  In this article we prove in the main theorem that, there is a bijection
between the isomorphism classes of a certain type of real hyperplane
arrangements on the one hand, and the antipodal pairs of convex cones of an
associated discriminantal arrangement on the other hand. The type of hyperplane
arrangements considered and the isomorphism classes have been defined
precisely. As a consequence we enumerate such isomorphism classes by computing
the characteristic polynomial of the discriminantal arrangement. With a certain
restriction, the enumerated value is shown to be independent of the
discriminantal arrangement. Later we observe that the restriction we impose on
the type of hyperplane arrangements is a mild one and that this conditional
restriction is quite generic. Moreover the restriction is defined in terms of a
normal system being concurrency free which is a generic condition. We also
discuss two examples of normal systems which are not concurrency free in the
last section and enumerate the number of isomorphism classes.
