derived from the experiments of Sir David Brewster. But, 

 in the absence of a real theory, it is important that we 

 should be able to represent the phenomena by means of em- 

 pirical formulae ; and, accordingly, the author has endea- 

 voured to obtain such formulas by a method analogous to 

 that which Fresnel employed in the case of total reflexion 

 at the surface of a rarer medium, and which, as is well 

 known, depends on a peculiar interpretation of the sign 

 V — 1. For the case of metallic reflexion, the author as- 

 sumes that the velocity of propagation in the metal, or the 

 reciprocal of the refractive index, is of the form 



m (cos X -h v^ — 1 sin x) ; 

 without attaching to this form any physical signification, 

 but using it rather as a means of introducing two constants 

 (for there must be two constants, m and x? for each metal) 

 into Fresnel's formulae for ordinary reflexion, which contain 

 only one constant, namely, the refractive index. " * 



Then if i be the angle of incidence on the metal, and t the 

 angle of refraction, we have 



sinz'=?w(cosx + \/ — 1 sinx)sin«, (1) 



and therefore we may put 



cos i'z=L m' (cos x'— V — 1 sin x') cos i, (2) 



if w'^cos^2 = l — gw^cos^Xsin^i + ^'^sin^i, (3) 



and tan^X = 1 2 — ^ — ^^^' (^') 



^ 1— m'^cos^XSi^^ 



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

 of reflexion, and if the preceding values of sin «', cos «', be 

 substituted in Fresnel's expression 



sin(i — «') 

 sin(i+0* 

 for the amphtude of the reflected vibration, the result may 



be reduced to the form 



a(cosg-V-lsinS), (5) 



b2 



