Escaping points of meromorphic functions with a finite number of poles

We establish several new properties of the escaping set I(f) = {z : f(n) (z) -> infinity and f(n) (z) not equal infinity for each n is an element of N} of a transcendental meromorphic function f with a finite number of poles. By considering a subset of 1(f) where the iterates escape about as fast as possible, we show that I(f) always contains at least one unbounded component. Also, if f has no Baker wandering domains, then the set I(f) boolean AND J(f), where J(f) is the Julia set of f, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.