V, 



1891.] vibrating electrical system, and its radiation. 167 



The displacement is proportional to the electric force, which 

 consists of a kinetic and a static part. The kinetic part is 



-!«<?, a). 



where V (F, G, H) = — ^tt/j, (u, v, w) 



because FGH is the vector potential of the current distribution ; 

 it is therefore continuous everywhere as the potentials of all 

 volume or surface distributions must be. The static part 



d d d 

 ^dx ' dy ' dzj 



can only be due to a surface density s on the conducting surface; 

 therefore its components along the surface must be continuous 

 when we cross it, while the discontinuity in the component normal 

 to the surface must be equal to 4<7rs. The tangential components 

 of the total electric force are therefore continuous. 



Now it is known that for rapid alternations the currents are 

 induced only in the skin of the conductor ; the vibrations of the 

 system and its free periods are in the limit quite independent of 

 the conductivity and of the nature of the interior parts of the 

 conductor : they are practically the same as if the conductivity 

 were infinite, and there is no sensible decay owing to any degrada- 

 tion into heat. 



We require the proper boundary conditions to express these 

 facts. 



The currents in the conductors may be treated as surface 

 sheets inside which the electric force is zero. 



Outside the sheet therefore the components of the electric 

 force along the surface must be zero also ; and therefore the 

 tangential electric displacements must be zero. 



The components of the electric force normal to the surface 

 will differ on the two sides of it by an amount determined by the 

 surface density s. 



In the elastic solid analogy we may therefore consider the 

 conductors to be cavities in the solid, provided we confer infinite 

 rigidity on the skin of each cavity so that no point of it can have 

 any tangential displacement. If we assume that the solid is 

 incompressible, its vibrations are completely determined by these 

 conditions ; the component of the displacement normal to the 

 surface must naturally adjust itself in such manner that the 

 condensation remains always null. This will be the analogue of 

 an electric surface density on the conductor which adjusts itself 

 instantaneously to the equilibrium value corresponding to the 

 actual phase of the disturbance in the dielectric, according to 



