Maximum entropy spectral analysis (Mesa) of a time series consisting of 107 annual mean values of the geomagnetic activity index as and analysis of subsets of this time series yield interesting results concerning both the maximum entropy method itself and periodicities in in geomagnetic activity. The influence of removing a trend from the original data is found to have an even more drastic influence on the low-frequency side of the spectrum than could be expected. Long-period peaks are found for most subsets but are very unstable (40--80 years) and cannot be safely related to any physical phenomena. With data such as the aa time series, Mesa cannot provide reliable estimates of periodicities (if any) having a period exceeding one third to one half of the record length. The '22-year' solar magnetic cycle and the '11-year' solar cycle together with some of their harmonics are observed, and their stability is discussed.