692 BELL SYSTEM TECHNICAL JOURNAL 



to different ellipses, and reversely. If we divide the phase-space 

 into compartments by a succession of ellipses centered at the origin, 

 each of these compartments corresponds to a specific range of energy- 

 values. If the dividing ellipses are so spaced that these compartments 

 are of equal area (equal volume of the phase-space), they correspond 

 to eqiial ranges of energy-values — an important difference between 

 this case and the one which was previously treated. 



If the dividing ellipses are spaced to form equal compartments, 

 they themselves correspond to energy- values forming a linear sequence : 

 call these a, a -{- h, a -\- 2h, - • • a -\- ih • • • as before. Whether we 

 call these the "permitted" energy-values and allow the oscillators 

 only the choice among them, or whether we sprinkle the oscillators 

 uniformly through the compartments, makes only a secondary differ- 

 ence. In this case, in fact, we can easily see exactly what difference 

 it makes. If the oscillators are sprinkled uniformly in each compart- 

 ment, then by applying the classical statistics we get just the same 

 distribution (35) as when we assume them restricted to the energy- 

 values {a + ih). But when we undertake to evaluate the average 

 energy of all the oscillators, then in the one case we must put down 

 the mean energy of those in the ith. compartment as the arithmetic 

 mean of the values a + ib and a + (z -f- \)b, while in the other case 

 we must put down the energy of those at the ith. permitted ellipse 

 as a + ib. Hence to change over from the picture of permitted 

 energy-values to the picture of compartments is the same thing as 

 to replace the original sequence of permitted energy-values by another 

 sequence of values located midway between them. I mention this 

 chiefly in order to emphasize that the subdivision of phase-space into 

 compartments is ipso facto quantum-theory. 



As every reader knows, Planck postulated that the quantity b — 

 the interval between the permitted energy-values, or the energy-range 

 within a compartment, whichever picture is chosen — is the product 

 of a universal constant {h) and the frequency of the oscillators {v). 

 The area of the equal compartments is then equal to the universal 

 constant ^ whatever the frequency of the oscillators. From this 

 latter statement the general principle is derived: To state it one must 

 first adopt a symbol (say n) and a name (say number of degrees of 



'' The point in the phase-space representing an oscillator of mass m, frequency v, 

 and amplitude C describes an ellipse having semi-axes Cand lirmvC a.nd area l-rr'^niuC'; 

 its energy is U = lir'^mv^C^; hence the relation between energy U and area F is 



U = vF 



and the area between two ellipses is equal to h if the energy-difference between them 

 is equal to hv. 



