686 BELL SYSTEM TECHNICAL JOURNAL 



It is now easily shown that such a distribution is the following 



Ni = aexp (- eilkT), (15) 



in which a stands for any constant and e^ for the average energy of 

 particles in the ^th compartment, which is related to the average 

 momenta of these particles by the equation 



' = j^^p" + p" + p"y (16) 



for we have only to write down the expression for bS as furnished 

 by equation (11), and introduce into it the value of log Ni as supplied 

 by equation (15) : 



bS = - kZil + log Ni)8Ni 



= - k(l + log a)Z8Ni + Z^i^Ni/T = SE/T, (17) 



the result which was desired. 



The value to be chosen for the constant a will be determined as 

 before by the total number of particles. Denote this number by N, 

 and conceive the compartments as tiny cubes of volume H, so that 

 there are 1/// of them per unit volume of the momentum-space. 

 The density p of the particles in momentum-space, which is no other 

 than the distribution-function in the momenta, is given anywhere by 

 the value of NijH computed for the value of energy there prevailing: 



p = ]y.iH = j^exp(- e/kT) 



(18) 



p—Pz-l2mkTf,—py^l2mkT^-p,^-l2mkT 



H 



=; — g—Px-l2mkTg-py^l2mkTg-p,'^l2mkT 



and it is the integral of this expression over the whole of momentum- 

 space which is equal to N: 



N 



/^X /»Q0 /^OO 



= 111 pdpjp,dp,. (19) 



ty_co I/— 00 'J —txi 



The integration is easily affected; the triple integral is the product of 

 three identical single integrals, and we have: 



N = jA {ImkTyi^^ ^ e-''''dw\ , (20) 



w being a symbol for each of the three momenta in turn; so that 



Nil (^,s 



" {iTvmkTyi^' ^ ^ 



