THERMODYNAMIC ANALOGIES. 175 



6j , and <f> 2 = QO for the least possible value of e 2 . Between 

 these limits (/> x and < 2 will be finite and continuous. Hence 

 $! + < 2 will have a maximum satisfying the equation (488). 



But if n^ < 2, or w 2 < 2, d(f) 1 /d l or d$ 2 /de 2 may be nega- 

 tive, or zero, for all values of e 1 or e 2 , and can hardly be 

 regarded as having properties analogous to temperature. 



It is also worthy of notice that if a system which is micro- 

 canonically distributed in phase has three parts with separate 

 energies, and each with more than two degrees of freedom, the 

 most probable division of energy between these parts satisfies 

 the equation 



That is, this equation gives the most probable set of values 

 of ej, 6 2 , and e 3 . But it does not give the most probable 

 value of e l , or of e 2 , or of e 3 . Thus, if the energies are quad- 

 ratic functions of the p 9 s and <?'s, the most probable division 

 of energy is given by the equation 



HI 1 _ n<2, 1 _ n z 1 



i i 



But the most probable value of ei is given by 



while the preceding equations give 

 KI 1 ^2 + 



1 



These distinctions vanish for very great values of n^ , n 2 , w 3 . 

 For small values of these numbers, they are important. Such 

 facts seem to indicate that the consideration of the most 

 probable division of energy among the parts of a system does 

 not afford a convenient foundation for the study of thermody- 

 namic analogies in the case of systems of a small number of 

 degrees of 'freedom. The fact that a certain division of energy 

 is the most probable has really no especial physical importance, 

 except when the ensemble of possible divisions are grouped so 



