ie) 
sin ACP sin BCP 
sin ACG sin BOG’ 
This is equivalent to the form in which the law was enun- 
ciated by Gauss (C. F. Gauss, Werke, Band 11. S. 308). 
4, It remains to be seen whether the symbol of a zone has 
any geometrical signification when the normals to the faces are 
referred to OA, OB, OC as axes. 
It appears from what precedes, interchanging a, 6, c and 
a, B, y, that 
oe and ha thotl=0, 
being the equations to a line and plane, one referred to the axes 
OX, OY, OZ, and the other to the axes OA, OB, OC, the line 
will be normal to the plane. But (Tract 200), the equations 
to the zone-axis u v w referred to the axes OX, OY, OZ, are 
GY eB 
UG VO ac. 
Hence, a plane through O, normal to the axis of the zone 
u v w, when referred to the axes OA, OB, OC, will have for its 
equation 
B 
Let a plane parallel to the zone plane uvw meet OA, OB, 
OC in U, V, W. Then it is evident that 
OO NOV OW 
.—— = i 
B Y 
Let the axis of the zone uv w meet the surface of the sphere 
in K. It is easily seen that 
oe Stee Bie 
u Vv 
eg og 
a oy 
- cos CK. 
5. It appears then that the notation for faces and zones 
suggested by the equations to the faces and zone-axes, when the 
crystal is referred to three zone-axes as axes of coordinates, is 
equally applicable when the crystal is represented by rays 
normal to its faces, and these are referred to three such rays as 
7—2 
