  We examine the relation between two known classes of solutions of the
sine--Gordon equation, which are expressed by theta functions on hyperelliptic
Riemann surfaces. The first one is a consequence of the Fay's trisecant
identity. The second class exists only for odd genus hyperelliptic Riemann
surfaces which admit a fixed--point--free automorphism of order two. We show
that these two classes of solutions coincide. The hyperelliptic surfaces
corresponding to the second class appear to be double unramified coverings of
the Riemann surfaces corresponding to the first class of solutions. We also
discuss the soliton limits of these solutions.
