in the Study of Crystallography . 157 



arc connecting the similar extremities of any pair of such 

 diameters measures the angle included between them, which 

 is equal to the plane angle included by them as edges of the 

 face which they assist to delimit. In this way we obtain a 

 catalogue of all the plane angles occurring on the faces of the 

 crystal ; and they are derived by a simple graphical con- 

 struction from the observed inclinations of the faces. Further, 

 the diameters representing the edges which bound one face 

 all lie in the same plane ; therefore the extremities of such 

 diameters lie in one great circle. By drawing great circles 

 through all the groups of diameters lying in the same plane, 

 we get a group of great circles, each of which represents the 

 intersection with the sphere of a plane which passes through 

 the centre of the sphere and is parallel to the face of the poly- 

 hedron, which is bounded by edges parallel to the diameters 

 which lie in the plane, and the extremities of which are 

 connected by the great circle. We have thus a catalogue of 

 the faces of the crystal which are inclined to one another, 

 and the inclination of any two of the great circles measures 

 the inclination of the pair of faces. Also the diameters 

 marked as meeting the surface of the sphere in the great circle 

 supply the number and direction of the edges which bound 

 the particular faces (where it must be remembered that 

 parallel edges are represented by the same diameters) and the 

 plane angles of the face are given by the angles between the 

 diameters. The representation of the faces of the crystal by 

 great circle planes and that of the edges by diameters, all of 

 which necessarily meet in the centre, facilitates the choice of 

 a suitable system of crystallographic axes. 



If we consider the poles of three adjacent faces, Nos. 1, 2, 

 and 3, and draw small circles round them, the radii of which 

 are equal, and of such a length that each circle cuts the other 

 two, then, as before, we have the projection of each edge 

 represented by the arc of the great circle connecting the 

 intersection of each pair of circles. These three arcs cut 

 one another in a point inside of the triangle formed by the 

 insection of all three email circles. This point is the pro- 

 jection or pole of the corner formed by the meeting of 

 planes 1, 2, and 3. If this corner be carried parallel to itself 

 to the centre, its bounding edges will coincide with their 

 parallel diameters, which thus form representative parallel 

 central corners. These diameters meet the sphere in the ex- 

 tremities, which have been already fixed. If these points be 

 connected by arcs of great circles, they determine a spherical 

 triangle whose area is a measure of the corner. 



