86 



sphaera siphonophora (Haeckel), etc., or the arrangement of their 

 spicula, or the type of symmetry of a flower or of some animal, may 

 often be found with small trouble, if the repeatedly occurring parts 

 of the object be projected in the way considered, upon a spherical 

 surface, and every projected part be denoted by a special sign. Even 



Fig. 90. 



Stereographical Projection of the Groups of the 

 Cubic System. 



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 G ado 1 in, we obtain the following 

 images. 1 ) 



The most unrestricted forms of any of these five classes are 

 reproduced in fig. pi. 



They have successively twelve, twenty- four, and fourty-eight 



*) 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: Q. 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, especially by German crystallographers. The above reproduced 

 figures will now be easily understood. 



