﻿Deduction of Wien's Displacement Law. 925 



Since this result is independent of the original value of the 

 wave-length X, it is valid for any value. Hence \-\-d\ and. 

 therefore d\, the interval within which the wave-lengths are 

 included, is changed in the same ratio as X. 



The result is also valid for any element of the surface of 

 the shell which is small enough to be treated as plane, and 

 for motion either in or out, so that equation (9) may be 

 integrated into the form 



X = const. X frv (10) 



The meaning of equation (10), reduced to its simplest terms, 

 is as follows : — If the shell changes its volume while re- 

 taining its shape, the dimensions of the waves change in the 

 same ratio as those of the shell. The whole system of waves 

 and shell remains geometrically similar to itself, the number 

 of waves present being unchanged. 



5. In the foregoing reasoning we have treated the radia- 

 tion as perfectly diffuse, for the cancellation of B, k dX from 

 the numerator and denominator of equation (4), and the 

 corresponding equation for the case of departure, involved 

 the assumption that R A was the same at all points and in all 

 directions. This assumption is not exact, for upon reflexion 

 from a moving mirror the angle of departure yfr' differs from 

 the angle yfr at which the ray would leave a fixed mirror by a 

 quantity of the order of magnitude of the ratio of the velocity 

 of the mirror to the velocity of light. The result of this is 

 that at any point within our shell the radiant vector in 

 directions away from the moving piece M has not exactly the 

 mean value R A , but a value 



R x ' = R x (l + e), 



where e is an infinitesimal of the same order as (3. This de- 

 parture from perfect diffuseness is not cumulative, but remains 

 of the same order of magnitude whatever the lapse of time, 

 for the disturbing effect of reflexion from M is continually 

 being damped out by the diffusing effect of the irregular 

 reflexions from the stationary walls. 



If we now review our reasoning, we find that R,\ appears 

 only in thft expressions for the total energy within the shell 

 and for the total energy which strikes or leaves M at a given 

 angle within the time t. If R A represents the average value 

 which satisfies the equation 



c , _ ( R\dX 

 \ p\dX .dc = l7r ■ . r, 



