The transmission of electric waves around the earth's surface

In a previous communication, an expression for the magnetic force at any point when a Hertzian oscillator is placed outside a perfectly conducting sphere was obtained, and the principal parts of this were evaluated ; in particular it was found that when the point is inside the geometrical shadow and at an angular distance from the lino of contact of the shadow with the surface which is of higher order than (ka )-⅓ , the ratio of the magnetic force to the magnetic force due to the oscillator alone is a Gilbert integral; when the point and the oscillator both approach the surface this integral tends to zero, but it may be expected to give information as to the way the magnetic force dies away in tins neighbourhood where it has diminished to order (ka )-½ ψ1. The magnetic force on the surface when the oscillator is also on the surface was reinvestigated by Poincaré. and he indicated that the magnetic force at angular distance θ from the oscillator is of the form Ae -aθ , where A and α depend on the zero of ∂/∂z 0 {z 0 ½ Kn +½(lz 0)}, for which |n +½-z0 | is least. The value of this zero was calculated by Nicholson,* who also calculated the value of A. The value given by him docs not agree with the coefficient of the corresponding term in the series found below ; this may be due to the fact that the process of summation employed by him is invalid as it assumes thatz ½ Kn+½ (lz )/∂/∂z {z 0 ½ Kn +½(lz )} is a meromorphic function ofn +½, which is not true. The problem has recently been discussed by March and Rybczynski; the discussion in the second of these papers is very interesting, but the singularity of the integrand which contributes ultimately alone to the result is one which appears to have been introduced in the process, and which is not a singularity of the original expression for which the integrand is an approximation. The author remarks on the impossibility of representing the facts by a single exponential as the oscillator is approached ; this is fully borne out below. In what follows a aeries is obtained which represents the magnetic force at any point on the surface when the oscillator is also on the surface ; the series converges rapidly for large values ofθ , and for largo values the first term is a sufficient approximation. For small values ofθ the series converges very slowly. Tables showing the fall in amplitude with increase of distance have been calculated for different wave-lengths.