  We have investigated the toroidal analog of ellipsoidal shells of matter,
which are of great significance in Astrophysics. The exact formula for the
gravitational potential $\Psi(R,Z)$ of a shell with a circular section at the
pole of toroidal coordinates is first established. It depends on the mass of
the shell, its main radius and axis-ratio $e$ (i.e. core-to-main radius ratio),
and involves the product of the complete elliptic integrals of the first and
second kinds. Next, we show that successive partial derivatives $\partial^{n
+m} \Psi/\partial_{R^n} \partial_{Z^m}$ are also accessible by analytical means
at that singular point, thereby enabling the expansion of the interior
potential as a bivariate series. Then, we have generated approximations at
orders $0$, $1$, $2$ and $3$, corresponding to increasing accuracy. Numerical
experiments confirm the great reliability of the approach, in particular for
small-to-moderate axis ratios ($e^2 \lesssim 0.1$ typically). In contrast with
the ellipsoidal case (Newton's theorem), the potential is not uniform inside
the shell cavity as a consequence of the curvature. We explain how to construct
the interior potential of toroidal shells with a thick edge (i.e. tubes), and
how a core stratification can be accounted for. This is a new step towards the
full description of the gravitating potential and forces of tori and rings.
Applications also concern electrically-charged systems, and thus go beyond the
context of gravitation.
