68 BELL SYSTEM TECHNICAL JOURNAL 



but to the degree of approximation which is now being used, the difference 

 may be neglected. The partition-function is then equal to 1/Ho times the 

 integral (25), Now the value of the integral (25) is given in all tables of 

 definite integrals, and in terms of our symbols it amounts to 



We are now to divide this by Ho, and proceed along the path which has been 

 indicated. 



The procedure is simple and straightforward. As a byproduct one finds 

 the result that the energy of the gas — which I have earlier symbolized by 

 U — is equal to {3/2)kNT. It follows that the specific heat at constant 

 volume is equal to {3/2)kNo for one mole, to n times this value for n moles 

 of gas. Utilizing this result, and putting nR for Nk wherever the latter 

 occurs, one duly arrives at (18) all filled out in the proper way. This 

 represents the contribution of the temperature to the entropy; now adding 

 the contribution of the volume from (19), one arrives at the entropy of n 

 moles of the gas as function of temperature and of volume, as derived by the 

 statistical method: 



(2-irmY'^ k^'^ 

 S = nRlnV + nC^lnT + nR In ^ !;^^ + (3/2)«i? (26) 



This is now to be compared with the equation (13) for entropy as function 

 of volume and of temperature, embodying the definition of entropy where- 

 from we started. 



So far as the dependence on T and on V is concerned, all is well! And 

 there seems even to be a prospect of finding a formula for the additive 

 constant. The prospect, though, is still befogged in two ways: by my lack 

 of precision till now as to the magnitudes of Fo and Ha, and by the absence 

 from (26) of any term convertible into the term nR ln{nR) which stands 

 out so prominently in (13), 



As to Fo and Hq: no assumption shall be made about either by itself, 

 but it will be assumed that their product is equal to Planck's constant h 

 raised to the third power (third power, because of the three dimensions of 

 space) : 



Fo^o = h^ (27) 



This I will attempt to justify from a fact not even divined when the formula 

 was made. 



To divide the momentum-space into cells of definite size, and to allot 

 to the partition-function just one term from each cell — this comes to the 

 same thing as allowing certain discrete momentum-values to the atoms in 

 question, and denying them all values intermediate to these "permitted" 



