690 BELL SYSTEM TECHNICAL JOURNAL 



ts + des. The value of the constant is fixed as heretofore by the 

 condition that the integral of F over the entire range of energy from 

 to 03 shall be equal to N: 



F(e)de = ^ I e-^'dw = N, (30) 



so that we arrive at the following distribution-in-energy function : 



F(e) =-^exp(-e/kT). (31) 



This function certainly does not display any feature which suggests 

 the achievements of Planck! It looks as smooth and continuous as 

 the Maxwell-Boltzmann function itself, and the constant b, the step 

 or interval between the successive permitted energy-values or the 

 boundaries of successive compartments, is nowhere to be seen. The 

 constant b however has slipped out for the same reason as the constant 

 i7 from the function (23), and the apparent continuity is due in both 

 cases to the same cause. In preparing and effecting the integration 

 (30) in order to obtain a value for the constant a, we assumed that 

 the various permitted energy-values within the range des are all 

 sufficiently nearly equal to be identified with the single value €s. 

 That is to say, we smoothed over the discontinuities which had 

 previously been brought in by the assumption of separate compart- 

 ments. No wonder that there is not a sign of them in the function 

 (31), even as there is not a sign of them in the Maxwell-Boltzmann 

 law! 



We might however avoid this smoothing-over, if we could attain 

 the value of a by an actual summation over the various compartments 

 instead of by integration. Now with Planck's postulate this is 

 mathematically feasible and indeed easy. For the number of particles 

 in the ith compartment being 



Ni = aexp (- ei/kT) (32) 



the total number of particles is computed thus: 



«=o 



^ 1 _ g-b/kT (•^■^) 



by virtue of the very convenient consequence of the binomial theorem 

 that {1 -j- X -\- x"^ -\- • • •) = (1 — x)~^; so that for a we obtain the 



