Geometrical Optics. 237 



any other point at distance d further from or nearer to the 

 centre, the formula for the new curvature &J being as 

 follows : — 



(i) 



The + sign must be taken where the new point is further 

 from the centre than the point for which the curvature <% 

 is specified ; the — sign when it is nearer the centre. This 

 proposition is of use in dealing with thick lenses, and with 

 thin lenses at a given distance apart. 



5. Refraction Formula. 



As a preliminary to lens formulae, it is convenient to con- 

 sider certain cases of refraction. 



A. Plane Surface : Curved Wave. 



Case (i.) a. Entrant Wave of Negative Curvature. 



Consider a retarding medium, such as glass, bounded on 

 the left (fig. 2) by a plane surface S S. Let P be a source of 



Fur. 2. 



waves incident on the surface, PM being a line perpendicular 

 to SS. The wave-fronts, at successive small intervals of time, 

 are represented by arcs of circles. At a certain moment 

 the wave, had it been going on in air, would have had for its 

 surface the position SAS ; the curvature being measured by 

 the sagitta AM. The medium, however, retards the wave, 

 and it will only have gone as far as B instead of penetrating 

 to A ; B being a point such that B M = h . AM, where h is the 

 velocity-constant of the medium into which the wave enters. 

 The curvature of the wave is flattened as the result of the 

 retardation. Now draw a circle through SBS, and find its 

 Phil. Maq. S. 5. Vol. 28. No. 173. Oct. 1889. T 



