A topology $\tau$ on a set $X$ is called a complementary topology if for each open set $U$ in $\tau$, its complement $X-U$ is also in $\tau$. These topologies and maximal ideals were characterized by this author. In this paper the relations between maximal ideals of $\tau$ as a Boolean ring and ultrafilters in $\tau$ as a complementary topology have been investigated. Finally these relations have been characterized.