Laws of Reflexion from Metals. 59 



and therefore we may put 



cos i' = tn' (cos x' - \/ - 1 sin x') cos *> (^) 



if 



w' 4 cos 4 / = 1 - 2m 2 cos2x sin 2 / + m 4 sin 4 /, (3) 



and 



tan 2x' = :; 5 ^ r-r-.. (4) 



1 - m 2 cos 2x sin 2 * 



Now, first, if the incident light be polarized in the plane of 

 reflexion, and if the preceding values of sin /', cos i', be substi- 

 tuted in Fresnel's expression 



sin (/ - /') 



~ 7~' -f\ > (" ) 



sin ((> + i) 



for the amplitude of the reflected vibration, the result may be 

 reduced to the form 



a (cos - \ - 1 sin ), 

 if we put 



tan * = * (6) 



tan 8 = tan 2i// sin (x + x')> 

 1 - sin 2\L cos (v + v') 



r \ /v /\ / 



1 + sin 2^ cos (x + xT 



a = 





Then, according to the interpretation, before alluded to, of 

 v / ^-~l, the angle 8 will denote the change of phase, or the retar- 

 dation of the reflected light ; and a will be the amplitude of the 

 reflected vibration, that of the incident vibration being unity. 

 The values of m', x' f r anv angle of incidence, are found by 

 formulae (3), (4), the quantities m, x> being given for each metal. 

 The angle x' is very small, and may in general be neglected. 



