INDEX OF ROTATORY POLARIZATION. 201 



E, parallel to MADB. It is conceivable that this plane may be made to pass 

 through the point where the spirals intersect each other. If I mark the point 

 of intersection, and we draw the tangents IP' and IQ' in the plane of the circle 

 EHl, then there will be a molecuh; at the point I which will be in the circum- 

 stances of the molecule in Fig. 49 at the- point M — t4iat is to say, solicited by 

 three forces, of which two, IP' and IQ', are equal and opposite, and the third is 

 directed in the line IG toward the centre. The molecule will, therefore, move 

 in this line, and not in a circle ; and if the plane of the circle EHIH' be the 

 bounding siu-face of the crystal, or the surface of emergence of the light, IG will 

 mark the azunuth of the molecular movement of the emergent ray. 



But if the plane of EHIH' do not pass through the point of intersection of 

 thf' spirals, it must cut each spiral in a different point. The figure is drawn to 

 represent this more general case, the points of intersection with the spirals being 

 severally L and K. By joining LK, and drawing the radius GI perpendicular 

 to it, GI will bisect the angle LGK, and M', at the intersection of GI and LK, 

 will b:^ the position of the molecule in the plane EHLIK, which, if the tangen- 

 tial force P only were acting, would be at L, and if the tangential force Q only 

 were acting, would be at K. The tangential forces acting at the moment on 

 this molecule will not be represented by IP' and IQ', but by tangents at K and 

 L, like RP' and RQ' in Fig. 49, in which figure the position of the molecuh; M' 

 corresponds to that marked by the same letter in the present ligure ; but in that 

 figure the resultant of the tangential forces is RC, directed to the centre, aud in 

 this it will, in like manner, be IG. 



Now, as DH, the distance between the planes ADB and EHI, ia a larger 

 part of the length of an entire turn of the spiral JMSNK than of the spiral 

 MFLN', the line GI will fall on the right of GH, the position it would 

 occupy if the two undulations were equal in length. We may therefore say, as 

 before, that if the plane EHI were the surface of emergence of a ray from a 

 crystal, in which it had been subject to the action of the forces supposed, its 

 plane of polarization, GI, would be turned toward the right from its original 

 azimuth. The plane of polarization turns, therefore, in the direction of the 

 windivg of the closest spiral, or of the ray of shortest undulation; but it turns 

 in the direction of the gyration of the ray of longest undulation. 



This rotation of the plane thus demonstrates that the tAvo rays advance with 

 unequal velocities in the axis of quartz — a remarkable fact which is not true of 

 any crystal which produces plane polarization only. 



It also enables us to determine the relative velocities, or to ascertain the index 

 of rotatory polarization. For since GI bisects the angle between the points K 

 and L, which mark the relative degrees of advancement of the two rays in their 

 respective rotations, if we take a thickness, 0, which produces a rotation of 90°, 

 we know that the difference of phase is then one half an undulation. If X de- 

 note the length of the longer undulation, and I' that of the shorter, then — 



0^=^m).=^{in + ijA' ; or -t=— =— . 



As --zizm, aud X may be determined by experiments on refraction, the value 



o^ m is known when is measured. By pursuing this method, j\[r. Babiuct 



found the value of —="-1.00003 ; a value which, small as it is, is the largest 



known i'or rotatory polarization. 



When light is transmitted through quartz at right angles to the axis, tht; 

 emergent rays are plane polarized. J\lr. Airy has proved that, for directions 

 oblique to the axis, the polarization is elliptic, the ellipticity increasing from 

 the direction perpendicular up to the direction parallel to the axis, where it 

 becomes circular. 



