It has already been shown that crystals are optically divisible into two classes characterised respectively by single and by double refraction. Singly refractive crystals belong exclusively to the system of highest symmetry, the cubic. They afford obviously only one index of refraction, which is generally symbolised by the Greek letter μ, the value of this constant being the same for all directions throughout the crystal. Crystals of the other six systems of symmetry are all doubly refractive. Those of the trigonal, tetragonal, and hexagonal systems have been shown in the last chapter to possess two refractive indices, a maximum and a minimum, one represented by ε corresponding to light vibrating parallel to the singular axis of the system, the trigonal, tetragonal, or hexagonal axis of symmetry, and another signified by ω corresponding to light vibrations perpendicular to that axis. For the properties are identical in all directions around this axis, which is thus the optic axis as well as the predominating crystallographic one. Such crystals are consequently known as “uniaxial.” When ε is the larger refractive index the crystal is positive, while if ω be the maximum the crystal is said to be negative. It has been shown in the last chapter that quartz belongs to the positive category, while calcite is negative. Along the one direction of the optic axis these uniaxial crystals behave like singly refractive crystals do in all directions. Crystals of the rhombic, monoclinic, and triclinic systems of symmetry have also a minimum refractive index, symbolised by α, and a maximum index indicated by γ, corresponding to light vibrating parallel to two directions at right angles to each other; the third direction perpendicular to both these and normal to their plane does not afford an index of refraction equal to either of these, however, as in the case of a uniaxial crystal, but one of an intermediate value, for which the second letter β of the Greek alphabet is reserved. Whether this value β is nearer to the minimum α or to the maximum γ determines the conventional optical sign of the crystal, whether positive or negative. In the case of the rhombic system the three rectangular directions in question are identical with the three rectangular crystallographic axes. In the monoclinic system the single symmetry axis normal to the unique plane of symmetry is identical in direction with either the α, β, or γ optical direction, but in the triclinic system there are no coincidences between the crystal axes and those of the optical ellipsoid. Along none of these axial directions of the optical ellipsoid which can be imagined to express graphically the refractive index—an ellipsoid known as the optical “indicatrix,” and which has been shown by Fletcher to be a more convenient mode of expressing the optical characters of a crystal than the vibration-velocity ellipsoid of Fresnel—do the optical properties