ELECTRIC WAVES OVEE THE SUEFACE OF THE EARTH. 109 



expressed in terms of a single function IT, the Hertzian function. The formulae which 

 hold in the dielectric are 



., an F a 2 ii ~ i 3 / am 



H,>=-U, , &,= -, & 2 = -- ^-(p-^-j ..... (5) 



Op Op CZ p Cp \ Op / 



where II satisfies the equation 



0; .......... (6) 



and those which hold in the conductor are 



P air -p 8 2 ii' v i a / air\ 



J-l* =-47ro-t--r , ^ = -5 T- , &.= --- r-U-^ , . ... (7) 



up op cz p cp \ cp I 



where II' satisfies the equation 



(V 2 + ' 2 ) II' = 0, .......... (8) 



in which 



, ........... (9) 



and II' has heen written instead of II in the formulae relating to the conductor. 



The special form of II which answers to a vibrating electric doublet, situated in the 

 dielectric on the axis of z, and having its axis directed along the axis of z, is II,, say, 

 where 







n= - ir -, .... . (10) 



11 denoting distance from the doublet. We may put 



11 = 110+11, ........... (11) 



Then II, satisfies the same equation (6) as II. 



The conditions to be satisfied by the functions II, and II' are the following : - 



(i.) They are solutions of the equations (V 2 + & 2 ) II, = and (V* + k' 2 ) II' = ; 

 (ii.) II, is free from singularities in the region outside the conductor, and II' is 



free from singularities in the region inside the conductor ; 

 (iii.) II, must represent waves travelling outwards ; 

 (iv.) The tangential components of electric and magnetic force derived, as above, 



from II arid II', nrnst be continuous at the bounding surface of the conductor. 



5. When the boundary is a plane, say z = 0, the conditions (iv.) become 



k 2 n = Fir 

 en air 



cz 



these equations being satisfied at z = 0. The condition (iii.) requires some modifica- 

 tion, for both regions now extend to infinite distances. It is now necessary that 11, 



