The cutoff wave numbers knm and the field of two-conductor, perfectly conducting wave guides are determined analytically. Three types of wave guides are considered: Eccentric circular conductors of radii R•, R2 and distance d between their axes, elliptic inner with circular outer conductor, and circular inner with elliptic outer conductor. The electromagnetic field is expressed in the first case in terms of circular cylindrical wave functions referred to both axes in combination with related addition theorems, while in the last two cases, both circular and elliptical cylindrical wave functions are used, which are further connected with one another by well-known expansion formulas. When the solutions are specialized to small eccentricities, kd in the first case and h = ka/2 in the last two cases (where a is the interfocal distance of the elliptic conductor), exact, closed-form expressions are obtained for the coefficients gnm in the resulting relations knm(d ) = knm(0)[1 + gnm(knmd) 2 + .-.] and knm(h ) = knm(0)[1 + grim h2 + '''] for the cutoff wave numbers of the corresponding wave guides. Similar expressions are obtained for the field. Numerical results for all types of modes, comparisons, and certain generalizations are also included.

Abstract-The scattering of a plane electromagnetic wave by a perfectly conducting prolate or oblate spheroid is considered analytically by a shape perturbation method. The electromagnetic field is expressed in terms of spherical eigenvectors only, while the equation of the spheroidal boundary is given in spherical coordinates. There is no need for using any spheroidal eigenvectors in our solution. Analytical expressions are obtained for the scattered field and the scattering cross-sections, when the solution is specialized to small values of the eccentricity h = d/(2a), (h 1), where d is the interfocal distance of the spheroid and 2a the length of its rotation axis. In this case exact, closed-form expressions, valid for each small h, are obtained for the expansion coefficients g (2) and g (4) in the relation

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Abstract. Power series expansions for the even and odd angular Mathieu functions Sem(h, cos θ) and Som(h, cos θ), with small argument h, are derived for general integer values of m. The expansion coefficients that we evaluate are also useful for the calculation of the corresponding radial functions of any kind.

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