Exact, explicit expressions are obtained for the field of a radiating periodic point dipole located in medium 1 at the center of a spherical shell of medium 2, which is bounded by medium 3. The values of &egr;, &mgr;, &sgr; and the shell radii are arbitrary. The reflected and transmitted fields are examined in various special cases. When the shell radii become infinite, the fields agree with those obtained by R. K. Luneberg for the case of plane waves normally incident on a flat plate of medium 2 between media 1 and 3. When the radii are infinite and media 2 and 3 identical, the Fresnel formulas for normal incidence of plane waves on a plane interface between two half‐infinite media are obtained. When the radii are finite, media 2 and 3 identical and &mgr;1= &mgr;3, the results of C. T. Tai are obtained. For finite radii, media 1 and 3 identical, and shell thickness small compared to a wave‐length, the fields check the approximate results of J. B. Keller for the fields reflected and transmitted by a thin shell of any shape, when his results are specialized to the present case.