A General Divergence Formula

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Abstract
A divergence expression for the ratio of energy per steradian reflected from a smooth curved surface to that incident on the surface is derived. It generalizes previous results in that the source and point of observation may both be at finite distances from the reflecting surface. No restrictions are placed on the angles of incidence and reflection except that they be equal. The only limitation placed on the analytical accuracy of the geometrical result is that the surface be sufficiently smooth so that the principal radii of curvature are defined at the point of reflection. It is required, of course, that the wave-length of the energy shall be small compared to the principal radii of curvature of the surface, in order that the geometrical result may be interpreted as a divergence formula. All of the previous results on this problem known to the authors are derived as special cases. Application of the result in connection with the spreading of radio rays by the curvature of the earth leads to somewhat simpler formulas than now available.