RECENT STATISTICAL THEORIES 691 



exact value: 



a = iVe(«-W/^-^(e&/^-^ - 1) (34) 



and for the populations of the various compartments, the formula: 



Ni = Ne-''"''^e-''>'^-^(e^"'^ - 1). (35) 



Here the discontinuity implied in the classical picture of a momentum- 

 space divided into compartments is admitted and accepted, as it 

 never was in the process of deriving the Maxwell-Boltzmann law. 

 Planck did not put discontinuity into the classical statistics; it was 

 there already; he refrained from disregarding it. Instead of confining 

 his studies to the circumstances in which it can safely be ignored, 

 he extended them to ranges where it had to be taken account of, 

 and he took account of it. 



As I intimated, the distribution (35) was proposed by Planck not 

 for freely-moving particles, but for oscillators. The "Planckian 

 oscillator" may be visualized as a particle which executes simple- 

 harmonic vibrations back and forth in a straight line across a position 

 of equilibrium, to which it is attracted by a force proportional to its 

 displacement. It is like a free particle, in that its state at any moment 

 is described by giving the values of its position q and momentum p, 

 q being measured from its point of equilibrium; but it is unlike a 

 free particle in that its energy depends not on p alone but on both p 

 and q, being a function of the form (Ap- -j- Bq~). Therefore we must 

 envisage not the momentum-space alone but the phase-space of the 

 variable p and q. In principle it would have been better, had we 

 envisaged the phase-space all along; but since for an assemblage of 

 free particles that space has six dimensions, it was impractical to 

 visualize more than the momentum-space, and since the energy 

 depended only on the momenta that compromise was not detrimental 

 except for one feature which I can later introduce. Here the compro- 

 mise would be ruinous, but it is unnecessary since the phase-space 

 has only two dimensions. 



Visualize then this two-dimensional phase-space as a plane with p 

 and q axes at right angles to each other. Suppose all the oscillators 

 to have the same mass and the same natural frequency, which is to 

 say, the same values of the constants A and B in the above-mentioned 

 formula for their energy; but let them differ in amplitude. The 

 point representing any oscillator in the phase-space runs round and 

 round in an elliptical orbit centered at the origin. Different ampli- 

 tudes correspond to different ellipses. The energy of an oscillator 

 depends on its amplitude; therefore different energy- values correspond 



