44 



r\ 



the period . By rotating the figure round ON through its charac- 

 teristic angle, ON remains unchanged in space, but OP coincides 



with an equivalent axis OP'. 

 180 ^ "~"~^7 If now the figure is turned 

 round OP, ON will coincide 

 with a similar axis ON'. The 

 successive rotations round ON 

 and OP have therefore the final 

 effect that ON coincides with 

 ON' and OP with OP'. Ob- 

 viously this result could also 

 have been obtained, if the 

 figure were turned round an 

 axis OS through 180; S is the 



point of intersection of two great circles joining N and N', and P 

 and P', on a sphere whose centre is at 0. 



We can from this draw the general conclusion: 



O O 



// axes of period - - and are simultaneously present, there exist 



also binary axes, bisecting the angle between two axes of the same period. 



The theorem is quite general and 

 holds good also of the three regular <, 

 polyhedra mentioned above. 



8. Reviewing the special sym- 

 metry of the three endospherical 

 groups, we may summarise our results 

 as follows: 



a. There are figures which have 

 the symmetry of the group T, including '80 

 three homopolar binary axes perpen- 

 dicular to each other, and having the 

 directions of the lines joining the 

 middles of every two opposite edges of 



a regular tetrahedron', and four heteropolar ternary axes, having the 

 direction of the four perpendiculars erected in the centres of each 

 tetrahedron- face (fig. 44). 



b. There are figures having the symmetry of the group K, 

 including three perpendicular, homopolar, quaternary axes, having 

 the direction of the perpendiculars in the centre of each cube-face', 

 moreover, four homopolar ternary axes having the direction of the 



180 



