If a dielectric ellipsoid or elliptical cylinder is placed in a uniform applied field it is well known that the field inside remains uniform, and is changed only by a depolarization factor that multiplies each applied field component. This paper generalizes this result. Namely, if for the three-dimensional case the potential Φapp of the applied field can be expanded in the neighborhood of the ellipsoid as Φapp=𝒥l=0L𝒥m=−llDlm rl Ylm(ϑ,φ) where l goes from zero to a maximum value L, then it is shown that the resultant potential inside the ellipsoid, Φint, is Φint=𝒥l=0L𝒥m=−ll ClmrlYlm(ϑ,φ) where the coefficients Clm are found explicitly and there is no Clm with l≳L. For a dielectric constant ε, the limits of the above solution as ε→∞ and ε→o are considered and are shown to yield respectively the solutions to the Dirichlet problem with potential zero on the boundary (grounded perfect conductor) and the Neumann problem with normal derivative of the potential zero on the boundary (ideal fluid flow). The homogeneous problem of free charge on an ellipsoidal perfect conductor is considered and it is shown to require a modification of the methods that yielded the results above. The modified method is applied to the problem of the oblate ellipsoid, and its limiting case of a disc, and it enables the easy derivation of various classical results due to Copson and others on the ’’problem of the electrified disc.’’ Finally, multiple dielectric ellipsoids or elliptic cylinders are considered and it is shown that problems involving such bodies can be solved in powers of ai/dij where ai is a typical length of the ith body and dij is the distance between the ith and the jth. This opens the way, in the proper limit of ε, to the solution of a variety of problems, such as flow around multiple strips and through the slots they may form, penetration of the electric field through perforated screens, and so on.