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A new approach to the scattering of electromagnetic radiation by dielectric scatterers and application of it to the case of scattering by homogeneous spheroidal and ellipsoidal raindrops is presented. We transform the (singular) integral equation for the scattering into an integral equation for the Fourier transform of the internal field, which has a nonsingular kernel. This equation is solved by reducing it by quadrature into a set of algebraic equations. The scattering amplitude so obtained is shown to satisfy the Schwinger variational principle, and the method is thus both numerically stable and known to be convergent. We present sample calculations for spheres, for spheroids, and for ellipsoids.