Again, when " 4 3π or then sin 2-1, and the maxima occur for the values of ♪ which gave the minima in the other sectors, and the minima for those which gave the maxima in the former case. The segments of the rings in the alternate sectors hence differ by a retardation of half an interval; or the distances from the center of the bright rings in one sector equal those of the dark ones in the neighbouring sectors, as seen in the experiments, and shown in figure 87, Part I. ART. 55. PROP. To find the phenomena when the light incident upon a plate of a uniaxal crystal is circularly polarized, and the light emergent from it is circularly analyzed. Let SCD (in fig. 50) be the plane of final analyzation making an angle = a - with Scd, any principal plane of the plate of uniaxal crystal ab. Then the planes inclined ± 45° to SCD will be those into which the first analyzations must be made of the bases equal to L √2, emergent from the plate ab, polarized in the principal plane and perpendicular to it; and we have from the principal plane the components These analyzed finally in the plane SCD, by multiplying by In the same way we have from the emergent light polarized perpendicular to the principal plane of the crystal the com λ These compounded with the retardation give for the base L. 4 as follows, after analyzation, by multiplying by cos (± 45o), Finally, L, and Le compounded with the retardation 8" give or the intensity does not change with, and there are no crosses bright or dark, but complete rings, the maxima occurring where the intensities of the bright rings being I'= (L)", and those of the dark ones zero, or I' = 0. ART. 56. PROP. To find the phenomena when a plate of quartz crystal cut perpendicularly to the axis is placed between the polarizing and analyzing plates. In figure 51, let SAa Ce be a ray entering the eye e, originally polarized in the plane SAB. Let Sab be a principal plane of the plate of quartz, making an angle with the plane of original polarization SAB, and let SCD be the plane of analyzation, making the angle a with SAB. From Art. 53 the base of an elliptically polarized beam after analyzation is expressed by the formula 12 = L2 (cos2 ß cos2 + sin2 ß sin2 ), where is the angle in fig. 48 between the chief plane of polarization of the elliptically polarized light and the plane of final analyzation SCD; but cos ẞ and sin ẞ are now to be considered coefficients only, which determine the degree of ellipticity of the two beams into which the light is refracted by the quartz plate, according to the experiments of M. Biot as interpreted by M. Fresnel, which are circularly polarized when they pass along the axis of the quartz plate, and elliptically polarized near it, but plane polarized when they pass at right angles to it; the two beams traversing the axis of the crystal with slightly different velocities; and the light becoming plane polarized to the senses at an inclination of 20 degrees to the axis, as found by M. Jamin. In the formula above we have B-45° at the axis, and B=0 at right angles to it, and ẞ also becoming sensibly zero at an inclination of 20 degrees to the axis. If L be the base of the light incident upon the quartz plate, polarized in the plane SAB, we must put L cos o for that of the ordinary beam, and L sin & 1 √/2 for that of the extraordinary beam when the transmitted light 12 = L2 cos2 + (cos2 ß cos2 + sin2 ß sin2 √), 12=L2 sin2 + (cos B sin' + sin ẞ cos2 ); the latter being formed for the plane Sac as the former was for the plane Sab, at right angles to it. These are to be compounded by the formula, with the retardation S', thus: L'” = L" cos3¿ (cos" cos3y+sin2ßsin')+sin2p (cos3ß sin3y+sin3ßcos3↓) +2sindcosp{(cos Bcos +sin'ẞsin) (cos Bsin+sin Bcos')}cos2π and the intensity I is I = (L'). Now at the axis we have ẞ= 45o, and cos B = sin ß, λ, Since has disappeared, the intensity is the same all around the axis, and near to it, whilst ß=45o nearly, or the light can be considered as consisting of two circularly polarized beams. The S' intensity depends upon cos 2π and is a function of a, the λ angle between the planes of polarization and analyzation, the phenomena recurring with every 180° that a is increased, so that we have That d' is a function of a we may see in the following manner. In fig. 52 let SE be a ray which has traversed the quartz plate AB, the transmitted light being circularly polarized with taken 45°. Let the lines drawn perpendicularly to SE through the points o, o', e, e' represent the sections of the luminiferous surfaces constituting the circularly polarized light, in contemporaneous positions with the distances oo' and ee', each equal 1⁄4, and of which the polarizations are indicated by the small letters, parallel and perpendicular to the plane of the figure. Now if the crystal were an ordinary uniaxal one, the retardation would be the distance oe, but in the circularly polarized light, to λ 4 π 4' putting a = a - 7, when the plane of analyzation coincides with the plane of the figure, or a' = 0, the light furnishes no compo λ nents from o' and e, and the retardation is oe'=oe+1; and when perpendicular to it, or a=7, the light furnishes no com ponents from o and e', and the retardation is o'e oe-: = λ 4. For intermediate positions the retardation will be intermediate and involve a function of a'. We may thus assume an expression fulfilling the above conditions in the following form: |