688 BELL SYSTEM TECHNICAL JOURNAL 



by transforming to a polar coordinate frame in the momentum- 

 space.^ 



I follow practically the same route. 



Divide up the momentum-space into spherical "shells" by means 

 of a sequence of spheres all centered at the origin. Each sphere 

 corresponds to a value of e, each shell to a range de of values of e. 

 Take one of the latter at random; call it shell s, denote by Cs and by 

 e^+i or es + de the energy-values at its boundary spheres, by Ta and 

 rs + dr the radii of these, by dV the volume of the shell. Then: 



/ ^ \l/2 



rs = (2me,y'\ ^'' ^ \2~ ) ^^' 



dV = 4Trrs'dr =-^^i2Trmy''(esy"de. (24) 



Suppose to begin with that each shell is large enough to contain very 

 many compartments. The number Q^ of compartments in shell 5 

 will then be: 



Qs=d V/H = ^^ {lirmyiHs'i'de (25) 



and the average number of particles per compartment in shell s, 

 call it Ns, will be : 



Na = aexp (- eslkT) (26) 



and the total number Ms of particles in the shell will be: 



Ms = QsNs = .^^Zsn u"'e-'^''''de = F{es)de. (27) 



This is the number of particles having energy-values in de at e^. 

 Hence the distribution-function-in-energy F is the factor multiplying 

 de (it would be well to discard the subscript s in writing it). I have 

 copied the value of a from (21), but it could have been derived by 

 integrating F from e = to e = oo and equating the integral to N. 



The separation of Ms or F{e)de into two factors — Qs the number of 

 compartments in the shell s, Ns the average number of particles per 

 compartment — is highly advantageous in searching for the distinctions 



^ Denote by p the quantity {pj^ -\- py^ -\- p/^^ which is the magnitude of the 

 momentum; and by d and the angles which with p constitute a spherical coordinate 

 system. We have 



pdpxdpydp, = pp'^ sin eded<t>dp = ~ e-p2/2mtr^2 gin ed9d<j>dp 



H 



and the distribution-in-momentum is obtained by integrating over all values of 6 

 and (j), the distribution-in-energy from it by means of the relation (5). 



