  We calculate the partition function $Z(G,Q,v)$ of the $Q$-state Potts model
exactly for self-dual cyclic square-lattice strips of various widths $L_y$ and
arbitrarily great lengths $L_x$, with $Q$ and $v$ restricted to satisfy the
relation $Q=v^2$. From these calculations, in the limit $L_x \to \infty$, we
determine the continuous accumulation locus ${\cal B}$ of the partition
function zeros in the $v$ and $Q$ planes. A number of interesting features of
this locus are discussed and a conjecture is given for properties applicable
for arbitrarily great width. Relations with the loci ${\cal B}$ for general $Q$
and $v$ are analyzed.
