46 SIR DAVID BREWSTER ON THE PHENOMENA OF THIN PLATES 



than 78° 4' the system of rings with the black circumference will again appear as at 

 incidences below 53° 1 1', and will be visible up to 90° of incidence, the interfering 

 pencils being now both reflected at angles above the polarizing angle of the surfaces 

 which reflect them. 



This experiment with vapour and'fluor spar I have not made ; and it may be diffi- 

 cult to see the rings at such an oblique incidence. If the rings are formed by soap 

 upon jo/a^e glassy or by alcohol upon Jluor spar, the second disappearance of the rings 

 may be seen : 



Plate glass m 1*510 n^r.A 



soap m' — ris; ~ 1"0154. 



Polarizing angle at second surface of the soap .... 45° 26' 



Angle of incidence on the first surface 71° 45' 



Fluor spar m 1*437 

 alcohol m' ~~ 1*370 "" ^'^^^' 



Polarizing angle at second surface of alcohol 46° 22' 



Angle of incidence on the first surface 82° 32' 



If we call my m! the indices of refraction of the two substances, viz. the Jilm and 

 the surface upon which it rests, m being the larger index, then a ray incident at 90° 

 will fall upon the common surface of the two media at the polarizing angle of that 

 surface, when the angle of refraction at the first surface is equal to the tangent, or 

 cotangent of the polarizing angle, according as the refractive power of the film is less 

 or greater than that of the body upon which it rests. 



Hence we have sin i' = — or — ,» 

 and 



tan ^' = —7, or cot t' = -»> 



and 



m! - , m 



^ = / ,0 , 3 and m' = . ^ ? 

 V m'2 — 1' Vm^ — 1 



when a ray incident at 90° is polarized at the second surface, or falls upon it at the 

 polarizing angle. 



These formulae enable us to discover between what limits of refractive power the 

 second disappearance of the rings can take place, and consequently what substances 

 we should employ in order to observe it. In this manner we obtain the following 

 results for the mean rays of the spectrum : — 



m' 



Values of m'. Values of , . or m. 



3-000 1*061 



2-500 1*090 



2*000 1*154 



1-900 1*176 



