Prof. MacCullagh on the Ldiaos of Reflexion from Metals. 38S 



flexion, and if the preceding values of sin 2', cos i', be substituted 

 in Fresnel's expression 



sin (i — i') 



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

 duced to the form 



a (cos 5 - V - 1 sin S), (5) 



if we put 



tan tf/ = ^, (6) 



tan 5 = tan 2^^/ sin (% + %') 0) 



^o^ l-sin2v|/cosfa+%^) (8) 



1 + sin 2^ cos (x + %') 



Then according to the interpretation, before alluded to, of \/ — 1 

 the angle S will denote the change of phase, or the retardation 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', for any angle of incidence, are found by formulae (3), (4), 

 the quantities tw, p^, being given for each metal. The angle %'is 

 very small, and may in general be neglected. 



Secondly, when the incident light is polarized perpendicularly to 

 the plane of reflexion, the expression 



tan (i — i') 

 tan (2 + i')' 

 treated in the same manner, will become 



a (cos S' - V"iri sin ^), (9) 

 if we make 



tan y}f' z=m m', (10) 



tan ($' = tan 2iJ/' sin (x - x')^ (10 



^,^_ l - sin 24^' cos (y - %^) . ,^2^ 



1 + sin 2^' cos (%-%')' ^ ^ 



and here, as before, S' will be the retardation of the reflected light, 



and a' the amplitude of its vibration. 



The number m = — may be called the modulus^ and the angle % 



the characteristic of the metal. The modulus is something less than 

 the tangent of the angle which Sir David Brewster has called the 

 maximum polarizing angle. After two reflexions at this angle a ray 

 originally polarized in a plane inclined 45° to that of reflexion will 

 again be plane polarized in a plane inclined at a certain angle <p 

 (which is 1 7° for steel) to the plane of reflexion ; and we must 

 have 



tan^=^*. (IS) 



Also, at the maximum polarizing angle we must have 



5' - 5 = 90°. (14) 



