This paper presents relatively unknown, though not new, theorems applicable to a real axially symmetrical optical system. It treats the situation where rays leaving a particular axial object pointOin object space are assumed to image perfectly at axial image pointO′. A ray throughOat angle α with the axis passes throughO′at angle α′ with the axis. The Abbe and Herschel conditions state the required functional relationship between α and α′ to ensure that rays fromPimage perfectly intoP′, whenPis infinitesimally displaced fromOperpendicular to, or parallel to the axis, respectively. The formulas derived here give the detailed variation ofβ1,β2, and γ in terms of the functional relation between α and α′, independent of the further specification of the system. They are derived using Fermat's theorem and the second law of thermodynamics. Hereβ1,β2, and γ represent, respectively, the meridional (primary) lateral magnification, the sagittal (secondary) lateral magnification, and the longitudinal magnification relative to small displacements fromO. The variation ofβ1andβ2with α specifies the coma figure, while the variation of γ gives the longitudinal spherical aberration for an axially displaced object point.