﻿the Partition of Energy. 473 



here *. There can, we think, be no question as to the correct- 

 ness of the weight 2r+l, bat most recent writers have used 

 a factor?*; our formula for the specific heat has therefore 

 a rather different value. 



We may now apply our general formulae to this case with 



p r =2r+l, e=rh, 6=^. . . (11-2) 



Then /(^) = l + 3S 6 + 5^ fc + 7S 9fc '+..., . . (H-3) 



E A = M$J^log/(a) (11-31) 



The contribution of the rotations to the molecular specific 

 heat, C rot , is dEJdT, where M must be taken as the number 

 of molecules in one gramme-molecule of gas. Thus, using 

 (7*1), we have 



M.k /. d 2 



5 (^)log/W, • • (H-4) 



Crot (log&J s 

 and M£ = R, the usual gas constant. If we write 



*-k5et' (11 ' 5) 



Crot^R^ — logil + Ze-' + Oe-^+le- 9 "-)-...). (11-51) 



then 



da* 



Equation (11*5) shows that, when T— >co, a— >0. It can 

 be shown by the application of standard theorems on series f 

 that when <r->0, 



C, *->B, (11-52) 



which is the correct limiting value as required by classical 

 dynamics. 



In the general case of any body we have three degrees 

 of rotational freedom, the motion is simply degenerate J, 

 and the energy enters as a sum of square numbers multi- 

 plying two units of energy. The motion of the axis of 

 symmetry and the motion about the axis of symmetry are 

 not independent, and it is impossible therefore for the parti- 

 tion function to split up into the product of two partition 

 functions which represent the separate contributions of the 

 two motions. The result is a double series of the same 

 general type as (11*3). 



* Assuming that no extraneous considerations rule out any of these 

 states. 



t JBromwich, Infinite Series, p. 132. The theorem is due to Cesaro. 

 X Epstein, Phys. Zeit xx. p. 289 (lyl9). 



