308 Mr. R. B. Sangster on some Consequences of 



But, by (1), AC = /x when BD = 1, and therefore also 



/*-AB : 1-DC : : S : S', wherefore 



sinQ'-r) _ /r DC-AB % p 

 sin (i + r) "" /I 7 DC + AB /a 



= 4fel (6) 



It may be of interest to first mention some minor conse- 

 quences of these formulae. At the incidence tan -1 //-, S = S', 

 therefore (5) vanishes and (6) becomes C/x 2 — l)/(/u, v 4- 1). 

 When S=/aS' (normal incidence) both (5) and (6) re- 

 duce to (fjb — 1)/(/a + 1). But, when /a8 = S', (6) becomes 

 l> 3 — i)/(/A s + lj and (5) reduces to (1 — ^/(1 + //). The 

 latter is an interesting ratio of S to S', for when the incident 

 light is plane polarized perpendicular to the plane of inci- 

 dence, we see that by simply reversing the normal incidence 

 ratio of S to S' the reflected amplitude is equal but opposite in 

 sign. The incidence where this occurs is found by writing 

 fi for % in (4), viz. : — 



-'V£i 



Mnts^yCj— (7) 



Another interesting ratio of S to S' occurs at z = 45°, for 

 when light incident at 45° is polarized ±45° to the plane of 

 incidence, then 



sin 2 (-i — r) _ tan (i—r) .„. 



sin 2 [i + r) tan (i -+■ r)' 



In the 8th edition of the Ency. Britannica. article " Optics," 

 Sir David Brewster wrote : "It is a curious circumstance 

 that .... in the action of all substances in turning round 

 the planes of polarization, [by reflexion] at an incidence of 

 45 , the angle of rotation, when the plane of the polarized 

 ray is ±45°, is equal to the angle of refraction, while the 

 new inclination [say 0] of the plane of polarization to the 

 plane of reflection .... is equal to the deviation i—i'." 

 Of course, Brewster's proposition is easily deduced from (8) 

 as the result of the particular angle of incidence. Now, when 

 2 = 45°, BA=AC= fi (see fig. 1) and BD*DC= (2/a 2 -1)^, 

 whence S : S' : : /jP : (2//, 2 — l)i and (5) and (6) become 

 respectively 



s- 



■(V-lj* and (*■-!)*-! 



^ + (2/X 2 -l)5 (V-l) 4 + 'l' 



the former expression equalling the square of the latter. 



