PRINCIPLES OF CRYSTALLOGEAPHY. 



243 



Subtracting now tlie products obtained by multiplying tbe index right 

 above with that left below, from tbat obtained by multiplying the index 

 left above with that right below, we obtain three whole numbers (u v to), 

 which are either positive or negative, are determined for the zone P Q, 

 and are called zone-indices. In order to distinguish these from the in- 

 dices of the faces, they are Inclosed in rectangular brackets. 



The zone-indices of a zone containing more than two faces can be cal- 

 culated from any two faces of the zone which are not parallel. The 

 same value is always obtained, abstraction being made of a constant 

 factor of all three indices, with which we can always multiply all of 

 them without changing the direction of the face or line represented. 



If, now, a third face, R(.»t/5;),isplaced in the above zone P O, we have 

 a simple criterion, whose expression is produced from the fact that the 

 zone-axis [P E] or [Q R] must have the same indices, even to a constant 

 factor, as [P QJ. This criterion is the existence of the equation — 



ucc + vy -\- wz =0 

 If this equation is realized, all the three faces P Q R are in the same zone. 

 If the symbols of two zones, [efg] and [w v lo], are given, the symbol 

 of a face (coyz) lying in both zones may again be found by crosswise 

 multiplication — 



^ f 9 efg 



XXX 

 U V W U V W 



fw — gv:gu — ew: ev ~ fu 

 in the same way as the zone-symbol from the indices of two faces. 



At the close of this section the most important special zone-laws and 

 some examples of the development of zones will be given. 



§ 3. — Spherical Projection. 



The method of spherical projection introduced by Naumann gives the 

 simplest mea*ns of representing the opposite faces of a crystal. It has 

 the advantage of showing, even p- q 

 in extremely rough executions 

 of it, a representation of the 

 zone-combinations of a crystal, 

 and allows of the determina- 

 tion of the indices of its faces, 

 on the assumption of a primi- 

 tive form, almost without any 

 measurements. 



For this purpose let us imag- 

 ine that from a point o, in the 

 interior of a crystal, (Fig. 6,) 

 perpendicular straight lines, 

 o a^ a', ob, o c, c', o d, o e, 

 be drawn to all of its faces. 

 From the point o, as a center, let us construct a sphere of any radius, and 



