RECENT STATISTICAL THEORIES 703 



assign the value 2 is introduced for a reason which will be stated 

 later. The corresponding distribution-function in the coordinates 

 and momenta is this: 



G 1 



fix, y, z, p,, Py, p,) = /^ jm^i^TI^qri • (62) 



The first step now is the same as in the classical statistics: to deter- 

 mine the constant A in terms of A'' by integrating F(e) over the whole 

 range of energy-values from to » , and equating the integral to N: 



f 



F{e)de = TV. (63) 



This was an easy step in the classical statistics, but here it is very 

 hard. The integral of /^(e) is not one of the common well-known 

 functions to be found in mathematical tables, nor a combination of 

 such ; and we do not get a simple equation to be solved for A in terms 

 of N. Sommerfeld indeed found it necessary to compromise by 

 deducing two series-expansions for the integral, one being available 

 for values of A smaller than unity, the other for the opposite extreme. 

 By a stroke of luck which seems almost too good to be true, the first 

 one or two terms of one or the other of these expansions form an 

 approximation amply good enough for all the cases where as yet 

 theory and experiment can be compared. 



I consider first the approximation which is of no importance in the 

 theory of electrons in metals — the one for values of A so very small 

 that the second term in the denominator of F{e) is negligible by 

 comparison with the first. Then the distribution-law approaches 

 that of Maxwell and Boltzmann, and of necessity the constant A 

 must possess the value which in the limit makes 7^(e) identical with 

 the classical expression written in equation (27): to wit, the value: 



A = -^{2TvmkT)-'i\ (64) 



It would however be a great error to suppose that this value of A 

 can be substituted into the function F under all circumstances. This 

 value is acceptable only if A is very small relatively to unity, which 

 is to say, if the quantity to which A is here equated is very small. 

 So the question arises: in any physical case, is the combination on 

 the right-hand side of equation (64) a small fraction of unity, or is 

 it not? 



Now for any material gas under any conditions usual in the labora- 



