It is shown that if the field directions (inclination and declination, or other combination of two independent angles) are completely known on the surface of the earth, the geomagnetic potential can be determined uniquely except an arbitrary multiplicative constant. On the other hand, when declinations only are specified on the surface there are infinitely many potentials which satisfy exactly the same boundary conditions. Such non-uniqueness seems also to be present for the cases of other incomplete data set composed of one angle only. The uniqueness theorem serves as the basis for spherical harmonic analyses of the paleomagnetic field.