At the extremity of the optic axis OQ, v=b, m=0; and therefore by (19)

x = lb (r2 — a2)/(b2 — a2),

z = nb (r2 — c2)/(b2 — c2).

The values of land n are given by (22), whence by substitution these equations become

b (r2 − a2) + x (a2 — b2)1 (a2 — c2)2 = 0,

b (r2 — c2) — z (a2 — c2)1 (b2 — c2)2 = 0.

These equations are satisfied by the coordinates of the points of contact of the tangent plane at the extremity of the optic axis with the wave surface, and since they represent two spheres, it follows that this tangent plane touches the wave along a circle.

The diameter of this circle is equal to QR (see fig. § 110). To find its value, let OD be that diameter of the elliptic section OCA, which is conjugate to OR. Then

115. To find the equations of the tangent and normal cones at

the singular points.

The coordinates of P (see fig. § 110) are,

x = c(a2 — b3)3 /(a2 — c2)1, z=a (b' — c2)1/(a2 — co)3.

Substituting in (19), we obtain

Now l, m, n are the direction cosines of any normal through P; whence eliminating v, we obtain

l2 (a2 − b2) + n2 (b2 − c2) + In (a2 + c2) (a2 — b2)3 (b2 — c2)3/ac = a2 — c2, whence the equation of the normal cone, referred to P as origin is a2 (b2 — c2) + y2 (a2 — c2) + z2 (a2 — b3) = xz (a2 + c2) (a2 — b3)1 (b2 — c2)1/ac (25).

Let λ, μ, v be the direction cosines of any generator of the tangent cone; then since this generator is parallel to the normal at some point of the normal cone, it follows that if F(x, y, z) be the equation of the normal cone,

and therefore since l, m, n are proportional to x, y, z in (25), we obtain

λ

=

μ
-

2l (b2 — c2) − n (a2 + c2) (u2 — b2)1 (b2 — c2)/ac ̄ ̄ 2m (a2 — c2)

116. There is a third cone which is also of importance, viz. the cone whose vertex is the origin, and whose generators pass through the circle of contact of the tangent plane at the extremity of the optic axes.

We have shown in § 114, that the circle of contact is the curve of intersection of the two spheres

b (r2 − a2) + x (a2 — b2)1 (a2 — c2)1 = 0............ (27),

b (μa2 — c2) — z (a2 — c2)1 (b2 — c2)3 = 0 ..............

......

... (28).

Hence if λ, μ, v be any generator of the required cone,

c2 (a2 — b2) λ2 + a2 (b2 − c2) v2 + (a2 + c2) (a2 — b2)1 (b2 — c2)3λv = (a2 — c2)b2

− (a2 + c2) (a2 — b2)1 (b2 — c2)1 xz = 0........(29).

Uniaxal Crystals.

117. If in equation (21) we put b = c, it becomes

(y2 − c2) {a2x2 + c2 (y2 + z2) − a2c2} = 0,

which is the form of the wave-surface for a uniaxal crystal. Hence the wave-surface consists of the sphere

x2 + y2 + z2 = c2,

and the ellipsoid

a2x2 + c2 (y2+ 2o) = a2c2,

the axis of x being the axis of revolution.

=

Also from (22) we see that when b c, 0= 0; whence the two optic axes coincide with the axis of x, which is therefore the axis of the crystal.

The ellipsoid is ovary or planetary, according as c> or <a. In the former case the crystal is positive, and in the latter case negative.

If a pencil of light be incident upon a uniaxal crystal, the ray corresponding to the spherical sheet of the wave-surface, will coincide with the wave normal, and refraction will take place according to the ordinary law discovered by Snell. Also if X, μ, v be the direction cosines of the direction of vibration, we obtain from (5),

c2 = a2X2 + c2 (μ2 + v2)

= a2λ2 + c2 (1 − x2);

whence λ = 0, which shows that the direction of vibration is perpendicular to the plane containing the normal to the wavefront and the optic axis.

The extraordinary ray is in the direction of the radius vector of the ellipsoidal sheet of the wave-surface, drawn to the point of contact of the tangent plane, which is perpendicular to the wave

CONICAL REFRACTION.

125

normal; and by § 112 the direction of vibration is the projection of the ray on the wave-front. Hence the direction of vibration in the extraordinary wave, lies in the plane containing the optic axis and the extraordinary wave-normal, and is perpendicular to the latter.

We have thus established the laws of the propagation of light in uniaxal crystals, which were discovered experimentally by Huygens.

Conical Refraction.

118. The existence of the tangent cone at the extremity of the ray axis was first demonstrated by Sir W. Hamilton, and this led to the discovery of two remarkable phenomena, known as external and internal conical refraction.

119. In order to explain external conical refraction, let us suppose that a small pencil of light is incident upon a plate of biaxal crystal, cut perpendicularly to the line bisecting the acute angle between the optic axes; and let the angle of incidence be such, that the direction of the refracted ray within the crystal coincides with the ray axis.

Let IO be the ray axis within the crystal, I being the point of incidence, and O the point of exit. At O draw the wave-surface for the crystal, and also the equivalent sphere in air. Produce IO to meet the crystalline wave-surface in P; then OP will be the ray axis. To obtain the directions of the refracted rays, drawtangent planes at P. These tangent planes will meet the face of the crystal in a series of straight lines T1, T...; through each of these straight lines T1, T... draw a tangent plane to the sphere,

and draw OP1, OP... joining the points of contact with 0. The points of contact of the infinite number of tangent planes to the sphere will lie on a certain spherical curve, and therefore the refracted rays on emerging from the crystal, will form a conical pencil whose vertex is 0, and whose generators are the lines OP1, OP....

120. In order to explain internal conical refraction, we must suppose that the angle of incidence is such, that the direction of the refracted wave coincides with the optic axis. Since the tangent plane at the extremity of the optic axis touches the wavesurface along a circle, the refracted pencil within the crystal, will consist of a cone of rays, whose vertex is the point of incidence, and all of whose generators pass through the above-mentioned circle. The equation of this cone is given by (29). On emerging from the crystal, each emergent ray will be parallel to the incident ray, and will form an emergent cylinder of rays.

121. The phenomena of external and internal conical refraction had never been observed nor even suspected, previously to the theoretical investigations of Sir W. Hamilton on the geometry of the singular points of the wave-surface; and at his suggestion, Dr Humphrey Lloyd' examined the subject experimentally, and found that both kinds of conical refraction actually existed.

122. The investigations of Sir W. Hamilton, coupled with the experiments of Dr Lloyd, are undoubtedly a striking confirmation of the accuracy of Fresnel's wave-surface; but it has been subsequently pointed out by Sir G. Stokes', that almost any theory which could be constructed, would lead to a wave surface having conical points, and would therefore account for the phenomenon of conical refraction. Also a series of very elaborate experiments by Glazebrook upon uniaxal and biaxal crystals, have shown that Fresnel's wave-surface does not quite accurately represent the true form of the wave-surface in such crystals, but is only a very close approximation.

123. The dynamical objections to Fresnel's theory may be classed under three heads.

1 Trans. Roy. Ir. Acad. vol. xvII. p. 145.

2 Brit. Assoc. Rep. 1862.

3 Phil. Trans. 1879, p. 287; 1880, p. 421.