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Equivalent, electric and magnetic, edge currents for arbitrary aspects of observation are derived for second-order diffraction by the edges of perfectly conducting, flat, polygonal surfaces. The physical model underlying the derivation is that each illuminated rectilinear edge segment excites a fringe current on the surface, acting as if it were part of an infinite straight edge. The surface wave associated with this fringe current traverses the surface along the grazing diffracted rays until it strikes an opposite edge segment, and its illumination area on the surface is delimited according to the finite length of the initiating edge segment. The "ray coordinate" measured along a grazing diffracted ray is chosen as the integration variable complementary to the edge coordinate in the fringe current radiation integral. The one-dimensional "radiation" integral over this coordinate is evaluated asymptotically in the high-frequency limit and reduced to the sum of two endpoint contributions. The upper integration limit contribution is east in the form of equivalent edge currents pertaining to the termination point of the grazing diffracted ray. These currents are responsible for the dominant part of the second-order edge diffraction. Their expressions incorporate the well-known Fresnel function (which in many eases may be replaced by its asymptotic approximation) and are finite for any combination of incidence and scattering directions, except when both of them coalesce with the grazing diffracted ray. The developed method applies both to flat plates and to plane faces of thick bodies. Examples of backscatter calculations for flat plates are given which exhibit certain improvements over previous calculations by Sikta, as compared with measured data.