ENTROPY 69 



ones. By using the words "permitted values" I am recalling the quantum- 

 theory, and it is in fact a part of the quantum-theory which we are now 

 employing, as betokened already by the entry of the symbol //. It is one 

 of the oldest parts of the quantum-theory; but the new fact — comparatively 

 new — is this. Atoms, like all other particles, are attended and governed 

 by waves. These waves, when with their atoms they exist in a limited 

 space such as that which the container offers to the gas, are constrained to 

 what in acoustics is known as "resonance." Not every frequency of vibra- 

 tion is allowed to the air within an organ-pipe, to the wire of a piano or to 

 the membrane of a drum, but only such as have wave-lengths fitting neatly 

 into the compass of the cavity, the wire or the drumhead. The dimensions 

 of these acoustical resonators control the permitted wave-lengths, and these 

 in their turn determine the frequencies. In the case which we are now 

 considering of a container filled with a gas, the dimensions of the container 

 control the wave-lengths associated with the atoms, which are the wave- 

 lengths of resonance. These in turn control the momenta of the atoms, 

 because of the relation between the momentum of a particle and the wave- 

 length of its associated waves — the "Rule of Correlation": 



p = h/X (28) 



To say that the momenta of the atoms are those and only those corre- 

 sponding to the resonant wave-lengths, and to say that VoHo in (26) is 

 equal to ¥ — these are equivalent statements. When the former is accepted, 

 so perforce is the latter, and the additive constant in (26) is fully determined. 

 But still it lacks the term —nR In nR or —Nk In Nk which figures in (13)! 

 To introduce this term into the theory in a way both logical and simple 

 is not an easy task. The formula at which we are about to arrive is fre- 

 quently known as the "Sackur-Tetrode formula" after the two physicists 

 of whom (before the first world war) one was the first to approach and the 

 other the first to reach it. Sackur assumed outright that VqHo is inversely 

 proportional to N, while Tetrode subtracted from (26) a term k In Nl — the 

 exclamation-point here not having its rhetorical meaning, but signifying 

 that Nl is "N factorial," the product 1-2-3---N. By Stirling's celebrated 

 formula, 



lnN\ = Nln{N/e) (29) 



an approximation amply valid for such enormous values of N as are normally 

 here considered. Be it noted that e here stands for the exponential base 

 and not for the electron-charge (in the latter sense it is never used in this 

 article). 



To make clear the basis for this subtraction oi klnNl I should have to 

 go far into the roots of the conception of entropy as probability. The best 



