different for different species of rays, which circumstance must affect the deviations they experience. Hence, in order to define rigorously all the phenomena, it is necessary to mention the particular species of rays with which they are supposed to be observed. Hereafter, when we use no limitation in this respect, it is to be understood as in the case of ordinary refraction, that regard is had to the mean rays of the spectrum, that is, to the green or yellow; or rather that we are speaking of a phenomenon in which the effect of dispersion is insensible or inconsiderable.

119. In general, for the same species of simple rays, the values of the coefficient k, or n'a-n2, are different in crystals of different forms and different natures; whence it follows that double refraction exerted by these bodies is of very unequal energies. We find also that substances of nearly the same composition, act in this respect very differently. Common sulphate of lime, for example, has a very weak double refraction; whereas the anhydrous sulphate of lime has a very strong one. It is true that the primitive forms of these two substances are very different. But the pure carbonate of lime, and the magnesian carbonate of lime, both of which have the primitive form of rhomboids, and rhomboids whose angles are so nearly equal that the difference, though real and measurable, has been doubted, have also double refractions sensibly unequal. For the mean value of k, which is 0,543 for the first, according to my experiments, varies for the second from 0,581 to 0,591; and it is probable that this last variation is owing to a small difference of composition. Some beryls which do not differ externally, except in colour, give also values for k very sensibly unequal; and several specimens have even exhibited phenomena which seemed to indicate the existence of two axes. Hence we may conclude generally that when two crystallized substances differ in their composition or in their primitive form, they differ also in their property of double refraction; and reciprocally, that a difference as to double refraction supposes always a difference in composition or structure, which renders this kind of phenomena very useful in characterizing minerals.

120. The coefficient k, not only experiences variations in its value, in passing from one substance to another, but also in its sign. It is positive for some substances and negative for others.

This circumstance leads us to apply to crystals of two axes the distinction already made of two sorts of double refraction, one attractive, and the other repulsive.

Dr Brewster having succeeded in distinguishing crystals of one axis from those of two, by means of certain phenomena of colour which they present and which will be hereafter explained, has subjected to this test a great number of crystallized substances; and he has found, in general, that all those which have only one axis, have such primitive forms, that the faces which surround the axis, are disposed about it in a similar manner. These forms are the rhomboid, the regular hexaedral prism, the isoceles octaedral prism with a square base, and the right prism with a square base. All the other forms have two axes, or do not exert a double refraction. M. Sorret, a skilful mineralogist of Geneva, who has made very extensive researches into the subject of the position of the axes, in double refraction, with respect to the faces of the primitive forms, has found that the plane which contains the axes, is always situated in a symmetrical manner in the solid adopted as the primitive form, or in one of its crystallographical derivatives; and the axes are so situated in this plane, as to make equal angles with the faces of the solid. These characteristics of symmetry leave ordinarily very little that is indeterminate as to the possible position of the axes in each form; and they are thus useful in limiting and directing the attempts that may be made to discover them.

121. In attending to the phenomena presented by crystals of one axis, we have remarked that they take place with perfect symmetry on the two sides of the plane which we have called the principal section, and which is drawn through the axis of the crystal and through the normal to the face of incidence, by which the rays enter. In crystals of two axes this symmetry does not generally take place; but we find it when the directions. of the two axes are equally inclined to the face of incidence. Conceive then a plane drawn through each of the axes of the crystal, and through the normal to this face; conceive also a third plane, drawn through the same normal, bisecting the angle formed by the two former planes. All the effects of the crystal are symmetrical on the two sides of this intermediate plane; and the two refractions take place without lateral deviation, as in the principal section of crystals of one axis. MoreOpt.

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over, the preceding definition embraces the construction which we have assigned to this section. For, if we suppose the two axes to approach each other until they form one line, these three planes will become one, which will be drawn through the single axis and through the normal to the face we are considering, which is precisely the construction of the principal section for crystals of one axis, according to the definition we have given.

Reflection at the second Surface of Crystals.

122. THE theory we have been explaining, is not confined to rays refracted extraordinarily by crystals. It applies also to those which are reflected inward at their second surface. But before entering into a detail of the consequences thus indicated, it is necessary to establish by experiment the principal characteristics of this kind of phenomena.

When a ray of light falls upon the first surface of a crystal, coming through a vacuum or any medium not crystallized, it is partially reflected in a single beam, in such a manner that the angle of reflection, reckoned from the normal, is equal to the angle of incidence. The attractive or repulsive force which emanates from the axes of the crystal, appears to have no influence on this phenomenon; for we may turn the crystal upon its plane in all possible directions without altering the intensity or the direction of the reflected ray. But it is not so with respect to the interior reflection which takes place at the second surface of the crystal. Each ray reflected at this surface, is generally divided into two portions which return into the crystal, the one experiencing the ordinary refraction, the other, the extraordinary, these terms being used in the sense already explained.

123. In order to comprehend the cause of this division, it must be understood that rays, refracted either ordinarily or extraordinarily, when they have penetrated into the interior of the crystal to a sensible depth, acquire such a mode of arrangement of their particles, that they can no longer be divided during their course through this crystal; and experiment proves that they would no longer be divided if they should traverse a second crystal contiguous to the first, and having its principal section

directed in the prolongation of the first. This particular mode of arrangement constitutes what Malus has termed the polarisation of light. Now when the particles which compose the same ray, refracted ordinarily or extraordinarily, approach the second surface of a crystal, at a distance sufficiently small to experience the influence of the reflecting forces proceeding from it, it happens, in general, that a certain number of particles are turned by these forces into directions different from those derived from refraction; so that in returning into the crystal by the effect of a total or partial reflection, they become anew susceptible of being divided into two refracted portions, ordinary and extraordinary. I say, in general, for there are certain particular positions, in which the reflecting forces do not alter the arrangement originally given by refraction to the luminous particles; and then the ray is reflected without being divided, or even escapes reflection entirely. It is sufficient for the present, to observe that the forces in question only influence the intensity of the reflected portion, and not the direction derived from reflection. A ray which is reflected single, or which emerges from the crystal without being reflected, would suffer double reflection, if the particles which compose it were otherwise disposed; as may be verified by experiment. Accordingly the direction of the reflection is the first thing to be determined.

124. This determination is easily made with respect to crystals of one axis, in which one of the velocities is constant. It is sufficient to rely on this fact, that the reflected ray must be affected on re-entering the crystal, as a ray would be, which came from without, and whose particles had not originally received any particular disposition. Now in case the crystal has only one axis, the direction of the return is determined by the known reflection of the ordinary ray having a constant velocity. Let l' be the point of Fig. 86 interior incidence, and O'I' the incident ray. If it has suffered ordinary refraction, construct the ordinary reflected ray I'O", which makes the angle of reflection equal to the angle of incidence, on the other side of the normal I'N'; then calculate by the theoretical formulas, the direction of the extraordinary ray I'E", which corresponds to it at its departure from the point of reflection I', that is, which has proceeded from the same exterior incident ray; we shall thus have the two reflected rays which result from the division of the incident ray O'l', after reflection.

If, on the contrary, this ray is extraordinary, draw it to the point Fig. 37. of incidence I'; then calculate by the theoretical formulas the

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ordinary ray I'O', which corresponds to it on the same side of the normal; and having obtained this, proceed in the calculation as before; thus we shall have the two reflected rays I'Q", I'E", into which the given ray is divided. In general, this is the rule; an ordinary and an extraordinary ray which accompany each other in their interior incidence, accompany each other also after reflection.

In the case where a crystal has two axes, the direction of the interior reflection can no longer be assigned, on the ground of symmetry alone, for either of the two rays; since, their velocities being variable, neither of them makes, in general, the angle of reflection equal to the angle of incidence. But in this case, the interior incident ray being given, whether ordinary or extraordinary, we calculate the emergent ray derived from it, then transfer this emergent ray to the other side of the normal, giving it an equal incidence, and calculate the refracted rays which it would produce on re-entering the crystal in this direction. These ought to be the same with the interior reflected rays; for it is every way manifest that the interior reflection takes place at a distance from the surface much smaller than that, to which the forces, dividing the ray, extend; so that the ray when reflected, ought, in its commencement, to be directed as it would be if the medium were not crystallized. It is only when it penetrates further into the crystal, that it is divided anew as if it came from without. Thus it appears that its primitive direction, after reflection, is to be deduced from the inversion of its emergence, as in the case of substances not crystallized. We see that this method comprehends, as a particular case, that just pointed out for crystals of one axis. But notwithstanding this analogy, I do not propose it without diffidence, and it is with the hope that experiments may be made to prove or disprove its correctness.

125. These are the general laws of reflection in the interior of crystals, both when a portion only of the luminous particles suffers interior reflection, and the rest exterior refraction; and when, the interior attraction being more powerful, all the incident particles are drawn inward by the forces which produce the refrac

tion.

Here, as in ordinary refraction, the incidence at which this