  We introduce a weighted de Rham operator which acts on arbitrary tensor
fields by considering their structure as r-fold forms. We can thereby define
associated superpotentials for all tensor fields in all dimensions and, from
any of these superpotentials, we deduce in a straightforward and natural manner
the existence of 2r potentials for any tensor field, where r is its
form-structure number. By specialising this result to symmetric double forms,
we are able to obtain a pair of potentials for the Riemann tensor, and a single
(2,3)-form potential for the Weyl tensor due to its tracelessness. This latter
potential is the n-dimensional version of the double dual of the classical four
dimensional (2,1)-form Lanczos potential. We also introduce a new concept of
harmonic tensor fields, demonstrate that the new weighted de Rham operator has
many other desirable properties and, in particular, it is the natural operator
to use in the Laplace-like equation for the Riemann tensor.
