310 Mr. E. B. Sangster on some Consequences of 



The equation is 



, 2 . __ sin 2 i jj? sin 2 r —/jl 2 



1 — sin 2 i "*~ 1 — yu, 2 sin 2 t* ~~ jj, 2 — 2' 



whence sin r= ± \/\. 



No doubts after knowing this, we can interpret the first 

 result, tan i= + 1, as meaning that condition (8) holds for 

 /n< 1 as well as for fi>l, and thence deduce that it can only 

 be true of /j,< 1 when jjl> 1/^2, because, otherwise, there is 

 total reflexion at i = 45°, but this view was not at first 

 apparent. 



And 6 (see quotation from Brewster) equals i — r in this 

 case also ; for 



, G tan (i — r) I sin (i — r) 



tan 6 = )-. {$— — ' 



tan (i -}- r) j sin (i 4- r) 



and (8) being the condition 



_ sin (i—r) _ tan i — tan r 

 sin (i + r) tan i + tan r' 

 but tan r=^l, 

 therefore 



tan = ^ ni r\ = tan (i-45°) = tan(i-r). 

 tani + 1 v y v ' 



The foregoing volume ratios lend themselves readily to 

 the solution of the important problem of determining the 

 Jingle of incidence required in order to reflect 1/nth of the 

 incident light both when one, and when two, surfaces of 

 separation are involved. The formula to be deduced for 

 the case when the incident light is unpolarized, or common, 

 light, is of necessity complex, but we shall find some com- 

 pensation for this in the interesting consequences of Fresnel's 

 Theory not otherwise easily brought to light. We shall, 

 first of all, dispose of the simpler cases where the incident 

 light is polarized in, and perpendicular to, the plane of 

 incidence, and of these, we may begin with Fi esnel's tangent 

 formula. 



When l/?ith the incident light is required to be reflected 

 we have 



1 tan 8 (i-r) _ /S-SV 

 n ** tan 2 {i + r) ~\jb + S')' 



and writing S ; = %S, 

 •* s rrh whence %= >v. .„< --v__ 01 , y 



