244 



PRINCIPLES OF CRYSTALLOGEAPHY. 



produce the perpendiculars until tbey cut the spliere AA' BCO'DE, 

 &c., which are called the poles of the faces, which they meet. 



lu this construction, in which, for the sake of distinctness, only the 

 front side is drawn, we see immediately that the poles of tautozoual 

 faces, A D B E A' for instance, lie in a great circle of the sphere, because 

 the normals of tautozoual faces lie in a plane, which must pass through 

 o, from which point all the normals are produced ; a plane passing 

 through the middle point, however, cuts the sphere in a great circle, 

 which consequently contains the pole of the tautozoual faces. 



In order to draw a sphere containing the poles of the faces of a crystal, 



we may select several different methods of projection. Of these the 



stereographic method, introduced by Miller, is the most convenient. 



As plane of projection let us take, for this purpose, a plane pass- 



_. . ing through the center of the 



sphere c, (Fig. 7) which, accord- 

 ing to the above, cuts the sphere 

 in a great circle, ABO; let us 

 draw a diameter of the sphere, 

 O C, perx^endicular to this, 

 whose extremities, O and 0, are 

 90° from every point of the 

 principal circle, so that the 

 lower pole O shall be the point 

 of sight; let us now join by a 

 straight line every pole of the 



sphere ABCDEF with 



the point of sight O. The inter- 

 sections AB cilef ... . of these 

 straight lines with the princi- 

 pal circle give the stereographic projection of the pole AB CD. 

 In general, the principal circle will be taken perpendicular to the faces 



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of a zone, so that the projection of these 

 l^oints of the faces will be the periphery 

 of the circle. 



The most important peculiarities of such 

 a projection are the following : 



1. Every circle will be projected on the 

 sphere, either as a circle or a diameter. 



2. Every great circle will be projected on 

 / the sphere as an arc, which cuts the prin- 

 cipal circle in the extremities of a diameter 

 of the zone, or as a diameter itself. In such 

 an arc, for that reason, also, the poles of the 



tautozoual faces lie, as, for instance, A e/A' ;BdeB';BcfB';A d c A'. 



3. Let every point, P, which, on the sphere, is at 90° from all points of 



this circle, be the pole of a zone-circle, H K, (Fig. 8,) which is also the 



