is the cube. Now in this, and its derived forms, we can assign three lines at right angles to one another, round which the whole figure is symmetrical; and we may, therefore, reasonably conclude that the density and elasticity of the crystal is the same in each of these directions, and consequently the same throughout. Again, crystals with one axis of double refraction belong either to the rhombohedral, or to the pyramidal system,-systems whose fundamental forms are the rhombohedron and the straight pyramid. In each of these forms there is one axis of figure, or one line round which the whole is symmetrical: and we may, therefore, assume that the density of the crystal is either greater or less in this direction than in others, while it is equal in all directions at right angles to it. The axis of form is, in this case, the axis of double refraction. Finally, in the oblique pyramid, which is the fundamental form of the prismatic systems, there is no one line, or axis of figure, round which the whole is symmetrical; and it is therefore probable that the density of the crystal is unequal in all the three directions. Such crystals are found to have two optic

axes.

It has been stated, that in uniaxal crystals the optic axis is also the axis of form. In biaxal crystals, it did not at first appear that the optic axes were in any manner related to the lines which bound the elementary crystal. Sir David Brewster, however, ascertained that if two lines be taken, one bisecting the acute, and the other the obtuse angle contained by the optic axes, these (together with a third line at right angles to both) are closely connected with the primitive form.

These relations between the optical properties of crystals and their external forms are so close and intimate, that any change (however produced) in one of them, is found to be accompanied by a corresponding change in the other. Thus, if the form of a crystal be altered by mechanical compression, or change of temperature, its refracting properties undergo a corresponding change.

(70) It was long supposed that one of the refracted rays, in every crystal, followed the ordinary law of the sines, while the other was refracted according to the Huygenian law. But Fresnel has proved, both from theory and by experiment, that this is not the case, and that in biaxal crystals, both rays are refracted in an extraordinary manner, and according to a new law. It is, in fact, a consequence of his beautiful theory of double refraction, that the form of the wave, which is propagated in the interior of such a crystal, is neither a sphere nor spheroid, as in uniaxal crystals, but a curved surface of the fourth order. This surface is composed of two sheets; and if tangent planes be drawn to these, after the same manner as to the sphere and spheroid in the Huygenian law, the points of contact determine the directions of the two refracted rays. These more general laws of double refraction will be more fully considered hereafter.

(71) We may now proceed to illustrate some of the more remarkable effects of double refraction.

If a rhomboid of Iceland spar, or any other double-refracting crystal, be placed close to a small object,—as, for example, a black spot on a sheet of paper,—it will be observed that one of the images is sensibly nearer than the other; and that the difference of their apparent distances changes with the thickness of the crystal, and with the obliquity of the ray.

This effect is easily accounted for. It is a well-known principle of optics, that when an object is viewed through a denser medium bounded by parallel planes,—as, for example, a cube of glass,-the image is nearer to the surface than the object; the difference of their distances being to the thickness of the medium, as the difference of the sines of incidence and refraction to the sine of incidence. This interval, through which the image is made to approach, increases therefore with the refractive power of the medium; thus in water it is onefourth of the thickness, in glass one-third, and so for other

media. Now as double-refracting crystals have two refractive indices, of different magnitudes, there will be two images, at different distances from the surface. In Iceland spar, the ordinary index is greater than the extraordinary, and therefore the ordinary image is nearer than the other. The reverse is the case in positive crystals, such as quartz, in which the extraordinary index is the greater.

(72) The refractions being equal at the two parallel surfaces of the rhomb, whether the refraction be ordinary or extraordinary, the two rays will emerge parallel to the incident ray, and therefore parallel to one another; and the distance between them will be proportional to the thickness of the crystal. But if the surfaces be inclined, so as to form a prism, the deviation of the two rays will be different, and they will emerge inclined to one another; consequently the separation will increase with the distance.

Such a separation is of use in many experiments. In order to render the divergence of the emergent pencils greatest, the prism should be cut with its edge parallel to the optic axis; so that the refraction may take place in a plane perpendicular to the axis. In this case the ordinary and extraordinary refractions differ by the greatest amount, and therefore the difference of the deviations of the two pencils is greatest. double-refracting prism, so cut, is usually achromatized by a prism of glass, with its refracting angle turned in the opposite way.

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A better arrangement has been suggested by Wollaston. Two prisms of the same substance, and of equal refracting angles, are cut in such a manner, that in one the refracting edge is parallel to the optic axis, and in the other perpendicular to it. They are then united, with their refracting angles turned in opposite directions, so as to form a parallelopiped; and the effect of this arrangement is to double the separation of the images produced by either singly. By this duplication

the weak double refraction of rock crystal is rendered very sensible.

(73) An achromatic prism of this kind is employed in the double image micrometer, an ingenious and valuable instrument invented by Rochon. It consists of a telescope, in which a prism, such as we have described, is introduced between the object-glass and its principal focus; and thus two images are

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formed in the principal focus, whose interval is greater or less, according to the distance of the prism from that point. When the instrument is used, the prism is moved until the two images appear in contact, and its distance from the focus is then read on a graduated scale. The two angles in this case having the same subtense, the visual angle of the object is to the deviation produced by the prism, as the distance of the prism from the focus is to the focal length. Now the divergence of the two rays is constant for a given prism, and may be determined either by calculation or experiment; consequently, the visual angle is deduced from the preceding proportion. By this instrument Arago has determined the apparent diameters of the planets with great precision.

The same instrument has been also employed in war, to determine the distance of an inapproachable object. Thus, if it be required to ascertain the distance of the walls of a besieged town, in order to know whether they are within the range of shot, it is only necessary to measure by this instrument the angle subtended by a man, or any other object whose height is known approximately. The height of the object, divided by the tangent of the angle, is the distance required.

CHAPTER V.

INTERFERENCE OF LIGHT.

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(74) HAVING considered the mode of propagation of a luminous wave, and the modifications which it undergoes on encountering the surface of a new medium, we may now proceed to inquire what will be the effect, when two series of waves are propagated simultaneously from two near luminous origins.

It is obvious that when two waves-one proceeding from each source—arrive at any instant at the same point of space, the particle of ether there will be thrown into vibration by both; and we are to consider what will be the result of this compound vibration. Now, it is demonstrated by analysis, that when two small vibrations are communicated at the same time to a material point, each of them will subsist independently of the other; and the motion of the point will, in consequence, be the resultant of the motions due to each vibration considered separately. This principle is denominated the superposition of small motions. Its nature may be made clear by a simple

instance.

Let a pendulous body receive an impulse in any plane passing through the point of suspension: it will then, of course, vibrate in that plane. Now, at the lowest point of the arc of vibration, let a second impulse be given to the moving body, in a direction perpendicular to the plane in which it already vibrates. This impulse, if communicated to the body at rest, would cause it to vibrate in a plane at right angles to the former, and through an arc depending on the magnitude of the impulse. Now it will be found, on trial, that the distance of the body from the vertical, measured in either of these