Crystalline Reflexion and Refraction. 71 



It is not easy to see why the principle should hold in the one case, and not in the 

 other ; but it is probably prevented from holding, in the case of metals, by the 



which shows that fi, is equal to unity at a perpendicular incidence, and that it vanishes at an incidence 

 90°, decreasing always during the interval. 



Now if plane polarised light be incident on the metal, we must distinguish two principal cases, 

 according as the light is polarised in the plane of incidence, or in the perpendicular plane. In the 

 first case, denoting the reflected and refracted transversals by f, and r, respectively, let us put A, 

 for the change of phase in the reflected ray, and Aj for the change of phase in the refracted ray. 

 Let the same symbols, marked with accents, be used in the second case with similar significations. 

 Then if the incident transversal be taken for unity, we shall have the following formulae : 



1. When the incident transversal is in the plane of incidence. 



' M'-)-^2-4-2Mpcosx' 



r^- !!■>! , 



tan A3 = —^ — f , tan A^ =-7 — — ■ 



2. When the incident transversal is perpendicular to the plane of incidence, 



, , 1+mV— 2m^cosx 1 



^ 1-|-m'/i'4-2m/'cosx' 



' l-j-M*/i^-|-2Mjncosx 

 ., 2Musinx ., silly 



tan A'3 = r^^ \ tan A' ^ = — r^^. 



(XV.) 



(xvi.) 



When %:=0, there is no change of phase, and the formulae become identical with those given in 

 the note, p. 43. When ;n;rr:90°, there is total reflexion at all incidences. The case of pure silver 

 approximates to this. For good speculum metal, ;^ is about 70°. The value of m ranges from 2i 

 to 5 in different metals. ' 



When the incident transversal is inclined to the plane of incidence, its components, parallel and 

 perpendicular to that plane, will give two reflected transversals with a difference of phase equal to 

 A'3 — A3. The reflected vibration will then be performed in an ellipse ; and the position and magni- 

 tude of the axes of the ellipse may be deduced from the preceding formulae. The consequences of 

 these formula! arc very simple and elegant, but I cannot dwell upon them here. Suffice it to 

 observe, that every angle of incidence has another angle corresponding to it, which I call its conjugate » 



angle of incidence ; and that the value of A'3 — A3 at one of these angles is the supplement of its 



value at the other, while the ratio — is the same at both angles ; whence it follows that, ceteris 



''3 



paribus, the elliptic vibrations, reflected at conjugate angles, are similar to each other, and have their 

 , homologous axes equally inclined to the plane of incidence, but on opposite sides of it. When 



