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Experimental results and a detailed analysis are presented of the transverse energy and charged particle azimuthal distributions measured by the E877 Collaboration for different centralities of Au+ Au collisions at a beam momentum of 10.8 A GeV/c. The anisotropy of these distributions is studied with respect to the reaction plane reconstructed on an event-by-event basis using the transverse energy distribution measured by calorimeters. Results are corrected for the reaction plane resolution. For semicentral events we observe directed flow signals of up to 10%. We observe a stronger anisotropy for slow charged particles. For both the charged particle and transverse energy distributions we observe a small but nonzero elliptic anisotropy with the major axis pointing into the reaction plane. Combining the information on transverse energy and charged particle flow we obtain information on the flow of nucleons and …