79 



cut off segments of such magnitude on the three coordinate- 

 OZ, ON, and ON lt that Hauy's law shall be fulfilled: thu- in 

 the case considered the proportion ^- must be a rational one. hut 



i j s* O^t 



, v being equal to^. -- because NS is perpendicular to ON lt - 

 none other than cos at,. Therefore if Hauy's law will hold, cos 

 must have a rational value, and the only allowable values of this 

 kind are: o, + or \, and + 1 or 1, the angle at. being then 

 60, 120, 0, and 180 respectively 1 ). From this it follows that 

 in crystallographical polyhedra no other symmetry-axes can occur 

 than those which are characterised by the values 1, 2, 3, 4, and 6 for 

 n. All other values of n are excluded in the case of crystals, because 

 :he validity of Hauy's law requires this. Hence we may conclude : 

 The symmetry-axes of crystallographical polyhedra can only be 

 binary, ternary, quaternary, and senary axes 2 ). 



4. The number of crystallographically possible symmetry-groups, 

 as deduced from the complete number of types already traced by 

 us, therefore proves to be limited to thirty-two. Their symbols are, 

 in the same order as the general groups found previously, the 

 following 3 ) : 



A. Groups of the first order: 



C lt C 2 , C 3 , C 4 , C 6 ; Z) 2 , D 3 , Z) 4 , Z) 6 ; T, and K. 

 All crystals appearing in two enantiomorphous forms belong to 

 one of these eleven classes. 



B. Groups of the second order: 



)f, D l l, D*; D D 2 , D D 3 > T 



All crystals which do not differ from their mirror-images, belong 

 to one of these twenty-one classes. 



N. B. Attention must be drawn again to the fact so often misunder- 

 stood, that the absence of a plane of symmetry need not necessarily make 

 the figure considered differ from its mirror-image. The reverse of this 



J ) For the complete demonstration, vid. N. Boudajef, in Ostw. Klass. 

 No. 75, p. 7883. (1896). 



) In crystallography these axes are usually named: digonal, trigonal, tetra- 

 gonal and hexagonal axes, with respect to the polygonal and polyhedral 

 forms occurring. 



8 ) The case of n = 1 (a = Sit) has been also considered here, although the 

 axis A , has, properly speaking, significance only as a symbol for identity. The 

 groups with such "unary" axes will therefore afterwards be indicated by the 

 special symbols A and S respectively. 



