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The pressure distribution on the surface of a long, thin prolate spheroid is found under the conditions:(1) The spheroid is illuminated by a harmonic point source of wavelength λ located on the axis of the spheroid.(2) If a and b are the semi-major and semi-minor axes of the spheroid, then(3) The boundary conditions on the spheroid are either the Neumann or the Dirichlet conditions.The distribution is found from the asymptotic solutions of the Helmholtz equation using the inequalities in (2). The distribution is interpreted on certain regions of the surface in terms of travelling waves.

Abstract. We consider the polygonal lines in the complex plane C whose Nth vertex is defined by S N = N n=0 exp(iωπn 2 ) (with ω ∈ R), where the prime means that the first and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small ω, to Cornu spirals (C-spirals), we prove the precise renormalization formulawhere, the nearest integer to n/ω and 1 < C < 3.14 . This formula, which sharpens Hardy and Littlewood's approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle mapwhose orbits are analyzed by expressing ω as an even continued fraction.