Steady-stat. solutions of the 2D incompressible Euler equations (which we call V-stat.s) are studied analytically. That is, a local expansion is done in the neighborhood of any point on the boundary of a V-stat. (which consists of piecewise constant regions of vorticity). The analytical results contained in paper I by Wu, Overman, and Zabusky are proven. In particular, the limiting V-stat.s for the numerically calculated translating and rotating V-stat.s have $90^\circ$ corners and not cusps. It is also proven that at a point which lies on the boundary of only one region and at which the tangent angle has a jump discontinuity then the difference in tangent angles can only be $90^\circ$ (a corner) or $180^\circ$ (a cusp). The analytical behavior of doubly and triply connected rotating V-stat.s is also investigated.