THE NEW STATISTICAL MECHANICS 385 



nice if I could say at once that the elementary cube of this lattice has the 

 volume h /L which is // /V. However, this cannot be said, for there is an 

 obstinate factor which makes the elementary cube have the volume li /2>L . 

 People get around this by remarking that since an atom reverberating to 

 and fro between the walls of the cube changes the sign of one of its mo- 

 mentum-components whenever it strikes against one of the walls, therefore 

 every dot is one of a group of eight dots all of which correspond to the same 

 motion of the atom, and all eight should be counted as though they were 

 one. Therefore in the region of momentum-space enclosed between any 

 two spheres centred at the origin (such as we used in determining the distri- 

 bution-in-momentum) we are to count one-eighth of the dots. The number 

 so obtained is the same as the number of cells of volume h /V contained in 

 the region. Thus it comes to the same thing whether one says that the 

 atoms are distributed among one-eighth of the dots or among cells of volume 

 /? /v. Now I recall my remark (page 383) that equation (66) down to the 

 last detail can be derived by playing the game of balls and baskets by the 

 rules of the new statistics in the momentum-space alone, provided that to the 

 elementary cell in this space we assign the volume // /V. For doing this last, 

 wave-mechanics has now offered a kind of retroactive basis. There seem 

 to be flaws in the basis, but they are of a kind which cannot be mended (if at 

 all) without a thorough study of a very hard subject, to wit, the art of 

 interpreting wave-mechanics in the ordinary language of space and time.' 

 I think it will be better to proceed at once to the test by experiment. 



Test by Experiment of the New Statistical Formula por Entropy 



Enough has been said already to cover the first three terms of the formula 

 (66), which correctly give the dependence of entropy 5 upon volume V, 

 temperature T, and number of atoms N. The present question is: what 

 does experiment say of the fourth term, the additive constant which involves 

 the mass m of the atom and the universal constants k and //? 



Having treated this question at length in the June 1942 issue of this 

 Journal, I will here give only the barest outline. For this purpose I rewrite 

 (66), by the aid of the equation of state of the perfect gas, 



PV = NkT (67) 



^ If (a, b, c) are the coordinates of one dot, those of the other seven of its group are: 

 {a, —b, c); (a, b, — c); {a, —b, —c); { — a, b, c); ( — a, —b, c); { — a, b, — c); { — a, —b, —c). 



' In previous pages I said that the proper way of playing the game of balls and baskets 

 is to play it in the six-dimensional space, with A'',- representing a definite number of atoms 

 located in a six-dimensional region which is composed of a narro^\l\-limited region in 

 ordinary space and another narrowly-limited region in momentum-space. Wave- 

 mechanics, in the current interpretation, will not allow this; it claims that, if the A', atoms 

 are located in a limited region of momentum-space, they are spread all over the box con- 

 taining the gas. 



