Geometrical Optics. 243 



position of curvatures. Thus, take the case of a wave 

 possessing initial curvature °ll entering from air into a medium 

 having velocity-constant A, and so curved that the focal 

 power of the curved surface is ft. Then, as the wave 

 enters the surface of the medium two effects will occur : 

 its initial curvature will be altered in the ratio of the velo- 

 cities, and there will be superposed upon it the focal curvature 

 of the surface ; or, in symbols, 



^J 1 = h^-hft l (8) 



For an emergent wave, possessing initial curvature <% in the 

 medium, the formula will be 



% = \<M+F 2 (9) 



Or, for the case of a wave passing from a medium of velocity- 

 constant A, to another of velocity-constant h i} the formula 

 will be 



V = J pW+ft. (io) 



It is easy, however, to prove any one of the several cases 

 that may arise, without in this way relying upon the prin- 

 ciple of superposition. Take the case of a positive wave enter- 

 ing a positively curved surface. 



Let SOS (rig. 8) be the surface of the medium, its cur- 

 sor. 8. 



vature being measured by the sagitta C M. There will be 

 a certain moment when the entrant wave, converging toward 

 P, would have had as its front S A S had its path lain wholly 

 in air. But the central portion has entered the retarding 

 medium at 0, hence will only have advanced as far as B 

 instead of A ; B being such that BO = A. AC. Hence the 

 resultant wave will have the form S B S, and the sagitta 

 of the resultant curvature is B M, 



