fraction will be that of E N to N P, because these lines are as the sines of the angles of incidence and refraction N P H, NE P.

Thus Huygens found that the ratio was as 5 to 3 in all incidences, as had previously been determined. The way in which he then proceeded to determine the ratio of the extraordinary refraction, was by withdrawing his eye to Q till the extraordinary image of C D coincided with K L. By marking the point R, he could then obtain by measurement, the relation of E R to E S, or the ratio of the sine of incidence to that of refraction. In pursuing this investigation, he found that this ratio was not constant; but varied according to the inclination of the incident ray of light.

In his explanation of the phenomena of double refraction by the undulatory theory, Huygens supposes, that as there are two different refractions in Iceland-crystals, so there must be two different emanations of light from the luminous body. The ordinary refraction is produced by rays propagated in spherical waves, while the extraordinary refraction depends upon undulations of an elliptical, or hemispheroidal, character. He considers that the regular arrangement of the particles of the transparent body contribute to the formation of the spheroidal waves; and that the form of the generating ellipse is determined by the ratio of the two refractions. The light is, by this hypothesis, supposed to be more quickly propagated in one direction, than in another.

For, suppose the surface of a crystal of Iceland spar, respresented by AB, Fig. 4. be exposed to a ray of light, the line R C, parallel and equal to A B, will be a portion of a wave of light which falls upon A B at a perpendicular incidence; and the points R H h C meet A B at AK k B. We must now suppose that, instead of hemispherical waves, as we have previously dealt with in ordinary refraction, these waves are hemispheroids, whose major semi-axes are oblique to the plane A B. Hence S V T will represent an individual wave coming from the point A, after R C has arrived at A B. Now, in the same time that the point A propagated the wave S V T, all the other points, K k B, will propagate similar ones, and therefore the common tangent, N Q, of all these semi-ellipses, will be the propagation of the waves RC in the transparent body, according to the Huygenian theory. But it will be observed, that the tangent, N Q, which is equal and parallel to A B, is not directly opposite to A B, but is comprehended between the lines A N and B Q, conjugate diameters to those which are in the line A B. Fig. 4.

Thus, by this supposition, Huygens was able to comprehend what previously appeared to him very difficult, how a perpendicular ray could suffer refraction in a transparent body? For the wave R C, instead of going on straight when it entered the transparent surface, A B, extends itself between the parallels A B and N Q.

It is trusted that this will be considered a sufficient elucidation of the Huygenian hypothesis of double refraction, for it is impossible to proceed farther in the enquiry, without having recourse to more intricate mathematical reasoning than would be agreeable to most persons. Since the time of Huygens, many eminent men have investigated the subject. Among others may be mentioned the names of Fresnal, Cauchy, Biot, Arago, and Airy. In the formulæ introduced for the explanation of the newly-discovered phenomena, they have reduced the laws of vibratory motion to differential equations of the second order, but have not, with them, been able to show that some of the later facts are the results of the undulatory theory.

At the Bristol meeting of the Association in 1836, Professor M'Cullagh proceeded still further with the subject, and showed that, by introducing differential coefficients of the third order into the equations of vibratory motion, the greater number of the laws discovered could be satisfactory explained. When these equations are applied to the elucidation of the phenomena observed in quartz and other binaxial crystals, there must be two waves of light elliptically polarised and moving with different velocities; the ratio of the greater and smaller diameters, or axes, of these ellipses being the same in each wave; but the greater axis of the one being turned towards the lesser axis of the other, and the difference of the sign of the two equal quantities corresponding to the ratio of these axes, it follows that, if the vibration be from left to right in one wave, it must be from right to left in the other.

Without attempting to follow the learned professor any farther in his speculations, let us now take a slight notice of the benefits which have resulted to science from the examination of the phenomena of double refraction. Before Sir David Brewster began his optical labours, all crystals were supposed to have but one axis, and the Huygenian was considered the universal law of double refraction. By the most ingenious and accurate experiments, he was soon led to believe that the greater number of crystals have two, some three, or even more axes of double refraction, while a few are totally irregular in this respect. Hence he was led to discover the general law which subsists between the primitive forms of crystals, and the number of their axes. Thus he was enabled to predict, on the faith of these principles, that different crystals would be found, eventually, to belong to particular systems of classification from which they had been excluded. In this manner he has been able to correct many errors in the systems of Hauy, and to establish, on a firmer basis, the characteristic of Mohs.

The assistance which has been thus afforded to mineralogy and crystallography, is of the most important nature, and will tend eventually to show the connexion which exists between the optical structure and the chemical composition of crystallised bodies.

In the course of his interesting experiments on light, Sir David found that many bodies received the polarising structure by compression, while

others were similarly affected by the application of heat. Hence he was led to imagine, that a satisfactory explanation could be given of the cause of the transmission of the two kinds of waves in doubly refracting crystals. He supposes that these bodies consist of two co-existent media of different densities, one of which transmits the ordinary ray, according to the law of Snellius, while the other, transmitting the extraordinary ray, gives origin to the secondary image.* By this hypothesis it may be demonstrated, that the undulations must necessarily be of the spherical and the spheroidal form; but it appears difficult to conceive two or three extraordinary media combined in the same sub

stance.

If we examine the rays of light after transmission through a crystal of Iceland spar, or any other doubly refracting substance, we find that they have acquired new properties, and are polarised in planes at right angles to each other. As some of our readers may not be exactly acquainted with the precise meaning of the term polarisation of light, it will be as well to give a familiar illustration of this most remarkable phenomenon.

Fig. 5.

A

D

C

B

Let a ray of light, B A, fall upon a plate of glass A, placed in a vertical position, at an angle of incidence of 56°. The ray B A will be reflected in the horizontal plane A D; and being then reflected from the glass D, so placed as to receive it at the same incidence, the ray A D, which should have been reflected in the vertical plane D C, is so weak as to be scarcely visible, and nearly the whole of the light will be found to have been transmitted through the glass D. Now, if we take A D as the axis of motion, and turn the glass D round 90o, we shall reflect the ray A D in a horizontal direction; and we shall find that, instead of going through the glass D as before, nearly the whole of the ray AD will be reflected. If we continue to turn the plate D round to another quadrant of the circle, the light will be again transmitted, and again reflected, when we arrive at the succeeding quarter. Thus we observe, that transmission and reflection take place alternately, and that the ray

* Phil. Trans., 1818.

BA has acquired a new property, after reflection from the first surface, and is then said to be polarised.

Or we may take a number of slips of thin window glass, and bind them together into a solid shape. If we now let a ray of light be incident upon the surface at the same angle of 56°, a portion of the light will be reflected, while the remainder will be transmitted according to the usual law of refraction. Upon examining these reflected and transmitted rays, we shall find that they are both polarized, but in a remarkable relation to each other. The reflected ray will, of course, follow the same laws as the polarized ray, BA, in the last figure. But if we receive the transmitted ray upon a plate of glass at the angle of polarization, it will refuse to be reflected, unless the glass be turned round 90°, or into a plane at right angles to that plane in which the reflected ray was again reflected; or, in other words, unless the planes are at right angles to each other: one ray will always be transmitted, whilst the other is reflected, and vice versa.

Applying the same kind of experiments to doubly reflecting crystals, the two rays will be found polarized in planes at right angles to each other, the ordinary rays being polarized like the ray transmitted through the bundle of plates, and the extraordinary ray like the ray reflected from the surface of the same. There are various other ways of polarizing light, which it is needless to enumerate, as the effect is always similar to those now mentioned.

The history of the progress of this new and absorbing branch of science is highly interesting. Many of the facts appear to have been twice discovered by separate individuals, living in parts of the world distant from each other, and therefore unconscious of each other's plans. The experiments have been of such a delicate nature, and produced such beautiful and unthought-of results, that it is with difficulty we can refrain from devoting too much space to their elucidation.

Huygens, the philosopher, whom we have so often mentioned, was at the head of these inventions; and, in the course of his observations on Iceland crystal, detected the change produced upon the original ray of light. He calls it a "wonderful phenomenon," and was led to its discovery in the following manner: After separating a ray of solar light, by transmitting it through a piece of Iceland spar, he made the two pencils fall upon the surface of another piece of the same crystal. He was greatly surprised to observe that when the principal sections of the two pieces were parallel, neither of the two pencils were divided in passing through the second rhomb; but that the pencil which had suffered the ordinary refraction in passing through the first crystal, was only refracted in the ordinary manner, in passing through the second; as also the one which had been extraordinarily refracted in being transmitted through the first, was now extraordinarily refracted in passing through the second crystal. Now, as he proceeded with his investigations, he found that when the principal sections of the doubly refracting crystals cut one another at right angles, the exact contrary took place; for, the ray which had been previously the extraordinary became the ordinary, and that which had been the ordinary ray was now refracted according to the extraordinary law alone. In all other positions of the crystal, when the principal sections did not bear these relations to each other, as in these

two instances,; each of the rays divided by the first crystal were again split into two, in passing through the second, by reason of its double refraction. So that out of the single ray of light incident on the first crystal, there were formed four pencils, usually of equal brightness; but the sum of whose light did not appear to exceed that of the original beam. Sir Isaac Newton, reasoning upon these data, which Huygens did not attempt to explain, concluded that every ray of light may be considered as having four sides or quarters. Two of these, opposite to each other, incline the ray to be refracted in the usual manner, as often as either of them are turned towards the surface or side of double refraction; while the other two incline the ray to be unusually refracted, whenever either of them are turned towards the coast of unusual refraction. Thus originated his theory of fits of easy reflection and transmission which is now almost universally exploded; but from them also arose the term polarization, which has been since adopted by all philosophers.

The next discovery of importance was that of M. Malus, Member of the National Institute of France, who had returned home to pass the remainder of his life in quiet, after suffering severely in Buonaparte's expedition to Egypt. He was in the course of a strict enquiry into the laws of double refraction, for the purpose of competing for the prize offered by the Institute. At that time, he resided in the Rue des Enfers, in Paris, and was speculating in his mind one afternoon on the phenomena observed by Huygens, when he happened to turn a doubly refracting prism towards the windows of the Luxembourg, which were at that moment highly illuminated by the setting sun. As he turned the prism round before his eyes, he was astonished to observe that one of the images of the windows vanished, alternately, from his sight.

He at first attributed this unexpected phenomenon to some change which, he supposed, the light had received during its oblique transit through the atmosphere. Then being unable to account for the change on this supposition, he was led to think whether the glass of the windows had not some effect upon it. To his inexpressible delight, he found that the rays of light had acquired the curious property which he observed, entirely from being reflected from the panes of glass of the windows of the Luxemburg. Thus was Malus led to the discovery of the polarization of light by reflection, which forms one of the most interesting epochs in the history of Optics.

This reflected ray has all the characters of an ordinary ray produced by the refraction of a crystal, whose principal section is parallel to the plane of reflection, or of an extraordinary ray formed by a crystal, whose principal section is perpendicular to the same plane. This remarkable property of polarization is produced by reflection from all solid and liquid transparent substances; but it must be observed that each of these polarizes light at an angle from the perpendicular peculiar to themselves, being, in general, in proportion to the refractive power. Thus: glass polarizes light at an angle of incidence of about 54° 35', and water

52° 45'.

Many other valuable and highly interesting facts were elucidated by Malus in the course of his experiments, and have been duly estimated by the few who are able to appreciate them. He was succeeded in the same branch of enquiry by M. Arago, who has given the result of some