. 133. OF THE CONNEXION OF FORMS. 1-il 



The first mode yields the tetrahedral trigonal-icositetra- 

 hedron. The solid angles of six faces of the original form 

 are likewise to be found in the half, because according to 

 this method all the faces contiguous to the principal points 

 are enlarged. The faces are intersected by two other faces 

 of the same principal point, and by one face contiguous 

 to another, exactly as in the preceding case ; the faces re- 

 main triangles, which are likewise scalene, but not similar 

 to those of the original form. The half therefore is a tri- 

 gonal-icositetrahedron, which owes its tetrahedral aspect to 

 the application of the first process. 



In the designation of these forms, it is necessary to indicate 

 the mode of resolution upon which they depend. The sign of 

 the tetrahedral trigonal-icositetrahedrons in both, the normal 



rp 



and the inverse positions, Figs. 25. 2G., will therefore be _ 



T 1 *! 



and r . The trigrammic tetragonal-icositetrahedron is 



the result of the second mode of resolution. The intersec- 

 tion of the enlarged faces takes place with two faces of the 

 same principal points, and with two faces contiguous to adja- 

 cent subordinate points. In comparing the pyramidal solid 

 angle of the tetracontaoctahedron with the corresponding 

 solid angle in its half, we find, that of the eight faces which 

 constitute the first, alternating Pairs of faces are enlarged, 

 and not the alternating single faces. From this it becomes 

 evident that the angle formed by four faces above the face 

 of the hexahedron cannot consist of equal edges, and that 

 consequently the faces of the tetragonal-icositetrahedron 

 thus formed cannot by a straight line be divided in two isosce- 

 les triangles. Besides the two kinds of edges meeting in the 

 solid angle of four faces, the form contains still another kind 

 of edges, which meet in the rhombohedral solid angle form- 

 ed by three planes ; it is therefore a trigrammic tetragonal- 

 icositetrahedron (. 75. 2.). 



The crystallographic signs of the trigrammic tetragonal 

 icositetrahedrons in both, the normal and the inverse posi- 

 tions, Figs. 27. 28., are and 

 2 ii 2 ii 



