and n, those of M1, - and -7. Draw M2H and M,K at right angles to the bisector PT. Then

This case is

If έ=n, OY is at an angle of 45° to the line joining M, and M2, and these two points are on the vertices of the curves. represented in Plate II., Fig. 4.

If either = 0 or n= 0, xy = 0, and the achromatic lines coincide with the axes of x and y. As in the case of uniaxal crystals, there are two sets of achromatic lines, one belonging to the polarizer and one to the analyser. Both have the same shape, and both pass through the same points M, and M. If the observing Nicols are crossed or parallel, the two sets of lines coincide; in the former case the hyperbolic brushes are dark, and in the latter, bright. Plate II., Fig. 5, gives a second example of dark hyperbolic brushes.

117. Measurement of angle between optic axes. The intersection of the isochromatic surface (Fig. 147) with planes drawn at different distances from O, shows that for small differences of path the interference rings surround the optic axes in closed curves. This affords a means of determining the angle between the optic axes. If a plate of a crystal cut symmetrically to the axes, as assumed in the last two articles, be mounted so that it can be rotated about an axis at right angles to the axes through an angle which can be measured, we may bring first one centre of the ring system belonging to one optic axis against a fixed mark in the observing telescope, and then the centre of the system belonging to the other axis. The angle of rotation is the so-called "apparent angle" between the optic axes, for it is clear that what is measured is the angle between the lines of vision ME and ME (Fig. 152). This angle is to be corrected for refraction to get the angles formed between L1M1 and LM 2.

1

118. Dispersion of Optic Axes. We have treated the problems of double refraction as if the position of the optic axes were independent of the wave-length. Though the position of the principal axes does not in most cases depend on the wave-length, the principal velocities are different for the different colours. Now V1, V2, V3, being the principal velocities for one wave-length, and ví, v2, v3', for another, the latter quantities are not in general proportional to the former, hence the positions of the optic axes change with the colour of the light. In some crystals the difference is very considerable.

A great

119. Two plates of a uniaxal crystal crossed. variety of effects may be produced by allowing light to traverse several plates in succession. We shall only consider one case, which is of some importance.

Let a plate be cut obliquely to the axis of a uniaxal crystal, and then divided into two halves which are therefore necessarily of the same thickness. Superpose the two halves and turn one of them through a right angle. We shall determine the shape of the isochromatic lines in this case.

The first plate produces a difference in optical length between two coincident wave normals, which as obtained from (9) is

the meaning of the letters being the same as that of Article (113). The second plate being turned through a right angle, the direction of vibration in the ordinary and extraordinary rays is interchanged, so that the phase difference in that plate is

Me

and μο is

The values of cos and cos d' are nearly equal for the double reason that is small, and that the difference between small. Hence the total phase difference is proportional to sin20,- sin2 0, cos2 01-cos2 0.

According to Art. 113

cos 01 = cos cos + sin o sin cos A.

To find the angle 0, which the optic axis makes with the plate normal in the upper plate, we have only to increase the angle A by a right angle, keeping all other quantities the same.

cos 02 = cos cos - sin & sin y sin A.

Neglecting higher powers of sin

cos2 01-cos2 02 = sin & sin 24 (cos A + sin A).

Hence

Introducing rectangular coordinates, so that

e sin cos A = x, e sin & sin A = y,

the equation to the isochromatic line for which the total difference in optical length 8,+ d2 is equal to nλ, becomes

This represents a series of parallel lines. The field of view is therefore crossed by a series of bands, the central one not being coloured. The bands are the wider apart the smaller, so that if the bands are to be broad, the plate should be cut nearly normally to the optic axis. It is found that in this case, the departure from straightness which depends on terms involving sin2 is also small.

Two plates combined together in the manner described, form the essential portions of the "Savart" polariscope, which is the most delicate means we possess for detecting polarized light. The double plate is provided with an analyser, consisting of a Nicol prism or a Tourmaline plate. In both cases, the plane of transmittance through the analyser should bisect the angle between the principal planes of the Savart plates in order to get the most sensitive conditions. If the incident light be polarized at right angles to the plane of transmittance, the eye sees a dark central band accompanied on both sides by parallel coloured fringes. If the incident light be polarized parallel to the direction which can pass through the analyser, the central band is bright, and the whole effect is complementary to that observed in the previous case. By examining the light reflected from the sky or from almost any surface, the coloured fringes are noticed, and by rotating the whole apparatus we may find the direction in which the fringes are most brilliant and hence determine the plane of polarization of the incident light.

120. The Half Wave-length Plate. If plane polarized light falls normally on a plate of a crystal cut to such a thickness that the two waves are retarded relatively to each other by half a wave-length, or a multiple thereof, the transmitted beam is plane polarized. Let OX and OY be the two principal directions of vibration in the crystal, and a the angle between OX and the direction of vibration of the incident beam. The displacements resolved along OX and OY may then be expressed by

α

Fig. 154.

X

u = a cos a cos wt,

v = a sin a cos wt.

Then if the thickness of the plate be such that its optical length

for the vibration along OY is half a wave-length greater, or half a wave-length less, than that for the vibration along OX, the displacements at emergence will be

u = a cos a cos wt,

v = - a sin a cos wt,

so that there is again plane polarization, but the angle of vibration forms an angle a with the axis of x. The same holds for a retardation equal to any odd multiple of two right angles. For even multiples, the plane is that of the original vibration. These plates, in which a relative retardation of the two waves amounting to half a wave-length takes place, are called "Half Wave-length Plates" and are used in some instruments in which it is desired to fix the plane of polarization accurately. The simple Nicol does not permit of very exact adjustment, for while it is moved about near the position of extinction, a broad dark patch is seen to travel across the field, and it is difficult to fix the exact position in which the centre of that patch is in the centre of the field of view. In the instrument in which a halfwave plate is used, that plate covers half the field of view. If ON and OM, Fig. 155, be the principal directions of the half-wave plate covering the left-hand portion of the field of view, and if the incident light vibrates along OP1, the field of view will be divided by the plate into two portions, the directions of vibration at emergence being along

P2

N

Fig. 155.

M

OP1, OP2, equally inclined to ON. An eye examining the field through an analysing Nicol will find the two halves unequally illuminated, except where its principal plane is coincident with ON or at right angles to it. In the latter position, the luminosity of the field is small if a is small, and the eye is then very sensitive to small differences of illumination, so that the position of the analysing Nicol may be fixed with great accuracy. A half wave-length plate used in this fashion is the distinguishing feature of "Laurent's Polarimeter." The weak point of the arrangement lies in the effect of refrangibility on the retardation, in consequence of which a retardation of half a wave-length can only be obtained for a very limited part of the spectrum. Hence homogeneous light must be used with instruments which contain these plates.

121. The Quarter Wave Plate. Plates in which the relative retardation of two waves is a quarter of a period, are called Quarter Wave Plates. They have the property of converting plane polarized light vibrating in a suitable direction into circularly polarized light. Let OX and OY be the two directions of vibration in the crystal, the vibration along OY being the one propagated most quickly.