In a steady-state, axisymmetric atmosphere surrounding a gravitating point mass, three constants of flow along lines of induction (equivalently, streamlines) are identified, collectively referred to as the Weber–Davis constants. The MHD Bernoulli function, the fourth constant along a line of induction, is derived from examining Euler’s equation in a rotating reference frame, and a link is made between the centrifugal terms and the magnetic terms found in an inertial reference frame. From the four constants, two types of magneto-rotational forces arise which, acting in tandem, can accelerate material from an accretion disc to escape velocities provided the line of induction emerges from the disc at an angle less than 60°. Two astrophysical examples are then described. The first is a quantitative account of Weber and Davis’ model for a stellar wind, including the derivation of specific fluid profiles along a poloidal line of induction. The second looks at how the four constants can arise naturally in an axisymmetric, non-steady-state simulation of an astrophysical jet.