PRINCIPLES OF CRYSTALLOGRAPHY. 



255 



right angles to it, and finally 1 as plane of X Y. For the determina- 

 tion of the lengths of the axes we select a face, 111, perpendicular to 

 one of the intermediate planes of symmetry. We thus have the ele- 

 ments — 



Thus we have only one unknown quantity, -. 



The intermediate planes 



of symmetry have the symbols 110, 110. The most general form is a 

 pyramid of sixteen faces. The similar faces of hJcl may be seen in 

 Fig. 24. 



6. Hexagonal system. — Seven planes of symmetry, six of which 

 are tautozonal and inclined at an %.2i. 

 angle of 30°; every other one, 

 A A' A", B B' B", (Fig. 25,) simi- 

 lar ; and the seventh, which is at 

 right angles to them, not similar. 

 We might here have selected for 

 the planes of the axes three planes 

 of symmetry, as C, and two others 

 from the zone, symmetrical to the 

 planes of the axes, but the sym- 

 metry of the notation would thus 

 be lost. We select, therefore, as 

 in the rhombohedric system, three 

 alternate faces, of a form perpen- 

 dicular to the six planes of sym- 

 metry, for tl 3 planes of the axes 10 0, 10, 1. We determine the 

 length of the axes, as in the rhombohedric system, by the face 111, 

 which is ijerpeudicular to the axis of the zone of symmetry, by means 

 of which we get, as before — 



Because in particular values of their elements there is no difference 

 between this and the rhombohedric system, they are often united, which 

 is contrary, however, to physical laws. 



In this system it is no longer possible to represent the united faces of 

 a form with the same indices with regard to the symbols of the planes 

 of symmetry, as 1 I, 1 1, 1 1 0, it is for the primary 112, 121, 211; 

 for the secondary planes, B B' B", whose sign follows from the zones, 

 we have, for the faces efg, belonging to those lying opposite to h k I, the 

 determinative equations — 



€= _/t + 2fc-f2Z 



f= 2 h- ]c + 2l 



g=2h + 2]c- I 



The most general form of this system is a tweuty-four-faced pyramid, 



the half of whose faces, as is seen on Fig. 25, are represented by the 



