ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. HI 



6. It appears to be desirable to write out the analysis for the problem of the 

 spherical conductor rather fully, in order to show how to determine the effect of 

 resistance, and to criticise the attempts that have been made to determine the effect 

 of curvature in the absence of resistance. 



We denote the radius of the sphere by , and use polar co-ordinates r, 0, <f>, with 

 the centre of the sphere as origin, and the radius vector on which the originating 

 doublet lies as the axis = 0. We write /u for cos 9, and note the formulae 



(13) 



We denote the components of electric force in the directions of increase of r, 6, </> 

 by E r , E s , E$, and those of magnetic force by H r , H 9 , H^,. In both media we have 



E, = H r = H 9 = 0. . ........ (14) 



In the dielectric, where r > a, we may put 



H il- ? ' 11 V l 3 ( ai M T^ l ?: ( ?II V 



t+=-ik-- } & e = ---U ), ^ r = -ip-\- ) . . (15) 



CP p cr\ op/ r^ dfj. \ cp/ 



and in the conductor, where r < a, we may put 



air i a / air\ T , i a / air 



H, pi <J 1 1 XT' i. V I Vli \ T.-I L U Vll \ I 1 c\ 



.'. = 47TO-O , JJij = - -r I p , Ej r = \p . . ( 1 ) 



(ip p or \ Cp I 7 >J d/x \ 3p / 



Then II and II' satisfy the differential equations (6) and (8). 

 The conditions which hold at the boundary r = a are 



cp cp 



a / aji'\ a / air\ ' ^ 17 ^ 



The function II answering to the primary waves is given by (10), and we take the 

 originating doublet, the origin of K, to be at the point for which r = r (> , = 0. 

 Then r a > a, but in practically interesting cases (r n a)/a is small. The functions 

 K! and II' are to be determined in accordance with the conditions laid down in 4. 



The proper forms for these functions can be expressed as series involving spherical 

 harmonics, viz. : 



-" 



where B n , B' B are constants to be determined by help of the boundary conditions, 

 P n (/u) denotes LEGENDRE'S n th coefficient, or the zonal surface harmonic of degree n. 



