The problem of a conducting sphere with a dielectric coating under the influence of an electrostatic field (primary) is solved by employing Kelvin transformation or the method of inversion. The primary field is assumed to be regular inside the body but otherwise arbitrary. The solution consists of linear combinations of the primary field, its inverted version, and weighted integrals thereof. Physical interpretation of the solution in terms of Neumann images reveals that there are point charges and line charges. The strengths of these charges depend on several parameters, including the orientation angle of the observer, a feature attributable to the layered nature of the object. This aspect of the solution is investigated in some detail, and numerical results are presented. The Kelvin solution presented here should converge faster than the conventional one consisting of an infinite sum of spherical harmonics. It is further believed that extension to other geometries such as a two-layered dielectric sphere should pose no new difficulties.