THE NEW STATISTICAL MECHANICS 383 



space. In pi-space, however, the corresponding integral is over the six 

 dimensions, and may be written thus: 



III dx dy dz f j f e "" dp. dpy dp. 



This sixfold integral is nothing but the product of V the volume of the 

 container (resulting from the first three integrations) by the integral here- 

 tofore denoted as / (resulting from the last three integrations). It is to be 

 multiplied by {\/Cjh ), Cy now standing for the number of cells in the six- 

 dimensional region and // for the volume of the six-dimensional cell (I 

 explain the curious symbol later). The product is the reciprocal of A, 

 and therefore: 



k In 11' = kN\nV +\nl + kBU - kN \n N -\- kN - kN In Ji (65) 



The term {—kN In N) appears just once, and not twice as it did in the sum 

 Sc and Sm'- all is well in this regard. The presentation of /, of B and of U 

 as functions of T follows just the same lines as above. 



(Notice, for future reference, that we should have attained to the same 

 result had we ignored the ordinary space, operated in the momentum-space 

 exclusively, and assigned the value h /V io the volume of the elementary 

 cell in momentum-space.) 



So, identifying k In W with S, we come to the consummation of the new- 

 statistical theory of entropy, the equation: 



S = kN In V + {3/2)kN \nT-kN\nN+kN In [(2 Trmkfe'yh^] (66) 



The dependence on volume is right; it was qualitatively so to start with, 

 was made exactly so by choice of the value of k as R/No. The dependence 

 on temperature is exactly right, since it is a fact of experience that for 

 monatomic ideal gases the specific heat at constant volume is (3/2)R per 

 gramme-molecule. The dependence on number of atoms is exactly right, 

 that is to say, it makes S proportional to N for given P and T. The additive 

 constant is fixed in value absolutely, or will be when we assign a numerical 

 value to //' ; for ^ is a universal constant, e the base of natural logarithms, and 

 ?n the mass of an atom of the gas. 



For the benefit of such as may still be interested in comparing the old 

 statistics with the new, I recall that the old statistics in its theory of entropy 

 furnished the first and the second terms of (66), and apparently furnished 

 also the fourth term though with e^'^ in place of e^ ~. The third term it 

 omitted, thereby lending itself to the untenable doctrine that entropy should 

 be proportional to iV for given V and T (and not for given P and T). Since 

 there was no term kN In N, I committed no error when in the previous article 

 I deduced Sc and Sm separately and then added them together to get ^. 



