MINERALOGY 355 whilst on the lower end it is bounded by the faces P of the primary alone. It has been found that all hemimorphic crystals become electri cally polar when heated, that is, exhibit opposite kinds of electricity at opposite ends of the crystal. The subject will be more fully con sidered under the electricity of minerals. The hemihedral forms of the cubic system are the following : 1. The tetrahedron (fig. 47), hemihedral of the octahedron, is bounded by four equilateral triangles. It has six equal edges with faces meeting at 70 32 , and four trigonal angles. The principal axes join the middle points of each two opposite edges. Examples : fahlore, boracite, and helvine. Fig. 47. Fig. 48. 2. The trigonal dodecahedrons (fig. 48), hemihedral of the icosi tetrahedron, are bounded by twelve isosceles triangles, and vary in general form from the tetrahedron to the cube. There are six longer edges corresponding to those of the inscribed tetrahedron, and twelve shorter, placed three and three over each of its faces, and four hexagonal and four trigonal angles. Example : tetrahedrite. 3. The deltoid dodecahedrons (fig. 49), hemihedral of the triakis- octahedron, are bounded by twelve deltoids, and vary in general form from the tetrahedron on the one hand to the rhombic dodecahedron on the other. They have twelve longer edges lying in pairs over the edges of the inscribed tetrahedron, and twelve shorter edges, three and three over each of its faces. There are six tetragonal (rhombic), four acute trigonal, and four obtuse trigonal angles. The principal axes join, two and two, opposite rhombic angles. Example : tetrahedrite. Fig. 49. Fig. 50. 4. The hexakistetrahedrons (fig. 50), hemihedral of the hexakis octahedron, are bounded by twenty-four scalene triangles, and most commonly have their faces grouped in four systems of six each. The edges are twelve shorter and twelve longer, lying in groups of three over each face of the inscribed tetrahedron, and twelve in termediate in pairs over its edges. The angles are six rhombic, joined in pairs by the principal axes, and four acuter and four obtuser hexagonal angles. Example : diamond. In these forms, often named "tetrahedral," the faces are oblique to each other. Their derivation and signs are as follows. The tetra hedron arises when four alternate faces of the octahedron, two opposite above and two intermediate below, are enlarged so as to obliterate the other four ; and its sign is hence . But, as either if four faces may be thus enlarged or obliterated, two tetrahedrons can be formed, similar in all respects except in position, and together making up the octahedron. These are distinguished by the signs + and - , added to the above symbol, but only the latter in general expressed, thus - - . In all hemihedric systems two forms simi larly related occur, which may thus be named complementary forms. The trigonal dodecahedron is derived from the icositetra hedron by the expansion of the alternate trigonal groups of faces. . mOm x , -xv- 202 rn. Its sign is - , the most common vanety being . Ine LI t deltoid dodecahedron is in like manner the result of the increase of the alternate trigonal groups of faces of the triakisoctahedron, and its sign is ^ . Lastly, the hexakistetrahedron arises in the m development of alternate hexagonal groups of faces in the hexa kisoctahedron, and its sign is Two semitesseral forms with parallel faces occur. (1) The pentagonal dodecahedrons (fig. 51), bounded by twelve symmetrical pentagons, vary in general aspect between the cube and the rhombic dodecahedron. They have six regular (and in general longer) edges, lying over the faces of the inscribed cube, and twenty- Fig. 51. Fig. 52. four, generally shorter (seldom longer), edges, usually lying in pairs over its edges. The solid angles are eight of three equal interfacial angles, and twelve of three interfacial angles, of which only two are equal. Each principal axis unites two opposite regular edges. This form is derived from the tetrakishexahedron, and its sign is . It is found frequently in iron pyrites and cobaltine. a (2) The dyakisdodecahedron (fig. 52), bounded by twenty-four trapezoids with two sides equal, has twelve short, twelve long, and twenty-four intermediate edges. The angles are six equiangular rhombic, united in pairs by the principal axes, eight trigonal, and twenty-four irregular tetragonal angles. It is derived from the hexakisoctahedron, and its sign is j??l2?n ( the brackets being used to distinguish it from the hexakistetrahedron, also derived from the same primary form. It occurs in iron pyrites and cobaltine. The two other semitesseral forms, the pentagonal Fig. 53. Fig. 54. dodecahedron (fig. 53), and the pentagonal icositetrahedron (fig. 54), both bounded by irregular pentagons, have not yet been observed in nature. Combinations. The above-mentioned forms of the tes- Combina- seral system (and this is true also of the five other systems tions. of crystallization) not only occur singly, but often two, three, or more occur united in the same crystal, forming what are named combinations. In this case it is evident that no one of the individual forms can be complete, because the faces of one form must interfere with, by diminishing, the faces of other forms. A combination therefore implies that the faces of one form shall appear symmetrically disposed between the faces of other forms, and consequently take the place of certain of their edges and angles. These edges and angles are thus, as it were, cut off, and a greater number of new ones produced in their place, which properly belong neither to the one form nor the other, but are angles of combina tion. These new faces are hence termed modifications, and the original or primary or simple form is said to be modified. Usually one form predominates more than the others, or has more influence on the general aspect of the crystal, and hence is distinguished as the predominant form, the others being considered subordinate. The sign of the combination consists of those of its constituent forms, written in the order of their influence or importance in the combination, with a point between each pair. It will be readily seen that such combinations may be exceedingly
numerous, or rather infinite ; and only a few of the more common