Theory of Radiation. 647 



the cf)' integration removes the middle factor, giving a mul- 

 tiplier c/v sin 6 cos <\> and changing <p' into cp. Thus 



e}c 2 /V 2 cos</>. (5-3) 



F=^-f"rfvf ,r ^f* /2 ^{G 1 T G 1 y -a 1 z a 



871 "Jo Jo J-ir/2 



Now substitute for the Gi's in terms of Ui and V x . 

 By (4'2) and (4*5) 



G l T G 1 y - Gx Z G/= sin cos <£[Ui 2 (<£) + Yi 2 (^) 



So if we denote U 1 2 + U 2 2 by U 2 , we have finally 



F=<M f sin*«tf #[U 2 (</>) + V 2 (<J>) 



OTTjo " JO J-tt/2 



-U 2 (7T-(/>)-V 2 (7r-^)]. (5'5) 



6. For the thermodynamic treatment of radiation * we 

 take an aperture do-, and from it, at angle e to the normal, 

 draw a cone of small solid angle dco. The energy passing 

 through the aperture into this cone in time dt is supposed 

 separated into its two components of polarization and 

 analysed into its frequencies, so that the specification of the 

 field is 



(K v + Kj)dvd(r cos edcodt. . . . (6*1) 

 Here we are to take 



da =pq, cos e = sin 6 cos <f>, dco = sin 6 dO dcfi, dt = T. 

 Let K v refer to the component polarized with electric force 

 in a plane through the z axis. Then the whole excess of 

 energy going in the positive direction is 



F=pqT Cdv {\in dO p' 2 #[K (</>) + K/(<£) 



-KXTr-^-K/CTT-^Jsin^cos^). (fr2) 



If we equate the terms of this integrand to those of (5*5), 

 we have 



K v =c d JJ 2 /87rv 2 .sm0coscf>.pqT) 



K p ' = cW 2 /Sttv 2 . sin 6 cos <f> . pq T J ' 



and these constitute the formal definitions of K and K '. 

 Just as in § 2 we saw that Z 2 involved T explicitly, though 



* See Planck, Vorlesungen ilber die Theorie der Warmestrahlung 

 (Barth). Many of the relations, which he proves with some difficulty 

 "by Fourier series, come out much easier with Fourier integrals. 



