Geometrical Optics. 



239 



P, situated in the retarding medium whose velocity-constant 

 is h. Had the wave been going on wholly through the denser 



Fig. 4. 



medium, the wave-front would have been at S A S; but, being 

 accelerated on emergence into air, it reaches B instead of A. 

 The new curve S B S has Q for its centre ; that is to say, the 

 wave emerges from Q as a virtual focus, its curvature being- 

 augmented. The sagitta B M is greater than A M in the 

 ratio of 1 to h. Hence in this case the formula is 



^ = ^. 



(3) 



Case (li.)b. Emergent Wave of Positive Curvature. 



There is no need to prove this case separately ; it leads to 

 the same formula 



#= \<U. 



(3) bis. 



In the case of either positive or negative initial curvature, 

 emergence from the retarding medium through the plane 

 surface into air augments the curvature. 



B. Curved Surface : Plane Wave. 

 Case (i.) a. Entrant Wave; Convex Surface. 



The surface S M S is convex toward the light ; the centre 

 of curvature being 0. Consider the plane wave entering the 

 surface. At a certain instant it would have arrived at S A S 

 had its path lain wholly in air. Because of the retarding 

 medium the central portion of the wave-front will only reach 

 B instead of A ; B being such a point that BM=/t.AM. 



T2 



