The ray-aberrations in axis-symmetrical systems are conventionally derived from wavefront functions or characteristic functions using classical approximate partial derivatives. However, the resulting aberrations typically have fifth-order errors, as described by Restrepo et al. (2017) [1]. Accordingly, in the present study, the secondary ray-aberration coefficients for object placed at finite distance are determined using the fifth-order Taylor series expansion of a skew ray. Notably, the derived expressions are exact since they are determined without any approximations. It is found that some of the aberration coefficients are not constants, but are functions of the polar angle of the entrance pupil. It is additionally found that, once the required derivative matrices have been generated, determination of the secondary aberration coefficients is straightforward without iteration, and incurs only a low computational cost.