SOME EXPERIMENTS ON THE ZONAL ABERRATION OF LENSES. 749 



And, in the language of the Gauss theory, it is quite easy for two 

 lenses to agrée as to the position of the principal focns while differing 

 as to the position of their principal planes. The principal plane of 

 Gauss is, however, an nnreality. Consider the diagram, Fig. 1, in which 

 the effects are pnrposely exaggerated. Let a ray parallel to the axis 

 enter the lens at A x . Instead of emerging in the direction B A it is 

 refracted first on entrance at A 1 and again on émergence at C 1 so that 

 it cnts the axis at F t . Let now the final part of the ray C 1 F 1 be pro- 

 dnced backwards till it meets the line A t in D x . The lens, in fact, 

 prodnces on this particnlar ray the same effect as if it had gone on from 

 A x to D x and had there abrnptly turned down to F A . Now if the same 

 construction be made for a number of parallel rays which meet the lens 

 at différent distances from its axis we obtain the points JJ 2 , etc. The 

 surface in which ail thèse points lie is the „ principal surface". The 

 place where it interrupts the axis is identical with the ^Haupt-punkt 11 

 or principal point of Gauss. It obviously passes through the outer edge 

 of the lens, since at the edge the points A, C, and J) corne indefinitely 

 near together. The form of this surface will obviously dépend on the 

 curvatures of the two faces of the lens : there will also be in every lens 

 two such surfaces, one for a parallel beam in one direction, the other 

 for a parallel beam in the other direction. It is also obvious that the 

 assumption of Gauss, so beautiful in the simple geoinetry to which it 

 leads, that in every lens each principal surface is always a plane, is 

 very far remote from actual facts. 



Now consider by the aid of this diagram the problems of zonal aber- 

 ration. The points F lf F 2) F 3 , are not coincident with the position F {) 

 of the limiting position for paraxial rays: this want of coïncidence being 

 the ordinary„spherical aberration/ 1 The focal lengths JJ i F 1 ,D 2 F 2 ,D 3 F 3 , 

 are not equal to one another nor to the limiting focal length for paraxial 

 rays, B 0 F Q : this want of equality is the cause of the „ zonal aberration". 

 Euler's condition for eliminating the former aberration brings F 0> 

 F ly F 2 ,F 3 together. Abbe's condition for eliminating the latter aber- 

 ration requires that then the points D 0 ,D 1} D 2 ,D 3 shall ail lie on a 

 spherical surface concentric round F 0 . To fulfil approximately the for- 

 mer condition if the lens be a simple lens of crown glass, it is known 

 that both surfaces must be convex, the curvature of the front surface 

 being about six times as great as that of the back surface. But to fulfil 

 the second condition, the form must be that of a convex meniscus, the 



