1905.] The Theory of Symmetrical Optical Objectives. 399 



c and d on OF at F. Through E draw EG parallel to ES, cutting OF 

 in G. Now the rays a b c and d can be chosen so that EG = GF = JPQ. 

 It is now evident from the figure that Fr 4 is parallel to E</ 4 and that 

 Fs 4 and Ep 4 intersect in k on the plane mid-way between 4 and 5, which 

 is the focal plane of the combined system. 



But by symmetry the rays b and c in the second medium correspond 

 to a and d in the first ; hence the parallel rays a and d intersect on the 

 focal plane of the combined system, but as ES is inclined at any angle 

 to PQ all other rays parallel to a and d will also intersect at the same 

 point. 



Thus we have shown that, subject to the errors introduced by the 

 want of correspondence of the stop and its image, the combined system 

 is completely corrected for astigmatism, curvature of field, and spherical 

 aberration, provided the back component is so corrected. This want of 

 correspondence introduces some slight errors, but in practical systems 

 these are almost negligible. 



These conclusions accord very well with the deductions made by 

 Dr. W. Zschokke, from the results of calculating the paths of rays 

 through the various Goerz double symmetrical lenses. 



