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We generalize the Rigid-Field Hydrodynamic equations to accommodate arbitrary magnetic field topologies, resulting in a new Arbitrary Rigid-Field hydrodynamic (ARFHD) formalism. We undertake a critical point calculation of the steady-state ARFHD equations with a CAK-type radiative acceleration and determine the effects of a dipole magnetic field on the usual CAK mass-loss rate and velocity structure. Enforcing the proper optically-thin limit for the radiative line-acceleration is found to decrease both the mass-loss and wind acceleration, while rotation boosts both properties. We define optically-thin-correction and rotation parameters to quantify these effects on the global mass-loss rate and develop scaling laws for the surface mass-flux as a function of surface colatitude. These scaling laws are found to agree with previous laws derived from magnetohydrodynamic simulations of magnetospheres. The dipole magnetosphere velocity structure is found to differ from a global beta-velocity law, which contradicts a central assumption of the previously-developed XADM model of X-ray emission from magnetospheres.