THE CANONICAL DISTRIBUTION. 113 



If V q is a continuous increasing function of e g , commencing 

 with V q = 0, the average value in a canonical ensemble of any 

 function of e^, either alone or with the modulus and the exter- 

 nal coordinates, is given by equation (275), which is identical 

 with (357) except that e, $, and \jr have the suffix ( ) ff . The 

 equation may be transformed so as to give an equation iden- 

 tical with (359) except for the suffixes. If we add the same 

 suffixes to equation (361), the finite value of its members will 

 determine the possibility of the canonical distribution. 



From these data, it is easy to derive equations similar to 

 (360), (362)-(372), except that the conditions of their valid- 

 ity must be differently stated. The equation 



requires only the condition already mentioned with respect to 

 V q . This equation corresponds to (362), which is subject to 

 no restriction with respect to the value of n. We may ob- 

 serve, however, that V will always satisfy a condition similar 

 to that mentioned with respect to V r 



If V q satisfies the condition mentioned, and e^ a similar 

 condition, i. e., if e^i is a continuous increasing function of e 3 , 

 commencing with the value (^ = 0, equations will hold sim- 

 ilar to those given for the case when n > 2, viz., similar to 

 (360), (364)-(368). Especially important is 



de q ~' 



If V q , 6*4 (or dV q /dq), d?V q /de* all satisfy similar conditions, 

 we shall have an equation similar to (369), which was subject 

 to the condition n > 4. And if cPVqjdef also satisfies a 

 similar condition, we shall have an equation similar to (372), 

 for which the condition was n > 6. Finally, if V q and h suc- 

 cessive differential coefficients satisfy conditions of the kind 

 mentioned, we shall have equations like (370) and (371) for 

 which the condition was n > 2 h. 



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