J. Phys. C: Solid State Physics, Vol. 11 (1978) 1635-1650 Non-uniform magnetisation and the vibrating-sample magnetometer C N Guy and W Howarth A general treatment is given of the response of vibrating-sample magnetometer pick-up coils to non-uniformly magnetised specimens. It is shown that the nth term in a multiple expansion of the sample magnetsiation give reis to a component of the induced EFM at the oscillation frequency, proportional to tne (n+1)th derivative of the 'pick-up field', h, along the direction of the sample motion (h is the field sucha that h*m is the flux produced in the coils by magnetic moment m). The analysis has been tested by observation of the signals produced by systems of small coils carrying DC current. To illustrate the method, experiments on the remanent states of several ordered magnetic materials are described. Application of the method, using a SQUID magnetometer, to the possible measurement of the sublattice magnetisation of antiferromagnetic materials is described.