  In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our
main result is that every two Artin's factorizations of $\Delta_n ^2$ of the
form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k,
j_k \in \{1,...,n-1 \}$), where $\{H_1,...,H_{n-1} \}, \{F_1,...,F_{n-1} \}$
are frames, are Hurwitz equivalent. This theorem is a generalization of the
theorem we have proved in \cite{Tz}, using an algebraic approach unlike the
proof in \cite{Tz} which is geometric. The application of the result is applied
to the classification of algebraic surfaces up to deformation. It is already
known that there exist surfaces that are diffeomorphic to each other but are
not deformations of each other (Manetti example). We construct a new invariant
based on a Hurwitz equivalence class of factorization, to distinguish among
diffeomorphic surfaces which are not deformations of each other. The precise
definition of the new invariant can be found in \cite{KuTe} or \cite{Te}. The
main result of this paper will help us to compute the new invariant.
