Form of Newton's Rings. 233 



Section V. Some Points about the Rings. 



From these details concerning the "surface of interference" 

 we may learn something about the rings. 



It is evident that since the plane xz cuts the rings in a 

 straight line inclined to the surfaces of the plate, we shall, by 

 proceeding far enough in either direction, at last reach rings 

 which lie partially above or partially below the plate. 



The upper surface of the plate is 



z— (d— £>) sec 



= d sec 0—z v 



The lower surface is z= —z r 



Hence the line (iii.) in the last section intersects the upper 

 surface of the plate where 



_ 2(d-z l cos 6) _ 2d cos* 

 sin cos fi sin cos 3 X ' 



and the lower surface where 



-2z x ,, ^-1 . 



#=— — \= —2d -5 e-rr sin u, 



sm /u? cos 3 l ' 



since the order of the ring in this direction (y = 0) is 

 given by 



4?=+«= + \/ for the dark rings, 



- - V C os & 9 



If h> 



+ \ / —ri -t\ — for the bright rings. 



— V 2 cos 



4d 2 cos 5 e 



\r /i* sin 2 cos 6 0, s 



the dark ring of order k has points above the upper surface of 

 the glass plate. 



4d 2 cos 5 



If h> 



Xr i* sin 2 cos 6 X ¥ ' 



order A lies parti 

 date. 



4d 2 (> 2 -l) 2 sm 2 0cos 



the bright ring of order h lies partially above the upper 

 surface of the glass plate. 



If h> _ 



Xr /j? cos b 0! 



the dark ring of order h lies partially below the lower surface 

 of the glass plate; and if /i + J be greater than the quantity 

 Phil. Mag. S. 5. Vol. 29. No. 178. March 1890. T 



