159 



whole surface is 4 ?'-«, since all the radiation is incident nurniall}'. Then if 

 the radius of the sphere decreases by a small amount dr 

 (2) dQ = dU + dW = 4x[d(r=u) + r=u dr] = I'jur^ (r du + 4u dr). 

 Proceeding as before wc may deduce the Stefan-Boltzmaim Law from this 

 ecjuation, showing that the law is true for radiation from a point source. 



The second law mentioned above is the Wien Displacement Law which 

 states that the product of the. ivave-length and the absolute temperature is a con- 

 stant, or XT = constant. In other words, if radiation of a particular wave- 

 length is adiabatically altered to another wave-length the temperature changes 

 in the inverse ratio. To prove this let us consider the sphere of the preced- 

 ing paragraph. Let it expand with a constant radial velocity v, and let the 

 velocity of the radiation be V. Then, by Doppler's principle, the wave- 

 length will be increased at each reflection. Let Aq = the original wave-length. 

 An = wave-length after n reflections, and t = time elapsing between the in- 

 stant when one wave is reflected and the instant when the next succeeding 

 wave is reflected. Then 



f Ao + Vtl 



Xi = Ao + 2vt = Ao + 2v I 1 , or eliminating t 



I V } 

 2vAo f V -f- V 1 



Ai = Ao H = I IAo. 



V — V I V — V J 



[V + vln 



.". Xn = Xo| 1 . 



IV-vJ 

 While the surface of the sphere moves out a distance dr the wave will travel 



V dr 

 a distance , and since the diameter is 2r, the number of reflections which 



^' dr r V + V 1 v^dL 

 will occur is n = . Conseciuently Xn = Xq 1 1 2rv ^ and the value 



2rv " I V — V J 



2rv , or X 



of X corresponding to an expansion dr is X = Limit of Xo 



V + vl 



V — V 



r drl V 



= Xn j 1 H 1 , when approaches infinitv and the scjuares and higher 



I r J 2v 



V dr 



powers of — and — are neglected. 



V r 



dX dr 

 Put dX = X — Xo, and — = — , or since dQ = 0, r du + 4u dr = 0, by 

 X r 

 du d/. 



equation (2), and h 4 — =0. On integration this gives 



u X 



