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plane of reflexion of the analyzing plate, will exhibit the phenomena of interference. But the interval of retardation will

differ by half a wave in the two cases; the tints produced will therefore be complementary, and the light resulting from their union will be of a uniform whiteness.*

(200) These laws of interference being kept in mind, the reason of all the phenomena is apparent. The ray is originally polarized in a single plane, by means of the polarizing plate. It is then divided into two within the crystal, which are polarized in opposite planes; and these are finally reduced to the same plane by means of the analyzing plate. The two pencils will therefore interfere; and the resulting tint will depend on the retardation of one of the rays behind the other, produced by the difference of the velocities with which they traverse the crystal.

It has been shown, that the difference between the reciprocals of the squares of the velocities, with which the two rays traverse the crystal, is proportional to the product of the sines of the angles which their direction makes with the optic axes; or, that if v and v' denote the velocities of the two rays, w and w' the angles which their direction makes with the two

axes,

v -2 – v′-2 = c sin w sin w'.

But if t and t' denote the times occupied by the two rays in traversing the crystal, and 0 the thickness actually traversed,

* We have here supposed the resulting light to be simply the sum of the lights derived from each of the portions into which the original light was supposed to be resolved, without reference to their phase. The justice of this assumption depends upon the fact adverted to in the preceding note,— namely, that the two oppositely polarized portions, into which we have supposed common light to be resolved, differ in phase,-that difference continually varying. The same thing is true, therefore, of the final components; so that these must be regarded as lights proceeding from different sources, and compound a light equal in intensity to the sum of the components.

--or the thickness of the plate multiplied by the secant of the angle of refraction,

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Now the first factor of this product is very nearly constant; we have, therefore,

t-t' const × 0 sin w sin w';

or, the interval of retardation is proportional to the product of the sines of the angles which the direction of the rays makes with the two axes, and to the thickness of the crystal traversed, jointly. When the two axes coalesce, or the crystal becomes uniaxal, the retardation is proportional to the square of the sine of the angle which the direction makes with the axis. But the tint developed is measured by the interval of retardation; accordingly the laws of the tints, discovered experimentally by M. Biot, flow immediately from the theory.

(201) It is plain that the light issuing from the crystal is, in general, elliptically polarized, inasmuch as it is the resultant of two waves, in which the vibrations are at right angles, and differ in phase. Hence, when homogeneous light is used, and the emergent beam is analyzed with a double-refracting prism, the two pencils into which it is divided vary in intensity as the prism is turned,—neither, in general, ever vanishing.

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When the thickness of the crystal is such, that the difference of phase of the two rays is an exact number of semi-undulations, they will compound a plane-polarized ray at emergence, the plane of polarization coinciding with the plane of primitive polarization, or making an equal angle with the principal section of the crystal on the other side, according as the difference of phase is an even or odd multiple of half a wave. Accordingly, one of the pencils into which the light is divided by the analyzing prism will vanish in two positions of its principal section; and it is manifest that the successive thick

nesses of the crystalline plate at which this takes place form a series in arithmetical progression.

On the other hand, when the difference of phase is a quarter of a wave-length, or an odd multiple of that quantity, —and when, at the same time, the principal section of the crystal is inclined at an angle of 45° to the plane of primitive polarization-the emergent light will be circularly polarized. This is one of the simplest means of obtaining a circularly-polarized beam; but it has the disadvantage, that the required interval of phase is only exact for waves of one particular length, and that, therefore, the circular polarization is perfect only for one colour.

(202) It has been stated (195) that the phenomena of colour are only produced when the crystalline plate is thin. In thick plates, where the difference of phase of the two pencils contains a great many wave-lengths, the tints of different orders come to be superposed (as in the case of Newton's rings, where the thickness of the plate of air is considerable), and the resulting light is white. The phenomena of colour may still, however, be produced in thick plates, by superposing two of them in such a manner, that the ray which has the greater velocity in the first shall have the less in the second. We have only to place the plates with their principal sections perpendicular or parallel, according as the crystals to which they belong are of the same, or of opposite denominations. Thus, if the crystals be uniaxal, and both positive, or both negative, they are to be placed with their principal sections perpendicular; and, on the other hand, these sections should be parallel, when one of the crystals is positive and the other negative. The reason of this is evident.

(203) Let us now consider the effects produced when a converging or diverging pencil of rays traverses a uniaxal crystal, in various directions inclined to the axis at small

angles; and let us suppose, for simplicity, that the crystalline plate is cut in a direction perpendicular to the axis.

Let ABCD be the plate, and E the place of the eye. The visible portion of the emergent beam will form a cone, AEB, whose summit coin

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sequently will be reflected, or not, from the analyzing plate, according as the plane of reflexion there coincides with, or is perpendicular to, the plane of the first reflexion. But the other rays composing the cone will be modified in their passage through the crystal; and the changes which they undergo will depend on their inclination to the optic axis, and on the position of the principal section with respect to the plane of primitive polarization.

M

Let the circle represent the section of the emergent cone of rays made by the second surface of the crystal; and let MM' and NN' be two lines drawn through its centre at right angles, being the intersections of the plane of primitive polarization, and of the perpendicular plane, respectively, with the surface. the rays which emerge at any

Now N

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point of these lines will not be di

vided into two within the crystal, nor will their planes of polarization

M

be altered; because the principal section of the crystal, for these rays, in the one case coincides with the plane of primitive polarization, and in the other is perpendicular to it.

These rays therefore will be reflected, or not, from the analyzing plate, according as the plane of reflexion there coincides with, or is perpendicular to, the plane of the first reflexion. In the latter case, therefore, a black cross will be displayed on the field, and in the former a white one,-as is represented in the annexed diagrams.

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But the case is different with the rays which emerge at any other point, such as L. The principal section of the crystal for this ray, OL, neither coincides with, nor is perpendicular to, the plane of primitive polarization; and consequently the incident polarized ray will be divided into two within the crystal, whose planes of polarization are parallel and perpendicular to OL, respectively. The vibrations in these two rays are reduced to the same plane by means of the analyzing plate: they will therefore interfere, and the extent of that interference will depend on their difference of phase.

Now the difference of phase of the two rays depends on the interval of retardation. When this interval is an odd multiple of half an undulation, the two rays are in complete discordance; and, on the other hand, they are in complete accordance when it is an even multiple of the same quantity. We have seen (201) that, for a given plate, the interval of retardation is proportional to the square of the sine of the angle which the ray makes with the optic axis within the crystal. It may be easily shown that the sine of this angle

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