PEINCIPLES OF CRYSTALLOGRAPHY. 



245 



projection of a face perpendicular to the zone-faces. The proposition 

 obtains that the normal angle of two faces, H and K, is equal to the arc 

 hlc^ which is cut off from the principal circle by the straight lines PH 

 and P K produced. 



From these three characteristics all the laws for the construction of 

 stereographic projection are derived. 



It is immediately apparent that the normal angle of all faces projected 

 by points on the principal circle are determined by the arcs contained 

 between the i)oles; that all zoues passing the center of the main circle 

 will be projected as diameters 5 p/gO 

 that, further, the pole of such a 

 zone falls again in the principal 

 circle, and will be on one of the 

 extremities of the diameter per- 

 pendicular to the zone. 



If the projection of a pole, P, 

 (Fig. 9,) is given, and that of the 

 opposite face parallel to it sought, 

 it is at once clear that it must lie 

 outside of the principal circle. If 

 a zone is determined by P and the center, 0, of the main circle, the oppo- 

 site pole P' must be in the same zone, because every zone in which a 

 face lies must also contain the opposite face which is parallel to it. In 

 the zone P O we have now only to look for the point at 180° from P in 

 order to determine P'. For this purpose we must, according to the third 

 chai'acteristic of projection mentioned above, draw from one of the 

 points R or Q, which, as before, represent the pole of the zone P O, the 

 point E for instance, a straight line, RPj?, to its intersection with the 

 principal circle ; find the point p' 

 of the principal circle, which is, 'V-'''^ 

 at the required angle, 180°, from 

 I) ) and then draw a straight line, 

 Ej/P', whose section with the 

 zone P O gives the pole opposite 

 to P. 



If two poles, P Q, (Fig. 10,) be 

 given, and the zone passing 

 through them be sought, we look 

 for the opposite pole of one of 

 them, P' for instance, which in 

 any case must lie in the zone P Q. 



Through the three points P Q P' we draw, according to the known method, 

 (erection of a perpendicular in the middle of a line joining any two 

 points,) an arc, which represents the required zone. 



In order to find the pole of a given zone, E, (Fig. 11,) we must con- 

 sider that it must be 90° distant from every point of the zone-circle. If, 



