240 



PRINCIPLES OF CRYSTALLOGRAPHY. 



The second section treats of tlie possible systems of crystallization and 

 their corresponding relations of symmetry ; it is taken as an abstract 

 from the work of von Lang. In the third section I have shown how, 

 with the foundation of the 0])tical relations of a crystal in general, the 

 optical characters for each individual system of crystallization are 

 derived from their symmetry. 



SEOTIOX I. 

 THE GEOMETRICAL RELATIONS OF CRYSTALS. 



J'ig.Z. 



§ 1. — Miller's Syinibols. 



It is well known that the situation of any plane is perfectly defined 

 when its sections, o H, o K, o L, (Fig. 2,) of three straight lines, o^, o Y, 



o 7i^ which are not parallel, and which 

 have a common origin, o, are known. 

 Tliese straight lines are called the 

 axes ; the point o the cejjter of the 

 axes ; the plane of every two axes, 

 ^> X o Y, Y Z, Z Y, the ]>lanes of the 

 axes; and the sections o H, o K, o L, 

 the parameters of the face H K L. 



Because every axis considered in 



regard to O has two sides, these are 



z distinguished as the positive and 



negative half-axes. For this reason the sections of the axes are used 



in the calculation as + o H or — o H. 



The lines joining every two sections of the axes of a i)lane,(H K, KL, 

 LH,)give the intersection of the plane HKL with the three planes of 

 the axes. 



If we multiply the three parameters of a face with the same number, 



the direction of the plane re- 

 ^'.'^'^ z mains unchanged ; it will only 



•-' be moved parallel to itself, 



(Fig. 3.) 



From the equality of the re- 

 lation — 



= m 



oJH' ^ oK' ^ oU 

 o H ~ oK o L 



results the similarity of the 

 triangles KOL, K'OL', &c., 

 and from this the parallel- 

 ism of HKL and H'K'L'. 



