Having thus evolved a scientific scheme of reference axes for the faces of a crystal, it is necessary in all the systems other than the cubic and trigonal, in which the axes are of equal lengths, to devise a mode of arriving at the relative lengths of the axes; for on this depends the mode of determining the positions of the various faces, other than the three parallel pairs (or four in the case of the hexagonal system) chosen as the axial planes. This is very simply done by choosing a fourth important face inclined to all three axes, when one of this character is developed, as very frequently happens, as the determinative face or plane fixing the unit lengths of the axes. When no such face is present on the crystal, two others can usually be found, each of which is inclined to two different axes, so that between them all three axial lengths are determined. The faces of the octahedron, of the primary tetragonal pyramid and the primary rhombic pyramid, and of the corresponding forms of the other systems, are such determinative planes, fixing the lengths of the axes. This fact will be clear from the typical illustration of the most general of these primary or “parametral” forms, the triclinic equivalent of the octahedron, given in Fig. 46, the faces being obviously obtained by joining the points marking unit lengths of the three axes. [Illustration: FIG. 46.—Triclinic Equivalent of the Octahedron. ] Having thus settled the directions of the crystallographic axes and their lengths, it is the next step which reveals the remarkable law to which reference was made at the opening of this chapter. For we find that all other faces on the crystal, however complicated and rich in faces it may be, cut off lengths from the axes which are represented by low whole numbers, that is, either 2, 3, 4, or possibly 5, and very rarely more than 6 unit lengths. By far the greater number of faces do not cut off more than three unit lengths from any axis. Prof. Miller of Cambridge, in the year 1839, gave us a most valuable mode of labelling and distinguishing the various faces by a symbol involving these three values, employed, however, not directly but in an indirect yet very simple manner. If _m_, _n_, _r_ be the three numbers expressing the intercepts cut off by a face on the three axes, _a_, _b_, _c_ respectively, and if the Millerian index numbers be represented by _h_, _k_, _l_, then— _m_ = _a_/_h_, _n_ = _b_/_k_, _r_ = _c_/_l_, or, _h_ = _a_/_m_, _k_ = _b_/_n_, _l_ = _c_/_r_. Each figure or “index” of the Millerian symbol is thus inversely proportional to the length of the intercept on the axis concerned. The intercepts themselves are used as symbols in another mode of labelling crystal faces, suggested by Weiss, but this method proves too cumbersome in practice.