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axed crystal, and is divided into two pencils having different directions, each of these pencils is polarized in one single direction; the ordinary one in the plane passing through its direction and a line parallel to the axis of the crystal, the extraordinary one perpendicularly to a plane similarly situated with respect to its direction. Either of these rays, when received on a plate of glass after its emergence, shows all the characters of polarization that we have described.
This law subsists equally, when the ray has been polarized by reflexion before its passage through the crystal. The two refracted pencils are always polarized, as if they had been composed of direct rays, but their relative intensities differ according to the direction of the primitive polarization given to them; this direction must therefore have predisposed the particles to undergo in preference one or other of the refractions.
These two laws were discovered by Malus. The analogy remarked above, between the single and double-axed crystals indicates sufficiently how it is to be extended to the latter; to find the direction of polarization for the ordinary pencil, draw a plane through its direction, and through each of the axes of the crystal. If either of these axes existed alone, the ordinary pencil would be polarized in the plane belonging to it. Now it is really found polarized in a plane intermediate to those two, and the extraordinary pencil perpendicularly to the analogous plane drawn through its direction between the two planes containing the axes. If the angle between these be equal to nothing, the crystal is single-axed, and the direction of polarization is conformable to Malus's indications: this law has been directly verified on the two pencils refracted by the topaz; as for other crystals in which it has not been possible to verify it directly, we may, by the consideration of some other phænomena that will shortly be mentioned, judge that it applies to them also.
These laws of polarization are applicable in all cases where the two pencils transmitted by a crystal are observed separately, but when they are received simultaneously, and in nearly the same direction, that of their apparent polarization is found to be modified, and at the same time their coincidence produces certain colours,
