182 BUCKINGHAM: WIEN's DISPLACEMENT LAW 



Similar reasoning on the effect of departures at the angle \p 

 leads to the same final expression, and we have for the combined 

 effect of all the alternate arrivals and departures in all possible 

 directions, 



T+AT = / 1 AvV 1 Av 



T \ 6 v) 3 v 



or, in terms of the wave-length, 



AX 1 Av 

 T 3 V 



(6) 



This equation holds for any value of X, hence the interval d\ 

 changes in the same ratio as X itself. The result is valid for any 

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

 for we have not assigned to M any special properties different 

 from those of the rest of the shell wall. Equation (6) may there- 

 fore be integrated into the form. 



X = const Xp (7) 



We have treated R\ as remaining perfectly diffuse, but this is 

 not exact because the motion of M disturbs the diffuseness. The 

 resulting error, however, may easily be shown to involve only 

 terms of lower orders of magnitude than those retained — which 

 are sufficient—- and equation (7) remains valid. There is, also, 

 no objection to making the long time t infinite, if we still make 

 pCts an infinitesimal of the first order by making /3s = const X tr 2 . 



Equation (7) gives the effect on wave-length caused by .change 

 of volume of diffuse monochromatic radiation enclosed within a 

 non-absorbing envelope. It may be put into words as follows: 

 if the shell retains its form during expansion or contraction, the 

 wave lengths change in the same ratio as the linear dimensions of 

 the shell, and the whole system of waves and shell remains 

 geometrically similar to itself.' 



In the extended form of this paper, to appear in the Bulletin of 

 the Bureau of Standards, more details are discussed and the 

 remaining steps required to complete the deduction of the dis- 

 placement law are given. 



