Three near geomagnetic tail current sheet crossings of the ISEE 1 and 2 satellites, on April 5, 1979, are examined in detail. All are associated with the passage of an interplanetary shock and the region of variable solar wind pressure behind it. The general geometry of field reversing current sheets is discussed, and this geometry is examined for the cases studied, by using the ISEE 1 and 2 coorbiting satellite data sets. A new technique is employed which removes the effects of a variable sheet normal velocity for the first time. This allows us to calculate firm upper bounds on the current sheet thicknesses, and by utilizing certain physically motivated assumptions, determine the most probable actual sheet thicknesses, and inclinations of the field lines within these sheets. Current density profiles derived with this technique show the main cross-tail current sheet to be a structure that is many thermal ion-gyroradii thick and which is sometimes imbedded in a region that is three or more times thicker and contains much smaller current densities. These profiles also exhibit a considerable amount of fine structure in the sheet which appears as narrow peaks in the current density distributions. Possible explanations for these structures, and for the overall sheet structure itself, are examined.