407 
24 poles of any twenty-four-faced trapezohedron will lie in 
each of the 24 arcs GO. 
93. The normals to any face of the three-faced octahedron (fig. 
13, Plate II.), such as 0 1 o 1 0 2 ,must, by symmetry of construction, 
pass through the line d 1 o 1 . Hence in the sphere of projection, 
(figs. 31 and 32, Plate IV.), the 24 poles of the three-faced 
octahedron will lie in each of the arcs DO. 
94. Hence in the same zone G 1 D 1 G 2 D Q 0 6 D 11 C 4 P 3 there will 
be four poles of the cube, G v G 2 , C 6 , 0 4 ; four poles of the 
rhombic dodecahedron, D v D Q , D 1V D 3 ; and eight poles of the 
four-faced cube. 
The same will be true of the two zones G 2 D 5 G 3 and G 3 D 2 G V 
Again in the zone G 3 0 1 D 1 0 4 C 5 0 Q D 11 0 7 G 5 , there will be two 
poles of the cube, 0 3 and 0 5 , two poles of the rhombic dodeca- 
hedron, H 1 and _D n , four of the octahedron, 0 1} 0 4 , 0 6 , and 0 7 , 
four of the three-faced octahedron, and also four of the twenty- 
four-faced trapezohedron, will lie. 
The same will also be true for the five other zones, C 3 0 5 D Q , 
G&D 6 , GjOJ) g, G 2 O x D 2) and G 2 0 4 D 4 . 
95. The 48 poles of any six-faced octahedron will, from the 
symmetry of its construction, occupy similar positions within 
the 48 spherical triangles GDO (figs. 31* and 32*, Plate IV.*).. 
96. In each of the 48 spherical triangles GDO (figs. 31 and 
32, Plate IV.*) is marked a notation for each of the 48 poles 
of the six-faced octahedron in terms of its three indices. The 
order in which the three indices 1, m, and n are written, mark 
the distances at which the face of the six-faced octahedron 
corresponding to the pole marked on the sphere of projection, 
cuts each of three cubical axes taken in the order AG 3 , AG 2 , 
and AG X (fig. 27, Plate IV.). When the index has a negative 
sign placed over it, it signifies that it cuts the axis AG 3 pro- 
duced in the direction AG 5 , AG 2 in AG 4 , or AG L in AG 6 . 
Thus the spherical triangle G 2 D s 0 1 (fig. 31*, Plate IV.*) has 
marked in it the indices m, 1, n, which indicates that the face 
G 2 d b o l of the six-faced octahedron (fig. 3, Plate I.) cuts the axis 
AG 3 produced at the distance m x AG 3 , the axis AG 2 at the point 
0 2 , and the third axis AG X produced, at nx AG V 
Again the indices n 1 m, in the triangle G 2 0 8 D Q (fig. 31*, Plate 
IV.*), show that the face 0 2 o 8 d Q of the six-faced octahedron 
(fig. 3, Plate I.) cuts the axis AG 5 produced at a distance n x AG 53 
the axis AC 2 at the point 0 2) and the axis AG 6 at a distance 
mx AG 6 . 
97. The indices marked on (figs. 31* and 32*, Plate IV.*), 
enable us readily to find the notation for any face of any form 
in Plate II. 
In (fig. 31*, Plate IV.*) the indices ml n in triangle 0 2 o 6 d 6 
