366 Dr. H. C. Pocklington on Rotatory 



squares of the velocities. Hence, from (6), 



Doc V(V 2 aXV 2 /3\-A. 2 S 2 7 \) 

 = A V {a 2 b 2 sin 2 L sin 2 2 + ?- 2 S 2 7 X), . . (8) 

 where A varies as the thickness of the plate. Hence 



i + 'h'*m* % % (9) 



6. We can now apply the whole of Airy's work on the 

 phenomena exhibited by quartz in convergent polarized 

 light to this case. In the case of a plate between crossed 

 n i cols, the intensity of the light is 



dj tf ./i-* 8 y • ,,,1, 



where yjr is the angle that the plane of polarization of the 

 incident light makes with the bisector of the angle between 

 the planes \a and \6. There are black curves where 

 D = 2n7r; and from (8) these curves are Cassinr's ovals. 



Since the other factor cannot vanish, the brush is not black. 

 For minimum brightness we must have sin 2-^ = 0, which 

 gives the same hyperbola that is obtained when the rotatory 

 power vanishes. The intensity at the central line of the 

 brash is 



sm 4 



4ft« . 2 P__ 2 



(l-h& 2 )* Sm 2 D*/D^ 



by (9), so that the darkness of the brush rapidly increases as 

 we pass from the axis. 



For light propagated along an axis, & = 1, the separate 

 waves are circularly polarized, and the emergent light is 

 plane-polarized with, however, a rotation of the plane of 

 polarization equal to half the relative retardation. 



If the polarizer and analyser are not crossed, let the angle 

 between them be 7r/2 — c9, then the intensity is 



/ 2k Y 2 



( sin 6 cos D/2 — - — p cos sin D/2 ) 



+ (^)%in*(2f-0)sm*-D/2. 



