159 



whole surface is 4 r-u, since all the radiation is incident normalh'. Then if 

 the radius of the sphere decreases by a small amount dr 

 (2) dQ = dU + dW = 4Tu[d(r=u) + r=u dr] = 4^r= (r du + 4u dr). 

 Proceeding as before we may deduce the Stefan-Boltzmann Law from this 

 equation, 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 adiabaticall}^ 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 exjjand with a constant radial velocity- v, and let tlio 

 velocity of the radiation be 1'. Then, by Doppler's principle, the wave- 

 length will be increased at each reflection. Let Ao = 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 

 ^1 = "Ao + 2vt = Ao + 2v I !, or eliminating t 



[ V J 

 2vAo [ V + V 1 



Ai = Ac H = |Ao. 



V — V i V — V 

 fV + vln 



.•.An = Ao 



V — V 



While the surface of the sphere mo^'es out a distance dr the wave will travel 



V dr 

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



y dr C V + v 1 v^dr^ 



will occur is n = . Consequentlv An = Ao I 1 2rv ^ and the value 



2rv ' I V — V J 



fV + vlVcl.^ 



of "A corresponding to an expansion dr is A = Limit of Ao | 1 2rv ^ or A 



iV-vJ 

 r drl V 



= "An 1 1 H !, when approaches infinitv and the scjuares and higher 



I r J 2v 



V dr 



powers of — and — are neglected. 



V r 



dA dr 

 Put dA = A — Ao, and — = — , or since dQ = 0, r du + 4u dr = 0, by 

 A r 

 du dA 



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



