A solution of the problem of Poiseuille slip flow through an eccentric cylindrical annulus is obtained in bipolar coordinates. The solution is in excellent agreement with the two published limiting cases of slip flow through concentric annuli and no-slip flow through eccentric annuli. It is shown that for a fixed aspect ratio, fully eccentric channels sustain the maximum average velocity (flow rate) under the same pressure gradient and slip conditions. For a given channel geometry, the average velocity varies linearly with Knudsen number except for small aspect ratio. It is also shown that the extrema of the friction factor Reynolds number product is determined by how this product is defined or scaled.