874 Prof. H. L. Callendar on 



which is proportional to T(dq/dT) v , but is not proportional 

 to q. It may be objected, with apparent reason, that Carnot's 

 principle cannot be applied to each particular frequency in 

 isothermal emission under equilibrium conditions on account 

 of the change of frequency caused by the Doppler effect at 

 the moving piston or expanding wall of the enclosure. It is 

 therefore necessary to show that the expression T(dp/dT) v for 

 the latent heat given by Carnot's principle, is not in conflict 

 with the Doppler effect in adiabatic expansion, but follows 

 directly from it. 



Latent Heat I of Isothermal Emission oj a Particular 

 Frequency. — Taking the perfectiy reflecting sphere already 

 considered, and supposed full of radiation in equilibrium at 

 a temperature T, let the radius of the sphere expand by a 

 small increment dr, so that the enclosed radiation falls to 

 a lower temperature T— dT, where —dT = Tdr/r as already 

 explained. The stream of energy q per unit range of a par- 

 ticular frequency v at the original temperature T will be 

 reduced at the lower temperature T — dT to the value 

 q — (dq/dT) v dT, where (dq/dT) v is the rate of change of q with 

 temperature for a constant frequency, which has a perfectly 

 definite value for each frequency in full radiation. If now 

 the perfectly reflecting surface is replaced by an emitting 

 surface at the original temperature T, equilibrium will be re- 

 stored by the absorption of the existing stream q — (dq/dT) v dT 

 and the emission of a stream q at constant volume. The 

 volume, which remains constant during this process, may be 

 taken as 47rr 3 /3. The final energy-density is 4g/c. The net 

 energy emitted will therefore be 167iT 3 (d<2/dT)|,dT/3c, which 

 reduces to 167rr 2 T(d^/dTj v d?*/3c, by substituting for dT its 

 value given above. The latent heat of emission I per unit 

 increase of volume is obtained by dividing this by the increase 

 of volume, namely, 4-7rr 2 dr, which gives l = 4:T(dq/dT) v j3c, or 

 T(dp dT) v , since ^ = 4^/3c. 



The above method may appear at first sight to be 

 unnecessarily circuitous, but it is really the most direct for 

 deducing the required result from the admitted properties of 

 the energy-stream and the Doppler effect in adiabatic expan- 

 sion. The same procedure is applied in the reverse direction 

 in elementary thermodynamics in deducing the fall of tem- 

 perature dT for a small adiabatic expansion dv in the case of 

 a perfect gas, by equating the heat, sdT, required to raise 

 the temperature at constant volume, to the work done pdv, or 

 the heat absorbed RTdv/v, in the same expansion performed 

 under isothermal conditions. 



Admitting the existence of the Doppler effect in isothermal 



