In 1967 a party from the Scott Polar Research Institute, Cambridge, England, carried out a radio echo sounding survey of the Antarctic ice sheet; their data have been used in the analyses discussed. The radio echo sounding apparatus, basically a pulse-modulated radar operating at 35 Mhz, was installed in a U.S. Navy Constellation aircraft with the aerials attached to the tail, giving a fan-shaped beam. The beam is broadest along the line of flight, causing problems of interpretation analogous with those encountered in marine echo sounding and seismic surveying. Also, bottom echoes are affected by refraction at the ice surface. Examples of hyperbolic variations of echo delay with horizontal movement may be seen at the edges of ice shelves, crevasses, or cracks at the bottom of the ice. Sometimes more complex surfaces can give roughly hyperbolic echo profiles. It is possible to distinguish specular from nonspecular reflections; we use a method of computing the position of specular reflecting points from the echo profile which allows the shape of the reflecting surface to be calculated. A computer program transforms the digitized film record into a real space profile. Errors may arise from faulty digitization, but these have been largely eliminated by checking the digitized points early in the program. Occasionally, because of their similar range and echo strength, strong side echoes may be confused with sub-ice features. A cross-section of Nimrod Glacier and the bottom of the ice sheet near Vostok have been plotted; in both cases there is a striking difference between the computed space profile and the echo profile.