MEMORIA L TO CLA SSICA L S TA TIS TICS 117 



but so long as the number of molecules N is many times as great as the number of 

 compartments M — a condition easy to realize — those distributions which are 

 markedly far from uniform have probabilities which are fantastically smaller 

 than the value of W for the uniform state. 



The Boltzmann statistics manages thus to derive the assertion aforesaid — 

 the assertion that the uniform distribution in ordinary space is of all the 

 most probable — from a principle which (at least in appearance) is more fun- 

 damental. It has indeed a couple of bothersome points — more than a couple 

 perhaps, but there are two in particular which the newer statistics will at- 

 tempt to assuage. One of these is the size to be assigned to the cells Fo ; but 

 we are borrowing trouble to think too much of that now, since whatever 

 choice be made so long as N/M be large will not affect the achievement just 

 cited. The other is, that one would much rather think of the molecules of a 

 gas (of a single chemical kind) as being alike absolutely, than as being dis- 

 tinguished one from another by a mysterious something-or-other represented 

 in this theory by numbers painted on balls. In the Boltzmann statistics, 

 however, the numbers must stay on the balls. 



We go over into the momentum-space, setting up a coordinate-frame and 

 representing the molecules by dots, the coordinates of which are the momen- 

 tum-components px, py, pz of the molecules in question. To each position of 

 a dot corresponds an energy-value, equal to (l/2w) {px -\- py -\- pz)', we will 

 call it E. E vanishes at the origin, and has a constant value over any spheri- 

 cal surface centered at the origin. To any distribution of the dots will cor- 

 respond a specific value for the total energy of the gas. For this we need a 

 symbol different from E; and as we shall have a good deal to do with thermo- 

 dynamics later on, I choose the thermodynamical symbol U. The average 

 energy of the molecules of the gas will then be U/N, to be denoted by U. 



The entr^^ of E and U into the situation is of the first importance. It is in 

 fact all that will save us from the highly unwanted conclusion that the most 

 probable distribution in the momentum-space is the uniform one, just as it 

 was in the coordinate-space. To see why it makes so great a difference is 

 not altogether easy. I think that the reflections which follow may give an 

 inkling of the reason. 



The momentum-space must be taken either as infinite or as finite. If we 

 take it as infinite and demand a distribution of uniform density, then the 

 density goes to zero and at the same time the energies of the molecules go to 

 infinity, producing an impossible situation. Let us then take it as finite, 

 blocking off all of the parts of it which lie beyond a certain sphere centered 

 at the origin. Assume a uniform distribution within the sphere. This will 

 correspond to a certain value of U. (The student may suppose, if it makes 

 him happier, that the L^-value was preassigned and the radius of the "certain 

 sphere" chosen accordingly.) The PF-value of this distribution will surely 



