71 



>! bin. n\ axes, .mil <>f OIK- pair dt tcin.iry axes, and tln-n-foic it 

 o N..IHC of the functions we have previously attributed to the 

 "diagonal" planes S D . On closer examination it appears also to be 

 rprmlicular to one of the binary axes of the system, and therefore 

 I i.i- in consequence the existence of a symmetry-centre. 

 Further it is obvious that it is impossible to add a horizontal 

 >laiir S a , perpendicular to the supposed vertical quinary axes; for 

 plane passing through five binary axes at the same time, does 

 it bring the axial system of the group to coincidence with 

 l self by a reflection in S H . The final result is therefore that only 

 - or what is in this case the same thing, the addition of a 

 imetry-centre will produce a new group of the second order. We 

 >hall call it P 1 , with respect to this last mentioned way of deduc- 

 tion; the new group is thus derived by combining P with the in- 

 .vrsion /. 



Summing up, we thus find altogether four new groups of 

 le second order, related to the endospherical groups T, K, 

 id P: 



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

 >roup T, three perpendicular planes of symmetry passing through every 



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 

 the group T, and six planes of symmetry passing through every 



*>air of ternary axes. They have no centre of symmetry, but every binary 

 ixis is at the same time a quaternary one of the second order. We shall 

 ime 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- 



ictry. 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-number 2n. We shall attribute the 

 symbol P 1 to this group. It represents the highest symmetry which 





