43 



180 



1 80 



120 



180 



Fig. 42. 



'iinulrs with ON', and OP with OP'. Obviously this result 

 >uld also be obtained if the fi^mv were turned round an axis OS 

 11 1 nigh 180; 5 is the point of intersection of two great circles 

 ining N and N', and P and 

 ', on a sphere whose centre 

 .it 0. 



can from this draw the 

 icral conclusion: 



// axes of Period and -r- 

 n p 



simultaneously present, there 

 ist also binary axes, right in 

 middle of two axes of the 

 me period. 

 The theorem is quite general, and holds good also of the three 



lar polyhedra mentioned above. 



8. If the special symmetry of the three endospherical groups 

 reviewed, we can state the following: 



a. There are figures which have the symmetry of the group T, 

 luding three homopolar binary axes perpendicular to each other 



and having the directions of the lines 

 joining the middles of every two oppo- 

 site edges of a regular tetrahedron; 

 and four heteropolar ternary axes, 

 having the direction of the four perpen- 

 diculars erected in the centres of each 

 tetrahedron-face (fig. 42). 



b. There are figures having the 

 symmetry of the group K, including 

 three perpendicular, homopolar, quater- 

 nary axes, having the direction of the 

 Perpendiculars in the centre of each 

 cube-face; moreover, four homopolar 

 ternary axes having the direction of the cube-diagonals; and six ho- 

 mopolar binary axes having the same direction as the lines joining 

 the middles of every two opposite edges of the cube. (fig. 7). 

 1 c. There are figures possessing the symmetry of a group P, including 

 six quinary axes having the directions of the perpendiculars in the 

 centre of each face of a regular pentagonal dodecahedron ; ten ternary axes 

 having the directions of the lines joining every two most distant corners of 



71' 



080 



oar 



,180 



J8C 



120 



72 



Fig. 43- 



<i? 



