The harmonic functions appropriate to the oblate spheroid, which are of the form P n (ζ), q n (ζ), or P n (ιζ), Q n (ιζ), when the large letters denote the usual Legendre functions, have received but little attention. Yet they provide, as we shall show in this memoir, a very elegant analysis of a variety of physical problems. We propose to exhibit a series of illustrations of their use, together with a large body of analysis whose applications extend very far, and lead to elegant solutions, in an analytical form, of problems which are in many cases new. In other cases—for example, the classical problems of electrified circular discs under influence—geometrical methods which lead to serious limitations have alone been effective hitherto. The analysis by spheroidal harmonics is shown to be intimately associated with that by other methods, such as the Fourier-Bessel integral method, and important theorems of analysis are involved. We may begin with a brief summary of the more important expressions already known for these functions. If a potential function ϕ satisfies ∇ 2ϕ = 0 and a transformation to cylindrical coordinates ( z, ρ, ω ) is made, ∂ 2ϕ / ∂ ρ2 + 1/ ρ ∂ ϕ / ∂ ρ ∂ 2ϕ / ∂ z2 + 1/ ρ2 ∂ 2ϕ / ∂ ω2 = 0, where ρ is distance from the axis.