Royal Irish Academy. 385 



sin <j> tan 2 cr = SL, (I.) 



or that the product on the left side of the last equation is indepen- 

 dent of the angle of incidence. It is to be observed that the relations 

 (G.) and (I.) are independent of the value of [/., and may hold good 

 though that value should require to be changed. 



All the preceding formulas are merely mathematical consequences 

 of those which I published long ago in the Transactions of the Aca- 

 demy (vol. xviii. p. 71). The formulae which I had previously given 

 in the Proceedings (vol. i. p. 2) are slightly different, and, I think, 

 less likely to be exact, because they are less simple, and do not lead 

 to any of the remarkable relations which may be deduced from the 

 others. 



Having had occasion, in the course of the few experiments which 

 I made with the instrument before mentioned, to study the nature 

 of Fresnel's rhomb, which constitutes an important part of it, I shall 

 here describe the method which must be followed in order to obtain 

 true results, when the rhomb is employed in observations on light 

 elliptically polarized. A ray in which the vibrations are supposed 

 to be elliptical is given, and what we want is to determine the ratio 

 of the axes of the elliptic vibration, and their directions with respect 

 to a fixed plane passing through the ray ; in other words, to deter- 

 mine the angles which we have denoted by /3 and 6 in the case of a 

 ray reflected from a metal. For this purpose the ray is admitted 

 perpendicularly to the surface at one end of the rhomb, and after 

 having suffered two total reflexions within, passes out perpendicu- 

 larly to the surface at the other end. Then causing the rhomb to 

 revolve about the ray, we shall find two positions of it in which the 

 emergent light will be plane-polarized, these positions being readily 

 indicated by a Nicol's prism interposed between the rhomb and the 

 eye ; for such a prism, by being turned round the ray, can make the 

 light totally disappear when it is plane-polarized, but not otherwise. 

 These two positions of the rhomb will be exactly 90° from each 

 other ; in one of them the principal plane of the rhomb (the plane of 

 reflexion within it) will be parallel to the major axis of the elliptic 

 vibration, and the angle which it makes with the plane of incidence 

 on the metal will be equal to d : while in the same position the angle 

 which the principal plane makes with the plane of polarization of the 

 emergent ray (as given by the Nicol's prism) will be equal to fi. In 

 the other position, the principal plane will be parallel to the minor 

 axis of the elliptic vibration, and the corresponding angles will be 

 equal to 90° — and 90° — /3 respectively. This, however, proceeds 

 on the supposition that the rhomb is exact. When it is not so, 

 which is of course the proper supposition, and a very necessary one 

 in the experiments with which we are concerned, there will still be, 

 generally speaking, two positions of it in which the emergent ray 

 will be plane-polarized, or in which a disappearance of the light may 

 be produced by the Nicol's prism ; but these positions will no longer 

 be 90° from each other, nor will the principal plane, in either of 

 them, coincide with an axis of the elliptic vibration. If we now 



Phil. Mag. S. 3. Vol. 24. No. 160. May 1844. 2 C 



