45 



cube-diagonals', and six homopolar binary axes having the same 

 direction as the lines joining the middles of every two opposite 

 edges of the cube (jig. p). 



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 

 it\ and fifteen binary axes having the directions of the lines joining the 

 middles of every two opposite edges ; all these axes are homopolar (fig. 4.5). 

 9. Finally it may be remarked that there exists an important 

 theorem dealing with the number of non-equivalent characteristic 

 operations, making all symmetrical figures of these groups coin- 

 cide with themselves. For the group T this number is evidently: 

 1+3+4x2 12 ; for the group K: 1+3x3+4x2 + =24; 

 for the group P: 1 + 6 x 4 + 10 x 2 + 15 = 60; the rotation 

 through 360 is of course only counted once here. 



Now the number of these non-equivalent operations is in every 

 case =2x, where x indicates the number of the edges of the tetrahe- 

 dron, cube, or pentagonal dodecahedron respectively. 



Indeed it appears to be a general property of each regular polyhedron 

 with x edges, that it can be brought to self -coincidence in 2x different ways. 

 This theorem is easily and quite generally demonstrable. It is 

 connected with the simple fact that every edge AB, by interchange, 

 can be placed so that its end A coincides with A or with B of any 

 other edge present. 



10. Figures and objects of this kind are represented in fig. 46, 

 and ^7. They relate to the crystalforms of a 



barium-nitrate: Ba(N0 3 ) 2 , and oi cuprite: 



Cu 2 0, from 

 Cornwall, as 

 illustrations of 

 the symmetry 

 of the groups 

 T and K re- 

 spectively. 

 The symmetry 

 of the group P 

 is not possible 

 in the province 



Fig. 46. 



Barium-nitrate. 



Fig. 47. 

 Cuprite. (Cornwall). 



of crystalline matter, for a reason to be explained later on. Of course it 



