It is shown that previous computations of catalog-to-catalog differences, whether by the bin-and-average method or by least-squares adjustment combined with an expansion in an incomplete set of basis functions, have all been performed at inappropriate epoch(s) of place. As a consequence, such results are frequently much more informative regarding the systematic differences between the proper motion systems than they are with respect to the systematic differences between the positional systems. Moreover, and for the same reason, some sets of proper motion differences really do not portray the true state of the evolution of the proper motion systems. Finally, this has practical effects in other astrometrical work, for instance, in the analysis of earth rotation time series. This error is corrected by arguing for a more meaningful epoch of place for catalog-to-catalog comparisons. This method of infinitely overlapping circles is applied at these other epochs to the FK sequence of catalogs, the N30, and the GC, and as a corollary the effectiveness of this new technique is shown.