Form of Newton's Rings. 223 



theretardation when one of the rays A B C B, C t . . . . 



C*_i D E G (fig. 2) is reflected k times at the surface of the 



Fig. 2. 



lens, and Jc — 1 times at the lower surface of the plate, the 

 other ray being only reflected at B' as before. 

 Take now 



J , 7), f as coordinates of F, 



Zr,Vr,$r ,, <>fC,_i [><&], 



«W,P - „ ofB 



u, v, „ of B'. 



Let the spherical coordinates of C, Ci . . . O r . . . be (/>, yfr ; 

 <f> ly ^ . . . ; <f> r , sjrr . . . ; 

 and let 



/r, mr, w r be the direction-cosines of C r _iB r . 

 Then 



— l r , —m r , nr are the „ of C r Br. 



Let 



V, m', ri be the direction-cosines of BA and B'A' ; 



-l l , -m', ri are those of B'E'. 



h f , m k ', m! „ ofDE. 



-I, -m, n „ ofFG'. 



4, w*, n k „ ofFG. 



Also 



h=/dk', m k =fim k ', s/ y? — l + n k 2 =fin k . 



If we write down the coordinates of B 1? deriving them from 

 the coordinates of C first, and then from those of C,, we obtain 



r sin $ cos \Jr + - 1 2r sin 2 ^ =r sin </>j cos ^ L 2?- sin 2 \\ 



? ?1 



m 



* 



r sin <£ sin ^4- — *2r sin 2 £ =rsin X sin \K -2rsin 



i 



.2 01 



2 



