The reality of some physical effects suggested for sunspot magnetic fields is still doubtful because of the low, or unknown, precision of the observations from which the effects are deduced. For the determination of magnetic field configurations Treanor's method, involving micrometer position measurements, is considerably more sensitive and more powerful than the usual methods depending upon line intensity measurements, but its full potential accuracy has not hitherto been achieved. Several modifications have now been made in the techniques of Treanor's method in order to obtain greater precision. The chief change is to convert the polarization-state analyser into a simple spectroscope slit device: this entails an extra mirror reflection before the analysis is made and the corrections for instrumental polarization have accordingly been reconsidered. The theory has now been extended to include the dependence of the corrections upon absorption line strength. The revised procedure has been used for magnetic field determinations across a medium sized spot and the consistency of the results for Fe I lines indicates that the direction of the magnetic vector is now obtained with a mean precision of . spot magnetic field, when the observations have this degree of certainty, is free from the anomalous effects suggested by earlier work.