408 
signify that the face of the six-faced octahedron marked G 2 o 5 cl 5 
(fig. 10, Plate II.) cuts the axis AG 3 at a distance m from A, 
the axis AG 2 at 0 2 , and AG Q at a distance n from A. 
The indices m 1 n in the triangle G 2 0 1 D 5 indicate that the face 
of the six-faced octahedron marked G 2 o 1 d 69 fig. 10, Plate II., 
cuts AG 3 at a distance m, AG 2 at G 2 , and AG 3 at a distance n 
from A. 
98. Hence n without any sign over it signifies that the face 
of the six-faced octahedron which it indicates cuts the cubic 
axis G-^AGq in the direction of AG X produced; if it has the 
sign — placed over it, it signifies that the face cuts the axis in 
the direction of AG 6 produced. 
Now if m be infinite, represented by the symbol oo , or 
this signifies that the face cuts the axis neither in the 
direction AG 1 nor AG 6 , and that if produced ever so far in 
either direction it will not cut the axis Gj^AGq, and is there- 
fore parallel to it. Hence when m— oo , m and m indicate 
that the face is parallel to the axis, to AG 3 if m is in the 
first place, to AG 2 if in the second, and to AG X if in the third 
place. 
99. Now, if in the triangle G 2 D 5 0 5 (fig. 31*, Plate IV.*), 
whose indices are m 1 n, we make both m and n infinite, since 
5o and oo are the same, we see that oo 1 oo is the index of 
the face 0 1 0 4l 0 3 0 5 of the cube (fig. 1, Plate I.); also that, 
substituting the sign oo for both m and n, the same notation 
oo 1 oo stands for each of the eight triangles G 2 0 1 D v 
G.fiJ) v C 2 O s B s , GPP,, Opp„ and Cpp,. 
100. When n alone is infinite in the index ml n, ml oo is 
the index of both G 2 o 5 d 5 and 0 2 o 1 ^ 5 , or of the face G 2 o 1 o 5 of 
the four-faced cube (fig. 9, Plate II.). 
101. When n= co , and m— 1, the index min becomes 
1 1 oo , which is the symbol for the four triangles G 2 cl 5 o 5 , 
^ 3 ^ 5 0 5 > an( ^ ^ 3 °i^ 5 > or the face G 2 o 1 G 3 o 5 of the rhombic 
dodecahedron (fig. 12, Plate II.). 
102. When n=m } the index min becomes m 1 m, which is 
that of the two triangles G 2 o 5 d 5 and G 2 d Q o 5 , or of the face 
G 2 d Q ° 5 d 5 of the twenty-four faced trapezohedron (fig. 11, 
Plate II.). 
103. When m=l, the index m 1 n becomes 1 1 n, which is 
that of the two triangles G 2 o 5 d 5 , G 3 o^d 5 , or of the face G 2 o 5 G 3 
of the three-faced octahedron (fig. 13, Plate II.). 
104. When m = 1 and n— 1, the index m 1 n becomes 111, 
which is the same for the six triangles, G 2 o 5 d 5 , G 3 o 5 d 5 , G 3 o 5 d 10J 
G 6 o 5 d 10 , G 6 o 5 d Q , and G 2 o 5 d Q , or of the face G 2 C 3 G Q of the octa- 
hedron (fig. 14, Plate II.). 
