428 
whose angular elements are |) 3 and X 3 , or whose indices are 
1 mn. 
131. To construct a map of all the forms of the octahedral 
system on a face of an octahedron comprised in an octant of 
the sphere of projection. 
(Fig. 43*, Plate IV.*) Describe any equilateral triangle 
0 , 0 , 0 ,. 
Bisect G X G 2 in d v 
0,0, in cl 2 , and C 2 C 3 in d 5 . 
Then G 3 is the pole of 1 oo oo, C 2 of oo 1 oo, and G ± of co co 1, 
three poles of the cube. 
d x is the pole of oo 1 1, d 2 of 1 oo 1, and d 5 of 1 1 oo, three 
poles of the rhombic dodecahedron. 
Join C x d 5 , C 2 d 2 , and C 3 d x meeting in o. Then o is the pole 
of the face of the octahedron whose symbol is 1 1 1. 
To place on this octant six poles of the six-faced octahedron 
whose indices are 1, -J, 2. 
In this case X 3 =36° 52', X 3 =26° 34', and X^S 0 41'. 
Graduate each of the lines C 3 d 2 , C 3 d 5 , C x d 2 , G x d v 
C 2 d 5 , from o° to 45° ; considering C x C 2i C 2 C 3 and 
chords of 90°, and making the three points G v G , 
zero, as described in § 132. 
Let OoF 1 =36° 52'=aF 2 =0 1 F 3 =aF 4 =0 2 F 5 =0 1 F 6 
C f 3 H 1 = 26° 34'= 0 3 H 2 = 0^= 0 2 H 4 = C 9 H 5 = G x IL g 
0^=33° 41'= C 3 Gc 2 = 0^3= G x G 4 = G 9 G 5 = G x G 6 
G 2 d v 
and 
GA as 
2> G 3 
each 
A 
\ 
A 
A 
& 
0 , F 1; 
OjL lt 
Ofi„ 
O P 
c i r n 
0,H,, 
E„ -®2> 
octahedron 
■®3> E V A> 
} ) 
and E. 
0, H„ 
o, ■*„ 
C.J.,, 
0-fi,, 
o 2 g 3 , 
G^&s) 
GA 
G 3 (j Q 
GA 
G s H 5 
5 
G s F. 
whose indices 
>o 
Then E x is the intersection of 
of 
of 
of 
of 
of GjOr# 
j will be six poles of the six-faced 
are 1, -§-, 2, and angular elements 
X 3 =36° 52', ^> 3 =68 0 12'. The lines of intersection are not 
shown in the plate. 
(Fig. 43*, Plate IY.*) has marked on it the poles on the 
octant of a sphere of nearly all the forms of the cubical system 
which have been observed ; all the faces whose poles lie in 
the same line having their poles on the sphere of projection 
on the same zone circle. 
The angular and linear indices of every form are given in 
the following table. 
Where _p 2 , and p 3 are the polar distances of each form 
from the three poles of the poles of the cube, G v 0 2 , and C v 
0 f (j), and ip the supplements of the angles of inclination over 
the edges of adjacent faces determined as in § 123, 124, 125, 
and 126. 
§ 124, 125, and 126 show how when these angles or any two 
