The magnetic field profiles in an RFP are calculated on the assumption that the plasma relaxes to a state of minimum Ohmic dissipation (Montgomery and Phillips (1988)) subject to fixed values of the toroidal flux and voltage (helicity dissipation). A prescription for the solution of the Euler-Lagrange equation in arbitrary geometry is given using a boundary condition derived from the variational principle rather than assuming zero current density at the wall. Explicit solutions are found for straight cylinders, first with circular boundaries and then with square boundaries. For the latter the equilibria obtained are not in pressure balance.