the axis of quartz with different velocities. This supposition is easily put to the test of experiment; since such a difference of velocity must produce a difference of refraction, when the surface of emergence is oblique to the direction of the ray. According to this hypothesis, therefore, a polarized ray transmitted through a prism of rock-crystal, in the direction of the optic axis, should undergo double refraction at emergence; and the two pencils into which it is divided should be circularly polarized. This has been completely verified by Fresnel, by means of an achromatic combination of right-handed and lefthanded prisms, arranged so as to double the separation; and he has shown that the two pencils are neither common nor plane-polarized light, but possess all the physical characters of light circularly polarized.

(214) The relation between the rotation and double refraction of rock-crystal, in the direction of its axis, has been very simply deduced by M. Babinet.

Letv and v' denote the velocities of the ordinary and extraordinary waves in the direction of the axis of the crystal; μ and the corresponding refractive indices; then

But, if be the thickness of the crystal, and 8 the interval of retardation of the two waves after traversing it, the second member of the preceding equation is obviously equal to

Now the angle of rotation is proportional to the interval of retardation of the two circularly polarized pencils; and when that interval is equal to the length of a wave in vacuo, the angle

of rotation is 180°.

Hence the interval of retardation of the emergent rays, corresponding to any angle of rotation, p, will

ρ

be, denoting the length of the wave; and the cor180°9

responding interval within the crystal is equal to this, multiplied by the velocity of propagation, or divided by the refractive index. Hence, if p be the rotation corresponding to the thickness of the crystal, 0, we have

metre, the angle of rotation, p, corresponding to the rays of mean refrangibility, = 30°. But for these rays, λ =·0005 of a millimetre; and therefore μ- μ = '00008.

(215) The phenomena hitherto described take place only in the direction of the axis of the crystal. Mr. Airy discovered that when a plane polarized ray is transmitted through rock-crystal in any direction inclined to the axis, it is divided into two pencils which are elliptically polarized; the elliptical vibrations in the two rays being in opposite directions, and the greater axes of the ellipses coinciding respectively with the principal plane, and with the perpendicular plane. The ratio of the axes, in these ellipses, varies with the inclination of the ray to the optic axis,-being a ratio of equality when the direction of the ray coincides with the axis, and increasing indefinitely with its inclination to that line. With respect to the course of the refracted rays, Mr. Airy found that it was still determined by the Huygenian law; but that the sphere and spheroid, which determine the velocities and directions of the

two rays, do not touch, as in all other known uniaxal crystals, -the latter surface being contained entirely within the former. This is a necessary consequence of the fact, that the interval of retardation of the two pencils does not vanish, with the inclination of the ray to the optic axis.

Mr. Airy has given an elaborate calculation, founded on these hypotheses, of the forms of the rings, &c., displayed by rock crystal in plane polarized and circularly polarized light ; and he has found the most striking agreement between the results of calculation and experiment. Among the most remarkable of the phenomena whose laws are thus developed, is that produced by the superposition of two plates of rockcrystal, of the same thickness, one of them being right-handed, and the other left-handed.

In order to complete the experimental investigation of this subject, it remained to determine the velocities of the two elliptically polarized rays, and the ratio of the axes of the ellipses, as dependent on the inclination of the rays to the axis of the crystal. This has been effected by M. Jamin, by measuring the amplitudes, and the differences of phase of the two component pencils, when the incident light is polarized in the plane of a principal section. From these data the quantities sought are deduced by calculation.

(216) All these complicated facts have been linked together, and their laws deduced, by Professor Mac Cullagh. In this remarkable investigation the author sets out by assuming the form of the differential equations of vibratory motion in rock-crystal; and from this assumed form he has deduced the elliptical polarization of the two pencils,—the law of the ellipticity as depending on the inclination of the ray to the axis, -the interval of retardation in the direction of the axis,-and the peculiar form of the wave-surface.

The ratio of the axes of the two ellipses is found to be equal to unity in the direction of the axis of the crystal. In

all other directions it is given by a quadratic equation whose constant term is equal to unity; so that this ratio has two values, one of which is the reciprocal of the other. Hence the ratio of the axes is the same in both ellipses; and the greater axis of one coincides with the smaller axis of the other.

When the ray traverses the axis of the crystal, the rotation of the plane of polarization is given by the formula

ᏟᎾ p = 12

which comprises all the experimental laws of M. Biot (210). The sign of the constant factor, C, determines the direction of the rotation.

It is a striking peculiarity of this theory, that it contains (in addition to the two refractive indices) but one constant, -and that this constant having been determined, from the known angles of rotation when the ray traverses the axis of the crystal, the ratio of the axes of the ellipses may be calculated, when the ray is inclined by any angle to the axis. The author has applied this calculation to the observations of Mr. Airy, and has found the calculated and observed results to agree.

(217) MM. Biot and Seebeck discovered that some of the liquids, and even of the vapours, possess the same property as quartz in the direction of its axis, and impress a rotation on the plane of polarization of the intromitted ray, which is proportional to the thickness of the substance traversed. The fact is easily observed by transmitting a polarized ray through a long tube filled with the liquid, and closed at each end by parallel plates of glass; and analyzing the emergent ray by a double-refracting prism. Among the liquids possessing this property are oil of turpentine, oil of lemon, solution of sugar in water, solution of camphor in alcohol, &c. The first-mentioned of these liquids is right-handed, and the others left-handed. They all possess the property in a much feebler degree than

quartz; so that the ray must traverse a much greater thickness of the substance, in order to have its plane of polarization altered by the same amount. Thus a plate of rock-crystal, whose thickness is one millimetre, rotates the plane of polarization of the red ray through an arc of about 18°; a plate of oil of turpentine, of the same thickness, turns the plane of polarization only through a quarter of a degree.

The rotatory liquids do not lose their peculiar power (except in degree) by dilution with other liquids not possessing the property; and they retain it, even in the state of vapour. From these and other facts, M. Biot concludes that this property, in liquids, is inherent in their ultimate particles. In this respect the rotatory liquids are essentially distinguished from rock-crystal, which is found to lose the property when it loses its crystalline arrangement. Thus, Sir John Herschel observed, that quartz held in solution by potash (liquor of flints) did not possess the rotatory power; and the same thing has been remarked by Sir David Brewster with respect to fused quartz.

(218) When two or more liquids possessing this property are mixed together, the rotation produced by the mixture is always the sum, or the difference, of the rotations produced by the ingredients, in thicknesses proportional to the volumes in which they enter the mixture, according as the liquids are of the same or of contrary denominations. The same law holds good in many cases in which the liquids are chemically united.

M. Biot has made an important application of this principle to the analysis of compounds, containing a substance possessing the rotatory power combined with others which are neutral, the quantity of which in the compound may (by the principle just stated) be determined, by observing the optical effects of the mixture. This application has been found of much industrial value, in the case of the sac