73 



Summing up, we thus find altogether four new groups of 

 the second order, related to the endospherical groups T, K, and P: 



a. There are symmetrical figures which have the axial system of the 

 group T, three perpendicular planes of symmetry passing through every 

 pair of binary axes, and a symmetry -centre. The ternary axes are at 

 the same time senary ones of the second order. The symbol of this group 

 shall be T H . 



b. There are symmetrical figures which possess the axial system of the 

 group T, and six planes of symmetry passing through every pair of ternary 

 axes. They have no centre of symmetry, but every binary axis is at the same 

 time a quaternary one of the second order. We shall name this group T D . 



c. There are symmetrical figures which possess the axial system 

 of the group K, three perpendicular planes of symmetry passing through 

 every pair of quaternary axes, and six planes of symmetry passing 

 through every pair of ternary axes. Moreover, they have a centre of sym- 

 metry. The ternary axes are at the same time senary ones of the second 

 order. We shall denote this group by the symbol K H . 



d. There are symmetrical figures which have all axes of the group P, 

 as well as fifteen planes of symmetry passing through two quinary, two 

 ternary, and two binary axes simultaneously; moreover, they have a 

 centre of symmetry, and every axis of odd period is at the same time one 

 of the second order with a period n. We shall attribute the symbol P 1 

 to this group. It represents the highest symmetry which a figure can 

 possess, if no axes with n = oo be taken into account. x ) 



Finally we may draw attention to the 

 fact that the group K H contains all 



operations which 

 are characteristic 

 as well of the 

 group T H , as of 

 T. These last 

 are therefore cal- 

 led sub-groups of 

 K H . In the same 

 way the groups T 

 and K themselves 



a 



Fig. 84. 

 Pyrite. 



Fig. 85. 

 Boracite. 



are sub-groups of K H . Also in the case of the other symmetry-groups 



1 ) Of course, if axes of isotropy are also concerned, the spherical symmetry 

 is the highest possible one. Indeed, just in the same way as the sphere is an 

 "endospherical" polyhedron with an infinite number of faces. 



