total reflection begins in each crystal, depends upon its greater or less refracting power, and that of the exterior medium; but we cannot in the same way calculate its limits theoretically, because we are ignorant how the attractive or repulsive force, which proceeds from the axes of the crystal, varies near its surface. It is, therefore, necessary to recur to experiment, and to determine the commencement of total reflection by the impossibility of obtaining an emergent ray. I have given a detailed account of this calculation* for crystals of one axis, and have developed the remarkable consequences which result from it, in relation to the variations which the forces, proceeding from the axes, undergo near the exterior surface of crystals. Crystals of two axes offer similar considerations. One of the best means of ascertaining the nature of these forces, is to place the surfaces of the crystal in contact with transparent media of a greater refracting power than themselves, for instance, with phosphorus in a state of fusion, and to note the limits, as well as all the other particulars of interior reflection in media thus terminated. Perhaps we may be able to measure by this process, the double reflection of opaque bodies, as Dr Wollaston has determined their simple refraction. This investigation requires some theoretical principles which will be detailed in the following section.

Passage of Light through several Contiguous Bodies possessing the Power of Double Refraction.

126. We have supposed the preceding experiments to be made in a vacuum, or in the air, whose proper action upon light is so feeble as to have no sensible effect. We are now to inquire what takes place when the rays which penetrate a doubly refracting crystal, instead of entering from the air, pass out of a medium possessing the power of double or simple refraction.

Let us begin with this latter case, which is less complicated, and for still greater simplicity, let us suppose also that the crystal, into which the ray is to pass, has only one axis, in which

* Traité de Physique.

Fig. 88. case the ordinary velocity will be constant.

Let AB represent the common surface of the medium and the crystal, and SI the incident ray. We calculate, or construct by the theoretical formulas, the ordinary refracted ray IO, according to the common ratios of refraction of the two substances, and precisely as if the second medium were not crystallized. Then, knowing 10, we deduce from the formulas the extraordinary ray, IE, which accompanies it, and which depends solely upon the position of 10, with respect to the axes of the crystal; the two rays, IO, IE, will be those into which the incident ray SI, will be resolved.

127. We pass now to the case in which the first medium is itself crystallized, still supposing it to have but one axis. The given incident ray SI, may be subjected, in this medium, either to ordinary or extraordinary refraction. If the first take place, we may proceed as above, employing the ratio of ordinary refraction of the first medium, and performing the calculation, as if were not crystallized. But if the given ray SI, is itself subjected to extraordinary refraction, we begin by calculating in the first medium, by the formulas, the direction of the ordinary ray Fig. 89. S'I, which accompanies it. When this ray is known, we make use of it, as above, to calculate the two rays 10, IE, which are derived from it in the second crystal. These will be the directions sought.

It will be seen from this analysis, that, except the case of a too great attractive force in the first medium, these constructions. give always, in the second medium, two rays, the one ordinary, the other extraordinary; whereas experiment seems to contradict the constancy of this result, at least when the first medium is crystallized, and endued with the power of double refraction; for then the subdivision of the rays which come from it, as well as the kind of refraction, ordinary or extraordinary, which they undergo in the second crystal, depends upon the position of this crystal with respect to the first. In order to fix these ratios with the more clearness, let us confine ourselves to crystals of one axis. In this case, if the principal sections of the two crystals are parallel, each ray from the first crystal, whether ordinary or extraordinary, remains single in passing into the second, and undergoes there, the same kind of refraction as it did in the first. If the principal sections, instead of being parallel are at right angles to each other, each ray coming from the first crys

tal, still remains single; but it changes its refraction, from ordinary in the first, to extraordinary in the second, and vice versa. Between these two limits of position, each ray, whether ordinary or extraordinary, coming from the first crystal, is divided into two when it enters the second, and these portions obey the laws indicated by the preceding constructions. But the intensity of each portion depends still upon the angle made by the two principal sections, increasing or diminishing with this angle, according as the motion of the principal sections causes the portion to diverge from or approach to the limit where it ought to disappear. Hence we must conclude, that the formation or non-formation of two portions in the second crystal, depends upon the physical modifications which the particles may have acquired in the first crystal, modifications which render them better fitted to undergo the one or the other refraction in the second, according to the directions of their faces, with respect to its axes; which does not prevent the theory from indicating with exactness, the directions of translation, which these particles would take, if their physical state permitted them to be shared between the two refractions. Similar cases have already been presented by refraction at the second surface of crystals; for, in fact, a ray reflected inward at the second surface of a crystal, experiences the same influence, as if it emerged entirely from the crystal to enter another or return into the same.

128. Hitherto we have supposed that the contiguous media were composed of crystals of only one axis. This supposition, by rendering one of the velocities constant, allowed us to calculate its propagation by Descartes' law of the sines being extended to contiguous media of different refracting powers. This expedient cannot be employed when one of the contiguous media is a crystal of two axes; since then its two velocities are generally variable; but we can supply the defect by considering that the rays, at the moment when they traverse the surface of contact, are without the sphere of action of the forces by which the double refraction is produced; so that they are really in the same condition they would be subjected to, if they had passed with their preceding velocity, acquired from an uncrystallized body, into another likewise uncrystallized, but of a different refraction. In this case, their definitive course would be still the same, if, instead of introducing them thus directly into the second

medium, we were to separate this from the first by a small void space parallel to their common surface. Consequently this same supposition applied to the crystals in contact, would give the definitive direction of the rays which would be transmitted from one to the other, in the case where they were contiguous. Thus, when a ray subjected to any law of velocity whatever, is given in the first medium, we can calculate the emergent ray which would result from it, if it were to emerge into a vacuum; which would be done by means of the principle of least action applied to the general expressions of the velocities. Then, by the aid of the same principle, we calculate the direction which this emergent ray would follow, if it penetrated, with the same continued incidence, into the second medium; and this will be the direction which it will actually take when it passes directly from the first medium without any intervening space. Yet, notwithstanding the clearness of the analogies on which the rule is founded, it would be well to institute experiments for the purpose of verifying it.

91.

Construction of Double Image Micrometers.

129. ROCHON employed the double refraction of crystals for measuring small angles, in a manner too useful in astronomy and the physical sciences, not to find a place here. Besides, the same apparatus affords a most simple means of ascertaining whether the double refraction exerted by a crystal, is attractive or repulsive.

Fig. 90, Suppose two prisms A, B, formed of the same crystal, of one axis, and cut in such a manner, that, in the first A, the exterior face AB is perpendicular to the axis AA', of the crystal; while, in the second B, this axis is the common intersection of the two faces A'B, A'B'. Suppose further, that the two prisms are equal to each other as to the dimensions of their parts and the size of their angles. Let them be brought into perfect contact with each other, their refracting angles being disposed in such a manner, that their union may form a plate whose exterior faces shall be parallel, as represented in the figures. Then suppose an inci

dent ray LI, directed perpendicularly to the surface of the first prism, and let us see what will take place.

Throughout the interior of the first prism, the ray will move in a straight line without being divided. Its course will not be broken at the first surface, because it is perpendicular to it; and it will not be divided in the first prism, because it is parallel to the axis of the crystal, and the force, whether attractive or repulsive, which emanates from this axis, can have no effect upon it.

When the luminous ray has arrived at I', the common surface of the two contiguous prisms, it will be divided into two rays, on its entrance into the second prism. The ordinary ray will not deviate, because it passes from one medium into another of the same refracting power; but will continue its course in a straight line, and emerge perpendicularly from the second face A'B'. It will be seen, therefore, in the prolongation of its primitive direction. But this will not be the case with respect to the extraordinary ray; for this, on entering the second prism, will experience, together with the ordinary refracting action, that of the attractive or repulsive force which emanates from the axis, and will undergo a new refraction resulting from it; this will take place, like the other, according to the simple law of Descartes, from the manner in which the prisms are cut. If the crystal is repulsive, as in figure 90, this second refraction will be less than the first, and the extraordinary ray II", in traversing the second prism, will diverge from N'N", the normal to the common surface, more than the ordinary ray I'O; and it will consequently, after its emergence into the air, be carried more towards the vertex of this prism. The reverse will take place when the crystal is attractive, as in figure 91; the extraordinary refraction in the second prism being stronger than the ordinary refraction in the first, the extraordinary ray I'I" will diverge from the normal less than the ordinary ray, and will consequently, after its emergence into the air, be carried more towards the base of this prism.

130. We have supposed the two prisms immediately contiguous; but as this perfect contact can never be realized in practice, we cement the two surfaces together, by means of a stratum of inspissated oil of turpentine, or mastic in tears, substances which are transparent, and whose refracting power is nearly Opt.

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