7O On the Laws of the Double Refraction of Quartz. 



revolution is also equal to , and parallel to the axis of the 

 crystal, while the radius of its equator is equal to &, the ordi- 

 nary nappe of the wave surface will fall entirely without the 

 sphere, and the extraordinary nappe entirely within the spheroid, 

 whether the crystal be right-handed or left-handed. With re- 

 spect to the little ellipse in which the vibrations are performed, 

 and of which the semiaxes parallel to x and y are represented by 

 p and q respectively, it is evident that p > q f or the ordinary 

 wave, since & < 1 ; and that p < q for the extraordinary wave. 

 When (7 vanishes, the minor axis of each ellipse also vanishes, 

 and the rays become plane-polarized, the ordinary vibrations 

 being then parallel to the direction of #, and the extraordinary 

 parallel to that of y. This is exactly what ought to happen on 

 the supposition that the vibrations of a plane-polarized ray* are 

 parallel to its plane of polarization a supposition which was 

 kept in view in framing the fundamental equations (1) and (2). 

 To show, with precision, how the two kinds of quartz are to 

 be distinguished by the sign of (7, we must give definite direc- 

 tions to the axes of co-ordinates. To this end, let us imagine the 

 plane of xy to be horizontal, and a circle to be described in it 

 with the origin for its centre ; and let the north, east, and 

 south points of this circle be marked respectively with the letters 

 N, E, S. Let the direction of + x be eastward, from to E ; 

 that of + y northward, from to JV; and that of -t- 2 vertically 

 downwards ; the progress of the light through the crystal being 



* On this point there are two very different opinions. Fresnel supposed, as is 

 well known, that the vibrations of a plane -polarized ray are perpendicular to its 

 plane of polarization ; whereas, according to M. Cauchy, whom I have followed, 

 they are parallel to that plane. I am induced to adopt the latter supposition, be- 

 cause I have succeeded, by means of hypotheses which are grounded on it, in dis- 

 covering the laws of reflexion from crystallized surfaces ; laws which include, as a 

 particular case, those discovered by Fresnel for ordinary media. The hypotheses 

 alluded to, along with some of their results, are published in the London and Edin- 

 burgh Philosophical Magazine, Vol. vin., p. 103, in a letter to Sir David Brewster 

 (supra, pp. 75, et seq.} See also Vol. vn., p. 295, of the same Journal (supra, 

 pp. 55, et seq.} I hope soon to offer the Academy a detailed account of my researches 

 on this subject. 



