1. The object of this investigation is to establish a formula for the longitudinal spherical aberration of rays which traverse a symmetrical optical system in an axial plane that shall be capable of fairly easy computation for any combination of lenses, and at the same time shall be accurate to the second order and free from certain important difficulties of convergency which occur in certain neighbourhoods when we attempt to use for the longitudinal aberration the method of aberration of successive orders. From the point of view of the optical designer, the usual theory of aberrations, which, for all practical purposes, is largely restricted to the first order, is known to give an unsatisfactory approximation. In practice, the designer adopts a semi-empirical method of tracing a number of rays through the system by means of the trigonometrical equations, a method which is laborious and lengthy, and which can at best give only incomplete information and very limited guidance for effecting improvements.
Ray transference of a system with radial gradi- ent index
