Electromagnetic Waves. 149 



U is to satisfy the hvo equations found by writing X=?0 in (1*4) 

 and (1*41), namely, 



ar c 2 d< 2 K J 



and 



8 P_VP'\ =0 .... (2-2) 



Now consider the integral 



;-JHG-SJ , t(ssyh<< ■ • • <*> 



taken over the rectangle on the surface £ = const, which is 

 bounded by the curves rj=ri 1 , rj^ and £=£ v £ 3 - 



Using equation (2-2) it will be seeu that the integral (2-3) can 

 be written i 





It is now convenient to introduce the special features of 

 spherical polar coordinates, where we have 



! i I = r, rj = 0, 'C = <j>, 



so that * A = 1, B = r, G = r sm 0. 



We shall take the area of integration for the integral I to be 

 the whole of a sphere r = const. Thus the limits for £ (or 0) will 

 be ^ = and £ 2 = 27r ; and since U and its differential coefficients 

 must be single -valued functions of position in space, it is evident 

 that now 



Thus (2-31) becomes 



I = f { [sin 6 Tiy - [sin 9 U^] } df. . (2-32) 



r./ '2 '1 



Now, in order to include the whole surface of the sphere, we 

 must make t/ 1 ->0 and >7o— >?r ; and both U and its differential 

 coefficients must remain finite at these limits, otherwise our 



* It may be noted that these coordinates do satisfy the hypotheses 

 used in (1-21) that A = l, and that B/C is independent of £ (=r, here). 



