382 BELL SYSTEM TECHNICAL JOURNAL 



adding the two and identifying their sum with the entropy of the gas. But 

 each of the summands contains the so-ardently-wanted term — A^^ In TV, 

 and therefore the sum must contain a term —2Nk In N, which is —Nk In 

 N'^. This term is not at all of the wanted form, and its mere presence in 

 {Sm + Sc) spoils the chance of identifying that sum with entropy. We 

 have in fact come to a result in contradiction with equation (46) and with 

 the assumption on which that equation was founded, viz. that if two samples 

 of a gas are at the same pressure and temperature their entropies are in 

 proportion to their volumes. The reasoning has not been suited to its aim. 



The origin of this final misadventure lies in the circumstance that in the 

 hope of making easier the exposition, I made what now has proved to be an 

 undue separation between the two "contributions" to the entropy. The gas 

 was mentally divided into groups of atoms, each occupying a certain region 

 of limited size and distribution among the cells of that region according to 

 the law of the new statistics. In computing Sc I defined the jth region as a 

 small piece of ordinary space, and then counted all the atoms in that region 

 regardless of the fact that they have very diversified momenta. In com- 

 puting Sm I defined thej'th region as a small piece of momentum-space, and 

 then counted all the atoms in that region regardless of the fact that they are 

 sprinkled all through the total volume of the container. I may properly 

 say that I used a six-dimensional region throughout, but in the first stage 

 it was a region limited in ordinary space and comprising the whole infinity 

 of momentum-space, while in the second stage it was a region limited in 

 momentum-space and comprising the whole volume of the box in ordinary 

 space. I should instead have carried through the operation in a single stage, 

 using a six-dimensional region limited in both ordinary space and momentum- 

 space. It may seem that this procedure must either lead to the same result 

 as the other, or must be much more difiicult, or both. Neither is the case. 



Instead of writing down a number of new equations which would look 

 precisely like the old ones, I invite the student to go back to page 368 and 

 recommence the argument at the words "We go into the momentum-space. 

 . . . ." If he will replace "momentum-space" by "ju-space," he need 

 make no other change as far along as equation (21) ; the argument is just the 

 same. Now let him turn ahead to page 379, and equation (52): this is 

 valid for the /x-space as it was for the momentum-space, and so are equations 

 (53). The novelty, however, is latent in the first of equations (53), which 

 reappears as (55), and which I now rewrite for one more time: 



- In yl = In S e~''^^ (64) 



On page 380, the summation was shown to be equal to (l/Cy^m) times a 

 certain integral denoted by /; the integral was over the three dimensions of 

 momentum-space; q^ was the size of the elementary cell in momentum- 



