788 Prof. J. H. Jeans on tlie 



the different free-path periods, 



Al= "~S U 2 $m 2 p{t + (s + i)r} sin 2 ipr 



Bj=*~2 4.P cos 2 pU + (s + i)r\ sin 2 ipr, 



and consequently 



Aj + Bj = 4m 2 sinHpr, .... (31) 



where i 2 now denotes the average value of i 2 in the different 

 intervals. 



We can easily evaluate i 2 . The ^-component of i is 

 given by 



i x = e(ii + u' + ...), 



so that 



v J m 



and hence 



N*? 2 

 i 2 = 3KT—Jv (32) 



Substituting in expression (28) from equations (31) and 

 (32), the emission in time t is found to be 



q-^Efto^BTdjp),*,. . . (33) 



This expression is proportional to t and to dv as it ought 

 to be. We notice also that it vanishes if t is either very 

 great or very small, so that when the motion is regarded as 

 made up of free-paths, the whole phenomenon of emission 

 depends on the free-path being finite. 



From the formula for the emission we can readily calculate 

 the partition of radiant energy in the matter in the steady 

 state in which emission and absorption are equal to one 

 another. Let the energy in this state be 



j Bpdp 



per unit volume. Then, as Thomson shows *, the absorption 

 of energy in the element dv in time t must be 



(J 



4 " C '„E 



«. ^pdpjt dv, .... (34) 



where c is the conductivity of the medium for currents of 

 * Phil. Mag. xiv. p. 223. 



