or as .6045 is to .6742, and through the points A, B, c, d draw the ellipse A c Bd. Then, if Rab is a ray of light incident on the rhomb at b, at an inclination to A B of ROA or 44° 36' 34", the radius Oa of the ellipse will be found either by projection or calculation to be .6361 Hence, the index of refraction for the extraordinary ray formed by or 1 1.5720* Rb will be 1.572. B T. MSS through the axis is called a principal seclar to the surface AC atr. Aray, Pr, incition of the crystal. Draw PQ perpendicudent perpendicularly at r, will be divided into two rays, the ordinary oner Q, which goes straight on, and the extraordinary ray r M. Then, if two rays Rr, R'r, fall on the same point r at equal inclinations to Pr, or at equal angles of incidence, but in the plane of the section ACBD, the extraordinary rays of each, viz. T. rS, will be so refracted that TM-SM, and these refracted rays, as well as the ordinary ones rt, rs, will be all in the same plane. The force which produces the extraordinary refraction exerts itself as if it emanated or proceeded from the axis AB of the rhomb; for when the plane of incidence passes through the axis, the extraordinary ray is always in the same inclined at any angle to the axis, the plane. But if the plane of incidence is extraordinary ray is pushed out of that plane by the force proceeding, as it were, from the axis; and hence it is tedious, either by a graphic projection or by cal position of the extraordinary ray. culation to determine, in that case, the When the plane of incidence is permay be called the equator of double pendicular to the axis, or is in what refraction, where the force is a maximum, the extraordinary ray is always in the plane of incidence, and its position may be determined at all angles of incidence in this plane, in the same manner as if it were acted upon with an ordinary force whose index of refraction is 1.6543. All these observations are equally applicable to all the other crystals with one negative axis. The following are the number of principal sections, or planes of refraction, passing through the axis in the different primitive forms shown in figs. 3 to 7. Rhomb Hexagonal prism Right prism, with a square 6 Infinite. 4 to 1 1 Infinite. 15484 1-5582' The secondary forms of these crystals have, of course, a different number of such planes, some more and some less. 3.--On the Law of Double Refraction in Crystals with one positive axis. The mineral called quartz, or rock crystal, crystallizes most commonly in six-sided prisms, terminated with sixsided pyramids, as shown in fig. 10. If we grind down and polish the summits A and B, we shall find that there is no separation of the images, or no double refraction when the refracted ray passes along the axis AB. Hence AB is the axis of double refraction. If in this case we measure the index of refraction, we shall find Fig. 10. or as 6458 is to '6418. If we, therefore, call m' the index of extraordinary refraction (or the velocity of the extraordinary ray,) and the inclination of that ray to the axis, it may be shown hat m = 154842 +•030261 sin. ; that is, the square of the index of the extraordinary ray m' at any inclination is equal to the square of the index of ordinary refraction increased by a quantity varying with the inclination to the axis. Hence, we see the propriety of calling such crystals positive, because the term which expresses the influence of the doubly refracting force is always positive. The above expression becomes m' = √2·3975 + *030261 sin.2 p. The existence of a positive axis of double refraction in quartz was discovered by M. Biot. 4.-On Crystals with two Axes of Dou ble Refraction. they are mineral or chemical substances, The great body of crystals, whether have two axes of double refraction. This discovery was made by Dr. Brewster, who traced the double image through the crystals, and found the double refraction to diminish as the ray approached two lines or axes, and at last to disappear wholly when the ray passed along either of these two axes. found also that these lines were not coincident with any prominent lines in the crystalline form, and that they formed various angles with each other from the smallest angle in glauberite up to 90° in sulphate of iron. He After examining more than one hundred of these crystals, Dr. Brewster also found that all crystals which belong to the prismatic system of Mohs, or whose primitive forms are the right prism with its base a rectangle, a rhomb, or an oblique parallelogram; the oblique prism, with its base a rectangle, a rhomb, or an oblique parallelogram, or the rectangular and rhomboidal octohedron, have two axes of double refraction. In these cases the double refraction follows a very complicated law (see Chap. VIII.); and M. Fresnel has made the important discovery that both the rays follow a law of extraordinary refraction. 5.-On Crystals which have two Axes for the most refrangible, and one Axis for the least refrangible rays. This singular property was discovered by Dr. Brewster in Glauberite, in which he found two resultant axes inclined to one another at an angle of 5° when red light was used, and only one negative axis when violet light was used. In this case, however, it may be shown, by principles which will afterwards be explained, that glauberite has more than one real axis even for the violet rays*. 6.-On Crystals with many Planes of double Refraction. In all the crystals hitherto mentioned the double refraction is related solely to one or more lines or axes; but Dr. Brewster has found that analcime has its double refraction related to various planes within the crystal, in all of which the 7.-On Crystals with circular double Refraction. M. Fresnel has discovered that a ray of light passing along the axis of quartz where its ordinary double refraction vanishes, is divided into two rays which have remarkable properties. The law of variation of the doubly refracting force is not known, but the properties of the two rays to which it gives rise will be afterwards described. 8.-On Bodies to which double Refrac tion may be communicated by Heat and Pressure. Bodies with one or more axes of double refraction may be formed artificially out of glass, &c. either by pressure or by the transmission of heat, or by rapid cooling. In these cases the double refraction depends on the external form of the body, and changes with a change of form. If the body is a cylinder, it may be made to have one negative or If it is a cylinder whose section is an one positive axis of double refraction. ellipse, or if it is a parallelopiped, it will have more than one axis; and if it is a sphere it will have an infinite number of axes of double refraction. In all these cases the double refraction may be accurately calculated, as will be shown in a subsequent part of this treatise. PART II. ON THE POLARISATION OF LIGHT. "THE Phenomena of the Polarisation of Light," to use the language of one of our most eminent mathematicians and natural philosophers+, "are so singular and various, that to one who has only studied the subject of physical optics under its ordinary relations, it is like entering into a new world, so splendid as to render it one of the most delightful branches of experimental inquiry; and so fertile in the views it lays open of the constitution of natural bodies, and the minuter mechanism of the universe, as to place it in the very first rank of the physico-mathematical sciences." the substance of any homogeneous uncrystallised body, the property of the reflected or transmitted light continues the same when we turn round the body, so that the light falls on the first surface always at the same angle; that is, the different sides of the rays exhibit no different properties in relation to the plane of its incidence. Such light is called common light. When light emitted from the sun, or from any self-luminous body, is reflected from the surface, or transmitted through Edinburgh Journal of Science, No. XIX. A kind of light, however, has been discovered which, when reflected from the surface, or transmitted through the substance of homogeneous uncrystallised bodies, exhibits different properties when the body is turned round in the manner above described. Hence it follows that different sides of the rays of such light must have different properties in relation to the plane of their incidence, and hence this light is called colarised light, because its rays have poles, or sides with different properties. Polarised light is never emitted from any self-luminous body, or from any artificial flame produced by combustion. Whenever it is obtained, it must have previously existed in the state of common light, from which it may be procured in three ways: 1. By reflexion from the surfaces of transparent and opaque bodies. 2. By transmission through a number of plates or planes of uncrystallised bodies. 3. By transmission through bodies regularly crystallised, and possessing the property of double refraction. CHAPTER II.-Polarisation of Light by Reflexion-Discoveries of MalusDr. Brewster's Law of the Tangents -Table of the polarising Angles of bodies-Polarisation of Light at the second surfaces of bodies-Polarisation of Light at the separating surfaces of two media-By successive Reflexions--State of partially polarised light-The polarising angle used to measure refractive powers. In order to explain the difference between common and polarised light, let A, fig. 12, be a plate of glass placed at the end of the tube MN, so that a ray of light R A, incident at A, may be reflected along the axis of the tube MN. E At the end of another smaller tube NP, which can turn round within MN, place a similar plate of glass, capable of reflecting a ray AC to the eye at E. Let a ray of light RA fall upon the vertical plate of glass A at an angle of incidence of 56°, so as to be reflected in the direction AC; and let this reflected ray A C fall at the same angle of incidence of 56° upon a plate of glass C, and be reflected from it to E. Then in the position shown in the figure, where the first reflexion is made in a horizontal plane RAC, and the second in a vertical plane ACF, the ray CE will be so weak as to be scarcely visible, the plate of glass CE having almost no power to reflect the light AC. If we now turn round the tube NP within NM, without shifting the tube MN, and reflector A, the ray CE will become stronger and stronger till it has been turned round 90°, or so that the plane of reflexion ACE is horizontal like R AC. In this position the light in the beam CE is the greatest possible. If we continue to turn the tube, C E will become fainter and fainter, till after being turned round 90° more, when the plane of re flexion ACE is again vertical, the ray CE will almost cease to be visible. After a farther motion of 90°, the ray CE will recover its strength; and by 90° more, which brings the plate C back into its first position, as shown in the figure, the ray C E will cease to be visible. From this experiment it clearly follows, that when the upper or the under side of the ray AC is towards or nearest the reflecting plate C, the plate is incapable of reflecting it, whereas when the right or left side of the ray is towards or nearest the reflecting plate, the plate reflects it as it would do common light; and at intermediate positions intermediate degrees of light are reflected. The ray AC has, therefore, properties different from common light; and as the common light RA, from which it has been obtained, has suffered no other change but that of reflexion, we are entitled to conclude that light becomes polarised by reflexion at an angle of 56 from glass. The simple test, therefore, of polarised light is, that it refuses to be reflected by the surface of a transparent body when it is incident at an angle of about 50°, and in two positions at right angles to one another, which will be discovered by turning the reflecting surface round the polarised ray. This beautiful property of light, in virtue of which it is polarised by reflexion, and refuses to be again reflected under the circumstances above described, was discovered, in 1810, by M. Malus, a French philosopher of distinguished eminence. In continuing his researches, Malus found that black marble, ebony, and other opaque bodies, polarised the light by reflexion like transparent ones; and that when the light RA was incident on A at an angle below or above 55°, only a part of the reflected ray was polarised; and that the light which fell upon the second surface of the glass plate was polarised at the same time with that which fell upon the first surface. He found the angle of incidence upon water, at which it polarised the light most completely, to be 52° 45', and the angle for glass to be 55°; and he concluded that the property by which bodies polarised light was independent of the other modes of action which they exert upon light. The experiment represented in fig. 12 is susceptible, as Dr. Brewster has shown, of a singular and pleasing variation. If, in the position shown in the figure, when the ray AC is not reflected, and the body from which it proceeds therefore not seen to an eye at E, we breathe gently upon the glass E, the ray CE will be, as it were, revived, and the candle or body from which RA proceeds will become instantly visible. The reason of this is, that a thin film of water is deposited upon the glass by breathing; and as water polarises light at an angle inclined at an angle of 52° 45' to AC, in of 52° 45', the glass C should have been order to be incapable of reflecting the polarised ray; but as it is inclined at an angle of 56°, it has the power of reflecting a portion of A C. If we now place the glass C at an angle of 52° 45' to AC, then it will reflect a portion of the polarised ray to the eye at E; but if we breathe upon the glass C, the reflected light will disappear, because the reflecting surface is now water, and is placed at an angle of 52° 45', the polarising angle for water. If we, therefore, place beside each other two sets of reflectors, arranged as above described, we may, by breathing upon two adiacent plates of glass, exhibit the If the original beam of light RA has that the reflected pencil C E does not considerable intensity, it will be observed wholly vanish, and that the remaining has a great dispersive power, or diamond portion is coloured. This effect is finely seen when we use oil of cassia, which or chromate of lead. With glass it is of it is a fine blue; these colours varying a purple colour, and with oil of cassia according as the angle of reflexion is This unpolarised light Dr. Brewster above or below the polarising angle. ascribed to the circumstance, that as the different rays had in every substance different indices of refraction, they would have also by the general law dif Malus, M. Arago, M. Biot, and Dr. Brewster. This is a mean of four observations by M + Mean of six observations. This and other crystals with powerful double refraction give different polarising angies in different azimuths. |