PRINCIPLES OF CRYSTALLOGRAPHY. 



235 



brevity mentioned above, bas tbe furtber advantage tbat, instead of tbe 

 symbol co, zero is used, because tbe figures of botb tbese systems are 

 reciprocal. How great tbe importance of tbis particular is in tbe calcu- 

 lation of zone-equations will be immediately sbowu. On tbe facility of 

 zone-development, bowever, depends tbe quick and sure solution of tbe 

 combination. 



Tbe metbod of establisbiug a zone-equation is, according to Miller, as 

 follows: Given two laces, efg andpyr, tbe sign of tbe zone formed by 

 botb can be obtained by crosswise multiplication and subtraction, as 

 follows : 



efg efg 



XXX 

 p q r p qr 



[fr-gq; gp-er; eq-fp] 



[}! VU-] is tbe symbol of tbe zone ; now, efg p q r are severally wbole num- 

 bers; tbe products, /r, gq^ gp, , are, for tbat reason, likewise so; 



tbe same is tberefore true of tbeir differences, wbicb represent tbe in- 

 dices u V 10 of tbe zone. 



If tbe face xyz lies in tbe zone represented by [vrrwj, tbe similarly-sit- 

 uated indices of face and zone multiplied, and all tbree added togetber, 



must be equal to : 



u X -\- V y -4- w ~ = 



A numerical example makes tbe brevity still more apparent : 



aJ)c 2 10 2 10 2 10 



p qr 



1 1 1 



2 1 2 1 

 _X X X_ 

 111111 



0.1-2.1; 2.1 — 1.1 

 0-2; -2-1 



1.1-O.i; 



IIV IV fl 2 oj 1 li o 



xyz 3 1 1.3+2. 0+3. 1=3 — 3=0 



Tbe face 3 01, tberefore, lies in tbe zone [1 2 3j, produced by 2 10 and 

 111. 



Let us observe tbe metbod of zone-calculation according to Weiss:* 

 Given two faces — 



a.a : (ih : nc\ and 



a : j3' b : no 



wbicb are already reduced to a similar co-efiScient of c. Tbe zone pro- 

 duced is — 



{nc; a"a+,i"h) 

 tberefore — 



a a' 



«" = - 



a' 3 — a 



'-fl'). ^3//_.^/5'(«-«0 



afi'—a' 



* Weiss, Berliner academische Abbandluugen, 1820-21, ^]i. 1G9-173. 



