380 BELL SYSTEM TECHNICAL JOURNAL 



This quantity, by (53), is given thus: 



- In y4 = In 2^-''^' (55) 



The so-called "partition-function," which is the sum appearing on the right, 

 is made of terms contributed one from each region, the term from the jth 

 region being exp( — 5£,) — herein Ej stands (it is proper to say) for the aver- 

 age value in thejth region of the function, 



E = (l/2m) (pi + pl + p\) (56) 



If the regions were of unit volume, this summation would be (approxi- 

 mately) equal to the ixv^le integral oi txp{ — BE j) over the whole of momen- 

 tum-space. But the volume of each region is C^qm, wherein qm stands for the 

 volume of the cell. This signifies that while the integral contains one term 

 for each unit of volume, the summation comprises \/Cjqm terms for each 

 unit of volume. The summation is accordingly \/Cjqm. times as great a? 

 the integral. Denoting the integral by 7, we have in place of (55), 



- In ^ = In 7 - In g„ - In C,- (57) 



and in place of (54), 



Sm= kN\nI - kN In qm + kBU - kN In N + kN (58) 



We note with satisfaction that the size of the region has disappeared, even 

 as it did while we were operating in ordinary space! 



The next step is to consult a table of definite integrals for the value of the 

 integral I (or to work it out one's self, if one's memory of the mathematical 

 technique is vivid). The tables give: 



7 = (2 TT m/Bf (59) 



Before returning the table of integrals to the library, the student should 



r -^ -ar 



also look up the value of the definite integral / x''e ""^ dx; for with its aid he 



will be able to find a very simple relation between B and U. I have already 

 said that either determines the other, and now for this special case we shall 

 find the relationship. The procedure consists in going back to the second of 

 equations (53), realizing that 



2 Ej exp(-BEj) = (1/C,-?J jj J E exp"^"^^ dp^ dpy dp, (60) 



and performing the triple integration over the whole of momentum-space, 

 a feat which is not so hard as it looks. MultipHcation by ^ , as indicated in 

 (53), removes the factor (l/Cjqm), and the simple conclusion is, 



U = 3N/2B (61) 



a relation valuable in two ways. 



