is very nearly proportional to the sine of the angle LEO (see first fig. p. 190), which the emergent ray makes with the axis; and this latter to LO, the distance of the point of emergence from the centre. The retardation therefore varies as the square of the distance LO; and consequently the successive dark and bright lines will be arranged in circles, (as represented in the preceding diagrams) the squares of whose radii are in arithmetical progression.

We have been speaking hitherto of homogeneous light. When white or compound light is used, the rings of different colours will be partially superposed, and the result will be a series of iris-coloured rings separated by dark intervals. All the phenomena, in fact, with the exception of the cross, are similar to those of Newton's rings; and we now see that they are both cases of the fertile principle of interference. These rings are exhibited even in thick crystals, because the difference of the velocities of the two pencils is very small for rays slightly inclined to the optic axis.

(204) Let us now consider briefly the case of biaxal crystals.

Let a plate of such a crystal be cut perpendicularly to the line bisecting the optic axes, and let it be interposed, as before, between the polarizing and analyzing plates. In this case the bright and dark bands will no longer be disposed in circles, as in the former, but will form curves which are symmetric with respect to the lines drawn from the eye in the direction of the two axes; the points of the same band being those for which the interval of retardation of the two rays, t-t', is the same. Now this interval is proportional to the product of the sines of the angles which the direc tion of the rays makes with the optic axes (200); and these sines are, very nearly, as the distances of the points of emergence (measured on the face of the crystal) from the projections of the optic axes. Hence the product of these distances

The

will be constant for all the points of the same curve. curve formed by each band is therefore the lemniscata of James Bernouilli,—the fundamental property of which is, that the product of the radii vectores, drawn from any point to two fixed poles, is a constant quantity.

The exactness of this law has been verified, in the most complete manner, by the measurements of Sir John Herschel. The constant varies from one curve to another,-being proportional to the interval of retardation, and increasing therefore as the numbers of the natural series for the successive dark bands. For different plates of the same substance, the constant is inversely as the thickness.

The annexed diagrams represent the systems of rings in a biaxal crystal whose axes form a small angle with one another, in two positions of the crystalline plate, the planes of polarization of the polarizing and analyzing plates being at right angles.

The form of the dark brushes, which cross the entire system of rings, is determined by the law which governs the planes of polarization of the emergent rays. There is no difficulty in showing, on the principles of Fresnel's theory, that two such dark curves, in general, pass through each pole;

and that they are rectangular hyperbolas, whose common centre is the middle point of the line which connects the projections of the two axes.

(205) The phenomena of depolarization and of colour, impressed by double-refracting substances upon the transmitted light, are, we have seen, the necessary results of the interference of the two pencils into which the light is divided within them. These properties, therefore, become distinctive characters of the double-refracting structure; and thus enable us to discover the existence, and to trace the laws, of that structure, even in substances in which the separation of the two pencils is too minute to be directly observed.

By such means it has been discovered that a double refracting structure may be communicated to bodies which do not possess it naturally, by mechanical compression or dilatation. Thus Sir David Brewster observed, that when pressure was applied to the opposite faces of a parallelopiped of glass, it developed a tint in polarized light, like a plate of double-refracting crystal; and the tint descended in the scale as the pressure was augmented. Single-refracting crystals, such as muriate of soda, and fluor spar,-acquired the property of double refraction by the same means.

The opposite effects of compression and dilatation may be very well seen, and studied, in a thick plate of glass bent by an external force. The entire mass of the plate is thus thrown into an altered state of density, the parts towards the convex side of the plate being dilated, and those towards the concave side compressed; while, about the middle of the thickness, there is a surface in which the particles are in their natural state.

Accordingly, when this body is interposed between the polarizing and analyzing plates, so as to form an angle of 45° with the plane of primitive polarization, two sets of coloured bands are seen, separated by a neutral line; and these vanish altogether when the compressing force is withdrawn.

The parts towards the convex, or dilated side of the neutral line, are found to have acquired a positive double-refracting structure, and those on the concave, or compressed side, a negative one.

In these cases of induced double refraction, the phenomena are related to the form of the entire mass; and the axes of double refraction are single lines within the substance, fixed in position, as well as direction. In this respect the phenomena are essentially different from those produced by regular crystals, in which the laws of the double refraction depend solely on the direction, and are the same in all parts of the substance.

(206) The phenomena described in the preceding article are in perfect accordance with the wave-theory. Owing to the connexion of the vibrating medium with the solid in which it is contained, its elasticity is rendered unequal in different directions by the effects of compression, the maximum and minimum of elasticity corresponding to the directions of greatest and least pressure. Accordingly the vibrations of the ray, on entering the substance, are resolved into two in these directions, and these are propagated with unequal velocities. The incident wave will therefore be separated into two within the medium, one of which will be in advance of the other, and these will be in different phases of vibration at emergence. The resolved parts of the vibrations, in the plane of reflexion of the analyzing plate, will accordingly interfere, and the tint developed will be determined by the interval of retardation.

These results of theory were experimentally confirmed by Fresnel; and he found that the velocity with which a ray traversed the glass was greater or less, according as its plane of polarization coincided with, or was perpendicular to, the line in which the pressure was exerted. The double refraction of the ray is a necessary consequence of this difference of velo

cities but this was also established by Fresnel by direct experiment. A series of glass prisms were placed together, with their refracting angles alternately in opposite directions, and the ends of the alternate prisms were powerfully pressed by screws. A ray transmitted through the combination was found to be divided into two oppositely polarized. The compressed prisms, in this arrangement, acquired a double-refracting structure, the axis of pressure being also the axis of double refraction; and their refracting angles being all turned in the same direction, the divergence of the two rays was increased in proportion to their number, and thus rendered sensible. The intermediate prisms served to correct the deviation, and to render the combination achromatic.

(207) The effects of unequal density and elasticity may be much more regularly produced by the application of heat. These effects may be studied by applying a bar of hot iron to the edge of a rectangular plate of glass, and placing it in the polarizing apparatus, so that the heated edge may form an angle of 45° with the plane of primitive polarization. At the end of some time, the whole surface of the plate will be observed to be covered with coloured bands, the parts near the opposite edges having acquired a positive double-refracting structure, and those near the centre a negative one. The effects are reversed when a plate of glass, uniformly heated, is rapidly cooled at one of its edges; and all the appearances vanish when the glass acquires the same temperature throughout.

If we transmit heat from the surface to the axis of a glass cylinder, by immersing it in heated oil, it will display a system of rings similar to those of a negative crystal with one axis, the axis of the cylinder being also the axis of double refraction. When the heat reaches the axis, the double refraction begins to weaken; and the colours disappear altogether when the glass is uniformly heated. Again, if the cylin