Current research progresses and the existing problems of terminological cycles in description logics are analyzed in this paper. Based on the works of F. Baader, the LCS (least common subsumer) and MSC (most specific concept) reasoning of hybrid terminological cycles in description logic εL is further studied. The syntax and semantics of hybrid terminological cycles in description logic εL are given. Under the requirement of the LCS and MSC reasoning of hybrid terminological cycles in description logic εL, TBox-completion is presented, and the description graph is redefined. The LCS and MSC reasoning algorithms of hybrid terminological cycles in description logic εL w.r.t. greatest fixpoint semantics are presented by using TBox-completion and description graph. The correctness of reasoning algorithms is proved, and it is also proved that the LCS and MSC reasoning w.r.t. greatest fixpoint semantics can be computed in the polynomial time. Theoretical foundation for the LCS and MSC reasoning for hybrid terminological cycles in description logic εL is provided through the reasoning algorithms.