872 Prof. H. L. Oallendar on 



every component in the mixture will increase in direct pro- 

 portion to the radius of the sphere, whatever the angle of 

 incidence. Each componont may be regarded as retaining 

 its identity while its frequency varies, and any arbitrary 

 distribution of components will be permanent as regards the 

 ratio of the energies of the different components*. 



Law of Adiabatic Expansion. — The simplest assumption to 

 make with regard to the variation of energy is that, when a 

 given quantity of radiation is adiabatically compressed or 

 expanded in a perfect reflector, the whole energy of each 

 component which retains its identity varies directly as its 

 frequency. This assumption is in agreement with electromag- 

 netic theory, and is equivalent to various other assumptions 

 which have been made for the deduction of the pressure. 

 The energy stream qdv of the component included between 

 limits v and v + dv of frequency in an expanding sphere of 

 radius r, is transformed into a stream qdv' when the radius 

 has increased to r', and is included in an interval dv' jv' which 

 is equal to dvjv, where v'/v = r// in virtue of the Doppler 

 effect. The whole energy of the component qdv at any stage 

 is the product of the volume 47rr 3 /3 and the energy-density 

 4:qdv/c. By the above assumption the whole energy varies 

 as 1/r, so that r 4 qdv is constant. The energy-stream qdv 

 of each component varies directly as the fourth power of its 

 frequency v, or inversely as the fourth power of the radius r. 



The Radiation Pressure. — The pressure pdv due to the 

 stream qdv is directly deducible by equating the work done 

 ■pdv X 4z7rr 2 dr in a small expansion dr to the loss of energy 

 of the stream. The expression for the whole energy of 

 the stream may be written (7* 4 qdv)167r/3rc. Since r 4 qdv is 

 constant, the loss of energy in a small expansion dr is 

 (r 4 qdv)lQdr/3r 2 c. Equating this to the work we obtain 

 p = £q/3c, which is true for each component separately; and 

 similarly P = 4Q/3c for the whole radiation. It will be 

 observed that the pressure and the work result essentially 

 from change of frequency caused by the Doppler effect. 



The Temperature of Fidl Radiation. — It is shown in 

 many textbooks (e. g. Poynting, p. 337) that " full radiation 

 remains full radiation in any adiabatic change." It follows 

 by a direct application of Carnot's cycle to full radiation , 

 that the temperature T, as defined by Carnot's principle, 

 varies directly as the frequency of each component, or 

 inversely as the radius of the expanding sphere. The energy- 

 density and the pressure vary as the fourth power of the tem- 

 perature for the radiation as a whole (the Stefan-Boltzmann 

 * Larmor, Brit. Assoc. Eep. 1900, p. 657. 



