + etc.) = 0. 



That is, d$ 0, 



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