  Tate sequences are an important tool for tackling problems related to the
(ill-understood) Galois structure of groups of $S$-units. The relatively recent
Tate sequence "for small $S$" of Ritter and Weiss allows one to use the
sequence without assuming the vanishing of the $S$-class-group, a significant
advance in the theory. Associated to Ritter and Weiss's version of the sequence
are connecting homomorphisms in Tate cohomology, involving the $S$-class-group,
that do not exist in the earlier theory. In the present article, we give
explicit descriptions of certain of these connecting homomorphisms under some
assumptions on the set $S$.
