Again, 



Form of Newton's Rings. 



u=i- 1 - (d-$)+ l ±d=%+ l -(!;-S l ), 



n K J n x n v 



221 



where 

 And 



.=•-=<*-«)+ =*-»+=<r-to, 





I. = f + -, (<*- - -^ - ^2r sin 2 %, 



, l= , + __ ( rf_ ?) __ < Z__2,.sm*!. 



Inserting the above expressions for the various direction- 

 cosines in terms of I, m, n, (j>, yjr, and writing 



^t 1- ^]^ 82 ' 



we have, to the 2nd order of (f>, 



e.=? + i (?-80-2* co S ^(5-8 t ) - 1(5-8 2 ) 



(Z$ cos yfr-\- m<p sin -^r) + - 2r sin 2 |- 



2£ 

 = u—2<f>costy(£—8 l ) g (Z0 cos i/r + m<£ sin t/t) 



(f-ao+ii*^; 



2m 



rj^v— 2<£sin ijr (f— 6\) 2~ (7$ cos i/r + m<£ sini/r) 



m 



(5-80 + i^^ 2 . 



W 



^ (iv.) 



Comparing these values with those in equations (i.), we find 

 that, to the first order, 



r<j> cosyjr = u, r<\> sin i/r = v (v.) 



Using this approximation in the terms of 2nd order. 



