I;YSTALL<K;|{.\I'IIY 



43 



FIG. 66. 



tin- northern hemisphere .-md ;t t north pole to view those in the 

 southern hemisphere. (Yvstal fares .-in- not ivpivM.ntr.l l, v the pro- 

 jection of their edges, but \>y the location on the plane of projec- 

 tion of the point of contact with the sphere of a radius perpen- 

 dicular (the normal) to the face, as 

 viewed from the south pole. In 

 Kitf. (Hi the plane of projection per- 

 pendicular to the paper is repre- 

 sented by bb', which also represents 

 the b axis; the c axis c'e ; the a 

 axis perpendicular to the paper will 

 he represented by a point at o. 

 Four faces c, f, d, b, belonging to 

 the zone of which the axis a is the 

 xonal axis, their normals when ex- 

 tended will intersect the sphere at 

 their respective poles c', f, d', b'; 

 these poles when viewed by the eye at e will appear on the plane 

 of projection, bb', as if they were actually located at o, f", d", b', 

 their projections. The distance from o at which the pole of any 

 face will appear in the projection, as f, is proportional to the tan- 

 gent of one half the angle foe ; for of" = oe tan oef ' ' ; oef " = \ cof ' ; 

 tan 26(f oc = 26) = .23, or the distance of f" from o is 23/100 of the 

 radius. It may be seen by the construction of Fig. 66, that all the 

 normals of any one zone will lie in one plane ; their poles will all 

 lie on the great circle in which this plane intersects the sphere of 

 projection ; therefore the arc between any two poles is a measure 

 of the angle between their normals. It is the supplement of the 

 angle between the crystal faces which the normals represent. 

 The arc f d' measures the angle d'of, and as the angle of a = oda 

 = 90, then d'of + daf = 180. As the angle between the normals 

 is the angle actually measured by the reflecting goniometer, it is 

 the angle reported and used in the descriptions of crystals. 



All poles in the northern hemisphere when viewed from the south 

 pole will fall upon the plane of projection within the equator or 

 primitive circle. When the plane of projection is a plane of sym- 

 metry, the projection of the northern hemisphere will also be a pro- 

 jection of the southern hemisphere as viewed from the north pole. 

 Similar poles above and below will coincide on the plane of pro- 

 jection. All zones or great circles perpendicular to the plane of 

 projection are projected in diameters of the primitive circle; all 



