PRINCIPLES OF CRYSTALLOGRAPHY. 235 



brevity mentioned above, has the further advantage that, instead of the 

 symbol co, zero is used, because the figures of both these systems are 

 reciprocal. How great the importance of this particular is in the calcu- 

 lation of zone-equations will be immediately shown. On the facility of 

 zone-development, however, depends the quick and sure solution of the 

 combination. 



The method of establishing a zone-equation is, according to Miller, as 

 follows : Given two faces, efg and p q r, the sign of the zone formed by 

 both can be obtained by crosswise multiplication and subtraction, as 

 follows : 



efu cfg 



XXX 

 p q r J} q r 



[fr — gq- gp — er; eq—fp] 



[u V ^v] 



[n vie] is the symbol of the zone ; now, cfg pqr are severally whole num- 

 bers; the products, /r, gq^ gp, , are, for that reason, likewise so; 



the same is therefore true of their differences, which represent the in- 

 dices u V ui of the zone. 



If the face xyz lies in the zone represented by [uviv\ the similarly-sit- 

 uated indices of face and zone multiplied, and all three added together, 

 must be equal to : 



n X -\- V y -|- w 2; = 



A numerical example makes the brevity still more apparent : 



ahc 310 210210 



_x X x_ 

 pqr Ill llllll 



1. 1-0. i; 0.1-2.1; 2. T-1.1 

 1-0; 0-2; -2-1 



uvw [12 ^] 1 2 3 



xyz 30 1 1.3+2.0+3.1=3-3=0 



The face 3 01, therefore, lies in the zone [1 2 3J, produced by 2 1 and 

 111. 



Let us observe the method of zone-calculation according to Weiss:* 

 Given two faces — 



aa : 3h : nc\ and 



a' a : i3' b : 11 c 



which are already reduced to a similar co-efficient of c. The zone pro- 

 duced is — 



{71g; a"a-\-,3"b) 



therefore — 



a' i3—a-i' ' ' aft' — a' 13 



* Weiss, Berliuer academische AbliuudluugeD, 1820-21, pp. 169-173. 



