OF NEWTON'S RINGS BEYOND THE CRITICAL ANGLE. 655 



It follows from (14) that the ring cannot vanish unless p, cos\f^^= p^^ cosxj/^ , and p sinxl/' = 

 P„ ^''^ "^i,- '^^'^ requires that pf = p^f, which is satisfied only when i = i, in which case as we have 

 seen the ring does not vanish. Consequently a ring is formed at all angles of incidence ; but it 

 should be remembered that the spot, and consequently the ring, vanishes when i becomes 90". 



25. When i = y, the expressions for P^, Q^, take the form - , and we find, putting for 



ttD 



shortness = p, 



p _ (m^ - 1)'' p ,/U— 1)-' 



p-' + (m^ - 1)-' ' * p^ + ^c'U^ - 1)-" 



If we take two subsidiary angles 1^, co, determined by the equations 



ttD /— 



— -Vm — 1 = tan -^ = /r tan cu, 



we get 



Fg = cos" T^, P^ = cos* w, Qe = - sin j^ cos 1^, Q^ = - sin w cos cu. 



Substituting in (15) and reducing we get, supposing a = 45°, 



/ = iversin (2i^ - '2to) (16). 



When i = i, cos 2<^ = - cos 29, sin 2(^ = .sin 29 ; and therefore P^- P„, Q^ = - Qj, which when 

 a = 45" reduces (15) to / = Q/. 



If we determine the angle nr from the equation 



1 - 5" = 25 sin 20 tan is; or tan Tsr = cot 2^. cosec 29, 



we get 



/=lsin'''27!r.cos=20 (17). 



In these equations 



2,rZ» //I'-l 

 log. tan ^ ^ -~~ , cot 9 = ,x. 



26. The following Table gives the intensity of the ring for the two angles of incidence i =7 and i = i, 

 and for values of D increasing by ^,\. The intensity is calculated by the formula; (16) and (17). 

 The intensity of the incident polarized light is taken at 100, and ^x is supposed equal to 1-63, 

 as before. 



4 p2 



