grammatically in Fig. 141. MM' is a mirror reflecting the light from the sky, N, and N, the polarizing and analysing Nicols, CC' is the crystalline plate. C AE Ne N The field of view of a Nicol prism is much restricted by the increased distance of the eye from the polarizer N1. Hence when light from a distant source, such as the sky, passes through both Nicols, only such waves reach the eye as subtend a small angle. The eye at E, focussed for infinity, receives light therefore which has passed through the crystal nearly in the normal direction, and the crystal appears coloured with a uniform tint. If the eye is focussed on the crystal, the colours are not so M' pure because the different rays leaving the same point of the crystal have traversed it at different inclinations, but when the crystal is thin, so that the relative retardation is only a few wave-lengths, a small variation in direction does not produce much effect on the colour, and therefore the colours are seen with the eye focussed on the plate, nearly as well as with the eye adjusted for parallel light. An interesting variation of the experiment may be made if the analysing Nicol is replaced by a double image prism; two partially overlapping images of the plate are then seen. The images are coloured where they are separate, but white where they overlap, showing that the colours are complementary. Fig. 141. Fig. 142. L2 K2 107. Observations with light incident at different angles. If the field of view is enlarged so as to include rays which have traversed the crystal at sensibly different angles, the effects are more complicated because they depend on the part of the crystal looked at, so that the plate appears to be covered with a pattern of coloured K bands. To realize experimentally the necessary increase of the field of view, we may look at the crystal plate through an inverted telescopic system consisting of two lenses L, and L, placed so as to diminish angular distances. The different parallel pencils which have passed through the crystal, pass out of this system with their axes more nearly parallel, so that they may now be sent through a Nicol. A similar telescopic system K2 K1 serves to increase the angular deviation of the rays which have passed through the polarizing Nicol. The thickness of the plate used ought now to be rather larger because it is desired to bring out the differences which are due to variations of length of paths and inclination. When crystals are examined in this fashion, it is generally said that convergent or divergent light is used, but it must be clearly understood that the rays of light which are brought together on the retina traversed the crystal as a parallel pencil. So long as the eye is focussed for infinity, the sole distinction between this case and the previous one, lies in the increase of the field of view. 108. Uniaxal Plate cut perpendicularly to the axis. In order to show how the equation (1) is to be applied to the explanation of the interference pattern under the experimental conditions of the last article, we may treat first the simple case of a plate cut normally to the axis of a uniaxal crystal. An eye E looking in an oblique direction through such a plate (Fig. 143) receives rays which have passed through lengths of path in the crystal, which only depend on the angle E between the line of vision and the normal to the plate. Hence the retardation 8 is the same along a circle drawn on the surface of the crystal, having its centre coincident with the foot of the perpendicular from the eye to the plate. As the colour effects depend on 8, the field of view is traversed by coloured circular rings. A line along which 8 is constant is called an isochromatic line, but the term isochromatic here includes the complementary colour. The illumination is not constant along an isochromatic line on account D A Fig. 143. Fig. 144. B of the variations of a and B. In Fig. 144 ABCD represents the plate, N the foot of the perpendicular from the eye to the plate. If the line of vision passes through the point 0, NO is the trace of the plane of incidence, and this plane also contains the optic axis. The two directions of vibration of the ray inside the crystal are therefore NO and the line at right angles to it, and to make equation (1) apply, we must put the axes of X and Y along those directions. The circle drawn through O with N as centre is an isochromatic line. The polarizing and analysing directions remain fixed in space, while the coordinate axes revolve with the point O round N. Whenever either sin 2a = 0 or sin 2ẞ= 0, the colour term disappears and we obtain therefore in general four diameters along which there is no coloration. The lines drawn along these directions are called achromatic lines. We consider three cases. Case 1. The Nicols are crossed, i.e. a В π The intensity as before is given by I = sin2 2a Σ (a2 sin2 There are two lines at right angles to each other, along which the intensity is zero, these lines coinciding with the directions of the planes of polarization of the analysing and polarizing Nicols. The intensity is greatest at an angle of 45° from these lines. The field is traversed by rings of varying colours, or in the case of homogeneous light, by coloured rings of varying intensity, the dark rings corresponding to the positions at which the phase retardation & is a multiple of four right angles. The whole appearance consists therefore of a number of concentric rings with a dark cross, as shown in the photograph reproduced in Plate II., Fig. 1. The cross widens out away from the centre and each of its branches is sometimes referred to as a "brush." and the whole effect is complementary to that observed in the first case. The rings are now crossed by bright brushes. Plate II., Fig. 2 shows the appearance. Case 3. This includes all positions of the analyser and polarizer in which these are neither parallel nor crossed. There are four achromatic lines corresponding to a = 0 and a = B=0 and B = π 2 ; π 2 Along an isochromatic circle, the colour changes into its complementary (or for homogeneous light, a minimum of light changes into a maximum) on crossing one of the achromatic lines. This is shown in Plate II., Fig. 3 which is also a reproduction of a photograph. When either the axis of x or the axis of y falls within the acute angle formed by the directions of the analyser and polarizer, the product sin 2a sin 2ẞ is negative so that the maxima of light are brighter and the minima less dark. The field is therefore separated into segments of unequal illumination and may at first sight give the fictitious appearance of a dark cross. The eight achromatic brushes in this case separate the bright and dark segments, and are not very conspicuous. 109. Relation between wave velocities. In order to discuss the form of the achromatic and isochromatic lines in more complicated cases, it is necessary to calculate the phase difference 8. The first step consists in finding an expression for 1 1 the difference between the , Οι V2 have passed through the polarizing Nicol. The thickness of the plate used ought now to be rather larger because it is desired to bring out the differences which are due to variations of length of paths and inclination. When crystals are examined in this fashion, it is generally said that convergent or divergent light is used, but it must be clearly understood that the rays of light which are brought together on the retina traversed the crystal as a parallel pencil. So long as the eye is focussed for infinity, the sole distinction between this case and the previous one, lies in the increase of the field of view. # 108. Uniaxal Plate cut perpendicularly to the axis. In order to show how the equation (1) is to be applied to the explanation of the interference pattern under the experimental conditions of the last article, we may treat first the simple case of a plate cut normally to the axis of a uniaxal crystal. An eye E looking in an oblique direction through such a plate (Fig. 143) receives rays which have passed through lengths of path in the crystal, which only depend on the angle E between the line of vision and the normal to the plate. Hence the retardation 8 is the same along a circle drawn on the surface of the crystal, having its centre coincident with the foot of the perpendicular from the eye to the plate. As the colour effects depend on 8, the field of view is traversed by coloured circular rings. A line along which is constant is called an isochromatic line, but the term isochromatic here includes the complementary colour. The illumination is not constant along an isochromatic line on account D Fig. 143. Fig. 144. B of the variations of a and B. In Fig. 144 ABCD represents the plate, N the foot of the perpendicular from the eye to the plate. If the line of vision passes through the point 0, NO is the trace of the plane of incidence, and this plane also contains the optic axis. The two directions of vibration of the ray inside the crystal are therefore NO and the line at right angles to it, and to make equation (1) apply, we must put the axes of X and Y along those directions. The circle drawn through O with N as centre is an isochromatic line. The polarizing and analysing directions. remain fixed in space, while the coordinate axes revolve with the point O round N. Whenever either sin 2a = 0 or sin 2ẞ= 0, the colour term disappears and we obtain therefore in general four diameters along which there is no coloration. The lines drawn along these directions are called achromatic lines. The intensity as before is given by I = sin2 2a ≥ (a2 sin2 There are two lines at right angles to each other, along which the intensity is zero, these lines coinciding with the directions of the planes of polarization of the analysing and polarizing Nicols. The intensity is greatest at an angle of 45° from these lines. The field is traversed by rings of varying colours, or in the case of homogeneous light, by coloured rings of varying intensity, the dark rings corresponding to the positions at which the phase retardation & is a multiple of four right angles. The whole appearance consists therefore of a number of concentric rings with a dark cross, as shown in the photograph reproduced in Plate II., Fig. 1. The cross widens out away from the centre and each of its branches is sometimes referred to as a "brush." Case 2. The Nicols are parallel, i.e. a= = B. and the whole effect is complementary to that observed in the first case. The rings are now crossed by bright brushes. Plate II., Fig. 2 shows the appearance. π O and a = ; B=0 and B= π 2 Case 3. This includes all positions of the analyser and polarizer in which these are neither parallel nor crossed. There are four achromatic lines corresponding to a = Along an isochromatic circle, the colour changes into its complementary (or for homogeneous light, a minimum of light changes into a maximum) on crossing one of the achromatic lines. This is shown. in Plate II., Fig. 3 which is also a reproduction of a photograph. When either the axis of x or the axis of y falls within the acute angle formed by the directions of the analyser and polarizer, the product sin 2a sin 2ẞ is negative so that the maxima of light are brighter and the minima less dark. The field is therefore separated into segments of unequal illumination and may at first sight give the fictitious appearance of a dark cross. The eight achromatic brushes in this case separate the bright and dark segments, and are not very conspicuous. 109. Relation between wave velocities. In order to discuss the form of the achromatic and isochromatic lines in more complicated cases, it is necessary to calculate the phase difference 8. The first step consists in finding an expression for 1 1 the difference between the |