424 
Hence on the whole we have the formulae 
sin i 
tan 3 a — ^ • - 
sin 30° sin - 
2 
tan L = tan 60° cos 2 a. 
. 0 
sm - 
sin P = - tan 8 = tan P cos (60°— L). 
sin L 
cos ff 3 = C0S ^ cos (54° 44' + j3) . 
cos p 
sin 0 
and sin (45° — A 3 ) = — — - 
smpa # 
for determining p 3 and A 3 in terms of 0 and 0 ; all the formulae 
being adapted for logarithmic computation. 
y> 3 and A 3 being determined from the values of 0, and \p, 
m and n can be expressed in terms of p 3 and A 3 . 
127. By the formulae given in § 124, § 125, and § 126, any 
two of the angles of inclination such as 0, 0, and \p, over the 
edges of a six-faced octahedron, having been observed by the 
goniometer, p% and A 3 can be determined. Again, by formulae in 
§ 118, 2h and A^ p 2 and A 3 can be obtained from the values 
of ^> 3 and A 3 . 
p 3 and A 3 being determined, m and n can be obtained. Now 
all the forms of the cubical system are derived from those of 
the six-faced octahedron. 
Hence by determining 0, (f>, and for any form of the cubical 
system, we can obtain the values both ofp 3 and A 3 , and also 
of the indices 1, m, and n. 
As we advance in this treatise we shall show good reasons 
for preferring the polar circular co-ordinates _p 3 and A 3 to the 
linear ratios or fractions m and n. 
128. The problems of crystallography being resolved for the 
most part into those of spherical trigonometry, may be solved 
by means of lines drawn on the surface of a solid sphere. 
This being inconvenient in practice, it is usual to project the 
points or poles on the surface of the sphere upon those of a plane, 
just as geographical and astronomical maps are projections 
from the surface of the sphere upon the plane of the paper on 
which the map is drawn. There are three principal projections 
of the sphere, — the steregrctphic, orthographic, and gnomic. 
The steregraphic when the eye is supposed to be placed on 
the surface of the sphere and the points in the hemisphere 
furthest from the eye are projected on the plane of the equator ; 
