1891.] vibrating electrical system, and its radiation. 173 



At the interface F is continuous, so that -^ is continuous ; 



and H is continuous, so that 



dV 1 dV . 



r^ 7 — is continuous ; 



ax tp dz 



these follow from the definition . of F, 0, H as the potentials of 



volume distributions. 



Further, the component magnetic induction along the normal 



must be continuous across the interface, which being zero it is ; 



and the magnetic force along the interface must be continuous, 



,, x 1 (dF dH\ . in,. 



so that - -, j— is continuous across the suriace, that is 



ix \dz doc J ' 



— V 2 ^' is continuous, 



r' 



or by the equations for ■%', 



K d*Xi = izf d X* 



1 di z <r 2 dt ' 



or finally, for this special case of harmonic waves 



- l P K iXi = — Xv 



Let 

 %/ = J.j exp i (Ix — n x z — pt) + B l exp i (Ix + n x z —pi), 



Xl = A 2 ex P l ( lx ~ n 2 z ~ ft)* 

 A t , B 1 thus representing the coefficients of the incident and 

 reflected waves, and A 2 that of the surface- wave in the conductor 

 for which n 2 must in the result be complex. 



The value of I must be the same for all these waves, as 

 their traces on the surface must move along it with the same 

 velocity. 



The differential equations satisfied by ■% give 



^(l 2 + <)=/, 





<7 2 + n *) = tp. 



The first of these gives the velocity of the dielectric wave 

 to be QiJZ^f , as it ought to be. The second gives the pene- 

 tration of the surface-wave into the conductor by the equation 



