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 formulas ; 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 + 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 ;y_, for each metal) 

 into Fresnel's formulas for ordinary reflexion, which contain 

 only one constant, namely, the refractive index. 



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

 angle of refraction, we have 



sin/= ?«(cosx + v 7 — 1 sin x) sin 2, (1) 



and therefore we may put 



cos i'— m (cos x'— V — 1 sm %') cos •* (^) 



if m' i cos 4 i = 1 — 2m 2 cos2xs\n 2 i-\-?n i sin 4 i, (3) 



„ , ?n 2 s'm2ysmH ,.. 



and tan2x — ; s — ~ — r-^. (4) 



*• 1 — nr cos %x sin i 



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

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

 substituted in Fresnel's expression 



sm (i — i') 

 sin^'-H')' 



for the amplitude of the reflected vibration, the result may 

 be reduced to the form 



a(cosS— \/ — 1 sinS), (5) 



B 2 



