of Reflexion and Refraction. 289 



spectively that B and B' do in the equations of motion for 

 transversal waves. 



4. The equations liere given for finding P, Q, fl and i are 

 apparently complicated, but in point of fact they are not so, 

 as will presently appear. The equations for determining Q 

 and I may be easily applied to determine the values of the 

 unknown constants involved in them by comparison with ex- 

 perimental results, and appear to be well-adapted, not only 

 for testing the truth of the theory here advanced, but also (sup- 

 posing that theory to turn out true) for deciding several im- 

 portant questions in the undulatory theory of light. We shall 

 now show how the equations for determining Q and i may be 

 compared with experiment. 



5. These equations apply to the case of reflexion at metallic 

 surfaces and surfaces of high refractive power. The formulae 

 for the incident light, 

 a cos [nt — k{2i x ■\- s%)} and h cos [nt — k {p x + sz) }, 



represent together a plane polarized ray incident at an angle 

 4) (jt? = sin <p, s = cos ^), the angle which the plane of pola- 

 rization makes with the plane of incidence being 



1 ^ 

 tan~i— . 

 a 



The corresponding formulas for the reflected light are 



— P « cos {71 1 — Jc{2^x — sz) — 9}} 



and V Qhcos [nt — k{px—sz)—^ — i], 



which, on account of the difference of phase (i), represent in 

 general, not a pi atie, but an ellipticalli/ polarized ray*. Ifi 

 be any multiple of 180°, the polarization becomes plane. 



If we suppose the ray to be reflected m times successively at 

 two parallel surfaces of the same metal or substance, the for- 

 mulae for the light after the last reflexion will be 



( — Py"flCos {nt—k[px—sz)—vi^], 

 and (PQ)"*icos {7it—k[px — sz)—m^—mi]. 



Let us suppose that mi — 180°, then these two expressions 

 represent together a plane polarized ray, the angle which the 

 plane of polarization makes with the plane of incidence being 



tan" 



{<)■ 



Now Sir D. Brewster has determined the angles at which 

 a plane polarized ray must be incident upon the surfaces of 



* Sir D. Brewster calls the polarization ellijitical when /=90^, znA par- 

 tially elliptical when ; is not equal to 90". It is easy to see that the pola- 

 rization is always completely elliptical, only the plane of incidence does not 

 contain either axis of the ellipse except when t=9(f. 



Phil. Mag. S. 3. Vol. 26. No. 173. April 18^5. X 



