PRINCIPLES OF CRYSTALLOGRAPHY. 237 



Tbe same must be done for the zone — 



from which, as a condition of tantozonality, follows the equality of both 

 relations. Queustedt and P. Klein* employ tjie zone-control in this form. 

 It is to be remarked that these zone-point formulae can be essentially 

 simplified, because the denominators of both sides are alike ; thus — 



\q nj \m pj \y nj \m xJ 



Also, the condition — 



\(i nJ \m ])J \y nJ \m xJ 



But this equation is much more complicated than Millers. In our 

 former example we had — 



210 = Ja : I : ooc; lil = «:Z>':c;301=:rJa:c/D&:c 



Exchanging the axes a and c in all the three faces, in order to be able to 

 make the coefficient of c equal to unity, which has no influence on the 

 tautogonality, we have — 



GO«:Z*:ic;«:&':c;a:co&:^c 

 or — 



. coa : 2h : c ; a : h' : c ; 3a : ooh : c 



It follows that — 



m n -5 J> (i X 3 y 



bv substitution — 



or — 



-3 :_i = _!:_: 



The proportion is correct, consequently the zones exist. The numerical 

 values of the letters must here, also, be substituted according to the 

 above-mentioned method, and the division carried out; while in Miller's 

 method the very simple and symmetrical calculation can be carried out 

 on the indices, without the help of letters, by means of the crosswise 

 multiplication and subtraction of whole numbers. 



* Kleiu ; Leonh. Jahrb., 1871, p. 480. 



