175 



diagonal. Let OA and oa', which are of course equal in length, 

 denote these two sides. Make a similar construction for the 

 other semiaxis oq, and let ob, ob', which are also equal, denote 

 the two sides of the corresponding parallelogram. Then will 

 the incident vibration be represented by the elHpse of which 

 OA and OB are conjugate semidiameters, and the reflected vi- 

 bration by the ellipse of which oa' and ob' are conjugate semi- 

 diameters. And the correspondence o^ phase in describing 

 the three elhpses will be such that the points a, a', p will be 

 simultaneous positions, as also the points b, b', q. 



The same construction precisely will answer for the case 

 of total reflexion at the surface of a uniaxal crystal, which is 

 covered with a fluid of greater refractive power than itself. 

 It is to be applied successively to the ordinary and extraordi- 

 nary refracted vibrations, and we thus get the uniradial inci- 

 dent and reflected vibrations, or rather the ellipses which are 

 similar to them. And as any incident vibration may be re- 

 solved into two which shall be similar to the uniradial ones, 

 we can find the reflected vibration which corresponds to it, 

 by compounding the uniradial reflected vibrations. 



It may be well to mention that, in a uniaxal crystal, the 

 plane of the extraordinary refracted vibration is always per- 

 pendicular to the axis, and therefore the ellipse in which the 

 vibration is performed may be easily determined by the re- 

 mark in p. 102. The plane of the ordinary vibration has no 

 fixed position in the crystal ; but if we conceive the auxi- 

 liary quantities 5i, rji, Z,i, (p. 98) to be compounded into an el- 

 lipse (as if they were displacements), the plane of this auxi- 

 liary ellipse will be perpendicular to the axis of the crystal. 



Whether the preceding very simple construction, for find- 

 ing the incident and reflected vibrations by means of the re- 

 fracted vibration, extends also to the case of biaxal crystals, 

 is a point which has not yet been determined, on account of 

 the complicated operations to which the investigation leads, 

 at least when attempted in any way that obviously suggests 

 itself. 



