84 



every projected part be denoted by a special sign. Even in rather 

 complicated cases the real symmetry can thus generally be found 

 without much difficulty. 



6. If now we review the special symmetry of the five classes 

 of the cubic system in the way of Gadolin, we obtain the following 



images. x ) 



The most 

 unrestricted 

 forms of any 

 of these five 

 classes are 

 reproduced 

 in fig. 8 9 . 



They have 

 successively 



- twelve, 

 twenty-four, 

 and fourty- 

 eight limi- 

 ting faces, 

 and are usu- 

 ally called : 

 tetrahedral- 



fientagonal-dodecahedron, pentagonal-icositetrahedron (gyroid), dyads- 

 dodecahedron (didodecahedron] diploid), hextetrahedron, and hexocta- 

 hedron respectively, and their general Miller ian symbol is { hkl } . 

 In the cubic system the three planes passing through every 

 pair of the perpendicular binary or quaternary axes, parallel to the 

 edges of a cube, are always taken as coordinate-axes. If now the 

 stereographical projection of a limiting face of the form considered, 

 should happen to coincide with the point of intersection of the sphere 



Fig. 88. 



Stereographical Projection of the Groups of the 

 Cubic System. 



l) As already stated, the faces on the upper half of the sphere are indi- 

 cated by X> on the lower half by O. A binary axis bears an ellipsoid , a 

 ternary one a triangle A, etc. at its ends. An axis of the second order is 

 indicated by an open polygon: (j\ The axes are represented by dotted lines; 

 if they are situated in a plane of symmetry, by a continuous line. If the circle 

 in the plane of projection is a continuous curve, it means that this plane of 

 projection is also a plane of symmetry; etc. These notations are now common- 

 ly adopted, expecially among German crystallographers. The above reproduced 

 figures will now be easily understood. 



