  We introduce a new class of solutions to Laplace equation, dubbed logopoles,
and use them to derive a new relation between solutions in prolate spheroidal
and spherical coordinates. The main novelty is that it involves spherical
harmonics of the second kind, which have rarely been considered in physical
problems because they are singular on the entire z axis. Logopoles, in
contrast, have a finite line singularity like solid spheroidal harmonics, but
are also closely related to solid spherical harmonics and can be viewed as an
extension of the standard multipole ladder toward the negative multipolar
orders. These new solutions may prove a fruitful alternative to either
spherical or spheroidal harmonics in physical problems.
