72 



12. The last groups which remain to be traced, are those which 

 relate immediately to the three possible endospherical groups T, K, 

 and P, previously dealt with. Again we have to investigate what 

 will be the result of their combination with S H , S v , S D , and /. 



In connection with our reasonings in the case of the analogous 

 deductions from the group D n , and bearing in mind that the groups T 

 and K also possess among their characteristic operations a number 

 of rotations round three binary axes which are perpendicular to 

 each other, we may conclude in the same way as before, that only 

 the combinations with S H and S D will produce two distinct new 

 groups in the case of T. 



For S H and S D combined are equivalent to a rotation through 1 80 

 round an axis which bisects the angle between two of the above 

 mentioned axes ; this new binary axis is not present in T, but in K its 

 direction is the line joining the middles of two opposite edges of 

 the cube. Therefore the three new groups appear to be: T H , T D , K H \ 

 other ones are not possible. 



With respect to the pentagonal-dodecahedral group P, we find 

 in quite the same way that, if the axial system of P should coincide 

 with itself by the added operations of the second order, this addi- 

 tion can be executed only in such a way that the plane of reflection 

 passes through two quinary, two ternary, and two binary axes 

 at the same time. If one of the quinary axes is put in a vertical 

 position ; we can regard this added plane as 5 v ; moreover, it will 

 bisect the angle of two pairs of other quinary axes, of two pairs 

 of binary axes, and of one pair of ternary axes, and therefore, it 

 has also some of the functions we have previously attributed to the 

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

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

 it has in consequence the existence of a symmetry-centre. 



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

 plane S H perpendicular to the supposed vertical quinary axes; for 

 this plane passing simultaneously through five binary axes at the 

 same time, does not bring the axial system of the group to coincidence 

 with itself by a reflection in S H . The final result is, therefore, that 

 only S v , or what is in this case the same thing, the addition 

 of a symmetry-centre, will produce a new group of the second order. 

 We shall call it P J , -- with respect to this last mentioned way of 

 deduction; the new group is thus derived by combining P with 

 the inversion /. 



