theoretical considerations, should at the present stage be considered simply as a representation of experimental facts.

10

P2

2

Take some fixed point O (Fig. 108) in the crystalline medium and imagine planes drawn through the point. In each plane take two lines OP, and OQ, which coincide with the two possible directions of vibration. If v be the velocity of the plane wave for a direction of vibration OP1 and 2 for a direction OQ, take the points P1, P2, Q1, Q2, so that OP1 = OP2 = V/v1 ; OQ1 = OQ2 = V/v2, V being the velocity of the wave propagation in vacuo*. If the plane through O be altered in direction and the points P and Q marked off for each, it is found that these points lie on an ellipsoid, which may be called the ellipsoid of plane wave propagation. It is also found that the points P and Q lie at the ends of the semiaxes of the central sections of this ellipsoid. If the ellipsoid is given, we may therefore find the direction of vibration and the corresponding velocities of waves parallel to any given plane, by drawing the central section which is parallel to that plane. The semiaxes of the ellipse in which the section cuts the ellipsoid give the directions of vibration, and the velocities are inversely proportional to the semiaxes.

Fig. 108.

Let the equation of the ellipsoid be

a2x2 + b2 y2 + c2 22 = V2..

.(1),

the quantities a, b, c, being in descending order. To simplify the equation, take the unit of time to be the time which a wave in vacuo takes to traverse unit distance, so that we may write V = 1. For x = 0, we obtain the intersection of the ellipsoid with the plane of yz, which is an ellipse having 1/6 and 1/c as semiaxes. Hence a wave-front may be propagated in the direction of the axis of either with a velocity b or with a velocity c, the direction of vibration in the former case being the axis of y, and in the latter the axis of . Similarly a and c are the velocities of propagation for a wave-front parallel to the plane az, and a, b the velocities for a plane parallel to ry. Fig. 109 illustrates how a plane wave separates into two, the directions of vibration in the two being at right angles to each other.

Fig. 109.

The problem of finding the velocities with which an inclined wave-front may be propagated, is one of geometry and may be solved

* We might also take OP1 = V/v, and OQ1 = V/r, and fit the observed phenomena equally well, but it is convenient to make our choice at once so as to fit in with the view that the direction of vibration is at right angles to the plane of polarization.

as follows. It is required to find the direction and magnitude of the principal axes OP and OQ of the intersection of the ellipsoid (1) by a central plane which is defined by the direction cosines l, m, n, of its normal ON, Fig. 110.

N

Let the length of the semiaxis OP be p. Then a sphere of radius p and having its centre at has one and only one point in common with Fig. 110. the ellipse PQ, because the ellipse can only have one radius vector which has a length equal to that of one of its semiaxes. The sphere intersects the ellipsoid in a curve which is called a sphero-conic, and this curve PS must touch the ellipse at P. If a cone be drawn having O as vertex, and passing through the sphero-conic PS, a similar reasoning shows that the cone cannot intersect, but must touch the plane of the section OPQ along the semiaxis OP.

The equation to the cone is obtained by combining the equations of the ellipsoid

and the sphere

a2x2 + b2y2 + c2z2 = 1,

x2 + y2+z2 = p2,

in such a way as to give the equation of a cone. Multiplying the first equation by p2 and subtracting, we obtain

(a2 p2 − 1) x2 + (b2 p2 − 1) y2 + (c2 p2 − 1 ) ≈2 = 0 .........(2).

If the velocity of propagation of the plane wave is v = 1/p,

(v2 − a2) x2 + (v2 − b2) y2 + (v2 − c2) ≈2 = 0.

The direction cosines l, m, n, of the normal to the section must coincide with the direction cosines of the tangent plane of the cone, the line OP being the line of contact. If x, y and z are the coordinates of P, we obtain, in the usual way,

D' being determined by the condition that + m2 + n2 = 1.

We may introduce in place of x, y, z, the direction cosines a, B, Y of the vector OP. The equations may then be written

and D may now be determined by the condition that a2 + ß2 + y2 = 1. This gives

As the vector (a, ß, y) lies in a plane normal to the vector (1, m, n) *

This equation allows the velocity v to be calculated. It is of the second degree in and has therefore two positive roots, which from the nature of the problem must be real.

Having obtained v, we may, by means of equations (3), determine a, B, y, the direction cosines of the directions of vibration, and we shall again have two solutions, one corresponding to OP and the other to OQ.

The velocities a, b, c, are called the three principal velocities, and as, with the unit of time adopted, the velocity of light in vacuo is one, the reciprocal velocities 1/a, 1/b, 1/c, measure quantities which in an isotropic medium would correspond to the refractive index. These quantities are therefore called the principal refractive indices. Denoting these by P1, P2, P3, we may write the equation of the ellipsoid (1) in the form

The coefficients of elasticity, which measure the resistance to distortion in the principal planes, are proportional to a2, b2, c2 respectively, so that these constants are intimately connected with the elastic properties of the medium. The ellipsoid (1) has therefore been called the ellipsoid of elasticity (see also Art. 103). In a homogeneous medium, 1 == 3, and the ellipsoid of elasticity becomes a sphere, having a radius numerically equal to the refractive index.

85. The Optic Axes. Every ellipsoid has two circular sections passing through that principal axis which is neither the largest nor the smallest. It follows that there are two directions in which a plane wave-front has only a single velocity. These two directions are called the "optic axes." The radius of the circular section is 1,6, and putting p = 1/6 in the equation (2) of the cone, it reduces to (a2 − b2) x2 + (c2 − b2)?2 = 0. —

This is the equation of the two planes which contain the two circular

sections. The two directions of single wave velocities are the normals to these planes, so that

are the direction cosines of the optic axes.

When a wave is propagated in the direction of one of the optic axes, the direction of vibration may be anywhere in the plane, as in the circular section, any direction may be considered to be an axis.

86. Uniaxal and Biaxal Crystals. In general a crystal has two optic axes and is then called "biaxal." If two of the principal axes are equal to each other, there is only one optic axis, which is the axis of x if bc, and the axis of z when a = b. The crystal is then said to be a "uniaxal" crystal.

The ellipsoid of elasticity for uniaxal crystals when a=b, is the spheroid

a2 (x2 + y2)+c22 = 1

and the equation (4) for determining the velocities of plane wave propagation becomes, writing for the angle between the optic axis and the normal,

Hence the velocity depends only on the angle which the normal to the wave-front makes with the axis of revolution of the spheroid.

87. Wave-Surface. The passage of waves through crystalline media is completely determined by the equation we have obtained for the propagation of plane waves, but it is often convenient to base our investigations on a surface which is the locus of equal optical distances measured from a point as centre. Such a surface, according to the definition of Art. 18, is called a "Wave-Surface." Its relation to the optical distance between parallel wave-fronts as deduced in the last paragraph may best be seen by applying Huygens' principle. Let a plane wave (Fig. 111), WF, be propagated upwards and with points P1, P2, P3, etc. as centres construct the surfaces of equal optical distance, corresponding to unit time, i.e. the wave-surfaces ST and S'T'. The furthest distance to which the wave-front WF can have gone in the time is the tangent plane W'F" to all the wavesurfaces, and by Huygens' principle, as explained in Art. 16, this plane will actually be the position of the wave-front after unit time. The lines which join the centres of disturbance P1, P2, P3, etc. with the

W' R R2

W Pi P2

Fig. 111.

F

points of contact R1, R2, R3, etc. of the wave-surfaces and wave-front, are the lines of shortest optical distance between WF and W'F". These lines we have called the "rays." If the wave-surfaces are not spheres, the rays are not in general at right angles to the wave-fronts, and this is an important distinction between crystalline and isotropic media.

If through any point P (Fig. 112), we draw a number of plane wave-fronts, we may, from the results of the last article, construct the

W4

W3 W2

WI

W5

P

Fig. 112.

positions WiF1, W2F2, WзF3, etc. of these wave-fronts after unit time. Each wave-front must be a tangent plane to the wave-surface drawn with P as centre. Hence the wave-surface is the envelope of all the plane wave-fronts. Its equation may thus be obtained by a purely mathematical process.

The equation to the wave-front is

lx + my + nz = V

.(6), where is the distance travelled in unit time, which itself is a function of l, m, n. Any point of the wave-surface is a point of intersection of planes, differing from each other in direction by infinitely small quantities. Hence a point x, y, z of the wave-surface must satisfy (6) and also the equation obtained by giving to l, m, n, v small increments dl, dm, dn, dv. Subtracting one of these equations from the other it follows that

xdl + ydm + zdn = dv.......

.....(7).

There are certain conditions to which the variations of the quantities. 1, m, n, v, are subject.

Thus

As there are only two independent parameters of the plane, i.e. I and m, it must be possible to express dn and do in terms of dl and dm. Two equations are sufficient for this purpose, and of the three equations (7), (8), (9) only two can be independent. To express the condition that one of these equations may be obtained as a consequence of the two others, we multiply (8) by A and (9) by B, and add.