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The incident converging cone was also formed by a lens of short focus, placed at the distance of its own focal length from the surface; and in this case, the lamp was removed to a distance, and the plate on the first surface dispensed with. The same experiments were repeated with the sun's light; and the emergent rays were even thrown on a screen, and thus the section of the cone observed at various distances from its summit.

In the first experiments there was a considerable discrepancy between the results of observation and theory, both as to the magnitude of the cone, and some other circumstances of its appearance. These discrepancies were found to arise from the sensible magnitude of the little aperture on the second surface of the crystal, which suffered rays to pass which were inclined to the line OM at small angles. Accordingly, the magnitude of the observed cone required a correction before it could be compared with the results of theory: when this correction was applied, the agreement of the observed and theoretical angles was found to be complete.

The rays which compose the emergent cone are all polarized in different planes. It was discovered by observation that these planes are connected by the following law,—namely, "the angle between the planes of polarization of any two rays of the cone is half the angle between the planes containing the rays themselves and the axis." This law was found to be in accordance with theory.

(190) A remarkable variation of the phenomenon took place, on substituting a narrow linear aperture for the small circular one, in the plate next the lamp, in the first-mentioned mode of performing the experiment,-the line being so adjusted, that the plane passing through it and the aperture on the second surface should coincide with the plane of the optic In this case, according to the hitherto received views, all the rays transmitted through the second aperture should be

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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

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the annexed diagram, in which NO is the incident ray, aOb the cone of refracted rays within the crystal, and aa'b'b 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°.

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