The transmission of electric waves around the earth's surface

In the ‘ Proceedings,’ Prof. G. N. Watson discusses the effect of a perfectly conducting layer in the atmosphere at a uniform height above the earth’s surface on the transmission of electric waves round the earth. The mathe­matical treatment adopted by him assumes that the time factoree -αCt can be removed from all the equations, and that the analytical results thus obtained represent the effect of a simple oscillator placed near the earth’s surface. The assumption that the time factor can be removed is equivalent to assuming that a steady state of oscillations exists in the space between the two spheres, such that at the end of a period the amplitudes of the electric and magnetic forces are identical at each point with the values they had at the beginning of the period. Now, when the surfaces of both spheres are perfectly conducting, no energy is transmitted across either surface, and therefore, if there is a steady state of oscillations in the space, the total energy in the space must be constant. When there is an oscillator in the space emitting electric waves, there is a finite amount of energy radiated from the oscillator in each period, and therefore the total energy in the space does not remain constant; it follows that in such a case there is no steady state of oscillations and the mathematical problem involved cannot be treated by assuming that there is a time factor which can be removed. The effect of any electric disturbance set up in the space between the two perfectly conducting spherical surfaces can be expressed in the usual way in terms of the natural periods of the space. Ifa andb are the radii of the two concentric spherical surfaces, the equation which determines the periods corresponding to the Legendre function of ordern , when the disturbance is such that the lines of magnetic force are circles with the axis of the harmonics as a common axis.