104 THE FUNCTION < AND 



as practically equivalent to the values relating to the most 

 common energy 



I j ( j j etc. 

 dtti JQ \ d&z J Q 



In this case also de is practically equivalent to de Q . We have 

 therefore, for very large values of n, 



dri d<f> Q (337) 



approximately. That is, except for an additive constant, 77 

 may be regarded as practically equivalent to < , when the 

 number of degrees of freedom of the system is very great. 

 It is not meant by this that the variable part of rj + < is 

 numerically of a lower order of magnitude than unity. For 

 when n is very great, 77 and $ are very great, and we can 

 only conclude that the variable part of 77 + < is insignifi- 

 cant compared with the variable part of rj or of < , taken 

 separately. 



Now we have already noticed a certain correspondence 

 between the quantities and 77 and those which in thermo- 

 dynamics are called temperature and entropy. The property 

 just demonstrated, with those expressed by equation (336), 

 therefore suggests that the quantities <f> and de/dQ may also 

 correspond to the thermodynamic notions of entropy and tem- 

 perature. We leave the discussion of this point to a sub- 

 sequent chapter, and only mention it here to justify the 

 somewhat detailed investigation of the relations of these 

 quantities. 



We may get a clearer view of the limiting form of the 

 relations when the number of degrees of freedom is indefi- 

 nitely increased, if we expand the function <j> in a series 

 arranged according to ascending powers of e e . This ex- 

 pansion may be written 



( f ) 



( ~ ^ 



(338) 



Adding the identical equation 



