The time of change N (S;t) of the 'number of intersections' N (S;t) of the lines of force of a magnetic field B with a closed surface S satisfies a certain theorem. This leads directly to a necessary and sufficient condition for self-exciting hydromagnetic dynamo action (involving inductive interactions between B and fluid motions) in the electrically conducting fluid 'core' of a 'planet,' in terms of the structure of B at the core surface S0. The condition shows the crucial importance of representing that structure with great accuracy in numerical or analytical models of dynamos, and it suggests that dynamo action might be associated with certain topological properties of the pattern of the normal component of B on S0. The condition also indicates that axisymmetric (but not necessarily steady) magnetic fields cannot be maintained by fluid motions; such fields must ultimately decay away, even in a compressible fluid (contrary to a recent claim in the literature to the contrary). The condition has to be modified when thermoelectric emf's are present, but when these emf's are proportional to B, they cannot prevent the decay of axisymmetric magnetic fields. This result refutes a recent claim that such fields can be maintained by Nernst--Ettinghausen effects. The theorem for N (S;t) also shows that while fluctuations in B can take place on time scales down to that (&tgr;a ) associated with core motions (decades to centuries in the case of the earth), the quantity N (S0;t) cannot change on time scales shorter than the Ohmic decay time &tgr;d (?104 years). This result provides the basis of a method for solving the inverse problem of determining the radius of the core of the planet from observations of secular changes in B near the surface of the planet. The method has been applied to the earth and Jupiter with encouraging results.