When Laplace's equation is expressed in terms of orthogonal curvilinear co-ordinates, it takes the familiar form
where (λ, μ, ν) are the co-ordinates, and dλ/h1, dμ/h2, dν/h3 the elements of arc in the directions in which only λ, μ, or ν respectively increase.
In the application to solids of revolution and potential problems associated with them, we invariably choose ν as the azimuthal angle φ of rotation about the axis of symmetry, and then h3 = 1/ρ, where ρ is distance from that axis. If λ and μ are chosen as identical with ξ and η in the complex substitution
z + iρ = F(ξ) = F(ξ + iη),
where F is any function, and the axis of revolution is that of z, we have further,
h1 = h2 = h
where h is the common value, and Laplace's equation becomes
and takes an especially convenient form for problems involving boundary conditions on the surfaces of revolution ξ = constant or η = constant.
The only solutions of (2) which have been effectively discussed hitherto are either (a) product solutions of the form
V = f1(ξ)f2(η)f3(φ),
where each f contains only one variable, or (b) of the form
V = f1(ξ)f2(η)f3(φ)F(ξ, η),
where F is a mixed function of two variables. Type (a) includes all the usual forms in terms of spherical and spheroidal harmonics in their various general aspects, while the toroidal functions and special functions relating to two spheres belong to (b).
We shall show that every problem in (a) can furnish the solution of another in (b), in an analytical sense, which is in fact a representation of the familiar geometrical process of inversion, but which is not readily derived from it in any direct manner. The precise analytical, as distinct from geometrical, correspondence between the two problems is of some importance.