every one of which, with the single exception of muriate of soda, has double refraction.

175. Believing, therefore, that all these five crystals had no double refraction, when they actually possessed it, Hauy deduced the conclusion that all cubical crystals had single refraction, whereas this property belonged only to one out of six. The conclusion, however, is still true, and has been established by more numerous and recent observations, both optical and crystallographical. Boracite belongs to the rhomboidal system of crystallisation; scheelin, calcaire, and oxide of tin, to the pyramidical system; amphigene to the prismatic system; and analcime to the composite system. Muriate of soda, therefore, is the only crystal left among the cubical forms to authorise the deduction of our author.

176. In the Bakerian lecture for 1801, our learned and ingenious countryman, Dr. Thomas Young, pointed out the advantages of the Huygenian theory of light, in affording an explanation of several phenomena which had not been accounted for by any other hypothesis. Dr. Wollaston, who had invented a new method of measuring the refractive powers of bodies, conceived the idea of employing this method to examine the accuracy of Huygens's theory of double refraction. Huygens had himself done this by direct experiment; but it was desirable to have the same examination repeated by a philosopher of Dr. Wollaston's accuracy, and by a method which promised to afford very nice results. In this way Dr. Wollaston obtained the following results :

177. Dr. Wollaston considers the result of this comparison as highly favorable to the Huygenian theory; and he adds that though the existence of two refractions at the same time, and in the same substance, be not well accounted for, and still less their interchange with each other, when a ray of light is made to pass through a second piece of spar, situated transversely to the first; yet the oblique refraction, when considered alone, seems nearly as well explained as any other optical phenomenon.

178. The attention of the illustrious La Place was no doubt directed to the subject of double refraction by the labors of Dr. Young and Dr. Wollaston, who had drawn the attention of the scientific world to this recondite branch of physical science.

179. These investigations are contained in a memoir, entitled Sur les Mouvemens de la Lumiere dans les Milieux Diaphanes, which was read at the Institute on the 30th of January, 1808, and published in their Memoirs for 1809, P. 300-342.

180. It occurred to M. La Place that it would be highly interesting to refer the law of Huygens to attractive and repulsive forces, as Newton had done the ordinary refraction. In employing the principle of least action for this purpose, he remarks that, in the case of the extraordinary refraction, the velocity of the light within the crystal must be independent of the manner in which it enters, and must depend only on the position of the ray with respect to the axis of the crystal, that is, on the angle which the ray forms with a line parallel to the axis.

181. In setting out from this datum, M. de

La Place arrives at two differential equations given by the principle of least action, and in which the interior velocity is an indeterminate function of the angle, which the refracted ray forms with the axis of the crystal. In the first case which he examines, the square of the velocity of the ray is increased in the interior of the medium by a constant quantity (which is the case of ordinary transparent mediums), and this constant quantity expresses the action of the medium upon light. The two equations, then, show that the incident and refracted ray are in the same plane, and that the ratio of the sines of their inclination to a vertical line is constant.

182. In the next case the action of the medium upon light is equal to a constant quantity, plus, a term proportional to the square of the cosine of the angle which the refracted ray forms with the axis; for as this action is equal on all sides of the axis, it must depend only on the even powers of the sine and cosine of that angle. The expression of the square of the interior velocity is thus of the same form as that of the action of the medium. By substituting this expression in the differential equation of the principle of least action, M. de La Place then determines the formula of refraction in relation to this case, and he finds that they are identically those which are given by the law of Huygens. Hence it follows, that the Huygenian law satisfies both the principle of least action, and the condition that the interior velocity depends only on the angle formed by the axis and the refracted ray.

183. M. de La Place then proceeds to remark, that the hypothesis of Huygens, that the velocity of the ray is expressed by the variable radius of the ellipsoid, does not satisfy the principle of least action, but that it satisfies the principle of Fermat, which consists in this, that the light arrives from a point taken without the crystal, to a point taken within it, in the least time possible. For it is obvious that this principle becomes the same as that of least action, by reversing the expression of the velocity. Hence both these principles conduct to the law of refraction discovered by Huygens, provided that, in the principle of Fermat, we assume with Huygens the radius of the ellipsoid as a measure of the velocity, and that, in the principle of least action, we assume this radius as representing the time employed by light in traversing a determinate space taken for unity.

184. The identity of the law of Huygens and the principle of Fermat's results, as M. de La Place has remarked, from the ingenious way in which Huygens considers the propagation of the waves of light, so that his way of considering it, though very hypothetical, represents nevertheless all the laws of refraction which may be due to attractive and repulsive forces, since the principle of Fermat gives the same laws as that of the least action, by reversing the expression of the velocity.

185. It will be evident that, in a work like the present, our space will not admit of further analysis of M. de La Place's Memoir, but merely to remark, that the formula which he deduces from the principle of least action, and that of Fermat, are found to be identical with the elegant formula which Malus deduced from the construction of Huygens. For a variety of full and interesting particulars, on this curious part of optical science, we must refer our readers to the Journal edited by Dr. Brewster.

186. POLARISED LIGHT.-It has been usual to commence this subject by considering the well' known phenomenon of double refraction, a property possessed by all crystals, the primitive form of which is neither the cube nor regular hexahedron. Of all known bodies the Iceland spar, or rhomboidal carbonate of lime, shows the fact with the greatest certainty; and as it is a mineral easily procured, and of sufficient size and transparency, it has been generally made use of. The crystals of this substance have the form of a rhomboid, having six acute solid angles, and two obtuse, These last, r and a, fig. 10, are formed by the junction of three equal plane angles, and equally inclined to each other. The line aa, joining these two angles, is therefore similarly situated with respect to the three planes forming each angle, and is called the axis of the crystal. A plane, perpendicular to the natural surface of the crystal, and coinciding with this line, is called its principal section, which term is also applied to any plane parallel

to it.

187. We have already stated that an object seen through the crystal in its natural form (that form to which it is easily brought by cleavage) will give two images, one of which will appear in its situation according to the common law of refrac

tion, while the other will be observed thrown towards the lower obtuse angle, but always in the plane of the principal section. (The most convenient method to examine these facts is to pierce a small hole through any opaque plate, which may be applied to the lower surface of the crystal, and directed to a sheet of white paper). Let x A, B, fig. 11, be the principal section of the crystal, and L a pencil of light falling on its surface, one part of the light will proceed in the ordinary direction (we will suppose perpendicularly), and is therefore called the ordinary ray, while the other portion of the light deviates considerably from this direction, and is called the extraordinary ray. Io will represent the ordinary, and I e the extraordinary ray.

188. Let the crystal be cut by two planes A B, and CD, fig. 12, parallel to the axis, and two other planes AC, and DB, perpendicular to the axis, to allow an object to be seen through it in the direction A C, or AB: it will be found that the two images will be farther separated, viewed in the direction A C, which is perpendicular to the axis, while in the direction of the axis there will be only one image. The inference from these experiments is, that there exists some peculiar force acting on the light passing through the crystal, producing a separation of the rays, and that this force emanates from the axis itself. As this produces a deviation of the second image towards, or from, the axis of the crystal, it is considered positive or negative, or, by Biot, attractive or repulsive.

189. The two rays into which a pencil of light is divided in passing through a crystal of Iceland spar are always of the same intensity, and always in the plane of the principal section. But the two emerging rays are not merely diminished in intensity by the division of the light between them, but have undergone a most important modification; for, if the rays be made to pass through another crystal, placed similarly to the first, there will be no subdivision of the light; the two images will be merely separated to a greater distance, from the increased thickness through which the light passes. If now the two crystals are so placed that the principal sections are at right angles to each other, there will still be only two images, but the ray ordinarily refracted in the first will become extraordinary in the second, and the extraordinary, ordinary. But at all intermediate positions of the two crystals there will be a subdivision of each ray, consequently four images: these four images will be of equal intensity when the principal sections of the two crystals are at an angle of 45° to each other; at all other angles one or other of the images diminish in intensity, as the principal sections approach to a perpendicular or parallelism; not by the coalescence of the two images, but by the gradual diminution of the intensity of one, and the augmentation of that of the other. In plate III. figs. 1, 2, and 3, we have supposed the rhomboids re duced to the form of cubes in all three, the axis is denoted by r r, and the direction of the rays by the lines passing through the figure, and the letters e and o the extraordinary and ordinary

rays. It is thus seen that each emerging ray is only subject to a farther division, in particular positions of the second crystal; whereas natural light is always divided into two portions of equal intensity. Each ray has suffered a physical change; it is not acted on by the force of the second crystal, as natural light would be, but requires that the force be applied in a particular direction relatively to the modification it has received from the first crystal. This has been called polarisation. We know nothing of the poles, nor even of the molecules to which these poles are said to belong; it must be considered merely as a conventional term to express a phenomenon; and, to avoid the repetition of the conditions producing it, it is usual to consider the poles of the ordinary ray as coinciding with the principal section of the crystal, and those of the extraordinary ray in a plane perpendicular to it.

190. In these experiments an effect is produced on the two rays, by the passing through a crystal, which so modifies them that it requires the action of a second crystal to be exerted in two directions, at right angles to each other, to produce a similar effect. As direct light is always divided into two rays of equal intensity by a double refracting crystal, it may be used as a test to discover and ascertain the direction of polarisation.

191. When light is reflected from the surface of transparent bodies it is found to be polarised in the plane of reflection, more or less completely, according to the angle formed with the surface. This angle varies for different substances, and to Dr. Brewster we are indebted for a law connecting the angle with the index of refraction of the given body, viz. that the tangent of the angle of polarisation, measured from the perpendicular, is equal to the index of refraction,' when complete polarisation, or the greatest the body is capable of, is produced.

192. The index of refraction is a number having the same proportion to 1 that the line of the angle of incidence has to the angle of refraction. It is, therefore, constant for all angles of incidence in the same body. It is generally calculated for light passing from a vacuum into the given body.

193. From this law are deduced three consequences :

1. That the complement of the angle of polarisation is equal to the angle of refraction.

2. That the reflected ray is perpendicular to the refracted ray.

3. That the angles of polarisation and refraction are together equal to a right angle.

194. Let CE, fig. 4, be the incident ray at the angle of complete polarisation, R E the reflected ray, and ÉG the refracted ray. The sine DC sine GH:: index: 1 But index tangent A B; and radius A E 1 As A B AE: DC: GH

which the index of refraction is known. Look in a table of tangents for the given index as a tangent, and the corresponding angle will be the angle of polarisation. If such a table be not at hand, let SS be the surface, and A E drawn perpendicularly; from a scale make AB = index, A E being 1. Join BE, and the angle A E B will be the required angle.

196. The index is generally given, for light passing from a vacuum; if it be required to determine the index under other circumstances, divide the index of the given body by the index of the medium from which it passes, and the quotient will be the index required. In bodies the opacity of which prevents the direct measurement of the angle of refraction, the angle of polarisation gives the index of refraction; if this law be rigorous, which it appears to be for all bodies which have been yet examined, though M. Fresnel seems to think it more probably an approximation.

197. We may here remark that some authors give the angle of polarisation measured from the surface and sometimes the index of refraction is calculated on the passage from air.

198. We may here mention that Malus and Biot state, that the peculiar influence of crystals is not excited on light reflected from their surfaces; and that the angle of polarisation is the same whether the plane of incidence be parallel, perpendicular, or oblique, to the principal section of the crystal. But Dr. Brewster found a difference of more than 2°. He found the angle of polarisation to be 57° 14' when the incidence was in the plane of the principal section, and 59° 32′ when in a plane perpendicular to it. The surface was that produced by a careful cleavage of the crystal.

199. One of the first laws of optics is, that light falling on a plane mirror is reflected at an angle equal to the angle of incidence;' but, if this mirror be transparent, the light so reflected is found to have received a polarity in the plane of reflection, more or less complete as the angle of incidence approaches the angle of polarisation for the substance employed. In fact there is some trace of polarisation discernible in light reflected from every body, at every angle of incidence except the perpendicular; but this in so feeble a degree, that we may consider it, at present, insensible, except from transparent

bodies.

200. We have already explained that there are, for some bodies, certain angles, at which complete polarisation takes place; in some others, though very transparent, yet of high refractive power, complete polarisation does not take place at any angle, as in the diamond; in others, as black marble, ebony, black varnishes, where the refractive density is less, complete polarisation takes place, though they be opaque. This angle for water is 52° 45' from the perpendicular, and about 55° for glass.

201. In our explanation we may have recourse to an instrument, which, though not indispensable, will be found useful in many experiments where accuracy is required. But, for the satisfaction of any one as to the general facts, two pieces of unsilvered mirror glass are all that will