FIG. 43.—Hexagonal Prism of the First Order and its Four Axes. ] [Illustration: FIG. 44.—Hexagonal Prism of the Second Order. ] [Illustration: FIG. 45.—The Rhombohedron and its Three Equal Axes. ] The cases of the hexagonal and trigonal systems are somewhat special. The hexagonal has four such axes, as represented in Fig. 43, the lines of intersection of the faces of the hexagonal prism closed by a pair of perpendicular terminal planes, namely, one vertical axis parallel to the vertical edges, and three horizontal axes inclined at 120° to each other, and parallel to the pair of basal plane faces, equal to each other in length, but different from the length of the vertical axis. The hexagonal axial-plane prism shown in Fig. 43 is known as one of the first order. The hexagonal prism corresponding to the tetragonal one of Fig. 39, in which the axes emerge in the centres of the faces, is said to be of the second order, and is shown in Fig. 44. The trigonal system of crystals is best described with reference to three equal but not rectangular axes, parallel to the faces of the rhombohedron, one of the principal forms of the system, so well seen in Iceland spar, and illustrated in Fig. 45. The rhombohedron may be regarded as a cube resting on one of its corners (solid angles), with the diagonal line joining this to the opposite corner vertical, and the cube then deformed by flattening or elongating it along the direction of this diagonal. The edges meeting at the ends of this vertical diagonal are then the directions of the three trigonal crystallographic axes. In this last illustration the vertical direction of the altered diagonal is that of the trigonal axis of symmetry, for the rhombohedron is brought into apparent coincidence with itself again if rotated for 120° round this direction. But although a symmetry axis, this is not a crystallographic axis of reference. It is not shown in Fig. 45, therefore, but is given in Fig. 19. On the other hand, the singular vertical axis of reference of the tetragonal and hexagonal systems is identical with the tetragonal or hexagonal axis of symmetry of these systems, and the three crystallographic axes of reference of the cube are identical with the three tetragonal axes of symmetry of the cubic system. In the rhombic system also, the three rectangular axes of reference are identical with the three digonal axes of symmetry, and in the monoclinic system the one axis of reference which is normal to the plane of the two inclined axes is the unique digonal axis of symmetry of that system.