CUBIC SYSTEM. 13 tion of irfild and. irpqr is essentially different from a combination of TvhJcl and irrqp. 42. If the distance between the poles of any two faces of either of the forms Mo, hick, be given, and we express the cosine of the given distance in terms of the indices of the faces, we ob tain an equation from which the indices may be found. ' If the distances between the pole of any face of the form TM, and the poles of each of two other faces of the same form, be given, and we express the cosines of the given distances in terms of the indices of the faces, we obtain two equations from which the indices may be found. 43. In the following description of the different simple forms of the cubic system, the letter placed upon the edge in which any two faces intersect, will be used to denote the angle between normals to the two faces. The edges at which equal angles are made- by the intersecting faces are denoted by the same letter. The arrangement of the poles is shown in figs. 8, 9, 28. The number of faces is given in (33). 44. The form 100 (fig. ll) has six faces, and is called a cube, its faces being parallel to those of the cube of geometry. e = 00°. 45. Tho form ill (fig. 12) has eight faces, and is called an octahedron, its faces being parallel to those of a regular octa hedron. D = 70° 3l'-7. 4G. The form mi (fig. 13) is a regular tetrahedron. t = 109° 28 ,- 3. TIG. 11. EIG. 12. FIG. 13. T in in in \ 010 100 a combination of the forms 100 and ill (fig. 14), he faces of one form truncate the solid angles of the other. 0 normal to any face o, of the form ill, makes an angle of 64 44 with a normal to any adjacent face a of the form 100. In a combination of the forms loo and mi (fig. 16), the faces oi loo truncate the edges of mi.