What is Centered system; Gauss conditions?

A centered system has an axis of rotational symmetry, the optical axis. Rays that stay close to the axis and make small angles with it are paraxial; a system used with paraxial rays only is said to work under the Gauss conditions. Under those conditions a centered system is approximately stigmatic for every point, and an object plane perpendicular to the axis has an image plane perpendicular to the axis (aplanatism).