Problem of Partition of Energy. G67 



5. To apply this expression to the case of radiation we hava 

 to evaluate E w r w = E\, the energy transmitted per second 

 per unit range of wave-length in the neighbourhood of wave- 

 length X. We must therefore either appeal to experiment 

 for the determination of the appropriate forms to be given 

 to N m and r m in terms of X, or we must determine these by 

 means of suitable assumptions regarding pether and matter 

 and their connexion. Thus if we assume that the fractional 

 rate of transmission of the energy content of each freedom 

 is identical per vibration, the fractional rate of transmission 

 per unit of time is proportional to the frequency, so that we 

 can write flr m = <y\~ 1 , y being an absolute constant. The 

 value of N m , when the frequency is not too small, is ijiven by 

 Rayleigh's reasoning (Sc. Papers, vol. iv. p. 484, or Phil. Mag. 

 xlix. p. 539, 1900) as AX -4 , where A is a universal constant. 

 Hence 



E= AX ^L_ (",) 



e p —1 



an expression which, with Wien's displacement law holding, 

 "gives the well-known experimental result that the maximum 

 energy is proportional to the fifth power of the absolute 

 temperature provided that the latter be identified with P. 

 The expression becomes identical with Planck's if we put 

 a = 0; it is practically identical with Planck's so long as aX 

 is negligible relatively to 7. We must therefore recognize 

 that this restriction holds throughout the range of wave-length 

 to which Planck's formula is applicable. Outside that range 

 the quantity E A becomes very small. 



If within that range PX becomes large relatively to 7, the 

 expression (5) reduces to Ay _1 .PX~ 4 , which is the form 

 given by Rayleigh as applicable when PX is sufficiently 

 large while X is not too large. When X is very large with 

 P not too small, (5) takes the form Aa" 1 . PX~ r> . 



6. The distinction between energy transmitted by, and 

 energy stored in, definite freedoms is of fundamental im- 

 portance. Thus (4') shows that there is not universal 

 eqnipartition of the energy allotted to all freedoms except 

 under the condition that ftr m is negligible relatively to a; 

 while, on the other hand, there is equipartition univer- 

 sally amongst the energies transmitted per unit of time 

 if fir m is larg3 relatively to a and small relatively 

 to P. 



