This paper reconsiders a fundamental ambiguity encountered during inversion of the radial component of the MHD induction equation evaluated at the top of a spherical, inviscid, frozen-flux core. Specifically, we demonstrate that when the input model of the radial main geomagnetic field at any fixed point on the core-mantle boundary evolves as an even degree polynomial in time, then there exists a unique unsteady fluid motion, that simultaneously evolves as a lowest degree polynomial in time and exactly generates the radial secular variation at the point. This result is illustrated by calculating the flow which evolves as a quartic in time and exactly fits the recent 1900--1980 field model of Bloxham and Jackson (1989), whose time dependence includes terms up through degree ten. ¿ American Geophysical Union 1991