240 



PRINCIPLES OF CRYSTALLOGEAPHY. 



The second section treats of tlie possible systems of crystallization and 

 their corresponding relations of symmeti\y ; 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. 



SECTION I. 



THE GEOMETRICAL RELATIONS OF CRYSTALS. 



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§ 1. — Miller's Symbols. 



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 X, o Y, 



Z, which are not parallel, and which 

 have a common origin, o, are known. 

 These straight lines are called the 

 axes ; the point o the center of the 

 axes ; the plane of every two axes, 

 )^. X Y, Y Z, Z Y, the planes 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 



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



The lines joining every two sections of the axes of a plane,(H K, K L, 

 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 j)lane re- 

 ^'9"^ 1 mains unchanged ; it will only 



be moved parallel to itself, 

 (Fig. 3.) 



From the equality of the re- 

 lation — 



= m 



H o K oh 



results the similarity of the 

 triangtes K O L, K' O L', &c., 

 and from this the parallel- 

 ism of HKL and H'K'L'. 



