  We perform a bivariate Taylor expansion of the axisymmetric Green function in
order to determine the exterior potential of a static thin toroidal shell
having a circular section, as given by the Laplace equation. This expansion,
performed at the centre of the section, consists in an infinite series in the
powers of the minor-to-major radius ratio $e$ of the shell. It is appropriate
for a solid, homogeneous torus, as well as for inhomogeneous bodies (the case
of a core stratification is considered). We show that the leading term is
identical to the potential of a loop having the same main radius and the same
mass | this "similarity" is shown to hold in the ${\cal O}(e^2)$ order. The
series converges very well, especially close to the surface of the toroid where
the average relative precision is $\sim 10^{-3}$ for $e\! = \!0.1$ at order
zero, and as low as a few $10^{-6}$ at second order. The Laplace equation is
satisfied {\em exactly} in every order, so no extra density is induced by
truncation. The gravitational acceleration, important in dynamical studies, is
reproduced with the same accuracy. The technique also applies to the magnetic
potential and field generated by azimuthal currents as met in terrestrial and
astrophysical plasmas.
