100 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



new origins. We introduce this method by remarking that the 

 addition of the second syllable [1] V2 of the symbol of the prism 

 D.in the last table of the preceding article has a deeper meaning 

 than might be supposed: in this form the coordinate symbol oïl) 

 is derived from the net symbol, and by examining how this process 

 runs in S fi we easily hit upon its generalization for S llt if necess- 

 ;ir\ by the assistance of the knowledge of the fifth constituent in 

 S 5 found in the manner described above. So we indicate for any 

 net in # 4 how the coordinate symbol of the constituents can be 

 derived from the net symbol. 



In fig. 20 we represent by O (X^ X 2 X 3 X 4 ) the system of 

 coordinates and by the shaded pentagon with the axis of symmetry 

 OM a fourth part of the section of the plane 0(X 1 X 2 ) with the central 

 polytope B. Then OP is the "period'" p of the net and the point 

 P 1 of OX l lying at twice that distance from O is the centre of 

 an adjacent polytope C filling a vertex gap, whilst P 2 with the 

 coordinates 2j), 2p, 0, is the centre of an other polytope B in 

 contact with the central one by a polyhedron of edge import. 

 Moreover 7 J 3 is the point 2p, 2/j>, 2p, and P 4 the point all the 

 coordinates of which are 2p; of these P 3 corresponds in character 

 with P 1? and P 4 with O and P 2 . So the midpoint Q 4 of OP 4 must 

 be the centre of a polytope in threedimensional contact of body 

 import with the two poly topes B with the centres O and P 4 , i.e. 

 of a polytope A. On the other hand the midpoint Q 3 of OP s must 

 be the centre of the prism interposed between the two poly topes 

 A of different orientation with the centres Q 4 , latter point being 

 the image of Q 4 with respect to the space # 4 = as mirror, as 

 these polytopes are derived from the two HM^ of the original net 

 {HM k , f> 4 ) which were in body contact in that space x k = 0. 



In this manner we find in general for all the cases in /S 4 for 

 the coordinates of the centres of the adjacent polytopes 



2p , , , , in the case of C , 



2p , 2p , , , „ „ „ „ an other B , 



V J P 5 P 5 P , 11 ,> ,, ,, A , 



P 3 P , P > , „ „ „ „ D , 



whilst the upright edges of the prism D are parallel to the 

 axis OX, r 



Now we consider the case e l e% e. à J¥H h in order to show how 

 the process runs. Here we have p=h -f- V 7 2, whilst the central 

 B is represented by [6 -f V2, 4 -+ V2, 2 -f- V2, V2~]. So we 

 obtain (J, A, D successivey as follows: 



