236 



PRINCIPLES OF CRYSTALLOGRAPHY. 



The values « a', /J/s', are tlierefore negative wlieu the axes a or h, for 

 which they staud, are primed («' h'). 

 If the face — 



lies iu this zone, one of the following propositions must be right; 



+ «' 



— P"- > = 



The simple inspection of this method shows how minute in detail this 

 method is. In the first place, the symbols of the faces, with respect to 

 an axis, (in the above case c,) must be reduced to similar co-eflBcients ; 

 then by mnltii>licatiou and addition, respective subtraction and division, 

 the values a" and fi" are to be determined. It is to be remarked that 

 both the numerator and the denominator of these quantities are frac- 

 tious, which nuist be reduced to a common denominator. The calcula- 

 tion, it is true, {loc. cit., p. 169,) cau be simplified when the symbols of 

 the faces are written — 



1 

 - a 



y 



This, however, is using Miller's symbols, which are the reciprocals of 

 Weiss's; and even then the calculation is more circumstantial, because 

 the three symbols are equated with reference to c, and are not symmet- 

 rical according to the three axes. 



The steps of the calculation in the hexagonal system are still more 

 incumbered, since, from a four-membered symbol a three-membered 

 parameter must be first calculated, aud then introduced into the previ- 

 ously-developed calculation. 



Quenstedt* employs these symbols in his so-called zone-point formnlte 

 in a somewhat more convenient, althongh in a much less concise, man- 

 ner than Miller. Let there be three faces — 



m a : nh : c\^ \p a : qb : c L and xa : yb : c 



whose tautozonality is to be proved. For ev^ery pair of these the zone- 

 point formuhe must be written, and the veriiicatiou as to whether the 

 zones are identical, made. Thus, for the zone — 



ma : nb : c\ to l p a : qb : c 



1 



m 



J_ 

 m q 



p n 



1 

 m q 



1 



}) n 



Quenstedt, Miueralogie, 1863, p. 44. 



