and the Partition of Energy in Continuous Media. 253 



gravity. The energy of the components of this last motion 

 in any given direction is equal (on the average) to RT. 

 Thus at the instant of closest approach in a collision or 

 encounter, an amount of kinetic energy equal to RT must 

 have been transformed into potential energy. Jf a is the 



e" 



distance of closest approach, the potential energy is ^- ,so 

 that K « a 



a= KM ( 3G) 



Mj J 



Defining a by this equation, we see that Planck's formula 

 differs from formula (35) as soon as \ becomes comparable 

 with a, a being the average distance of closest approach in 

 a collision. 



When we consider matter in which, in the limit, we take 

 e = 0, we have a = 0, and formula (35) gives the partition of 

 energy in the issuing radiation. This is as it should be ; 

 collisions in which the electrons approach indefinitely close 

 to one another will in this matter be of frequent occurrence, 

 and radiation, even of very small wave-lengths, is produced 

 at a rapid rate. 



30. We now come to the final problem. Is formula (35), 

 which gives the " steady state " for large values of X, identical 

 with formula (3), which gives the "normal state " ? Or, in 

 other words, are k and R identical in § 5 ? 



The work of Lorentz, already referred to (§§ 8, 27), 

 provides incidentally an answer to this question. Lorentz 

 finds, as the result of actual calculation, that what we have 

 called the " steady state " formula, has for large values of A 



the limiting form 



8ttRTX-^ (37) 



and so establishes the identity of k and R. Lorentz's 

 analysis, however, proceeds on certain definite assumptions, 

 such as that the free electrons in a metal undergo instan- 

 taneous encounters, and that between these encounters they 

 describe undisturbed free paths. It is therefore still necessary 

 to examine whether, in order to compensate for the inaccuracy 

 of these simplifying assumptions, it may not be necessary to 

 correct Lorentz's expression by multiplication by a certain 

 numerical factor*. If this is found to be necessary, then 

 the identity of k and R will not have been proved. 



* This is the only kind of correction which is possible without 

 violating the physical dimensions of the formula in question. 



