A 5-axis milling machine incorporates three translational and two rotational axes. To cut a surface with concavities, a spherical-tipped tool must be employed, with a tip radius less than the smallest concave principal radius of curvature of the surface. To maintain a constant cutting speed, the tool axis A must maintain a constant angle psi relative to the unit surface normal N along the tool path. Hence, A must lie on a cone of angle psi about N, but its azimuthal position on this cone is indeterminate. It is shown that, to minimize actuation of the machine rotary axes, the tangent-plane component of A must be parallel-transported along the surface tool path. Equivalently, the axis A must be rotation-minimizing with respect to the surface normal N along the tool path. The inverse kinematics problem of determining the rotary axis inputs that satisfy this condition is also addressed.