SPHERICAL A HKII NATION. 



57 



eliminated the aberrations as far as possible, the magnifying 

 power may vary considerably from what was intended. Ex- 

 perience teaches how to keep these unavoidable discrepancies 

 within tolerably narrow limits. 



The combination of lenses, which we have previously assumed 

 for the determination of the cardinal points, has still to be ex- 

 amined with respect to both the aberrations. This examination 

 can, of course, be only a purely theoretical one ; it is not intended 

 to discuss the different assumptions upon which this determina- 

 tion is based, but rather to furnish an example of an approximate 

 calculation for similar combinations of lenses. 



The data for the estimation of chromatic aberration are, how- 

 ever, wanting, as the dispersive ratios of flint- and crown-glass 

 for the assumed indices of refraction are not known. We will 

 therefore confine ourselves to spherical aberration, and will in- 

 troduce the further simplification, that the marginal rays, whose 

 course is to be traced, proceed from one point of the optic axis. 



Let A B (Fig. 19) be the first objective-lens, a an axial point 

 of the object whose distance from the lens is S, a p a marginal 



(X, 



FIG. 19. 



ray, which forms the angle < with the axis, and therefore also 

 with the perpendicular to the first refracting surface, x p its 

 direction after the first refraction ; then the angle P which 

 the refracted ray makes with the axis, is determined by the 

 refractive index of the flint-glass. Consequently, the distance of 

 the point a^ from the refracting surface, or the focal length corre- 

 sponding to the object-distance 8, which we will indicate by /j, 

 may be regarded as given. We have therefore 



tan 



