Formation by Crystalline Media. 255 



Draw the ray PABG whose successive directions make 

 the angles u, /3, 7 with the axis ; the line QA makes the 

 angle 8 with the axis. These angles, much exaggerated in 

 the diagram, are sufficiently small to admit of the ordinary 

 approximations of the standard u first order " discussions of 

 the principles of image-formation. 



Fig. 3. 



The line SB is parallel to QA. For the time required to 

 travel from any one wave-position to any other is the same 

 for all rays ; and for any given ray this time is proportional 

 to the distance between the waves. Hence, since A lies on 

 the wave-front B/M from R and B lies on the corresponding 

 wave-front through N, it follows that BA : RB :: RM : RN. 

 But QA : SB in this same ratio by (10), so that the triangles 

 RQA and RSB are similar. 



The distances from A and B to the axis are 



y=PM. a =QM.S=RM.& 



and y' = B,N.£ = SN.8 = ra\. 7 , (11); 



and, in view of these values, standard relations for the first 

 and second refractions may be written from (8) as follows : 



n l h-u = k'y\ (12), 



and 7— wi8 = *V, (13). 



The problem now resolves itself into the elimination of 

 the quantities which are involved in the use of particular 

 rays and of the intermediate image formed within the medium 

 of the lens. By this process the law for the image-position 



