Theory of Radiation, 643 



Let Z 2 be the mean of the square of Z taken over the range 

 ±T/2. Then 



Z 2 = r U Zfdt 



1 J -T/2 



1 /"T/2 Z 100 {*"° 



= pp I <ft 1 civ 1 dz/(<£ cos 2irvt + y^ v sin 277 w) 



-•-J -T/2 Jo Jo 



x (<f> v , cos 2ttv' t -\- yjr v , sin 27rV£). 



With the restrictions on cj> v and yfr v it is permissible to invert 

 the order of integration, and to take that for t first. Then 



2TJ Jo 1 ^" YvYv) tt(v'-v) 



lW ^ vYvJ . 7T{V'+V) J 



Following; the usual reasoning for Fourier integrals, with 

 the conditions imposed on T the important part of this is the 

 first term, and for it v' can be changed into v in the first 

 factor. Performing' the integration for v' , we then have 



zr= irw + +>' • . . . (2-3) 



w - 1 - J 

 which is the theorem. 



There is one point in this result that deserves mention, 

 and that is the presence of the factor 1/T. For at first sight 

 this suggests that if the time considered were long enough, 

 the average would tend to vanish. This is not so, because 

 the magnitudes of <j> v and yjr v will vary with T by (2*1). Still 



if Z were quite arbitrary, Z 2 would naturally depend on the 

 exact value of T. The utility of (2'3) depends on its appli- 

 cation to rather more specialized typ«-*s of function. For use 

 in radiation theory we must attribute to Z the property which 

 the radiation, in fact, has — that Z 2 has a value independent of 

 the exact value of T, and </> v and yjr v will then be proportional 

 to T • But it does not seem possible to make T disappear 

 from the expression (2*3). 



3. The radiation field is specified in the optical manner 

 by means of the six components of electric and magnetic 

 force. To derive from these the thermodynamic description 

 it is necessary first to express the optical quantities as a set 

 of plane waves going in ail directions ; then the Poynting 

 vector must be calculated from these, and must be integrated 



2 T 2 



