413 
and cos q x — 
cos q 0 — . 
/ Vh + h + h + h 
and cos q 3 = 
VI 
+— +— 
«r V V 
Substituting these values in the expression 
1 . 1.1 
cos 0= 
act. 
+ 
b\ + 
cc. 
V (^ + p+^)(^- 
1 
“ + 6 1 3 + c." 
we have 
cos 0= cos p-L cos ^4 + cos p 2 cos q 2 + cos p 3 cos q 3 . 
110. If, in (figs. 31* and 32*), We substitute for 1, m, n ; cos 
cosp 2i and cos p 3 in the order in which they occur, we have a 
notation for every face of the six-faced octahedron in terms of 
p v p 2 > an( l Ps> fhe polar distances of the face from the three 
adjacent poles of the cube; —1, — m, and — n being replaced 
by — cos^, — cosp 2} and •— cos j; 3 . 
Thus if 6 be the angle between the normals of the faces whose 
poles lie in the spherical triangles C 1 D 1 0 1 and G^D^ or the 
supplement of the angle of their inclination over the edge G,o, 
(fig. 3, Plate I.), 
1 , 1 2 
f -f 1 -f 1 
COS 0: mn mn mn 
V(i 
1 
+4 (A+4+ 1 ) i+- 2 +4 
m" J \m M n z J m 2 n 2 
if expressed by the indices of the six-faced octahedron, 
cos 6 = cos y> 3 cos p 2 + cos p 2 cos p 3 + cos p x cos p x 
= 2 cos p 3 cos p 2 + cos 2 p 1 
if expressed by the three polar distances of the pole of any 
face from the three adjacent poles of the cube. 
111. The notation for each face of a crystal, or of its pole 
on the sphere of projection, is expressed in the terms of the 
three indices at which a plane drawn through a point in one 
of the cubical axes, taken at an arbitrary distance called unity 
from the centre where the axes meet, cuts the other two axes 
which are at right angles to the former ; the indices being 
reckoned positive or negative as the points of intersection are 
right or left of A along the three axes AG 1 
VOL. II. 2 G- 
AG 2 , and AG 3 . 
