382 Royal Irish Academy. 



it makes with the plane of incidence ; and as the reflected light will 

 he elliptically polarized, or, in other words, will perform its vibra- 

 tions in ellipses all similar and equal to each other, as well as simi- 

 larly placed, put 6 for the angle which either axis of any one of these 

 ellipses makes with the plane of incidence, and let ft be another 

 angle, such that its tangent may represent the ratio of one axis of 

 the ellipse to the other. Then when the optical constants M and % 

 (of which I suppose the first to be a number greater than unity, and 

 the second an angle less than 90°) are known for the particular 

 metal, the angles and ft may be computed for any value of a, at 

 any given angle of incidence, by the following formula? : — 



tan20 = / (, '- y)8fa2g , sin2/3 = , 2 g si " 2 " . (A.) 

 2/+ (v' + v)cos2a H v' + y + 2/cos2a v ' 



in which /and g are constant quantities given by the expressions 



/=(M-^-)cos % , 9 =(M + ^ym X ,. . (B.) 



and v, v' are quantities depending on the angle of incidence i, in the 

 following way. Let i' be an angle such that 



sin i M /n . 

 (W 



and put 

 then will 



cos^' 



cos % 



v = J_-^, *m£±j£ ( e.) 



The angles 6 and ft are given by immediate observation with the in- 

 strument ; and from their values at any incidence, and for any azi- 

 muth a of the plane of primitive polarization, we can find the con- 

 stants M and %, which we may afterwards use to calculate the values 

 of and ft for all other incidences and azimuths, in order to compare 

 them with the observed values. It is indifferent, in the formulae, 

 whether be referred to the major or the minor axis of the elliptic 

 vibration, as also whether tan ft be the ratio of the minor to the 

 major axis, or the reciprocal of that ratio ; but in what follows we 

 shall suppose 6 to be the inclination of the plane of incidence to that 

 axis, which, when a is 45° or less, is always the major axis ; and ft 

 shall be supposed less than 45°, in order that its tangent may repre- 

 sent the ratio of the minor axis to the major. 



When the azimuth a is equal to 45°, the formulae (A.) become 



tan20 = ^, sin 2/3 = -*£-; .... (F.) 

 2/ v 4- y 



from which we may deduce the remarkable relation 



tan 2/3 __ g_ , G . 



cos 2 6 f' 



showing that, in the case supposed, the ratio of tan 2 ft to cos 2 d is 



independent of the angle of incidence. In the experiments which I 



made with Mr. Grubb this azimuth was always 45° ; and the follow- 



