160 



f X ] * uo To^ 



(3) - = - = . or 



.AoJ u T^ 



(4) XT = AoTo = constant. 



Wien's displacement law may be extended so as to give a general formula 

 for the distribution of energy in the spectrum, viz., E"a = C"a "'^f (XT), where EX 

 is the emissive power for radiation of wave-length X and C is a constant. To 

 prove this let ro change to r, then radiations of wave-lengths between Xo and 

 (Xo — dXo) will be changed to those of wave-lengths between X and (X — dX). 



r r dX r X 

 Also X = — Xo, and X -|- dX = — (Xo + dXo), whence = — = — . 



Vi) ro ClAo To Ao 



du fXol" 



From (3) = I — | . But du is j)r(iport ioiial to Kx dX. Therefore 



duo i X I 



Ex dX du rXo]^ Ex Xo^ 



= = I — I , or = . Since ('(|uali()n f4) holds wo may write 



EXodXo duo i,X j EXo X* 



(5) Ex = X-^EXoX^o = CX''f(XT). 



All general distribution formula' must satisfy this ("(luation. It renuiins 

 to determine the form of the function fiXT). 'I lie i)articular form will de- 

 pend upon the assumi)tions made. Wieii found 



f(XT) = e XT, or Ex = CX'^c XT. 



For large values of X and T this formula fails. Lord Kayleigh proposed the 



C, 

 fornuila EX = CiX-^Te XT_ This formula fails for snuill values of X and 



7'. About 1901 Planck proposed the formula 



C,X- 

 EX = 



eXT_i 



This formula agrees with experiment and approaches the fornuilse of Wien and 

 Raylcigh for the range of values for which each holds best. It has already 

 supplanted Wien's to a considerable extent in commercial practice with high 

 temperature furnaces. 



This formula is based on a new and startling hypothesis which has come 

 to be known as the "Quantum Hypothesis," to which reference has already 

 been made. The importance of the new hypothesis is made apparent by the 

 following quotations. Xernst* says, "If Newton, when he created modern 

 mechanics, paved the way to the results of theoretical physics, if Dalton in 

 the atomic theory gave |)hysics and chemistry their most fruitful logical 

 <Preu33. .\lia;l. Wis-., Be.-lin, .Sit/,. Ber. 4, pp. 6.5-90, 1911. 



