Self-induction of Wires. 421 



former outer boundary, i. e. when it surrounded the inner 

 wire concentrically. 



The quantity V will still be the line-integral of the electric 

 force across the dielectric by any path that keeps in one plane 

 perpendicular to the axes of the conductors, in which plane 

 lie the lines of magnetic force. Also, the product VC will 

 still represent the total longitudinal transfer of energy per 

 second in the dielectric at that plane, or, in short, the energy- 

 current. As regards the modified forms of S and L , there is, 

 in strictness, some little difficulty, on account of the dielectric 

 being necessarily bounded by other conductors than the pair 

 under consideration, in which others energy is wasted, to a 

 certain extent. This can only be allowed for by the equations 

 of mutual induction of the various conductors, which are not 

 now in question. But if our pair, for instance, be suspended 

 alone at a uniform height above the ground, so that only the 

 very small dissipation of energy in the earth interferes, it 

 would seem, so far as the wire current is concerned, to be an 

 unnecessary refinement to take the earth into consideration. 

 There are, then, two or three practical courses open to us ; as 

 to suppose the earth to be a perfect nonconductor and behave 

 as if it were replaced by air, or to treat it as a perfect con- 

 ductor. In neither case will there be dissipation of energy 

 except in our looped wires, which have no connection with 

 the earth, but there will be a different estimation of the quan- 

 tities L and S required. For when we suppose the earth is 

 perfectly conducting, we shut it out from the magnetic field 

 as well as from the electric field. The electrostatic capacity 

 S is that of the condenser formed by the two wires and inter- 

 mediate dielectric, as modified by the presence of the earth 

 (the method of images gives the formula at once), and the 

 value of L is such that L S = ^ic = v~ 2 , where v is the velocity 

 of undissipated waves through the dielectric ; that is, as 

 before, L is simply the inductance of the dielectric, per unit 

 length of line. On the ground there will be both electrifica- 

 tion and electric current, due to the discontinuity in the 

 electric displacement and the magnetic force respectively ; 

 but with these we have no concern. In the other case, 

 with extension of the magnetic and electric fields, the product 

 L S still equals v~ 2 . Neither course is quite satisfactory ; 

 perhaps it would be best to sacrifice consistency and let the 

 magnetic field extend unimpeded into the earth, considered 

 as nonconducting, with consequently no electric current and 

 waste of energy, whilst, as regards the external electric field, 

 we treat it as a conductor. We must compromise in some 

 way, unless we take the earth into account fully as an ordi- 



Phil. Mag. S. 5. Vol. 22. No. 138. Nov. 1886. 2 G 



