In many branches of mathematical physics it is of some importance to know how a given potential field is disturbed when a finite sphere is introduced into it. The new potential is usually obtained as the superposition of the former one and a suitable “perturbation potential” which becomes vanishingly small at large distances from the sphere. Such problems arise in electrostatics, hydrodynamics and steady heat flow. The nature of the perturbation potential depends on the boundary conditions on the surface of the sphere. In this note a method is given for finding this perturbation potential for several different types of boundary condition. The field need not be axially symmetric. The method is useful in several cases for determining image systems directly and without using special geometric properties of inverse points in a sphere.