RECENT STATISTICAL THEORIES 687 



The expression for the number of particles in any compartment thus 

 becomes : 



involving the four constants m, N, k and //. The first three are 

 determinable by experiment, the third is the universal constant 

 known in Boltzmann's honor by his name, though he himself never 

 evaluated it. The fourth, the volume H assigned to the compart- 

 ments, drops out of the distribution-functions — out of the function 

 p, out of the distribution-in-energy soon to be deduced, out of the 

 fundamental distribution-function / in the coordinates and momenta 

 defined by equation (1), and which I now set down in place of p: 



V{2TrmkTy'~ \ 2mkT / ^ ^ 



V standing for the volume in ordinary space of the enclosure which 

 contains the assemblage. This evasion of iJ is very deceptive ; for it 

 suggests not merely that the exact volume of the compartments is of 

 no importance, but that the compartments themselves were invented 

 only as a momentary stepping-stone to the distribution-functions, 

 and should be allowed to shrink to zero like the infinitesimals of the 

 calculus. This however is precisely what is not allowed. It is of the 

 essence of the argument that there are compartments of finite size. 

 As will presently transpire, I suspect that the division of momentum- 

 space into compartments should be regarded as a quantum postulate, 

 even in this case of the derivation of the Maxwell-Boltzmann law 

 which seems to be at the opposite extreme from all the notions of 

 quantum-theory. 



The next step is the derivation of the distribution-in-energy. In 

 preface I point out that the distribution which we are considering is, 

 in respect to the directions of motion of the particles in ordinary 

 space, isotropic. Mathematically, this occurs because px, py and p^ 

 enter symmetrically into all the distribution functions; physically it 

 occurs because we have made no assumption leading to a preference 

 of any direction over any other. Later on we may establish a preferred 

 direction by introducing a field of force, and then the impending 

 steps may have to be reconsidered. Until then the distribution 

 which we shall study will be described completely by saying that 

 they are isotropic and giving the distribution-function-in-energy. 

 This may be obtained from the distribution-function-in-the-momenta 



