Based　on　the　multipole　expansion　theory　of　the　potential,　a　satisfactory　interpretation　is　put　forward　of　the　exact　nature　of　the　approximations　of　asymptotic　boundary　condition　(called　the　ABC)　techniques　for　the　numerical　solutions　of　open-boundary　static　electromagnetic-field　problems,　and　a　definite　physical　meaning　is　bestowed　on　ABC,　which　provide　a　powerful　theoretical　background　for　laying　down　the　operating　rules　and　the　key　to　the　derivation　of　asymptotic　boundary　conditions.　This　paper　is　also　intended　to　reveal　the　shortcomings　of　the　conventional　higher-order　ABC,　and　at　the　same　time　to　give　the　concept　of　a　new　type　of　higher-order　ABC,　and　to　present　a　somewhat　different　formulation　of　the　new　nth-order　ABC.　In　order　to　test　its　feasibility,　several　simple　problems　of　electrostatic　potentials　are　analyzed.　The　results　are　found　to　be　much　better　than　those　of　conventional　higher-order　ABCs.