Electromagnetic Theory of Radiation. 109" 



3. Let us now consider the electromagnetic field specified 

 by the vectors E and H whose components are given by 

 equations of type 



M = (« - f ) f ' + (y -,) V + (* - Of - c? (t-r), 



where a, /3. 7, 8, X, yu-, v are functions of t, and c is the 

 velocity of light. Writing 



I m n 1 



we see that if 



£ - = cu+u£'—vr} / + m(cv — ur) f -i-l3i; f ) — n(cfi--yf;+oi£') 



• • (7) 

 we may write 



E = c |l i [cs-v-c(«-t)s'], H = sxE, . (8) 



where a, w, v, and s denote the vectors with components 

 (a, £, 7), (X, /a, v), (£', 77', ?')> (/, ra, n) respectively. 



The rate of radiation of energy in the direction s at a very 

 great distance from S is approximately 



and this is positive except when s does not vary with t. 

 Both a and w must be zero for there to be no radiation. 

 The equation (7) is identical with (4) if 



cu + fiiZ' — vrj' =-hp, cX — j3£ + 777' = —Sr\ 



These equations generally determine the ratios of a, p\ 7, S, 



X, fi, v uniquely in terms of f, </, A, /), q, r. To make the 



electric charge associated with IS a constant quantity. t \ 



we write ' * . 2 H , , 2 y , 2 , 



•Itto = e{c- — J — ?; - — L, ' ), 



