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s, \vh. tin r tin ir symmetry be a higher or a lower one. As a conse- 

 n in < of thi>, the parameters of the forms of all classes belonging 

 tin same crystal-system, are fully determined by the same number 

 independent measurements: the higher the special symmetry of 

 lowest-symmetrical sub-groups of the system is, the smaller 

 tlu 11 umber of such independent data required for the determi- 

 ition of the coordinate-system and the parameters of a crystal. 

 If now we investigate, which groups of the thirty-two mentioned 

 )ve are sub-groups of others, we get the following seven crystal- 

 -tcms. The principal group in every system, of which the others 

 -ub-groups, is always mentioned as the first one: 



I. The triclinic system includes the groups: 7 and A (= C t ). 

 The polyhedra of every class of this system can be absolu- 

 tely fixed by five independent data 1 ). 



II. The monoclinic system includes the groups: C*%, S, and C 2 . 

 The forms of this system are fully determined when three 

 independent data are given. 



III. The rhombic system includes the groups D*j , C V 2 , and Z) 2 . 

 All forms of the whole system are known if two independent 



data are given. 



IV. The tetragonal system includes the groups: D*f , D^, C*f , 



C V 4> C 4 , D 4 , and'C 4 . 



All polyhedra of this system are determined by one single 

 measurement. 



V. The trigonal system includes the groups: D*j , D%, C* t 

 C V 3 , and Z) 3 , C J 3 , and C 3 . 



VI. The hexagonal system includes the groups: D",D 9 ,C", C% 

 and C 6 . 



Three independent data are generally sufficient to fix a coordinate- 

 stem, whether there be given three angles between every pair of coordinate- 

 s, or the three dihedral angles between every pair of coordinate-planes, or 

 any arbitrary combination of three such elements. For the determination of 

 a fourth plane of the crystal, two other data are necessary and sufficient. But 

 if this plane be determined, all other planes of the crystal follow from it 

 according to Hauy's law. If now the coordinate-system is not arbitrary, but 

 a higher symmetrical one, whose angles have fixed and known values (90, 

 60, 45, etc.), then of course the number of data required to define it, is 

 reduced more and more, while the same will be the case with respect to 

 the fixing of the fundamental fourth crystal-plane mentioned before. 



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