In quartz, the phenomena manifested by polarised light passing nearly along the axis appeared peculiar, resembling a combination of those belonging to an ordinary uniaxal crystal with those belonging to syrup of sugar ; but the same have recently been discovered in some other uniaxal crystals.
In investigating the colours exhibited by crystalline plates, Sir David Brewster was led to the discovery that in many—in fact, in the greater number—of crystals, there exist not one, but two, directions about which coloured rings are seen in polarised light. Such crystals are called biaxal, and the two directions in question are called the optic axes. Sometimes, as in sulphate of lime, sugar, &c., the optic axes are widely separated ; and in this case, if a crystal cut perpendicularly to either axis be introduced into the previously dark field of a polarising apparatus, a series of nearly circular coloured rings is seen interrupted by a single dark brush, in place of the pair of brushes forming a cross which are seen in uniaxal crystals. If the crystal be turned round an axis coinciding with the optic axis, the black brush turns round in a contrary direction at an equal rate relatively to space, or a double rate relatively to the crystal, whereas in a uniaxal crystal similarly treated the black cross remains stationary. The rings are ordinarily nearly equidistant, whereas in a uniaxal crystal they obey the law of increase of Nev‘ ton's rings, the squares'of their radii increasing in arithmetical progression. In other crystals, like nitre, the optic axes are near each other, and may be seen together, especially if the plate be cut in a direction perpendicular to their middle line. In this case, on intro ducing the crystal into the dark field a set of coloured curves are seen resembling lernniscatcs, having the optic axes for poles ; and each optic axis is traversed by a dark hyperbolic brush; and at certain azimuths of the crystal, 90° apart, the two brushes unite and form a cross, one arm of which passes through the optic axes.
Sir David Brewster also discovered the relation between the optical characters of crystals and their crystallographic forms. It was found that the system of rectangular axes formed by lines bisecting the acute and obtuse angle between the optic axes, and a line perpendicular to their plane, were intimately connected with the crystalline form, so that whenever there existed a plane of crystalline symmetry, two of these axes lay in it. It is found that crystals of the cubic system are singly refracting, those of the rhombohedral and pyramidal systems uniaxal, and those of the prismatic, oblique, and anorthic systems biaxal. No account is here taken of properties like those of syrup of
sugar, nor of what Biot has termed lamellar polarisation.
The explanation of these beautiful coloured rings and curves follows at once from combining the observed laws of double refraction, including therein the polarisation of the refracted rays, with the laws of the interference of polarised light. The latter, as we have seen, admit of a perfectly simple explanation on the hypothesis of transverse vibrations; it remains to be seen what account that hypothesis can give of the former.
For some time after the discovery of biaxal crystals, it was supposed that one of the refracted rays followed the law of ordinary refraction, while the other followed some unknown Law more complicated than the Huygenian. It was theory which first pointed out to Fresnel, that neither ray followed the ordinary law, an anticipation which he found to be confirmed by experiment.
Our limits would not permit us to enter into the theory of double refraction as given by Fresnel ; we shall content ourselves with a brief notice of the principles of the investigation, and a statement of the results to which it conducted him.
In any theory of double refraction, there are two kinds of laws which have to be accounted for ; those which regulate the velocity of propagation, and those which regulate the state of polarisation. For the two are evidently so bound up together, that any true theory ought to explain both at the same time.
With regard to the former, if we only knew the form of the wave surface, all the rest would follow from Huygens's construction. To determine, however, the propagation of a disturbance spreading out ou all sides, is a problem presenting many difficulties, some of which may be evaded by the following consideration. Imagine an infinite number of plane waves, the effect of which, severally, is infinitely small, to pass initially through the point from which the disturbance is supposed to emanate. . These will serve to represent initially the disturbance in the neighbourhood of that point, and their effect will elsewhere be insensible. As the time progresAes, they will travel along with the velocities belonging to plane waves in their respective directions, and their effect will be insensible except along the surface of their ultimate intersections, which, therefore, will be the wave surface required. Hence, everything is reduced to the determination of the mode of propagation of a plane wave in an arbitrary direction.