FremeVs Reflexion of Light Theory. 311 



We shall refer to the values of % as the -\- or — % value 

 according as the + or — , respectively, of the double sign is 

 indicated. 



Substituting these values of % in (3) and (4), we get 



When n = oo (as at the polarizing angle), % = 1, and tan i—fi, 

 whichever of the pair of % values be employed. 



Now, first consider fi>l. At normal incidence the ratio 

 of light reflected is (/x — l) 2 /(/x-fl) 2 ; therefore, make 

 n = (/A + l) 2 /(/j,— l) 2 , then, the -f% value =/ul, and the — % 

 value =/jb~ 1 . Substituting these % values in the expression 

 for sin i we get respectively 



sin i = fi\ / ^ ~ and 0. 



V fM b — 1 



The latter sine value indicates normal incidence, and the 

 former is identical with (7), which was shown to be the sine 

 value of i when the ratio of reflected light is again equal to 

 that reflected at normal incidence. But both ^ values were 

 found to be applicable at the incidence tan -1 // ; therefore, 

 we should write the — % value in (3) or (4) for the range 

 of i from normal incidence to tan -1 //,, and the + x value for 

 the range from the latter angle to that whose sine is expressed 



by (7). 



The + Rvalue has also to be adopted for the remaining 

 range oft to 90°, a range where the value of n must lie 

 between (/x-f l)'-'/(/x — l) 2 and 1. If n lie within these limits, 

 the adoption of the — ^ value leads to an imaginary expres- 

 sion for tan i, and to impossible values of sin i. The unreality 

 of tan i under these conditions may be shown as follows : — 

 Let n = (//,-}-l — h) 2 /(fi- l-i- h) 2 where h may have any 

 value between and 1, thus covering the case of any 

 value of n between (/x+l) 2 /(/x— l) 2 and 1. Substituting 

 (fju+ 1 -h)l(iJb -1 + h) for y/n in the — % value, we find 



% =(i-/0/V 



and tant = /^/M^ 



The numerator, h(h—2), is negative for any value of h<2, 

 and jjl 2 — 1 is positive when /*>1, wherefore the expression 

 is imaginary. In similar manner it can be shown that the 



