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Almost since its first observation in 1925, it has been clear that the $A1\Sigma +$ state of AgH is irregular, in that its level energy patterns and isotope effects resisted conventional systematic description.} {\bf 72}, 163 (1931).} A proposal by Ger{o} and Schmid} and R.\ Schmidt, {\em Z.\ Physik} {\bf 121}, 459 (1943).} that this irregular behaviour was due to perturbation by the nearby $B1\Sigma +$ state was disputed by Learner,} {\bf A,269}, 327 (1962).} who argued that the problem was due to an avoided crossing which gives rise to a shelf'' on the outer part of the potential well. Recent {\em ab initio} studies by Witek {\em et al.}} {\bf 113}, 8015 (2000).}$^,$} {\bf 116}, 8386 (2002).} suggest that the irregular behaviour is actually due to two avoided crossings, and that the resulting effective potential has what we call anti-shelf'' behaviour which is qualitatively different than that described by Learner. The present paper describes our efforts to determine an accurate effective potential function and delineate the degree and nature of residual non-mechanical behaviour for this state from a combined-isotopologue analysis of all available experimental data.