PRINCIPLES OF CEYSTALLOGRAPHY. 237 



The same must be done for the zone — 



from wlaich, as a condition of tautozonality, follows the equality of both 

 relations. Quenstedt and P. Klein* employ the zone-control in this form. 

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

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



\q uj \vi pj \y nj \m xj 



Also, the condition — 



V^i nJ \m pj \y nJ \m xJ 



But this equation is much more complicated than Miller's. In our 

 former example we had — 



210 = ^ a : h : coc; \\\ = a -.1' : c;?>^\^\ a \ (j^h : c 



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

 make the co-efficient of c equal to unity, which has no influence on the 

 tautogonality, we have — 



CO a : Z> : i c ; rt : &' : c ; « : CO & : ^ c 

 or — 



CO a : 2 Z> : c ,• fl : &' : c ; 3 a : cob : c 

 It follows tbat— 



i=0; i=l,l=l;l=_l;l=l;l=0 



vi n 'J. p <i ^ X '6 y 



by substitntion — 



or — 



_3 . _ I ^ _\ . _\ 



2 ' 2 ■ 3 



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. 



Klein ; Leonh. Jahrb., 1871, p. 480. 



