﻿tlie Partition of Energy. 471 



large quantum numbers. Condition (v.) is satisfied if not 

 all the e's and v's have a common factor. There remains 

 (ill/), and here there are trivial analytical difficulties when, 

 as in general, M/E and N/E are fractional. 



It is, however, easy to generalize § 6 by replacing [</>(c )] E 

 h J 



and letting E, M, N all tend to infinity independently. 

 Condition (iii.) is then satisfied, as can be seen by multiplying 

 out, and so all the conditions are satisfied, and the final 

 results stated above are unaffected. 



Finally, we may observe that all our results can be 

 extended at once to an assembly containing any number of 

 types of system. If there are M c systems of type C, for 

 which the partition function is/ c (3), then 



B.=lW^-log/.(»). 



where 3 is determined by 



E=S M^log/ c ($). 



The formal validity of the proof will require all the 

 quantities e c to be commensurable. It will be shown in § 12 

 how this restriction may be removed. 



§ 10. Vibrators of two and three degrees of freedom. 



As a first example we take a set of vibrators each of which 

 is free to vibrate in a plane under a central force proportional 

 to the distance. The sequence of energies is again 0, e, 

 2e, ..., but the weights are no longer unity, as the system is 

 degenerate. Following the principle laid down in § 2, we 

 may evaluate the weights by treating the system as non-de- 

 generate and counting the number of different motions 

 which have the same total quantum number. Now we can 

 quantize the plane vibrators in directions x and y, and as an 

 example for the case 2e, we have three alternatives (2e, 0), 

 (e, e), (0, 2e). This is easily generalized, and gives to re 

 the weight r-f-1. The partition function for these vibrators 

 is thus 



/(c) = l + 2* + 3; 2 +4s 3 +..., 



