Form of Newton's Rings. 239 



Again, 



V-- (rf-?) + §- (d-?)=rf=r<\j sin o/r'. 

 Therefore, writing 



«=f+^(?-8 1 ), «=9 + A?-S 1 ),. . . (iii.) 



where 

 we have 



*H>-£]» *'- <, [ 1 -7^]» 



w — — r £'=7-<£'cosifr', 



■"■ * = ; T ; " ¥ *■ * ^ 7 ' ~~ T 5 " ; 



.'. </>' COS ^ = 1 — — + i -si! 2—? 



,, . ,, v . u 2 + v 2 , . au + {3vu* + v 2 

 r r r fxj r 2 fi ry u r r 2 



We shall now proceed to connect <£, -ty with <//, i/r'. 

 Writing 



7 n 



Now 7 fc+ 2, w fc+2 , n^+2 are derived from l k , m k , n k as a, /3, 7 are 

 from I, m, n, <j>' yfr' being replaced by fa ty* . 



Now <f> k cos -yfr k = (f> cos yfr — k - <£ 2 , 



<£*. sin yjr k , = <f) sin ^ — & — <£ 2 . 



> (iv.) 



