130 TERMINOLOGY. . 123. 



cent solid angle in a plane which touches the edge of the 

 hexahedron, and joins these two solid angles. Hence, of the 

 forty-eight faces, four and four will coincide, and the solid 

 obtained will be limited by twelve faces. The prismatic axes 

 pass through the centres of the edges of the hexahedron, 

 and consequently also through the centres of the faces ow 

 obtained. Thus the faces of the derived form become 

 perpendicular to the prismatic axes, and are themselves 

 the faces of the monogrammic Tetragonal-dodecahedron 

 (. G3. 3.). 



. 1&3. THE OCTAHEDRAL TRIGONAL-ICOSITETRA- 

 HEDRON. 



In the third position the moveable plane is the 

 face of an octahedral Trigo?ial-icositetrahedron 

 (. 72.)- 



In this case there are no pairs of faces from one solid 

 angle, coinciding with pairs from another ; but of the six 

 faces contiguous to one and the same solid angle, two and 

 two faces will coincide. Hence the number of faces of this 

 form is twenty -four. Each of these faces is intersected by 

 the two other faces contiguous to the same, and by one 

 contiguous to the adjacent solid angle ; with the last of 

 these faces it produces an edge in the direction of the 

 greater diagonal of the dodecahedron, or in the direction of 

 the edge of the octahedron. Its faces therefore assume the 

 figure of isosceles triangles ; the rhombohedral solid angles 

 of the fonn consist of three faces, and they are monogram, 

 mic ; the pyramidal solid angles are formed by eight faces, 

 and they are digrammic ; the form itself is an octahedral 

 Trigonal-icositetrahedron (. 72. 1. 2.). 



The different varieties of octahedral trigonal-icositetra- 

 hedrons may be considered as forms intermediate between 

 the dodecahedron and the octahedron. If the angle 

 measuring the inclination of the moveable plane to the axis 

 AX, Fig, 36., becomes greater than CAX, the face of the 



