  In his seminal work, Taylor (1963) argued that the geophysically relevant
limit for dynamo action within the outer core is one of negligibly small
inertia and viscosity in the magnetohydrodynamic equations. Within this limit,
he showed the existence of a necessary condition, now well known as Taylor's
constraint, which requires that the cylindrically-averaged Lorentz torque must
everywhere vanish; magnetic fields that satisfy this condition are termed
Taylor states. Taylor further showed that the requirement of this constraint
being continuously satisfied through time prescribes the evolution of the
geostrophic flow, the cylindrically-averaged azimuthal flow. We show that
Taylor's original prescription for the geostrophic flow, as satisfying a given
second order ordinary differential equation, is only valid for a small subset
of Taylor states. An incomplete treatment of the boundary conditions renders
his equation generally incorrect. Here, by taking proper account of the
boundaries, we describe a generalisation of Taylor's method that enables
correct evaluation of the instantaneous geostrophic flow for any 3D Taylor
state. We present the first full-sphere examples of geostrophic flows driven by
non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow
admits a mild logarithmic singularity on the rotation axis, in the fully 3D
case we show that this is absent and indeed the geostrophic flow appears to be
everywhere regular.
