We consider a random binary alloy. The fundamental equations satisfied by the effective scattering amplitude t;(&) in the CPA for non-overlapping muffin-tin potential wells is simplified and a new formula for the density of states, n(e), in terms of t&), is derived. It is shown that in the low density limit n(e) is given by the averaged Friedel sum, where S'f and 6'? are the energy derivatives of the I-th phase-shifts corresponding to the A and B potentials respectively. We then propose a particularly suggestive representation for t: in terms of Argand plots and demonstrate the utility of this representation by calculating n(e) for an approximate t;(&) appropriate to a Ni-Cu alloy. Based on the insight gained in the foregoing discussion we introduce a new approximation for solving the fundamental equations of CPA for t: and argue that this scheme is feasible and will allow us to treat d-bands with arbitrary band widths, arbitrary separation of resonant energies and any amounts of s-d hybridization. 1. Introduction. - By now, there are a vast number of calculations in the literature (I) using the CPA in the context of simplified, usually tight binding, model Hamiltonians. Together, these appear to indicate that the basic principles involved are sufficiently sound to make the CPA prescription, in its model indepen- dent form, a reasonable starting point for more elaborate treatments of the electronic structure in random substitutional, metallic, alloys. Our purpose here is to discuss the feasibility and the desirability of carrying out the CPA program for a crystal potential which while varying randomly from site to site is of the non-over-lapping muffin-tin form. We have in mind potentials constructed according to the Mattheiss (2) prescription as is customary in the band theory of ordered metallic alloys (3). We shall discuss the problem in the language of multiple scattering theory and, therefore, the theory will be analogous to and on the scale of a KKR (4) band structure calcu- lation.