242 Prof. Silvanus P. Thompson on 



As before, for any two media having respective velocity- 

 constants hi and h 2 , the formula becomes 



^=^}irh (5) bis. 



hi 



which, in the present case where \ < h 2 , will give $ of opposite 

 sign to «%. 



Case (ii.)Z>. Emergent Wave; Surface Concave toward light 

 (i. e. convex toward air into which wave emerges). 



This is similar to the preceding, and yields the same formula. 



Comparison of Case (i.)a with Case (ii.)ft. 



. Comparing formula (4) with formula (6), we get for the 

 two primary focal curvatures impressed respectively on plane 

 waves passing in opposite directions through the curved 

 surface, 



■%- /..'•••••■ (7) 



whence, for the two primary focal lengths, 



fa K 



The focal lengths are, as the negative sign indicates, to be 

 measured in opposite directions with respect to the surface. 

 Also, taking the algebraic sum of the two primary focal 

 lengths, we get 



fa=r 



h,-h 



fa = 



h 9 — h 2 





h 2 — h\ hi — h 2 9 



h x —h 2 



and, as one of the focal lengths is negative, it follows that 

 the difference of their lengths is equal to the radius of the 

 curved surface. 



C. Curved Surface : Curved Wave. 



The cases in which a wave possessing initial curvature 

 passes through a curved surface and acquires a resultant 

 curvature may be dealt with, apart from any further geo- 

 metrical constructions, by applying the principle of super- 



