Difficulties associated with the use of Buchdahl's retardation coefficients in image assessment are examined. It is shown that, by a series of approximations and corresponding transformations, the set of coordinates of transmitted rays from any object point can be expressed as a circular region perpendicular to the optical axis. Furthermore, it is shown that, under these transformations, the form of the retardation expansion remains constant and only the coefficients need be altered. These changes are independent of the field angle, but depend on thef-number of the system. The coefficients thus derived are field-independent in contrast to those specified by most authors. Expressions for the coefficients under each of the transformations introduced are presented. Also a brief discussion of the convergence of the retardation expansion is presented and the results indicate that the above approximations are sound over the region of convergence of the truncated aberration expansion of order eight.