386 Royal Irish Academy. 



measure the angles between the different planes as before, and denote 

 them by &, /3' in the first position, and by 90° — 0", 90° — /3" in 

 the second, we shall find that 0' and 0" are unequal, but we shall 

 have /3' equal to /3". The values of and /3 will then be given by the 

 formulae 



8 = 6 1±JH, cos 2/3= C ° 9 , 2 ^' . . . (K.) 



2 r cos(0'-0") v J 



The error of the rhomb may easily be found. Supposing the vi- 

 brations to be resolved in directions parallel and perpendicular to its 

 principal plane, the rhomb is intended to produce a difference of 90° 

 between the phases of the resolved vibrations, or to alter by that 

 amount the difference of phase which may already exist ; but the 

 effect really produced is usually different from 90°, and this differ- 

 ence, which I call s, is the error of the rhomb. The value of s is 

 given by the formula 



sin(0'-0") n y. 



tane = i — ^ . ' \ (L.) 



tan 2/3 v J 



and as the error of the rhomb is a constant quantity, we have thus 

 an equation of condition which must always subsist between the 

 angles 0' — 0" and /3. For any given rhomb the sine of the first of 

 these angles is proportional to the tangent of twice the second, and 

 therefore 0' — 0" constantly increases as /3 increases towards 45°, that 

 is, as the axes of the elliptic vibration approach to equality. When 

 j3 is equal to 45° — \ s, we have 0' — 0" ss 90° ; and for values of 

 /3 still nearer to 45°, the value of sin (0' — 0") becomes greater than 

 unity, that is to say, it becomes impossible, by means of the rhomb, 

 to reduce the light to the state of plane-polarization. This is a case 

 that may easily happen with an ordinary rhomb in making experi- 

 ments on the light reflected from metals ; because at a certain inci- 

 dence, and for a certain azimuth of the plane of primitive polariza- 

 tion, the reflected light will be circularly polarized. 



The rhomb which I used in the experiments tabulated above, was 

 made by Mr. Dollond, and was perhaps as accurate as rhombs usu- 

 ally are ; it was cut at an angle of 54^°, as prescribed by Fresnel. 

 Its error was about 3°, and the value of 0' — 0", at the incidence of 

 75°, was about 7°. But in another rhomb, also procured from Mr. 

 Dollond, and cut at the same angle, the value of 0' — 0", under the 

 same circumstances, was about 20°, and the value of s was therefore 

 about 8°. The angle given by Fresnel was calculated for glass of 

 which the refractive index is 1*51 ; and the errors of the rhombs are 

 to be attributed to differences in the refractive powers of the glass. 

 I was not at all prepared to expect errors so large as these when I 

 began to work with the rhomb, and they perplexed me a good deal 

 at first, until I found the means of taking them into account, and of 

 making the rhomb itself serve to measure and to eliminate them. 

 The value of the rhomb as an instrument of research is much in- 

 creased by the circumstance that it can thus determine its own effect, 

 and that it is not at all necessary to adapt its angle exactly to the re- 

 fractive index of the glass. It may also be remarked, that this cir- 



