Laws of Reflexion from Metals. 59 



and therefore we may put 



cos i f = m' (cos \ - y - 1 sin x ') cos , (2) 



if 



m* cos 4 / = 1 - 2m* cos2 x sin 2 / + m* sin 4 /, (3) 



and 



m 2 sin2v sin 2 / 



tan 2 X = 1 5 4 (4) 



1 - m 2 cos 2 X sm 2 



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

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

 tuted in Fresnel's expression 



sin (*' - 



sin (/ + /')' 



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

 reduced to the form 



a (cos 8 - */ - 1 sin ), 

 if we put 



tan i// = (6) 



tan 8 = tan 2i// sin ( x + x '), (7) 



a 2 = 1 ~ sin 2$ cos ( x + \) , g , 



1 + sin 2t// cos ( x + x ')" 



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

 -t/~i, 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' 9 x ', for any angle of incidence, are found by 

 formulae (3), (4), the quantities m 9 x , being given for each metal. 

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



