Possible evidence for a variable fine-structure constant from QSO absorption lines: systematic errors

Comparison of quasar (QSO) absorption spectra with laboratory spectra allow s us to probe possible variations in the fundamental constants over cosmolo gical time-scales. In a companion paper we present an analysis of Keck/HIRE S spectra and report possible evidence suggesting that the fine-structure c onstant, alpha, may have been smaller in the past: Delta alpha/alpha = (-0. 72 +/- 0.18) x 10(-5) over the redshift range 0.5 < z < 3.5. In this paper we describe a comprehensive investigation into possible systematic effects. Most of these do not significantly influence our results. When we correct for those which do produce a significant systematic effect in the data, the deviation of Delta alpha/alpha from zero becomes more significaut. We are led increasingly to the interpretation that alpha was slightly smaller in t he past.