﻿Deduction of Wien's Displacement Law. 929 



8. The remainder of the deduction contains nothing new, 

 but may be given for the sake of completeness. Let the 

 shell be rilled with black radiation of temperature 6 and 

 density p, by first covering a hole in the shell with an 

 ordinary body of temperature 0, and then, after equilibrium 

 has been established, closing the hole with a cover which has 

 the same reflecting properties as the rest of the shell wall. 

 Let the volume of the shell, which may, if we prefer, have the 

 form of a cylinder closed by a piston, be decreased a finite 

 amount by an infinitely slow compression. The fact that this 

 requires an infinite time need not concern us. The density 

 of the energy is increased on account both of the work done 

 and of the decrease of volume of the energy already present. 

 The spectral distribution also changes in the manner already 

 given. 



At the end of the compression we introduce into the shell 

 a particle of ordinary matter so small as to be of negligible 

 thermal capacity. If the spectral distribution after com- 

 pression was not that of black radiation of the same integral 

 density, absorption and re-emission by the particle cause a 

 re-distribution and a " blackening " of the radiation without 

 change of density. This establishment of stable equilibrium 

 by re-distribution of the energy among the different periods 

 is spontaneous and therefore irreversible. 



\V r e now expand to the original volume. The radiation 

 remains black on account of the presence of the particle, and 

 the work given out is the same as that put in during the 

 compression, because the pressure depends only on the total 

 density of the energy present and not on its spectral distri- 

 bution. At the end of the cycle, which may be completed 

 by removing the particle, we have therefore re-established 

 the original state exactly. No heat has been added to or 

 taken from any outside body, the work done has been 

 regained, and no changes remain. The cycle is therefore 

 reversible and cannot have included any irreversible element. 

 Hence the introduction of the particle after compression did 

 not cause any change in the spectral distribution of the 

 energy, which must therefore already have been that of 

 radiation from a black body. Hence we conclude that 

 during infinitely slow adiabatic change of density, radiation 

 which was initially black remains black. 



9. We may now apply equation (12) to an adiabatic change 

 of the volume and density of black radiation. The integral 

 density changes from that needed for equilibrium with an 

 absorbing shell at the absolute temperature 6 X to that needed 



Phil Mag. S. G. Vol. 23. No. 138. June 1912. 3 P 



