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refracted doubly in the plane of the optic axes, so that no part of the line should appear enlarged in breadth, on looking through this aperture; while, according to Sir William Hamilton, the ray which proceeds in the direction OM should be refracted in every plane. The latter was found to be the case in the neighbourhood of each of the optic axes, the luminous line was bent, on either side of the plane of the axes, into an oval curve. This curve, it is easy to show, is the conchoid of Nicomedes, whose asymptot is the line on the first surface.

(191) The other case of conical refraction-called internal conical refraction by Sir William Hamilton—was expected to take place when a single ray has been incident externally upon a biaxal crystal, in such a manner that one of the refracted rays may coincide with an optic axis. The incident ray in this case should be divided into a cone of rays within the crystal, the angle of which, in the case of arragonite, is equal to 1° 55'. The rays composing this cone will be refracted at the second surface of the cryscal, in directions parallel to the ray incident on the first, so as to form a small cylinder of rays in air, whose base is the section of the cone made by the surface of emergence. This is represented in

a

N

the annexed diagram, in which NO is the incident ray, aOb the cone of refracted rays within the crystal, and aabb the emergent cylinder.

The minuteness of this phenomenon, and the perfect accuracy required in the incidence, rendered it much more difficult to observe than the former. A thin pencil of light, proceeding from a distant lamp, was suffered to fall upon the crystal, and the position of the latter was altered with extreme slowness, so as to change the incidence very gradually. When

the required position was attained, the two rays suddenly spread out into a continuous circle, whose diameter was apparently equal to their former interval. The same experiment was repeated with the sun's light, and the emergent cylinder was received on a small screen of silver paper, at various distances from the crystal; and no sensible enlargement of the section was observable on increasing the distance. The angle of this minute cone within the crystal was found to agree, within very narrow limits, with that deduced from theory,the observed angle being 1° 50′, and the theoretical angle 1° 55'.

The rays composing the internal cone are all polarized in different planes; and the law connecting these planes is the same as in the case of external conical refraction.

(192) We have seen that the problem to find the direction and magnitude of the reflected and refracted vibrations, when those of the incident vibration are given, was solved by Fresnel in the case of ordinary media. In the year 1831, M. Seebeck generalized, to a certain extent, the solution of Fresnel, and extended it to the case of reflexion by uniaxal crystals in the principal plane. The general solution of the problem of reflexion and refraction by crystalline media was obtained, a few years later, by Professor Mac Cullagh and M. Neumann upon other principles (156); and the memoir of the former is distinguished for the beauty and elegance of its geometrical laws. This solution, like that of Fresnel for ordinary media, does not include the change of phase, which is now proved to take place in reflexion at the bounding surfaces of all media (174). Its results, accordingly, are only approximately true, the degree of approximation being probably the same as in the case of Fresnel's laws themselves.

CHAPTER XII.

INTERFERENCE OF POLARIZED LIGHT.

(193) HAVING considered the theory and laws of double refraction, we are prepared to study the phenomena of interference which take place when polarized light is transmitted through crystalline substances. The effects displayed in such cases are probably the most splendid in optics; and when it is considered that, through them, an insight is afforded into the very laboratory of Nature itself, and that we are thus enabled almost to view the interior structure and molecular arrangement of bodies, the subject will hardly be thought inferior in importance to any other in the science.

The first discoveries in this attractive region were made by Arago, who presented a memoir to the Institute, in the year 1811, on the colours of crystalline plates when exposed to polarized light. The subject has since been prosecuted with unremitting ardour by the first physical philosophers of Europe, and among the foremost by Biot, Brewster, and Fresnel.

(194) It has been already shown (142), that when a beam of light, polarized by reflexion, is received upon a second reflecting plate at the polarizing angle, the twice-reflected light will vanish, when the plane of the second reflexion is perpendicular to that of the first. The first reflector, in any apparatus of this kind, is called the polarizing plate, and the second (for reasons which will presently appear), the analyzing plate. Now, if between the two reflectors we interpose a plate of any double-refracting substance, the capability of reflexion at the analyzing plate is suddenly restored, and a por

tion of the light is reflected, whose quantity depends on the position of the interposed crystal. The light is thus said (though improperly) to be depolarized by the crystal; and it was by this property that the double-refracting structure was detected by Malus in a vast variety of substances, in which the separation of the two rays was too small to be directly perceived.

In order to analyze this phenomenon, let the crystalline plate be placed so as to receive the polarized beam perpendicularly, and let it be turned round in its own plane. We shall then observe that there are two positions of the plate in which the light totally disappears, and the reflected ray vanishes, just as if no crystal had been interposed. These two positions are at right angles to one another; and they are those in which the principal section of the crystal coincides with the plane of the first reflexion, or is perpendicular to it. When the plate is turned round from either of these positions, the light gradually increases; and it becomes a maximum, when the principal section is inclined at an angle of 45° to the plane of the first reflexion.

(195) In these experiments the reflected light is white. But if the interposed crystalline plate be very thin, the most gorgeous colours appear, which vary with every change of inclination of the plate to the polarized beam.

Mica and sulphate of lime are very fit for the exhibition of these beautiful phenomena, being readily divided by cleavage into laminæ of extreme thinness. If a thin plate of either of these substances be placed so as to receive the polarized beam perpendicularly, and be then turned round in its own plane, the tint does not change, but varies only in intensity; the colour vanishing altogether when the principal section of the crystal coincides with the plane of primitive polarization, or is perpendicular to it,—and, reaching a maximum, when it is inclined to the plane of primitive polarization at an angle of 45°.

If, on the other hand, the crystal be fixed, and the analyzing plate turned, so as to vary the inclination of the plane of the second reflexion to that of the first,-the colour will be observed to pass, through every grade of tint, into the complementary colour; it being always found that the light reflected in any one position of the analyzing plate is complementary, both in colour and intensity, to that which it reflects in a position 90° from the former. This curious relation will appear more evidently, if we substitute a double-refracting prism for the second reflector; for the two pencils refracted by the prism have their planes of polarization-one coincident with the principal section of the prism, and the other at right angles to it, and are therefore in the same condition as the light reflected by the analyzing plate, with its plane of reflexion successively in these two positions. In this manner the complementary lights are seen together, and may be easily compared. But the accuracy of the relation is completely established by making the two pencils partially overlap; for, whatever be their separate tints, it will be found that the part in which they are superposed is absolutely white.

(196) When laminæ of different thicknesses are interposed between the polarizing and analyzing plates, so as to receive the polarized beam perpendicularly, the tints are found to vary with the thickness. The colours produced by plates of the same crystal, of different thicknesses, follow the same law as the colours reflected from thin plates of air, the tints rising in the scale as the thickness is diminished; until finally, when this thickness is reduced below a certain limit, the colours disappear altogether, and the central space appears black, as when the crystal is removed. The thickness producing corresponding tints is, however, much greater in crystalline plates exposed to polarized light, than in thin plates of air, or any other uniform medium. The black of the first order appears in a plate of sulphate of lime, when the thickness is the

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