BUCKINGHAM: WIEN's DISPLACEMENT LAW 181 



between and 71-/2, within a long time t; and we shall evidently 

 have to evaluate a product of the form 



r a r d . r' a r' d . r" a r"d etc. 



But since multiplication is commutative, if we treat all the arrivals 

 by themselves, then all the departures, and finally multiply the 

 two results together, we shall arrive at the same value as if we 

 considered the effects alternately, as they occur in fact. 



We start from the consideration that in a sufficiently long time, 

 every element of the energy within the shell must undergo reflex- 

 ion from M at any particular angles (<p, \p) just as often as every 

 other element, so that the number of times n that any element is 

 affected in a given way within a long time t is the quotient of the 

 total amount affected in that way by the total amount present 

 within the shell to be affected. 



Taking the arrivals first, the whole energy which strikes M 

 at angles between <p and <? + dtp, within t, is t • R\ d\ scos <p • 2x sin 

 <pd<p, and the total energy present within the shell of volume v is 

 v ■ 4:TR\d\/C, so that we have 



CJs 

 2v 



n = z— cos <p sin <p d <p (3) 



All these n arrivals together increase the period in the ratio 



rl = (1 + /3cos^) n = 1 + ft/3cos <p (4) 



Inserting the value of n from (3) and noting that pCts = Av 

 is the infinitesimal increase of volume which has occurred during 

 t, we have 



rl = 1 + cos 2 <p smcp d cp (5) 



2 v 



The effect of arrivals during t at all possible angles is found by 

 taking the product of all the factors of this form from <p = to 

 <p = 7r/2 or, neglecting higher powers of @, 



cos 2 (p sin <pd <p = \ + - — . 



6 v 



