366 A. C. LANE — THE ANGLES OF CRYSTALS. 



most commonly arising. I do not pretend to work out all applications, but 

 I hope to give the method clearly enough to enable any one to modify it to 

 suit special cases. 



The StereographiG Projection. 



§ 2. As the stereographic projection will be much used in illustrations 

 and in solutions, a few notes on this projection and its properties, designed 

 to introduce shortened terminology and simplified constructions, may be 

 pardoned. 



The surface planes, or a plane section, of a crystal may be represented by 

 normals drawn to them from a given point, (figure 1 ;* see Dana's "Text- 

 book," appendix, figure 756). These normals may in turn be represented 

 by the points N^, N^, N^, etc., at which they cut the surface of a sphere about 

 0. This is a spherical projection. These points (N^, K, Nj may in turn 

 be represented in a plane by the points P^, F.^, P3, where the lines JVi^, 

 N.^A, j^.^A, which join them to any point of the sphere (A) cut the diametral 

 plane perpendicular to the radius OA. These points P^, P^, P3 represent in 

 stereographic projection the faces of the crystal. We call them face-points, 

 and by the angle between them we mean the normal angle between the faces 

 to which they, correspond (i. e., the angle through which one face must be 

 turned to become parallel to another, or the supplement of the angle given 

 by the ordinary hand goniometer). f 



Points in the same hemisphere as A may be projected by producing the 

 lines of projection. Their face-points lie outside the circle, which may be 

 called the primitive, in which the diametral plane cuts the sphere. P1P2 P^P^^ 

 in figure 2, is such a primitive. 



The properties of this projection that we shall need to use may be sum- 

 marized as follows : 



(1) The distance of a given face-point P^ from the center G (figure 2) is 



tan i << C : P^, the radius of the primitive being unity. 



(2) Angles between edges are projected into equal angles ; whence it may 



be shown that — 



(3) Circles are projected from the sphere into circles ; so that — 



(4) Face-points making a given angle with a given face-point lie on a circle ; 



and in particular — 



(5) Face-points making an angle of 90° with a given face-point (i. e., be- 



longing to a zone) lie on a circle that has a diameter of the primitive 

 for chord. Call such circles zone-circles {e. g., P^S, figure 2). 



*The figures referred to in this memoir are grouped on plate 14, faeing page 382. 

 t It is a pity that the scale of this instrument is not reversed, so that the angle read by " reflection- 

 goniometers" and "application-goniometers" would become the same. 



