252 



PKINCIPLES OF CRYSTALLOGRAPHY. 



5. Five planes of symmetry, of which four are taiitozonal and inclined 



to ea<ih other 45° and 90°, every two 90° apart si^uilar, A A', 

 A' A", (Fig. 17.) The fifth, C, at right angles to all the others, and 

 not similar Tetragonal system. 



6. Seven planes of symmetry, six of which are tautozonal and inclined 



30° and G0°, every three 60° apart, similar, A A' A", B B' B", (Fig. 

 18.) The seventh, C, at right angles to all the others, not 

 similar Hexagonal system. 



Fi3.i8. 



FigiS 



7. Nine iDlanes of symmetry, three of which, A A' A", (Fig. 19,) are at 

 right angles to each other, and similar. The other six, similar to 

 each other, B B' B" . . . . B"^, intercalated between every two tautoz- 

 onal A, and at an angle of 45°. Tesseral (isometric) system. 



§ 2. — Characteristics of the Systems. 



From the above statement of the relations of symmetry in each crystal- 

 line system we shall next derive the single faces belonging to each form 

 as well as the most practical method of selecting the axes of the crystal. 



For axes we may select any three edges or zone-axes which are formed 

 by three possible faces of the crystal not tautozonal with each other. 



We shall, however, on account of the existence of planes of symmetry, 

 so select the axes that, wherever it is possible, they are placed sym- 

 metrically to the planes of symmetry, by which we shall at once see that 

 all the faces of a form will take the same numerical indices, but arranged 

 in different orders. We understand by form the combination of all those 

 faces which are symmetrical with each other, according to the planes of 

 symmetry of the given crystal, and which, together, possess the same 

 physical peculiarities. 



With regard to the selection of the axes, we only remark that it ap- 

 pears necessary, on theoretical grounds, which were first developed by 

 Frankenstein, so to select the axes that every acute axis-angle shall be 

 greater than 60°, and that every obtuse one shall be less than 120°, 

 which is always possible. 



"1. Triclinic system. — No plane of symmetry. The choice of the 

 axes is arbitrary, as also the face 111, by which the jDlane of the axes 

 is determined — 



