81 



NCN 2 must cut off segments of such magnitude on the three coordi- 

 nate axes OZ, ON, and ON^, that Hauy's law shall be fulfilled: thus 

 in the case considered, the proportion ^-must be a rational one. But 

 j^f- being equal to ^~, because NS is perpendicular to ON lf - 

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

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

 kind are: o, -f J or -- \, and + 1 or 1, the angle a being then 

 90, 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 

 the 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 8 , C 4 , C 6 ; D 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: 



r r r r n r H r H r H r v r v r v r v r 1 



<^l> ^2> ^4> ^2> ^ ,i > ^4> ^ 6 > ^ 2 > ^ 3 > ^ 4 > ^6> ^ 3 > 



D f j, D H 3 , D H 4 , D H 6 ; D D 2 , D D 3 ; T*, KH-, 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 



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

 No. 75, p. 78 S3, (1896). 



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

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

 forms occurring. 



3 ) The case of n = 1 (a = 2*) has been also considered here, although the 

 axis A l 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. 



6 



