An eikonal expansion method of the potential scattering transition amplitude in impact parameter representation is considered and evaluated, without approximation, through third order in the inverse momentum. A sequence of four approximations to the exact impact parameter representation is obtained which consists of the eikonal representation of Glauber and three systematic corrections to the Glauber approximation. The correction terms agree with the form conjectured by Wallace. Numerical

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... s are given for the exponential and Yukawa potential. The sequence of eikonal amplitudes show4 systematic improvement at all angles by comparison with partial wave calculations. cl i • , i 2 . 40) which holds for a spherically symmetric function ¿tc ( ~j/ J and follows from eq. (2 ,3 5 ) and an integration by part. The result 1 s ( ¿ & u ) ) 2 1 2 , J ( 2 . 4 1 )