﻿Theory of Long Wave Radiation. 163 



Since the motion of the electrons between the metallic 

 atoms is highly irregular and of such a nature that it is 

 impossible to follow it in detail, we must rather content 

 ourselves with mean values of the variable quantities cal- 

 culated for a sufficiently long interval of time. We shall, 

 therefore, always consider only the mean values of our 

 quantities taken over the large time between the instants 

 t = and t = 0. For example, the flow of energy through 

 10' is, on the average, equal to 



cw'~ 1 E x 2 dt = ?E7V say. 



Now whatever be the way in which E x changes from one 

 instant to the next, we can always expand it in a series by 

 the formula 



lii x — 2, a s sin -£-, 

 s=i V 



where s is a positive integer and 



sirt 



2 re 

 a s = -A sin — ~ ~E x dt. 



The frequency in the sth term of this series is ~ so that 



u 



the wave-length of the vibration represented in it is 



A- — 



If is very large the part of the spectrum corresponding to 

 the small interval of length d\ between wave-lengths X and 



X-t-dX will contain the large number —— dX of spectral 



X - " 



lines represented by terms of this series. 



If now we substitute the Fourier series for E, into the 

 expression for the mean energy flux through w f , we shall And 

 in the usual manner that it is equal to 



c&2w' = icw' 1 a* (2) 



*=i 

 To obtain the portion of this flux corresponding to wave- 

 lengths between X and X-\-dX we have only to observe that 



2c0 

 the "^ dX spectral lines, lying within that interval, may be 



considered to have equal intensities. Tn other words, the 

 value a s may be regarded as equal for each of tliem. so thai 

 they contribute to the sum 2 in (2) an amount 



2c0a*dX 



X ' M 2 



