We define the Fatou and Julia sets for two classes of meromorphic functions. The Julia set is the chaotic set where the fractals appear. The chaotic set can have points and components which are buried. The set of these points and components is called the residual Julia set, denoted by , and is defined to be the subset of those points of the Julia set, chaotic set, which do not belong to the boundary of any component of the Fatou set (stable set). The points of are called buried points and the components of are called buried components. In this paper we extend some results related with the residual Julia set of transcendental meromorphic functions to functions which are meromorphic outside a compact countable set of essential singularities. We give some conditions where .