A general theory of a vibrating magnetometer pick-up system employing extended circular coils is outlined in the point magnetic dipole approximation. Systems based upon cylindrical and toroidal coils are discussed in more detail with some attention focused on the ability of the systems to measure simultaneously all components of the magnetic moment. The EMF detectability limits due to overall sensitivity, thermal noise and sample displacement are analysed for several existing and proposed coil arrangements. A Smith-type arrangement has been found which is very stable with respect to axial displacement. The tables essential for the rapid computation of the vectorial geometry factor (VGF) for arrangements composed of cylindrical coils have been compiled.