Curvature and Refractive Index. 35 



A similar proof will show that 



BsCA + Bi) 



^-/XRi+B,) (2) 



By the property of lenses, 



^ "=F(ffTy +1 > (3) 



where F is the principal focal length. 

 By combining (1) and (3), 



111 -d F /i fA\ 



% = jrY , or R 1=f ^ (4) 



By combining (2) and (3), 



By combining (1) and (2), 



---=4-4 . (6) 



Ri E2 /\ f 2 



It is not a little surprising that, whatever the refractive 

 index of the material of the glass, or the curvature of the front 

 surface, the curvature of the back surface can always be ob- 

 tained ^from an expression in which both apparently are 

 omitted. They are both of course involved in each observa- 

 tion, F and/, which accounts for the possibility of their being 

 eliminated. 



It is interesting to follow the changes which occur between 

 the two extreme limits of form — a double convex and a double 

 concave lens. Take a double convex lens, and suppose one of 

 the surfaces to be gradually pushed in ; when it has become 

 plane we have the first particular case — a plano-convex lens. 

 Call the flat surface 1 and the convex surface 2 ; then, by (4), 



i_-i__JL 



*> " /; f ; 



.'./i=F, or the apparent centre of curvature of the flat sur- 

 face seen through the round surface is at the principal focus. 



By (5), 



1 _ 1 n-1 

 B« f 2 R2 



T> 



since F = — -^; .-. R 2 =/J : f 2 ,ov the apparent radius of 2 is less 



than the true radius in the ratio of /x to 1 . 



If the pushing-in process is continued, the surface 1 will 

 become concave. Four observations can then be made — F, 

 fit fit) an( l Ei ; therefore E 2 mav be found bv either of the 

 D2 ' 



