248 PRINCIPLES OF CRYSTALLOGRAPHY. 



Let the zones to be determined be — 

 l)man; bdce; afc; apd; hpfq; cspm; dsfii; cqn; aqe 

 The existence of these will be seen principally from the parallelism oi 

 the respective edges. Where there is no real edge, as is the case in the 

 angle a q, the hypothetical zone-axis can be found by turning the crystal 

 round. All faces which, in turning round the same axis, reflect the 

 light are tautozonal. 



In order to determine the combination, it is first necessary to select a 

 system of axes. Eegard will be had to the real or apparent symmetry of 

 the crystal, in this way, that when a system less symmetrical in com- 

 pleteness and inclination of the faces approaches one of higher sym- 

 metry, this analogy is retained. 



We select three faces, ahc, for the planes of the axes ; their lines of 

 section give the crystallographic axes. We project these in such a way 

 that the zone a 6 is contained in the principal circle. 



The exactitude of the relation of the angles makes naturally no dif- 

 ference, if it is only a question of the solution of the combination. The 

 faces are introduced into the projection in the order in which they are 

 to be determined, first a bo. 



The faces ahc then are designated by the symbol belonging to the 

 pinacoids, 1 0, 1 0, 1. 



In order to fix a ground-form, we have yet to determine the relations 

 of the axes ; this, according to the relation ofp, may be (111); the axes- 

 sections of the face p give also the value o A, o B, o 0, from which the 

 parameters of every other face will be determined. 



That the indices of p must be 1 1 1 follows from the equatron (p. 9) in 

 which the indices of a face are determined, as — 



,_^oA 7_oB ] _oC 



Substituting the section o A, o B, o G, in this equation, we have — 



h=lc=l=l 



After the outline and the axes of the crystal are determined, the 

 drawing of the faces can be developed. 



Determination of m. m lies in the zones 6 mail and c 1> m. In or- 

 der that a face may lie in the first zone, it is a necessary and sufficient 

 condition that it has the symbol h Tc o, that is, is parallel to the axis c, 

 as also follows from the derivation of the zone-equation. 



For the second zone we have the condition — 



because, as we have seen, the equality of the same index-relations, in 

 two faces of a zone, determines their equality for all the faces of that 

 zone ; thus — 



h ^ 1_^0_ 



& 1 



