FresaeVs Reflexion of Light Theory. 309 



Thus for all values o£ /a, when the vibration of the light 

 incident at 45° is symmetrically distributed as between 

 being in and perpendicular to the plane of incidence, the 

 reflected component in the plane of incidence is equal to the 

 square of the reflected component perpendicular to that 

 plane. 



But the problem suggests itself, Is there not another angle 

 of incidence, greater than tan _1 /x, where the same relation 

 holds good ? We know Fresnel's tangent formula is a 

 diminishing ratio so far as the polarizing angle, and there 

 had need be an incidence where (8) is the condition. But 

 when i exceeds tan*" 1 //, the tangent ratio increases with i until 

 at grazing incidence the tangent and sine ratios are equal. 

 Therefore, in some part of the range of i beyond the 

 polarizing angle we might reasonably look to find a recur- 

 rence of (8). Referring to fig. 1, AC : AB : : fju : /*/ tan?, 

 whence 



S : S' : : /r/tan ? : \/(//, 2 -f yu, 2 cot i — 1). 



Substituting the latter ratio in (5) and (6) and equating 

 these as in (8) we have 



/ Q 2 - tan 2 / + ,a 2 tan 2 Q * - 1 \ 2 _ ff-Qi*- tanS+M a tairt')* 

 \ (fM 2 - tan 2 i + fx 2 tan-i) * + 1 / fi 2 + (ji?- tan 2 i + ft 2 tan 2 ^' 



whence tan?' = +1. This result comes from what seems a 

 perfectly general statement, yet it does not appear to give 

 us any alternative to ? = 45 , and one might be tempted to 

 conclude that there was in fact no alternative. It was possible, 

 however, that the statement could be made more general by 

 raising it to a higher order, and so it proved, for on first 

 squaring both members of the last equation and resolving 

 for tan? (the abridged operation involving a statement 

 containing sixty-two terms) the result is tan i = ±1 or 



± [— ^ 2 /(^ 2 ~p)] 2 ' The latter pair of roots are what was 

 sought. It will be seen at once that these new values of tan i 

 are real only when //,< \/2, thus we may conclude that there 

 does not exist an alternative value of i for the condition (8) 

 when fi exceeds \/2. Nevertheless, although we find an 

 alternative angle of incidence for condition (8) when p<\/2, 

 yet we still have 45° as an adjunct, for when 



tan i = [-/*7fo f -r2)"]*j then sin r= ± x /i, i, e., r = 45°. 



